Light +Volume+I

LIGHT

Volume 1

WAVES, PHOTONS, ATOMS

r H. HAKENa-

Institut fur Theoretische Physik, Stuttgart00

NORTH-HOLLANDAMSTERDAM • OXFORD • NEW YORK • TOKYO

0

•

•

Elsevier Science Publishers B.V., 1981

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical photocopying,recording or otherwise, without the prior permission of the publisher, Elsevier SciencePublishers B. V. (North-Holland Physics Publishing Division), P.O. Box 103, 1000 ACAmsterdam, The Netherlands.Special regulations for readers in the U.S:A.: This publication has been registered with theCopyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can beobtained from the CCC about conditions under which photocopies of parts of this publicationmay be made in the USA. All other copyright questions, including photocopying outside ofthe USA, should be referred to the publisher.

ISBN: 0 444 86020 7

Published by:North-Holland Physics Publishinga division ofElsevier Science Publishers B.V.P.O. Box 1031000 AC AmsterdamThe Netherlands

Sole distributors for the U.S.A. and Canada:Elsevier Science Publishing Company, Inc.52 Vanderbilt AvenueNew York, N.Y. 10017U.S.A.

First Edition 1981First Reprint 1986

Library of Congress Cataloging in Publication Data

Haken, HLight.

Bibliography: p.Includes index.CONTENTS: v. 1. Waves, photons, atoms.1. Light. 2. Lasers. 3. Nonlinear optics.

4. Quantum optics. I. Title.QC355.2.H33 535 80-22397ISBN 0-444-86020-7 (v. 1)

Printed in The Netherlands

Preface

In theof two inclight, ancintroduce(fundamento generadiscoverytram -ghw,this 11.1,physics, atfield in p.physical papplicatioimakes thisthe samefundamenof laser acThe presetmeet this

At thequantumthe quanttelementar)tum mechzdevelop qcgreat deta:more or le:light fieldsstudy requpresent texresults are

•

stored in a retrievaltanical photocopying,!er, Elsevier ScienceBox 103, 1000 AC

Preface

In the 20th century, the classical discipline of optics has been the subjectof two incisive revolutions, namely the discovery of the quantum nature oflight, and the invention of the laser. The concept of energy quantaintroduced by Planck at the turn of this century has deeply influenced ourfundamental understanding of light and matter. The laser made it possibleto generate light with entirely new properties. These in turn led to thediscovery of completely new types of optical processes such as frequencytransformations in matter and many other "non-linear" phenomena. Inthis way, whole new branches of physics called quantum optics, laserphysics, and non-linear optics rapidly developed. There is hardly any otherfield in physics in which a profound understanding of the fundamentalphysical processes is so intimately interwoven with technical and physicalapplications of great importance, as in modern optics. This connectionmakes this branch of physics particularly attractive for scientific study. Atthe same time, the need for a coherent text arises which, starting fromfundamental principles of the physical nature of light, presents the physicsof laser action, and finally gives a transparent account of non-linear optics.The present text, which will be subdivided into three volumes, is meant tomeet this need.

At the same time, this text offers a new pedagogical approach toquantum optics by giving a self-contained and straightforward access tothe quantum theory of light. The present Volume 1 begins at a ratherelementary undergraduate level and requires no prior knowledge of quan-tum mechanics. It thus (and in other ways) differs from usual texts whichdevelop quantum mechanics with its applications to the physics of atoms ingreat detail. There the quantization of the light field is often presentedmore or less in the form of an appendix. A detailed treatment of quantizedlight fields is mostly left to texts on relativistic quantum field theory whosestudy requires a good deal of mathematical knowledge. By contrast, thepresent text develops the quantum mechanics of matter only insofar as theresults are directly relevant to the interaction between light and matter, but

n registered with theInformation can be,ts of this publication)tocopying outside of

o

vi Preface

leaves aside all the superfluous material of atomic or relativistic phy,ics.From the very beginning, this text focusses its attention on the pi- ysicalnature of light. In particular, the present volume deals with the coherenceproperties of light, its seemingly conflicting wave and particle aspects andits interaction with individual atoms. This interaction gives rise to absorp-tion and spontaneous and stimulated emission of light (the latter processbeing fundamental to laser action), and to numerous other effects. Thisbook will also be of interest to graduate students and research workers. Itincludes, among others, most recent results on quantum beats and thedynamic Stark effect and it clearly mirrors a shift of emphasis which hasbeen taking place in quantum physics in recent years. While originallyquantum theory emphasized stationary states, its interest is becoming moreand more concentrated on processes. In addition, we recognize that isolatedquantum systems often represent too great an idealization. Rather, quan-tum systems interact all the time with their surrounding. This leads to anumber of quantum statistical effects. Because of their fundamental impor-tance to laser physics and nonlinear optics, we give a detailed presentationof methods to cope with these phenomena.

Volume 2 will deal with the laser. Here we will get to know theproperties of laser light and how it is produced within the laser.

Volume 3 will then be dedicated to the action of intense coherent lighton matter, where we will find a whole new world of non-linear phenomena.

I wish to thank my coworker, Dip!. Phys. H. Ohno, for his continuousand valuable assistance in the preparation of the manuscript. In particular,he carefully checked the formulas and exercises, contributed some inaddition, and drew the figures. Dr. Chaturvedi and Prof. Gardiner criti-cally read the manuscript. I am indebted to Prof. Gardiner for numeroushighly valuable suggestions on how to improve the text. My particularthanks go to my secretary, Mrs. U. Funke, who in spite of her heavyadministrative work always managed to type the various versions of thismanuscript both rapidly and perfectly. Her indefatigable zeal constantlyspurred me on to bring it to a finish.

The writing of this book (and of others) was made possible by a programof the Deutsche Forschungsgemeinschaft. This program was initiated byProf. Dr. Maier-Leibnitz. The Bundesministerium für Bildung und Wissen-schaft provided the funds and the Baden-Wiirttembergische Ministeriumfiir Wissenschaft und Kunst and the University of Stuttgart awarded me asabbatical year. I wish to thank all of them for their unbureaucratic andefficient support of this endeavor.

H. Haken

Contenb

Pr

Li

A

of

1.1. ThGeArk,Th

1.6. Qui

1.7. An1.8. Qu,

eke1.9. The1.10. The1.11. Col1.12. Spo1.13. Dar1.14. Pho1.15. The1.16. HoN

1.17. Last

istic physics.the physical

he coherenceaspects and

se to absorp-Latter processeffects. Thish workers. Iteats and the Contents

sis which hasIle originallycoming more Prefacethat isolated Contents vii

tather, List of symbols xius leads to aaental impor-preibtation

• What is light? 1to know theer. A brief excursion into history and a preview on the content

oherent light of this book 1

phenomena. 1.1. The wave—particle controversy 1

s continuous 1.2. Geometrical optics 1

In particular, 1.3. Waves 3

ted some in 1.4. The oscillator model of matter 4

ardiner criti- 1.5. The early quantum theory of matter and light 8

or numerous 1.6. Quantum mechanics 12

ly particular 1.7. An important intermediate step: The semiclassical approach 14

if her heavy 1.8. Quantization of the electromagnetic field: Quantum

sions of this electrodynamics (QED) 16

11 constantly 1.9. The wave—particle dualism in quantum mechanics 181.10. The wave—particle dualism in quantum optics 21

y a program 1.11. Coherence in classical optics and in quantum optics 24

initiated by 1.12. Spontaneous emission and quantum noise 28

und Wissen- 1.13. Damping and fluctuations of quantum systems 29

Ministerium 1.14. Photon numbers and phases. Coherent states. 31

yarded me a 1.15. The crisis of quantum electrodynamics and how it was solved 32

.ucratic and 1.16. How this book is organized 341.17. Laser and nonlinear optics 34

H. Haken

•

viii Contents

2. The nature of light: Waves or particles?

2.1. Waves2.2. Classical coherence functions2.3. Planck's radiation law2.4. Particles of light: Photons2.5 Einstein's derivation of Planck's law

V/3. The nature of matter. Particles or waves? 63

3.1. A wave equation for matter: The Schrodinger equation 63 7. Ti

3.2. Measurements in quantum mechanics and expectation values 71 7.1. Intro3.3. The harmonic oscillator 80 7.2. Inter.3.4. The hydrogen atom 93 open3.5. Some other quantum systems 102 7.3. Inter.3.6. Electrons in crystalline solids 104 7.4. The i3.7. Nuclei 110 7.5. The c3.8. Quantum theory of electron and proton spin 112 7.6. Spon

7.7. Perth4. Response of quantum systems to classical electromagnetic 7.8. Lamt

oscillations 119 7.9. Once4.1. An example. A two-level atom exposed to an oscillating 7.10. How

electric field 119 A sin4.2. Interaction of a two-level system with incoherent light. The 7.11. The c

Einstein coefficients 1224.3. Higher-order perturbation theory 126 8. Q1

4.4. Multi-quantum transitions. Two-photon absorption 130 8.1. Quan4.5. Non-resonant perturbations. Forced oscillations of the atomic 8.2. Exam

dipole moment. Frequency mixing 133 coher4.6. Interaction of a two-level system with resonant coherent light 137 8.3. Cohe4.7. The response of a spin to crossed constant and time 8.4. Quan

dependent magnetic fields 1414.8. The analogy between a two-level atom and a spin 1 145 9. Di4.9. Coherent and incoherent processes 155 9.1. Dami

J 5. Quantization of the light field 157equat

9.2. Dami5.1. Example: A single mode. Maxwell's equations 157 Field5.2. Schrodinger equation for a single mode 164 9.3. Quan5.3. Some useful relations between creation and annihilation quani

operators 165 9.4. Lang(5.4. Solution of the time dependent Schr6dinger equation for a 9.5. The d

single field mode. Wave packets 169 Mathematic5.5. Coherent states 170 References 15.6. Time-dependent operators. The Heisenberg picture 174 Subject inde

5.7. The f37 5.8. Quan44 5.9. Unce

37

505658

6.Q

6.1. Moti6.2. Quar

Contents ix

373744505658

6363718093

102104

10/2°

119

119

122126130

nic133

it 137

141145155

157157164

165

0174

5.7. The forced harmonic oscillator in the Heisenberg picture 1785.8. Quantization of light field: The general multimode case 1795.9. Uncertainty relations and limits of measurability 189

6. Quantization of electron wave field 195

6.1. Motivation6.2. Quantization procedure

\./ 7. The interaction between light field and matter

7.1. Introduction: Different levels of description 2017.2. Interaction field— matter: Classical Hamiltonian, Hamiltonian

operator, Schrodinger equation 2047.3. Interaction light field—electron wave field 2087.4. The interaction representation 2127.5. The dipole approximation 2187.6. Spontaneous and stimulated emission and absorption 2227.7. Perturbation theory and Feynman graphs 2307.8. Lamb shift 2427.9. Once again spontaneous emission: Damping and line-width 2517.10. How to return to the semiclassical approach. Example:

A single mode, absorption and emission 2547.11. The dynamic Stark effect 256

8. Quantum theory of coherence 265

8.1. Quantum mechanical coherence functions 2658.2. Examples of the evaluation of quantum mechanical

coherence functions 2718.3. Coherence properties of spontaneously emitted light 2758.4. Quantum beats 277

sJ 9. Dissipation and fluctuations in quantum optics 285

9.1. Damping and fluctuations of classical quantities: Langevinequation and Fokker—Planck equation 285

9.2. Damping and fluctuations of quantum mechanical variables:Field modes 295

9.3. Quantum mechanical Langevin equations. The origin ofquantum mechanical fluctuating forces 297

9.4. Langevin equations for atoms and general quantum systems 304, 9.5. The density matrix 316,5( Mathematical Appendix 335

References and further readings 339Subject index 349

195196

201

•

List of symbols

A Einstein coefficient of spontaneous emission_

A(x, t) vector potential-A(+)( x, t), A"(x, t) positive and negative frequency parts of vector poten-

tialA(t) expansion coefficient

a.. a.1. fermion annihilation and creation operators

J

o Bohr-radiusa lattice constant

Einstein coefficient for spontaneous emission

jk Einstein coefficient for stimulated transition j —0 kB(x, t) magnetic inductionBzz-component Of magnetic inductionB 0 spatially and temporally constant magnetic inductionBP oscillating magnetic induction

B + heatbath annihilation and creation operatorsEinstein coefficient for stimulated transitions

b,b +annihilation and creation operators of harmonicoscillatorphoton annihilation and creation operators of mode

integration constantvelocity of light in vacuum

c(t) expansion coefficientc(t) k th iterate in perturbation expansionD(x, t) dielectric displacement

expectation value of dipole momentD(v)dy number of modes per unit cavity volume

xii List of symbols

D, ( 1 ); Di,x(t)do;

d(t)d3x, dVE(x, t)Ei(x,t)E(+)(x, t);E(-)(x,t)

ex

F(t)f(v, t)

f(t)

G(1,2)hn

g; gX; g jk; gA,jk

Ho

HP

11(T)HB

'In()

H(x, t)

h, h = -27T

I; I(v), I(w)

j(x, t)K(t,T)

expansion coefficientsLaplace transforms of expansion coefficientsexpansion coefficients in the interaction picturevolume elementselectric field strengthpartial electric wavepositive and negative frequency part of electric fieldstrengthelementary chargeunit vector of polarization of mode Adriving force, fluctuating forceprobability distribution functionstationary solution of Fokker-Planck equationexternal forceHooke's constantmutual coherence functioncorrelation coefficients of fluctuating forcescoupling constantsHamiltonianunperturbed Hamiltonianperturbation Hamiltonianmatrix elements of perturbation HamiltonianHamiltonian in the interaction pictureHamiltonian for single cavity modeinteraction Hamiltonian in 2nd quantizationHamiltonian of heatbathHermitian polynomialmagnetic field strength

Planck's constant

intensityindex, integerimaginary unit, i 2 = - 1index, integercurrent densitykernel of integral

{n} = {nfi

n( p ), n(6.) ,P(t)P(x,t)P„

Pjk

Pn

Px

P = -h v

Pjk

Mt)

111

•

List of symbols

k(t) external force acting on a particleindex, integerBoltzmann's constantwave vectorlength of cavity

Li periodicity interval in j-direction[x] Laplace transform

E, operators of angular momentum1 index, integer

index, integermass of particle (electron)

m o electron rest massm5 effective mass of electron

number of atoms

1,1;W occupation number of state jgc, normalization factor (of mode X)

index, integer; index of refractionn number of photons in mode./1n(T) average number of photons with frequency w at tem-

perature T{n) = {n 1 , n 2 ' } set of photon numbers

average photon numbern(v), n(w) photon numbersP(t) probabilityP(x, t) electric polarization densityP„ occupation probability for state n

Pjk projection operatoroccupation probability for state nmomentum of particle

Px probability of finding a photon in mode X

P -7 V momentum operator

Pjk matrix element of momentum operator

px(t) coefficient in cavity mode expansion (electric field)diffusion constant in Fokker—Planck equationdisplacement of particle

itsicture

ectric field

Won

s

an

XiV List of symbols

q(t) coefficient in cavity mode expansion (magnetic in-duction)spatial distance

r, i, op spherical polar coordinatesindependent variable in Laplace transformspseudo spin of two-level atomcomponents of pseudo spinspin vector

sx ,sy ,sz components of spin vector, spin matricess + ,s_ spin-flip operators

period of oscillationtemperature

Ti , T2 longitudinal and transverse relaxation timesTr traceTrB trace over heatbath variables

timeU = exp[ —iHot/h] unitary operatorU(x) energy density of radiation field

energy of radiation field in volume Vu(v) spectral energy density of radiationuk,n( x) Bloch wave functionuA(x) cavity mode function3 volumeV( x) potential energy3 visibility of fringe pattern

velocity of particleV(X) cavity mode function

energyenergy eigenvalue of state n

Wjk transition rates kenergy expectation value, averaged energy

x,y,z Cartesian coordinatesXjk matrix element of vector x

coordinatepartition functionCartesian coordinatenumber of elementary charges per nucleus

Zk

a

fi

fl=

r)

1jk

Ap, Ax

•8

8jk

8(x — x'

eoen = Wni

9

tk

ICA

A, A.;eh

's Tin

1= —

T

•

List of symbols xv

Zk zero of polynomiala angle, complex variable

/3amplitude of driving mode1

= normalized inverse temperature

damping constantF(t,r) correlation function

FA( t ); r;(t ) fluctuating forces for atomic variablesatomic linewidth, phase factor

Yo decay rate, damping constanty(1,2) complex degree of coherence

7,/k damping constant for transition j 4— kA LaplacianAp, Ax uncertainty in momentum, positionAc energy shiftV gradient8 complex constant

8jk Kronecker's symbol

8(x — x') Dirac's delta functiondielectric constant of matter

eo dielectric constant of vacuum

en = h frequency corresponding to energy flin0 angular variable

angular variable0 dipole moment

matrix element of dipole momenttc damping constantKx damping constant of mode

wavelengthA, Aimode index

eh= 2m

Bohr's magneton

permeability of vacuum1

IP T, frequency of oscillation

complex constant, dimensionless coordinate ofharmonic oscillator3.14159 •

ietic in-

•

•

xvi List of symbols

7T dimensionless momentum of harmonic oscillatorp(x,t) charge density

density matrixP( P ) energy density of radiation

P s density matrix of heatbath

13,1177matrix element of density matrixreduced density matrix

a electric conductivitytime variable, lifetime of excited stateintegration variable

9) phasecon ; I con > eigenfunction of unperturbed HamiltonianciP t; 9) 1spin functions4)(x, t ); I OW> general solution of Schrodinger equation(1)./random phase lag

on; Ion> eigenfunction of unperturbed Hamiltonian95o; Ich> vacuum stateco.; 10.> coherent state4,(x , t); 11,40> general solution of Schrodinger equation1,44 IP + (x ) annihilation and creation operators for fermion fields

1 14, > eigenfunction of unperturbed Hamiltonianhydrogen wave function

x> state vectorfrequencyplasma frequencygeneral operator

4; 2nn, matrix element of general operatorS-2 +adjoint operator

= 2irv circular frequencyw

9.• w(j) eigenvalues of general operators

J

(7); ("*I0 center frequency(4 ), circular frequency of modewik (W) — Wk)/h transition frequency j —+ k

1. What is

(A brief exof this boo

1.1. The wa

Milk theour eyes. Tthat the ph3many centt:Newton (16.light consistwhich propFthe other lirSince thesedisputes conthat both ccof the object

1.2. Geomet

Our daily exWhen we pua point-likeexperimentstraight line:

*It is not thdevelopments wI always presentthe way to our

•

•

lator

1. What is light?

(A brief excursion into history and a preview on the contentof this book)

1.1. The wave-particle controversy

Most of the information we receive from our surroundings passes throughour eyes. This information is carried by light. Thus it is not astonishingthat the physical nature of light has been a subject of scientific study formany centuries.* Two important and contrasting concepts are due toNewton (1643-1727) and to Huygens (1629-1695). According to Newton,light consists of individual particles which are emitted by light sources andwhich propagate through space in straight lines. According to Huygens onthe other hand, light is described by waves quite similar to water waves.Since these two concepts seemed so different there were serious scientificdisputes concerning them. Quite surprisingly, however, we know nowadaysthat both concepts are correct in a well-defined way, and indeed it is oneof the objectives of our book to elucidate this point.

1.2. Geometrical optics

Our daily experience is rather adequately described by geometrical optics.When we put an opaque obstacle into the pathway of light stemming froma point-like light source, we obtain a clear cut shadow (fig. 1.1). Thisexperiment and similar ones lead us to the idea that light propagates instraight lines in the same way as freely moving particles do. When a light

*It is not the purpose of my presentation to give a complete outline of the historicaldevelopments which eventually led to the modern quantum theory of light and matter, nor doI always present the development of ideas in exact historical order. I rather wish to illuminatethe way to our present understanding of the nature of light.

on fields

2 1. What is Light?

•

Light source object screen

Fig. 1.1. Propagation of light along straight lines; an example from every day life.

tarbeam is retieincident bearsame plane,equals that bclose to a swirefraction (filthe normal tiand the raticrefracted bea

As we knccase of the vference. Rotaobstacles or I

13. Waves

mirrorFig. 1.2. The law of reflection allows us to construct the mirror image.

The decisive

scree d otake Yo

(fig. 1.4). IEresult in dad,non well knoin detail in sc

Fresnel (Ifto phenomerthese light wzequations oflowed for soIn particular.from purely t

a

observeddepth

realdepth

Fig. 1.3. The law of refraction. A swimming pool appears shallower.

(DVA(PL

Fig. 1.4. YoungPhil. Trans. Ro

•

>-

Fig. 1.4. Young's double slit experiment. Figure from his original paper. [1'. Young, 1802,Phil. Trans. Roy. Soc. 12, 387.]

rhallower.

§1.3 Waves 3

beam is reflected by a mirror, we may easily check the law of reflection:incident beam, reflected beam, and the normal to the mirror lie in thesame plane, and the angle between the reflected beam and the normalequals that between the incident beam and the normal (fig. 1.2). Standingclose to a swimming pool we immediately realize the validity of the law ofrefraction (fig. 1.3). According to this well-known law, the incident beam,the normal to the surface, and the refracted beam lie in the same plane,and the ratio of the sine-functions of the angles of the incident and therefracted beam equals that of the indices of refraction of the two media.

As we know, geometrical optics can only be considered as a limitingcase of the wave picture insofar as we can neglect diffraction and inter-ference. Roughly speaking this approximation holds if the dimensions ofobstacles or mirrors etc. are large compared to the wavelength of light.

13. Waves

The decisive step towards a decision in favour of the wave picture wastaken by Young (1801). He let light pass through two slits in an opaquescreen and observed the distribution of light intensity on a second screen(fig. 1.4). His observation showed clearly that light added to light mayresult in darkness, or in other words, he observed interference, a phenome-non well known in the case of water waves. (We will discuss his experimentin detail in sections 2.1 and 2.2.)

Fresnel (1819) applied Huygen's principle (fig. 1.5) in an improved formto phenomena of diffraction and interference. But what was the nature ofthese light waves? When Maxwell (1831-1879) established his fundamentalequations of electromagnetism it soon turned out that his equations al-lowed for solutions describing the propagation of electromagnetic waves.In particular, the propagation velocity of these waves could be calculatedfrom purely electromagnetic quantities and this velocity was exactly that of

)rn every day life.

-or image.

O0 I 2 3 4

SEPARATION, orb units0 HOPAQUE

PL ATE SCREEN

i0FRINGEPATTERN

0 60

a 06

= o4

7..; 0.2SOURCE

4 1. What is Light?

Fig. 1.5. Visualization of Huygen's principle. Each space point hit by a wave becomes astarting point of a new spherical wave. By interference of these spherical waves a new totalwave is constructed. For visualization we have chosen only a few discrete points as startingpoints.

light. A detailed study of these equations also revealed how such electro-magnetic waves could be generated, namely by accelerated (or decelerated)charges or in particular by oscillating charges and currents (fig. 1.6).Following up these ideas, Hertz (1888) soon made his fundamental dis-covery that electromagnetic waves were produced by an electric oscillatorcircuit. The idea that light is composed of electromagnetic waves hadalready been adopted by the end of last century. Light represents just anarrow band of a frequency range, which extends from the highest fre-quencies of y-rays down to comparatively low frequencies of radio waves.

1.4. The oscillator model of matter

While it had been known for a long time that the laws of reflection andrefraction could be derived from wave theory by means of Huygen'sprinciple, the index of refraction was still to be derived theoretically. Here,the model of Drude and Lorentz proved most useful: A piece of trans-parent matter was assumed to be composed of atoms whose electrons wereelastically bound to their nuclei (fig. 1.7). Thus each electron was describedas an oscillator which was forced to oscillate under the action ofthe incident electromagnetic wave. Depending on the ratio between thefrequency p of the incident wave and the natural frequencies of theoscillators, the latter could follow the driving force to a greater or lesserdegree. Or, in other words, the oscillator amplitudes became v-dependent.As can be seen from fig. 1.8, each oscillator represents an oscillating dipolemoment. According to Maxwell's equations, each of these oscillatingdipoles emits electromagnetic waves with the frequency v with which theoscillator is driven by the incident field. All these newly emitted wavesmust be superimposed on the incident wave. In this way a new waveresults which moves more slowly with a velocity v = c/n, where n is the

Fig. 1.6. RadiThe individualfield strength.dipole.

index of reshows, thedependenceextended todifferent elttion of disp

§1.4 The oscillator model of matter 5

wave becomes araves a new totaljoints as starting

such electro-r decelerated)nts (fig. 1.6).damental dis-:tric oscillatoric waves hadresents just ae highest fre-• dio waves.

I/2

eflection andof Huygen's;fically. Here,ece of trans-lectrons werevas describedle action ofbetween thencies of theater or lesser',dependent.llating dipole,e oscillatingth which thenitted wavesa new wavehere n is the

b I

Fig. 1.6. Radiation field of the Hertzian dipole. The dipole is indicated by the vertical bar. (a)The individual pictures show from top to bottom the evolution of the field lines of the electricfield strength. (b) Field lines of magnetic induction in horizontal plane through center ofdipole.

index of refraction and c the velocity of light in vacuum. As the theoryshows, the ',dependence of the oscillator amplitudes causes the v-dependence of n: n(v) (fig. 1.9). This oscillator model can be easilyextended Jo anisotropic crystals: One has only to assume that there aredifferent electronic oscillators with elastic forces depending on the direc-tion of displacement of the electrons. Such a microscopic model, together

6 1. What is Light?

Fig. 1.7. Oscillator model of matter. Each atom of the crystal is represented by an individualoscillator.

with Maxwell's equations, allowed for an adequate theoretical treatment oflight propagation in anisotropic transparent media. The emission andabsorption of light by atoms was also described within the framework ofthe oscillator model. To treat emission, one calculates the field generatedby an oscillating dipole moment. To treat absorption, one considers theenergy transfer of the field to an oscillatory dipole. The term "oscillatorstrength", still used in spectroscopy, stems from such a model. But why isthe same oscillator model capable of simultaneously describing dispersion,i.e. the response of a transparent medium, to the incident field, andabsorption, i.e. the response of an opaque medium to the field? The answeris easily given: In the case of dispersion, the frequency of the incidentwave is different from the natural frequencies of the oscillators, in theother case, the incident wave is in resonance with the oscillators. As

dipole moment

D .-eq

Fig. 1.9. Thew0 is the natudamped. (to

spectroscorcourse, als(spread, or,width cantors, whose

?774 +!

(q: displaodamping a

After sucamplitude

Fig. 1.8. Model of individual atom. Left part — indicates the electric field strength of a lightwave impinging on the atom; Right part — displacement of the electron from the positivecenter is denoted by q. Fig. 1

§I.4 The oscillator model of matter 7

Fig. 1.9. The index of refraction plotted as a function of the frequency of the incident field.coo is the natural frequency of the oscillator. In this picture it is assumed that the oscillator isdamped. (o 2vv)

spectroscopy shows the frequencies of light emitted from atoms, (and, ofcourse, also from other light sources) are not sharp, but show a certainspread, or, in other words, a finite linewidth is observed (fig. 1.10). Thiswidth can often be accounted for by a damping of the individual oscilla-tors, whose equations of motion thus read:

mq + r4 + fq = —eE (1.1)

(q: displacement of an electron, m: mass of electron, —e: its charge, T:damping constant, f: Hooke's constant, E: field strength).

After such a damped oscillator is excited, it generates a light wave whoseamplitude decreases exponentially. When such a wave is decomposed into

WoFig. 1.10. Example of a spectral line. The intensity 1(4.) is plotted versus w.

xi by an individual

al treatment ofemisgon and

3 framework offield generated3 considers theerm "oscillatordel. But why is)ing dispersion,ent field, andId? The answer)f the incidentillators, in theoscillators. As

lent

X

strength of a lightfrom the positive

8 1. What is Light?

A E (t)

Fig. 1.12. Thehypothesis.

describing tiand which ishe representinto the box.an individulthermal enerBut to obtaiian assumptiquantize theW of an oscspecifically,n = 0, 1, 2, ..oscillator (fitthe classical

Let us ncRutherford vthe followintwhich is ne.distances insatoms are attelectrons ch.(ing this to btand are acceemit radiaticmotion. But

Fig. 1.11. The electric field strength of light from thermal sources. The field is composed ofdamped wave trains with random phases.

waves with sharp frequencies z', e.g. by a spectrograph, the distribution ofthe intensity l( p) over v is measured, and the width Ay is proportional tothe damping constant F. Since any light source consists of many atoms, (orin our model oscillators), we have to superimpose on each other the lighttrains emitted by the individual oscillators. In the conventional lightsources the individual oscillators are -excited in an uncorrelated fashion, sotheir wave trains are also uncorrelated, i.e. the field amplitudes possessrandom phases (cf. fig. 1.11). The light we observe resembles a plate ofspaghetti (or a box of chinese noodles), it is by no means just a pure,infinitely extended sine-wave. Of course, it is important to find a quantita-tive measure of the degree of "interruptedness" or, in physical terminologyof "incoherence". Expressed more positively we wish to find a measure forthe "degree of coherence". We will come back to this problem extensivelylater in this book (cf. sections 1.11, 2.2 and ch. 8).

In spite of the success of the oscillator model, serious difficulties soonarose. Just to mention two of them: (i) Planck (1900) found that theconventional oscillator model led to results in conflict with the observedthermal properties of light; (ii) Rutherford (1911) discovered a physicalstructure of atoms which differs basically from that of oscillators. Let usdiscuss these problems more closely.

15. The early quantum theory of matter and light

Planck tried to unify the theory of light with that of another great branchof physics, namely thermodynamics. He wanted to derive a formula

§1.5 The early quantum theory of matter and light 9

t

ield is composed of

distribution ofproportional tonany atoms, (orother the light

wentional lightated fashion, sollitudes possessibles a plate of.ns just a pure,find a quantita-cal terminologyd a measure for)1em extensively

difficulties soonfound that theth the observedered a physicalcillators. Let us

er great branchrive a formula

W3 = 3hv

= 2hv

= hv

Wo = 0

Fig. 1.12. The quantized energy levels of a harmonic oscillator according to Planck'shypothesis.

describing the energy density of the light field which is enclosed in a boxand which is in thermal equilibrium with the walls of that box. To this endhe represented the light field as a superposition of standing waves fittinginto the box. He made a model in which he coupled each of these waves toan individual oscillator kept at temperature T and derived the meanthermal energy of the light wave by means of thermodynamic arguments.But to obtain a formula consistent with experimental data he had to makean assumption which was quite revolutionary at his time. He had toquantize the energy of the oscillators, i.e. he had to assume that the energyW of an oscillator can have only discrete values, Wo, WI , W2, . Morespecifically, he had to assume that these energies are given by W„ = nhv,n = 0, 1, 2, ... , where h is Planck's constant and v the frequency Of theoscillator (fig. 1.12). (We will present Planck's theory in section 2.3.) Thusthe classical oscillator model had to be abandoned.

Let us now talk about the second failure of the oscillator model.Rutherford was led by his experiments on a-particle scattering by atoms tothe following conclusion. An atom contains a positively charged nucleuswhich is nearly pointlike and Coulomb's law is valid till very smalldistances inside the atoms. Therefore he assumed that the electrons ofatoms are attracted by the nucleus by Coulomb's force. As a consequence,electrons circle around nuclei as planets circle around the sun. But assum-ing this to be so, further difficulties arose: Since the electrons are charged,and are accelerated all the time due to their circular motion they shouldemit radiation with a frequency equal to the frequency of their circularmotion. But since, according to classical mechanics, any frequency is

hv

hv

hv

10 1. What is Light?

possible, the atoms should emit a continuum of frequencies. We all knowthat the contrary is true. Atoms emit quite characteristic discrete spectrallines. Furthermore, as they give up their energy to the radiation field', theelectrons should come indefinitely close to the nucleus and eventuallycollide with it (fig. 1.13). This is in contrast to the experimental finding thatatoms have a definite size. To overcome these difficulties, Bohr (1913)extended Planck's ideas on quantized energy levels of oscillators to quan-tized energy levels of atoms. He assumed like Rutherford that an electronin an atom orbits its nucleus according to classical mechanics, i.e. like aplanet around the sun, but attracted by Coulomb's force, —e2/(4re0r),r = electron—nucleus distance. However, in addition he made his revolu-tionary postulates. The electron can move only along certain well-definedorbits with discrete energies Wn , and does so without radiating light. Itradiates light when it jumps from one orbit n to another one, n', emittinglight with frequency

(1.2)

where h is again Planck's constant (fig. 1.14). To bridge the gap betweenquantum physics and classical physics he formulated his correspondenceprinciple: At high enough quantum numbers, n, the effects predicted byquantum theory must approach those of classical physics. This, togetherwith his postulates, allowed him to calculate the energy levels Wa of thehydrogen atom explicitly and his model was consistent with the observedspectral series of hydrogen.

So far in this section we have been concerned with atoms. But simultan-eously, at the beginning of this century, new ideas on the nature of lightevolved. An important step was the explanation of Einstein (1905) of the

Fig. 1.13. An electron orbiting the nucleus performs an accelerated motion and should all thetime emit light according to classical electrodynamics. Because it must give up its energy itcomes indefinitely close to the nucleus.

Fig. 1.14. Anstationary orbita photon is em

photoelectri(consisted ofphotons of ep. But whattation? Weintroduction

Later, Eii"quantum oderivation irelectron. Eacorrespondiithe electron

= W2 —three diff ere(a) Initiallyemit a phot(and n photcadditional p2 stimulated

Fig. 1.15. Scheelectrons. Thefrequency of th

§1.5 The early quantum theory of matter and light 11

icies. We all knowic discrete spectralradiation field', the.us and eventuallymental finding thatlilies, Bohr (1913)ncillators to quan-rd that an electron‘chanics, i.e. like arce, — e2/(4Treor),3 made his revolu-ertain well-definedradiating light. It

one, n', emitting

3ht

ition and should all theat give up its energy it

Fig. 1.14. An illustration of Bohr's postulate. An electron moving around its nucleus on astationary orbit with energy W2 may suddenly jump to another orbit with energy W1 wherebya photon is emitted.

photoelectric effect (fig. 1.15 and section 2.4). He assumed that lightconsisted of particles, photons. According to these ideas, one associatesphotons of energy hi' and momentum hv/c with a light wave with frequencyv. But what does the word "associate" mean in a precise physical interpre-tation? We will come back to this highly crucial point later in thisintroduction and again and again in this book.

Later, Einstein (1917) gave his second fundamental contribution to"quantum optics" when he rederived Planck's law. (We shall present thisderivation in section 2.5.) Einstein considered a set of atoms each with oneelectron. Each electron can move in either of two orbits 1, or 2, with thecorresponding energies W1 and W2 . When jumping from 1 to 2 or 2 to 1,the electron is assumed to absorb or to emit a photon with energyhi' = W2 - W1 . To derive Planck's law properly, Einstein had to introducethree different processes:(a) Initially the electron is in its upper level and no photon present: It canemit a photon spontaneously. (b) Initially the electron is in its upper leveland n photons of the "same kind" are present: The electron can emit anadditional photon by stimulated emission. (As we will see later in Volume2 stimulated emission is fundamental for the laser process.) (c) Initially the

light electrons

metal surface

Fig. 1.15. Scheme of photo-electric effect. Light impinging on a metal surface can freeelectrons. The kinetic energy of the electrons does not depend on the light intensity but on thefrequency of the light.

(1.2)--

;e the gap betweenus correspondenceffects predicted bysics. This, togethery levels wn of thewith the observed

oms. But simultan-the nature of lightstein (1905) of the


electron is in its lower state and n photons are present: The electron canabsorb a photon and jump to its upper level.

To describe the number of jumps the electrons make per second betweenlevels 1 and 2 in connection with the different processes (a), (b), (c),Einstein introduced certain rate constants, now called the Einstein coeffi-cients, phenomenologically.

Thus, within less than two decades, the theoretical description of atomsand the light field changed dramatically. However, theory was still not in asatisfactory state. On the one hand, many of the laws of "classical" physicswere still considered valid down to the atomic level, for instance when theelectron orbits were calculated. But the new quantization rules had to beapplied in addition. It turned out that more atomic quantum numbers than"n" were needed, and Sommerfeld developed additional quantizationrules. Similarly, Einstein's postulates were added to the laws of classicalphysics. Eventually it even became evident that entirely new types ofquantum numbers (half-integers) had to be introduced. Thus physicistsbecame more and more aware of the fact that this kind of procedure couldonly be an intermediate step and that the "true" quantum theory had stillto be discovered. (In a way the present situation in elementary particlephysics is reminiscent of this picture. Here quantum theory (see below) isapplied, but has again to be supplemented by rules which carry a similar"ad hoc" character.)

and momenp h/

to a light w.the ingenio,electrons. HA wave witlof a free ele

= E/

We will cona later secti(ger gave aDebye said:enough, Sci.the introducto the explastructure ofcrystalline s,applied to tstationary stof electronshad to be abwave functk

1.6. Quantum mechanics electronic "(We will g,

The breakthrough was achieved by Heisenberg (1925) and Schrodinger in this book(1926) working independently of each other. Heisenberg invented "matrix IP dependsmechanics", while Schrodinger formulated quantum mechanics by means = t).of the by now fundamental Schrodinger equation. Shortly afterwards functions. SSchrodinger showed that his and Heisenberg's formulations were mathe- energy level:matically equivalent. We shall base our approach mainly on the Schro- In many (dinger equation, which we will derive and explain in detail (cf. ch. 3). As wave functiwe will see, the operator techniques introduced by Heisenberg also play an importanceimportant role in quantum optics. The Schrodimger equation is non- quantum oprelativistic but its extension to the relativistic case was given by Dirac only a few(1928). In order not to overload our presentation we will leave aside his for this purapproach in our book. harmonic os

Schrodinger's discovery is connected with an interesting anecdote. In differ fromorder to elaborate, I must first mention another important intermediate material. Restep: as we have seen above, Einstein attributed photons with energy mechanical

E = hv (1.3) gen atom, ca

§1.6 Quantum mechanics 13

electron can

'ond between(a), (b), (c),nstein coeffi-

ion of atoms; still not in a;ical" physicsnce when theles had to belumbers thanquantizations of classicallew types ofus •.cists)ced ouldeory had still-itary particle'see below) islay a similar

Schrodingerented "matrixlies by meansly afterwardswere mathe-

m the Schro-cf. ch. 3). As

g also play anIlion is non-'en by Dirac

3ave aside his

anecdote. Inintermediateenergy6,1.3)

and momentump h/X

to a light wave with frequency v and wavelength X. De Broglie (1924) hadthe ingenious idea to apply these relations to particles, for instance toelectrons. He interpreted these relations so to speak, in the "opposite" way.A wave with frequency 1, and wavelength X is now attributed to the motionof a free electron with energy E and momentum p by the relations

= E/h, X = h/p. (1.5, 6)

We will come back to de Broglie's idea and its experimental verification ina later section (cf. section 3.1). But here is the anecdote: When Schrodin-ger gave a seminar talk on de Broglie's theory at the ETH in Zurich,Debye said: "When there is a wave, there must be a wave equation!" Sureenough, Schrodinger derived this equation a short time later. Soon afterthe introduction of the Schr6dinger equation, quantum theory was appliedto the explanation of many fundamental properties of matter, such as thestructure of atoms and molecules, and the electronic and atomic states incrystalline solids. Later it turned out that similar approaches could also beapplied to nuclei and amorphous solids. The emphasis was primarily onstationary states which are the quantum mechanical analogues of the orbitsof electrons in the sense of Bohr. It turned out that the concept of an orbithad to be abandoned. It was replaced by a new mathematical quantity — thewave function 4 which, loosely speaking, can be visualized as describing anelectronic "cloud" (fig. 1.16).

We will get to know the precise meaning of these wave functions 1p laterin this book (ch. 3). In the case of a single particle, for instance an electron,IP depends on the coordinate x of that particle and on time t, so that

= 4/(x, t). The solutions of the Schrodinger equation are just these wavefunctions. Simultaneously, the SchrOdinger equation fixes the quantizedenergy levels Wn.

In many cases it is important to know the details of such energies andwave functions. On the other hand, those details are not of centralimportance for an understanding of the basic methods and results ofquantum optics to which this book is devoted. Indeed we shall see thatonly a few general properties of energies and wave functions are neededfor this purpose, with one exception, namely the quantum mechanicalharmonic oscillator which we will treat in detail. Therefore, our text willdiffer from usual texts on quantum mechanics by leaving aside suchmaterial. Readers who wish to learn about this or that detail of quantummechanical wave functions and energies, for instance those of the hydro-gen atom, can find them in any standard textbook on quantum mechanics.

(1.4)

14 I. What is Light?

the center of grioscillates back r1.17). Thus thesurprising way,book.

The new quastimulated emiss,("spaghetti-like"or 2 of an ense,electrons in the (give a precise deand 4).] In thisEinstein coeffici(incident light watomic transitiooscillator modelin the coilkofwave pa ccelectron.

Since laser ligto observe a varthe "photon ccdescribed in thisit treats the elewhile matter, irdinger equation.

Fig. 1.16. How to visualize an electron cloud in quantum mechanics. The density of dotsmeasures the density of the electron. For a more correct interpretation see below when wediscuss Born's probability interpretation.

1.7. An important intermediate step: The semiclassical approach

As I mentioned above, the old classical oscillator model of atoms had beenquite successful in treating the absorption or emission of light by atoms aswell as dispersion, though classical mechanics eventually could not provideany sound justification of that model. So it was one of the first tasks of thenewly developed quantum mechanics to treat this problem again. To thisend, one studied the response of electrons, described by the Schrodingerequation, to a prescribed classical electromagnetic field, in particular, to awave. As we will demonstrate in chapter 4, the answer can be described asfollows. In the case of dispersion, i.e. when the frequency v of the incidentwave differs from the frequency v' =- (W2 — W1 )/ h of the electronic transi-tion, the following happens: Under the influence of the field, the electronleaves its stationary state 1, which is represented by a certain wavefunction p i . The motion of the electron is now described by a superposi-tion of wave functions, e.g. by

Ip(x, t) = c l (t)so i (x) + c2(t)so2(x), (1.7)

i.e. by a "wave packet", where Cp2 is the wave function of the excited stateof the electron, and c 1 and c2 are time dependent factors. As we will see,

Fig. 1.17. Examplegenerally a complerthe dot-dashed cursNO is cleairieen.

§1.7 An important intermediate step: The semiclassical approach 15

the center of gravity of the charge density described by this wave packetoscillates back and forth as if the charge were that of an oscillator (fig.1.17). Thus the original oscillator model finds its justification in quite asurprising way, whose details we will learn about in the course of thisbook.

The new quantum mechanics were also able to treat absorption andstimulated emission of light by atoms. Under the influence of incoherent("spaghetti-like") light, the mean number of electrons in the initial state 1or 2 of an ensemble of atoms decreases, while the mean number of theelectrons in the other state 2 or 1, increases proportionally to time. [We willgive a precise definition of these "mean numbers" later in this book (chs. 3and 4).] In this way it became possible to calculate the correspondingEinstein coefficients from first principles. Quite new effects occur when theincident light wave is completely coherent and in resonance with theatomic transitions, i.e. vfield •='- (W2 - W1 )/h. According to the classicaloscillator model, the oscillator amplitude should increase more and morein the course of time. Quite on the contrary, in quantum mechanics againwave packets eq. (1.7) are formed describing an oscillatory motion of theelectron.

Since laser light has a high degree of coherence, it has become possibleto observe a variety of these highly interesting new phenomena, especiallythe "photon echo" which we will treat in section 4.8. The approachdescribed in this section is often called the semiclassical approach. Indeed,it treats the electromagnetic field in the framework of classical physics,while matter, in particular electrons, is treated by means of the Schro-dinger equation, i.e. quantum mechanically. In this way we can adequately

Fig. 1.17. Example of the oscillation of a wave packet 4, = cop, + c 2p2 . Because 4, isgenerally a complex function we have plotted 1%1,1 2 . The solid curve applies to the initial time,the dot-dashed curve refers to some later time. The displacement of the center of gravity of1 11, 1 2 is clearly seen. The position of this center oscillates as a function of time.

•The density of dotssee below when we

'roach

f atoms had beenlight by atoms as:ould not provide; first tasks of the.m again. To thisthe Schrodinger

n particular, to abe described as

v of the incidentelectronic transi-ield, the electrona certain wave

1 by a superposi-

(1.7)

the excited state

A5will see,

Aak,//7-\\\,,/,,-..,,, ,v V, \ ,,,,,,\\ ...,

mirror mirror

\ I\ /\ /\ /n


describe absorption and stimulated emission. However, this kind of ap-proach does not allow us to theoretically describe any spontaneous emis-sion of atoms. This strongly indicates that a major ingredient of the theoryis still missing. We will find this missing ingredient when we quantize theelectromagnetic field.

1.8. Quantization of the electromagnetic field: Quantum electrodynamics(QED)

We have mentioned above that some experiments, such as the photo-electriceffect, can only be explained by the assumption that light consists ofparticles, photons, which carry quanta of energy. Quantum mechanicsmade it possible to calculate quantized energy levels for particles, e.g. theelectrons. We may therefore ask whether these theoretical methods allowus also to quantize the light field. The basic idea for the solution of thisproblem is the following.

As usual we describe the light field by its electric and magnetic fieldvectors. Let us consider the electric field vector in a given direction, say thez-direction, as an example, i.e. Ez(x, t) which is attached to each spacepoint x and time t. We may decompose Ez into a superposition of waves,for instance of waves of the form of standing waves (fig. 1.18)

E(x , t) = q(1) sin kx (1.8)

in a cavity, i.e. a metal box. We know that for free fields q(t) represents aharmonic oscillation

q(1) = A sin(cot) (1.9)

Fig. 1.18. Example of a standing wave of the electric field strength between two mirrors. Thesolid line and the dashed line represent the field strength at different times.

or in otherHowever, qoscillator b

Wn =

where n isphotons wirepresentspresent, tindimensionsquantizingch. 5.) No,which bothare quantizannihilatioithe former

mwe ll"

•csexperiences"feel" thea transitiorsame time.emission arquantized;duction orbe composwaves seen

The pictuntil thelong overstprocesses.aspects ofthat evenwhich aremade it pcwith the ftthe particlgreat triurrmore detaitoo).

•

§1.8 Quantization of the electromagnetic field: Quantum electrodynamics 17

is kind of ap-ntaneous emis--it of the theoryye quantize the

iectrodynamics

photo-electric;ht consists ofurn mechanicsrticles, e.g. themethods allow,olution of this

.ectilllty thema field

to each spacetion of waves,

(1.8)

) represents a

(1.9)

two mirrors. The

or in other words, q(t) obeys the equation of the harmonic oscillator.However, quantum mechanics has taught us how to quantize the harmonicoscillator by means of the Schrodinger equation. The energy levels are

ffin = nhv + const (1.10)

where n is an integer, n = 0, 1, 2, ... . n can be interpreted as the number ofphotons with which the wave sin kx is occupied. The "constant" in (1.10)represents the zero-point fluctuations of the field: even if no photons arepresent, the field fluctuates all the time. If we quantize all waves in threedimensions, and include the magnetic field as well, we shall succeed inquantizing the light field. (We will present the quantization procedure inch. 5.) Now we have reached a rather satisfactory level of approach inwhich both the motions of electrons and the oscillations of the light fieldare quantized. In particular, the new formalism describes the creation andannihilation of photons. By this new approach we can reproduce not onlythe former results of absorption and stimulated emission but in additionwe find an adequate treatment of spontaneous emission. In quantummechanics the light field never vanishes entirely, rather it constantlyexperiences zero-point fluctuations. Because of these, electrons always"feel" the light field, which may then cause an electron in an atom to makea transition to its ground state, emitting a photon spontaneously at thesame time. This theory allows one to calculate all Einstein coefficients ofemission and absorption from first principles. From the point of view ofquantized fields, these coefficients are rate constants describing the pro-duction or annihilation rate of photons. In this way light is considered tobe composed of particles and no space for any coherence properties ofwaves seems to be left.

The picture of light as photons dominated the quantum theory of lightuntil the advent of the laser. It appears nowadays that, in fact, physicistslong overstressed the notion of individual photons in the treatment of suchprocesses. The problem of reconciling the photon point of view with theaspects of wave optics and coherence was largely ignored. In fact we knowthat even spontaneously emitted light has certain coherence propertieswhich are manifestations of its wave properties. Furthermore, the laser hasmade it possible to produce coherent light. Thus we are again confrontedwith the fundamental question of how to reconcile the wave concept withthe particle concept. The solution of this problem is indeed one of thegreat triumphs of quantum theory which we will now discuss (and in much •

more detail in our book taking into account the more recent developments,too).

•


1.9. The wave—particle dualism in quantum mechanics

Since the answer to this problem was first given in quantum mechanics, letus return to the seemingly quite simple problem of the motion of a freeparticle. When we describe it according to classical physics we visualize it,of course, as an extensionless point mass, i.e. the particle can be localizedup to any degree of accuracy. At the same time we can measure its velocity(or momentum) precisely. This assumption is indeed the basis of Newto-nian or Hamiltonian mechanics. On the other hand, describing a particlequantum mechanically by means of a wave function means quite adifferent thing. According to de Broglie's fundamental assumption, (cf.section 1.6) in quantum mechanics we attribute an infinitely extendedwave

Ip(x) = A exp(2Trix/X — 2 .7rivt) (1.11)

to a freely moving particle with momentum p so that X = h/p. The wavemust be of infinite extent, otherwise it would not have a definite wave-length, but an infinitely extended wave does not determine any definitespace points (fig. 1.19). The localization of the particle has been lostentirely. As is known from wave theory, by a superposition of plane waveswe may form "wave packets" which are concentrated around certain spacepoints, thus describing the localization of a particle. But when building upa wave packet we must use a variety of wave lengths or wave vectorsk = 2Tr/X and thus, according to X = h/p, a variety of momenta p.

As Heisenberg has shown there is a fundamental limit connecting thespread Ax, called the uncertainty of the space coordinate x and Ap, theuncertainty of the momentum p. Heisenberg's uncertainty relation reads

AxAp h/2. (1.12)

This tells us that any measurement which determines the particle coordi-nate x within a certain accuracy ix excludes a measurement of themomentum p to a better accuracy Ap than given by eq. (1.12) (cf. fig. 1.20).

Fig. 1.19. The absolute square of the plane wave function plotted versus x. No space point xis distinguished from any other.

•

Fig. 1.20. Hoparts show thupper and lopackets. (a) Uuncertainty A

The uncalso comeit impliesinstance, bwe inducetainty ofknowledgemotion caldiscussionwe can de•

§1.9 The wave—particle dualism in quantum mechanics 19

(a)

x. No space point x

Ap

-Ak

(b)

Ap

-Ak

Fig. 1.20. How to visualize Heisenberg's uncertainty relation (1.12). The left upper and lowerparts show the absolute square, 101 2 , of the wave function 11., of the wave packet. The rightupper and lower parts show the spread of k-values which is needed to build up the wavepackets. (a) Upper part: large uncertainty ix and small uncertainty tip. (b) Lower part: smalluncertainty x and large uncertainty Ap.

The uncertainty relation has far reaching consequences which we willalso come across again and again in the quantum theory of light. Namelyit implies that the observer perturbs the system by observing it. Forinstance, by a precise observation of the localization point x of the particle,we induce a complete uncertainty in its momentum. The resulting uncer-tainty of the particle's momentum is by no means just a lack of ourknowledge. After the measurement of the particle's position its futuremotion cannot in principle be predicted. We can also put the results of thisdiscussion in other words. Namely by choosing an appropriate experimentwe can determine, for instance, the position of the particle within the

im mechanics, letmotion of a freecs we visualize it,can be localized

;asure its velocitybasis of Newto-

cribing a particlemeans quite a

assumption, (cf.finitely extended

(1.11)

= h/p. The wavea definite wave-lin* definitele een lost,n of plane wavesInd certain spacevhen building upor wave vectors,mentap.t connecting thee x and Ap, therelation reads

(1.12)

particle coordi-surement of the12) (cf. fig. 1.20).

•

•


desired degree of accuracy, i.e. at our will. At the same time, however, weinterfere with the further temporal development of the quantum systemand thus cannot make exact predictions about the momentum which is avariable complementary (or "conjugate") to the space variable. Is it reallytrue that we cannot make any statement on that complementary (con-jugate) variable? Born found the correct answer to this problem. Accordingto him we can make only probability statements. Let me explain this inmore detail — the simplest access to this probability statement is by meansof the wave function (x, t) itself. According to Born,

10(x,t)1 2 dx (1.13)(one dimensional example) gives us the probability of finding the particlein the region x x + dx (fig. 1.21). In a similar way we can give a preciserule for the probability of finding that the particle has a certain momen-tum. We shall derive and apply this probability interpretation in detaillater in this book (ch. 3, especially section 3.2, and many other sections).According to this interpretation, making quantum mechanical measure-ments is quite similar to throwing a die, and we will exploit this analogy inour later chapters.

The philosophical consequences of Bom's probabilistic interpretation ofquantum mechanics and of Heisenberg's uncertainty relation are probablymore incisive than even Einstein's theory of relativity. Owing to this,quantum theory versus a "classical interpretation" is still being checked bymore and more sophisticated experiments. A set of such experiments hasbeen devised to check "Bell's inequality". Its discussion is beyond thescope of our book, but the interested reader may find references to thecorresponding articles on page 333. So far all known experiments are in

accordancereason to dcments will cbut this is n

1.10. The w:

Let us retuncomposed oend, let us cwe observe zcomposed afully. We rerFurthermoreonly one ph(At a certainphoto Thisenera ph;W:r;the photonwith certaint.number of ation which is

d xFig. 1.21. The probability interpretation of quantum mechanics. The shaded area 1412dxgives us the probability of finding the particle in the range dx around that specific spacepoint.

Fig. 1.22. Left lution behind the stphoton-counters -number of events

IiSO

2 holes counters

number number

of events of events(normalized)

few events many events

§1.10 The wave—particle dualism in quantum optics 21

)wever, we1111 systemwhich is aIs it reallytary (con-Accordingamn this inby means

(1.13)

he particlee a precisein momen-n in detailr sect ns).1 e-ana in

, retation ofe probablyAg to this,2hecked byiments has)eyond theices to theents are in

area 102dxspecific space

•

accordance with quantum theory and in this author's opinion there is noreason to doubt its correctness at the atomic level. Of course, new develop-ments will certainly take place at the level of elementary particle physics,but this is not our concern here.

1.10. The wave-particle dualism in quantum optics

Let us return to our original problem, namely to the question, what light iscomposed of "in reality". Is it composed of particles or of waves? To thisend, let us consider again Young's double-slit experiment (fig. 1.22). Sincewe observe an interference pattern on the screen, we conclude that light iscomposed of waves. But let us study the interference pattern more care-fully. We replace the screen by a set of photon-counters (photomultipliers).Furthermore, we lower the light intensity so much, that a maximum ofonly one photon could be present at a time. Then the following happens:At a certain moment, one of the counters indicates that it was hit by aphoton. This event takes place in a very localized region, and indicates theenergy transfer of a single quantum. Repeating this experiment, each timea photon arrives, the counter is triggered. However, the position at whichthe photon arrives is statistically distributed, i.e. it cannot be predictedwith certainty. When we repeat this experiment many times, and plot thenumber of counts versus space, we eventually obtain an intensity distribu-tion which is the same as that of the classical interference fringes. Thus we

Fig. 1.22. Left half of figure: experimental set-up for measuring the spatial photon distribu-tion behind the screen. Right half of figure: The number of events indicated by the individualphoton-counters in case of a small total number of events and in the case of a very big totalnumber of events (schematic).

screen with

0

0

0

0

0photon

11•n••nnlightsource

•


are inclined to say: clearly, the particle aspect is more fundamental! But iflight is composed of particles, we must be able to follow up the path ofeach particle.

So far we have had two holes in the screen (double slit experiment). Totrack the path of the particles, i.e. the photons, we just close one of the twoholes and repeat the experiment. Again and again one of the photoncounters clicks, but when we eventually plot the intensity distribution,quite a different picture results, namely no interference pattern occurs (fig.1.23). Thus by our different experimental arrangement (closing one hole)we have entirely changed the outcome of the experiment! On the otherhand, to obtain the interference pattern with two holes we must not closeeither of the two holes, i.e. in that case we cannot in principle track thepath of the photon. Here the wave character of light is decisive. Theseresults demonstrate that we cannot track the path of a photon, i.e. of aparticle, and simultaneously perform the double-slit experiment to provethe wave character of light. It can show the one or the other aspectdepending on the kind of experiment we perform. Note that in this contextthe notion of an experiment must be taken in a wide sense. For instance, aswe have seen, even closing a hole in the screen before we start sendinglight on it, must be considered as part of the experiment. By building upany experimental setup, for instance by installing a screen or photo-counters, we predetermine the outcome of the experiment! Note, however,that the outcome cannot be predicted with certainty, at least not ingeneral, but only in the probabilistic sense.

Fig. 1.24. TIevents.

In viewspontaneotheory, byemitted piemiiii) pim exan otmeasured,spontaneoquantumthe experi:case we ebelow). T

screen with1 hole

Fig. 1.23. Experimental set up to track the path of a single photon. In this case the formerinterference pattern has disappeared.

Fig. 1.25. Ticorrespondin

lightsource

•

§1.10 The wave—particle dualism in quantum optics 23

numberof events

measured photon energy hv

Fig. 1.24. The photon energy of spontaneously emitted light. Measurement of only a fewevents.

In view of these results we may also reconsider the physical properties ofspontaneously emitted light. In an experiment corresponding to Einstein'stheory, by which he rederived Planck's formula, we want to measure theemitted photons. More specifically, we want to measure the energy of theemitted photon for each emission act. This energy may differ from experi-ment to experiment (fig. 1.24). When we repeat this experiment many timesand plot the numbers of the events when a definite quantum energy ismeasured, versus energy, we find the classical frequency distribution ofspontaneously emitted light (fig. 1.25). Thus to make contact between thequantum mechanical picture and the classical theory we must either repeatthe experiment many times or, use an ensemble of identical atoms. In thiscase we assume that the atoms do not influence each other (see alsobelow). The ensemble then gives the classical intensity distribution. We

number of events

( normalized)

measured photon energy hv

Fig. 1.25. The same as in fig. 1.24 but in the case of very many events. The ordinate wascorrespondingly reduced.

mental! But ifp the path of

periment). Tone of the two

,f the photondistribution,

rn occurs (fig.ing one hole)On the othernust not closeiple track theecisive. These)ton, i.e. of anent to proveother aspectterntext

)ri ce, asstart sendingy building upen or photo-ote, however,least not in

case the former

•


shall present the detailed theoretical treatment of this problem in sections7.9 and 8.3.

Let us return for a moment to the classical oscillator model describingthe emission of radiation. To describe the observed linewidth (the naturallinewidth of the emission line), a damping of the classical oscillator waspostulated. As a result, a damped light wave was emitted. When such awave is decomposed spectroscopically into its frequency spectrum, a finitewidth results. From the spectroscopic point of view, such a wave has acertain degree of coherence.

Can we still speak of damping in quantum theory and in what sense?Can we still speak of coherence in quantum optics? To answer the secondquestion first, let us briefly discuss how we treat coherence in classicalphysics.

1.11. Coherence in classical optics and in quantum optics

The concept of coherence is closely related to the phenomenon of inter-ference. Let us therefore start this discussion by studying the way in whichinterference patterns arise. We consider an experimental arrangement bywhich we superimpose two waves on each other at a certain point x inspace (compare fig. 1.26). Let the waves 1 and 2 have the amplitudes Eland E2 , respectively. The individual intensities are I = E? and 12 =respectively. Interference patterns result from the fact that we should notadd the individual intensities of the two waves but rather their amplitudes,i.e. we have to first form E = El + E2 . The intensity of the total wave,which is measured by a photographic plate or a photomultiplier, is givenby

I = E 2 = + El + 2E,E2 . (1.14)

E 2

Fig. 1.26. Two waves E 1 and E2 being superimposed on each other within a region indicatedby the circle. Fig. 1.27. An exam}

which vanishes ever

In the case tit;may result indouble slit ex;ment for a s(course of tim,independentlyrelation E2 =intensity I nochanges or flunot exactly cothe quantity 2_

This leads uE2

can be considccoherence. Weis correct excelof thiliptiorexpress (1.1E E2 experime,can measure tForming thendone over a cactually measudenote by <E1,

What is impcthe degree of c,

region indicated Fig. 1.27. An example of negative interference. The wave amplitudes of Ei and E2 result in Ewhich vanishes everywhere.•

§1.11 Coherence in classical optics and in quantum optics 25

In the case that E2 = — E 1 , we evidently find I = 0, i.e. light added to lightmay result in darkness (fig. 1.27). At those points on the screen in Young'sdouble slit experiment we find dark fringes. Consider now such an experi-ment for a somewhat extended period of time and assume that in thecourse of time the waves E l and E2 change their amplitudes E, and E2independently of each other within a certain range. Then clearly therelation E2 = — E i cannot be maintained all the time and the totalintensity I no longer vanishes (figs. 1.28a, b). The independent temporalchanges or fluctuations of the amplitudes just mean that the two waves arenot exactly coherent. This incoherence is connected with the decrease ofthe quantity 2E 1 E2 as compared to E7 +

This leads us to the idea that the quantityE2 (1.15)

can be considered as an appropriate quantity for measuring the degree ofcoherence. We shall see in section 2.1 of this book that this interpretationis correct except for a slight modification which we will discuss at the endof this section. For our purpose it suffices, however, to consider theexpression (1.15) as the relevant physical quantity. How can we measureE1 E2 experimentally? To this end we have to use a beam splitter so that wecan measure the intensities /1 , 12 and I separately (compare fig. 1.29).Forming then —I — I2 ) we obtain E 1 E2 . Because any measurement isdone over a certain time, for instance by the photographic plate, weactually measure in classical physics a time average over E 1 E2 , which wedenote by <E1E2

What is important in our present context is the fact that we can measurethe degree of coherence simply by measuring intensities. However, we can

E„„ ------- ,„

„s%„.i./ •Xo

N.N.

./.

E 2

m in sections

lel describing(the natural

)scillator wasWhen such atrum, a finite

wave has a

what sense?er the seconde in classical

no inter-wa whichrangement byin point x inimplitudesand 12 = E,

ye should notir amplitudes,ie total wave,plier, is given

(1.14)

X

•

(a)

(13)Fig. 1.28a, b. If in the course of time E l and E2 are not completely correlated, a nonvanish-ing sum E results.

measure intensities at the "quantum level" also, namely by measuringphoton numbers and their energies. This allows us to theoretically defineand experimentally measure the degree of coherence also in quantumphysics. The former time average of classical physics <E 1 E2 >, is then to bereplaced by an "ensemble average" which we shall come across later in thisbook in more detail (see especially ch. 8).

Our considerations can also be used to measure or to calculate thecoherence of a single field amplitude E(t). To this end we correlate thefield amplitude E at a time, to with the amplitude E at a later time, t2.Experimentally, this can be achieved, for instance, by a beam splitter and adelay line (cf. fig. 1.30). Identifying E(t i ) with E1 and E(t2 ) with E2 , whereE1 and E2 are the two amplitudes considered above, we can repeat all steps

Fig. 1.29. Schemsplitters, m mirroE1 , E2 and E occ

and introduc.coherence fur,neously emittifind theilrthe d h"ensemble av4

Is this agrtquantum phy.1.13.

As we haveare only preli2.1, 2.2) will swe decompos

Fig. 1.30. Possibltimes t i and t2.into two partial

§1.11 Coherence in classical optics and in quantum optics 27

photoncounters

Fig. 1.29. Scheme of a possible experimental arrangement to measure E1 E2 . 3 denotes beamsplitters, m mirrors. Note, that the optical path must be arranged so that no time lag betweenE1 , E2 and E occurs.

•

• a nonvanish-

measuring:ally definea quantum; then to belater in this

ilculate theorrelate theer time, t2.'litter and ah E2 , whereeat lir

and introduce E(t 1 )E(12 ) or, more precisely speaking, its average, ascoherence function. We shall compute this coherence function for sponta-neously emitted light fully quantum theoretically (cf. section 8.3). We willfind that the result agrees with that of the classical calculation based onthe damped harmonic oscillator model! Therefore, as long as we deal with"ensemble averages" we recover the classical results.

Is this agreement purely accidental or can we speak of damping inquantum physics also? We shall discuss this important question in section1.13.

As we have indicated above, the expressions for the coherence functionsare only preliminary. As a later detailed discussion in our book (sections2.1, 2.2) will show, we have to modify the definition somewhat. To this endwe decompose the electric field strength E into a positive frequency part

E 2

Fig. 1.30. Possible arrangement to measure the coherence of a single wave E at two differenttimes t l and 1 2 . The original wave is split by a beam splitter s, a delay line and a mirror minto two partial waves El and E2 which now represent E at different times.

•


E(+) and a negative frequency part E". For instance, in the case that Eoscillates harmonically in time with frequency w, Et+) and E" are of theform

Et = B e - "" , E= B* e''. (1.16, 17)

Now we can give the precise definition of the coherence function whichreplaces the former definition:

<E1(x,t)E2(x,t)>—* <Er)(x,t)E1+)(x,t)>

+ <E1-)(x,t)Ef+)(x,t)> (1.18)

Instead of the sum we introduce as a more fundamental quantity thefollowing coherence function:

<Er)(x,t)E1+)(x,t)>. (1.19)

1.12. Spontaneous emission and quantum noise

The spontaneous emission of photons by atoms is a beautiful example ofquantum noise. After we have excited an atom and then want to measurethe photon spontaneously emitted by it, for instance by means of aquantum counter, we cannot precisely predict the arrival time of thephoton. We can only make probability statements of the following kind:The photon will be counted (or will be present) at a given time t withprobability P(t) (which we will calculate later in our book). An ensembleof initially excited atoms which are assumed not to interact with each other(not even via the light field) so that they cannot emit light in a correlatedfashion will therefore emit a "rain" of photons whose arrival times arerandom (fig. 1.31).* Such a process is a typical example of noise. Because itis connected with the emission of quanta (i.e. the photons) we call itquantum noise.

Two features of this process are of particular importance. This noise isunavoidable and its stochastic nature is inherent in quantum physics. Thispoint must be particularly stressed because in some other branches ofphysics or mathematics it is sometimes assumed that noise is not basicallystochastic but only seemingly so, due to complicated underlying processeswhich in reality are deterministic. Though spontaneous emission is a causalprocess, it is by no means deterministic.

Furthermore, not only students, but even quite a number of scientistsare inclined to ignore the phenomenon of noise entirely or to assume that

'There are a number of very important cooperative effects when photons are emitted. Weshall discuss them in detail in the second volume.

Fig. 1.31.light. Show

it plays (

agntrht

laser. ItsThis is cphysicalvolume.

1.13. Dm

Let us cc1.11. Canconsidertron hascannot piwe canncthe lowei1.32). It isection 7.of the upmany atcoccupaticthe usual

N2(t

case that E-) are of the

(1.16, 17)

action which

measured

events

•

§1.12 Spontaneous emission and quantum noise 29

t 2 3 4 5 6

Fig. 1.31. An ensemble of atoms all initially excited at the same time spontaneously emitslight. Shown are the arrival times of photons measured by a photon counter.

it plays only a minor role. Spontaneous emission noise tells us that quite

11 Ale of the contrary is true. If that noise were not present we would sit in the dark.

it to measure All light sources emit this kind of noisy light, with one exception - the

means of a laser. Its light has features quite different from those of "ordinary" light.

time of the This is one of the reasons that makes the laser so interesting from the

!lowing kind: physical point of view and we will treat these properties in the second

time t with volume.

1.13. Damping and fluctuations of quantum systems

Let us come back to the question which we asked at the end of section1.11. Can we speak of damping in quantum physics also? To this end let usconsider again the spontaneous emission process of an atom whose elec-tron has been excited to its upper level. We have seen above that wecannot predict precisely at what time the photon will be emitted. Similarlywe cannot predict precisely when the electron will jump from the upper tothe lower level, but again we can make probability statements (cf. fig.1.32). It is indeed possible (and we will perform such calculations later insection 7.9) to calculate the probability with which the occupation numberof the upper level decreases. This means that considering an ensemble ofmany atoms allows us to study the decrease of the mean value of theoccupation number of the upper level. It turns out that this decrease obeysthe usual exponential law (fig. 1.33)

N2(1) = N2(0) e -27`. (1.20)

(1.18)

quantity the

(1.19)

An ensembleth each othera correlated

val times arese. Because it,$) we call it

This noise isphysics. Thisbranches ofnot basicallying processeson is a causal

of scientists) assume that

are emitted. We

Fig. 1.34. Scheminstance a light fi

heatbath at zcreases exponemitted at ramour present ccreases expon

The phenol-to thAlOces:that phfor a proper Ilaser physics .hardly any s:interacts witlheatbaths (hihave to takematical treat)growth of afrom thermal

1.14. Photon

The problemfundamentalconsider a sp,the form of a

E(x,t)

In the case o

q(t) =

(A and cp reaIn classical

intensity I cc

Fig. 1.32. Each time a photon is measured we conclude that the occupation number N2 , i.e.the number of occupied states of the atoms has decreased.

This "atomic" damping is transferred to the emitted field. In this way thequantum mechanical average values ("expectation values") of light emittedby atoms give the same results as we obtain them for classical fieldsemitted by damped oscillators. What causes this damping? In the case ofspontaneous emission the answer is easily found. Because by each sponta-neous emission act the photon carries away the energy from an atom, themean occupation number of the ensemble of atoms decreases. Thus thedamping of the atomic system is caused by its coupling to the quantizedfield. In the terminology of thermodynamics the field acts like a heat bathor reservoir on the atoms. In the special case of spontaneous emission,where initially no photons are present, the lightfield corresponds to a

<N2(t))

N 2( 0)

tFig. 1.33. Exponential decrease of occupation number N2 (corresponding to the case of verymany emission acts).

•

§1.14 Photon numbers and phases. Coherent states 31

quantum

system

(e.g. atom)

reservoir

(heatbath)

(e.g. lightfield)

Fig. 1.34. Scheme of coupling of a quantum system such as an atom to a reservoir, forinstance a light field.

heatbath at zero temperature. While the mean occupation number de-creases exponentially, i.e. the system is damped, the individual photons areemitted at random. These random processes are also called fluctuations. Inour present case the occupation number fluctuates while its mean de-creases exponentially.

The phenomena of damping and fluctuations are by no means restrictedto the process of spontaneous emission. Indeed we will see in chapter 9that these phenomena are quite general and of fundamental importancefor a proper treatment of many processes in quantum optics, especially inlaser physics and non-linear optics. The reason for this lies in the fact thathardly any system is entirely isolated from its surroundings but ratherinteracts with its environment which acts as a heatbath or a set ofheatbaths (fig. 1.34). In this way for most realistic quantum systems wehave to take into account damping and fluctuations. The correct mathe-matical treatment of these effects has substantially contributed to thegrowth of a new discipline, namely the quantum statistics of systems farfrom thermal equilibrium.

1.14. Photon numbers and phases. Coherent states

The problem of the relation between the particle- and wave-picture is sofundamental to quantum optics that we will discuss it still further. Let usconsider a specific example in which the classical field strength E(x, t) hasthe form of a standing wave

E(x , t) = q(t) sin kx . (1.21)

In the case of a freely oscillating field, q(t) can be written as

q(t) = A sin(tot + (p) (1.22)

(A and cp real).In classical physics, we can simultaneously measure the phase (p and the

intensity I a A 2 with absolute precision, at least in principle. As we have

cupation number N2 , i.e.

eld. In this way thees") *ght emittedf o sical fields

)ing? n the case ofuse by each sponta-. from an atom, thelecreases. Thus theig to the quantizedcts like a heat bathntaneous emission,

corresponds to a

t

.ing to the case of very

•


photon number known unknown

phase unknown known

Fig. 1.35. Scheme showing possible results of simultaneous measurements of photon numberand phase.

mentioned above, in quantum mechanics the photon number n appears asa quantity analogous to the intensity I cc A 2 . While the concept of "phaseq9" is typical for wave-phenomena, i.e. for the wave-picture of light, theconcept of "photon number n" is typical for the particle-picture of light. Inthe course of this book we will show that q) and n play in quantum opticsroles strongly reminiscent of the roles particle coordinate x and momen-tum p play in quantum mechanics. It is impossible to simultaneouslymeasure both with absolute precision (cf. fig. 1.35).

In quantum optics we shall find "wave functions" describing a quantumstate of the electromagnetic field in which a definite number of photons ispresent. In such a case, the phase is entirely undetermined. (Note thatthese "wave functions" have a meaning entirely different from that of theclassical wave (1.21)!) On the other hand, we can construct wave-packetsof wave functions containing different numbers of photons. As we willdemonstrate (cf. ch. 5), these wave-packets can be constructed in such away that they describe an electromagnetic wave with a well definedprescribed phase. Yet at the same time, the photon number is not fixed butshows a spread, An. These wave-packets are called coherent states. Theywere originally introduced by Schrodinger into quantum mechanics and, inthe sixties, by Glauber into quantum optics.

1.15. The crisis of quantum electrodynamics and how it was solved

In the preceding sections we have seen that quantum mechanics togetherwith the quantization of the light field has brought about a very satisfac-tory theory. It solved all the difficulties classical theory had been con-fronted with. This new theory could adequately treat dispersion, emissionand absorption and the question of coherence and above all it gave asolution to the century-old puzzle of the question whether light consists ofparticles or waves. Of course, there was a price to be paid for thereconciliation of the wave and particle picture, namely the probabilisticinterpretation of quantum theory. As the reader will learn later in thisbook the new theory has a beautiful symmetry in itself and he will learnvery quickly how to use these new concepts.

However,difficulties.treats the iran atom, qtan atom witelectron, wlits lower le%is possiblephoton whienergy consin a waythat a simispeaking, t1law can be sreabsorbednot violated

When th.alsolltedene farealistic atowe should dnot able tcobserved 133idea by Betdevised a relectron frcquantities vition theory.

Fig. 1.36. PossReal transitionW2 . It goes toconservation hthe electron issimultaneousl)

•

§I.15 The crisis of quantum electrodynamics and how it was solved 33

However, in the forties this beautiful theory was marred by very greatdifficulties. These difficulties stem from the following fact. When onetreats the interaction between the quantized light field and an electron inan atom, quite unexpected processes happen. Let us consider for simplicityan atom with two levels only. So far we have treated processes in which anelectron, which is initially in its upper level emits a photon while going toits lower level. However, in quantiun electrodynamics also another processis possible (fig. 1.36b). In it the electron in the lower level can emit aphoton while going to the upper level. This process obviously violates theenergy conservation law. However, it can be shown in quantum mechanics,in a way quite similar to Heisenberg's uncertainty relation, AxAp > h/2,that a similar uncertainty relation holds for energy and time. Looselyspeaking, this means that for short enough times the energy conservationlaw can be violated. Thus the photon can indeed be emitted but it must bereabsorbed after a sufficiently short time so that the conservation law isnot violated for too long a time.

When theoretical physicists investigated the effect of these processes,also called virtual processes, they found that such processes change theenergy of an electron. In the cases of a free electron or an electron of arealistic atom this energy shift turns out to be infinite! We might think thatwe should do away with these strange processes altogether. But then we arenot able to explain energy shifts of certain atomic levels, which wereobserved by Lamb and Retherford. Theoretical physicists following up anidea by Bethe found an ingenious way to cope with this problem. Theydevised a method by which one can subtract the energy shift of a freeelectron from that of a bound electron in spite of the fact that bothquantities were infinite. This well defined procedure is called renormaliza-tion theory. It has been also applied to other problems than the level shift.

-0- W2 W2

h v

rlAnnfirW'

h\/

(a) (b)Fig. 1.36. Possible example of a real and a virtual transition in quantum electrodynamics. (a)Real transition: Initially no photon is present and the electron is in its upper state with energyW2. It goes to its lower state with energy W 1 simultaneously emitting a photon. The energyconservation holds: W2 + (b) Virtual transition: Initially no photon is present andthe electron is in its lower state with energy W. It goes to its upper state with energy W2simultaneously emitting a photon. Clearly the energy conservation law is violated.

of photon number

n appears ascept of "phase-e of light, theure of light. Intuantum opticsr and momen-iimultaneously

ing a uantumr of ons isd. te that3In that of thewave-packets

Is. As we will;ted in such awell defined

; not fixed butt states. Theyhanics and, in

iolved

anics togethervery satisfac-ad been con-sion, emissionall it gave a

;lit consists ofpaid for theprobabilisticlater in this

. he will learn


In all cases, quite excellent agreement between theory and the results ofhigh-precision measurements was found.

A detailed presentation of renormalization theory is beyond the scope ofour book. We will give, however, its basic ideas in section 7.8. As we shallsee later, these problems can be almost entirely neglected in laser theoryand nonlinear optics, so that we need not worry about them.

1.16. How this book is organized

The book is devoted to the light field, to single atoms, and to their mutualinteraction. We are primarily interested in processes rather than in thetreatment of stationary states. We first deepen the discussion of thisintroductory chapter by treating the particle and wave aspects of lightincluding the concepts of coherence. First we will outline quantum me-chanics and then study the interaction between a classical field and a"quantum mechanical" atom treating absorption and stimulated emission.We will also see how a classical field can cause a forced oscillation ofelectrons in an atom. This is the basis for dispersion effects. Finally we willget acquainted with an example of multiphoton absorption. The quantiza-tion of the field, which will be presented in all details, will allow us to treatthe spontaneous emission as well as stimulated emission and absorption byatoms. It will lead us to a treatment of such fundamental problems as theatomic linewidth, the coherence properties of light fields and their photonstatistics. We will then treat atoms which have been excited by a shortpulse of light or which are steadily driven by a coherent field. This leads usto a discussion of quantum beats and the dynamic Stark effect which havebeen experimentally observed only recently.

The last chapter is devoted to a thorough presentation of the mathemati-cal tools used in treating damping and fluctuations of quantum systems.These methods are indispensable for an adequate theory of the laser andfor many processes of quantum optics.

1.17. Laser and nonlinear optics

In our book we shall become aware of a very strange situation. Togenerate fully coherent fields we must use coherently oscillating atomic orother dipoles. But coherently oscillating dipoles can be generated onl y bycoherent fields. This leads us to a seemingly unresolvable vicious circle. Aswe shall see the laser has managed to solve this problem.

The laser is not only a very important new type of light source but is, aswell, a rather simple system which exhibits the typical behavior of order on

macroscopiquantum sya system thnew brandshape theequilibriumsystem andshall come

The uniqhigh intensnon-linearname "non.magnetic fitmaterial la\with the fie

D EE,

whet isconstant ofrelation, ATI-thermal sou

When E 1On the pure(1.23) by rei

D = eol

where, for sUsing six

entirely newimpinge onassociated v.

w2 = 2

For instanctgenerated.

When wecrystal, new

==

§1.17 Laser and nonlinear optics 35

results of macroscopic scale which is often found in more complicated classical orquantum systems. The proper treatment of the laser as an open system — i.e.

ie scope of a system through which energy is pumped all the time — has given birth to

is we shall new branches of physics and even science. It has appreciably helped toser theory shape the field of quantum statistics of systems far from thermal

equilibrium. As we now know it is a beautiful example of a self-organizingsystem and it has inspired a new field of science called synergetics. Weshall come back to these exciting questions in the second volume.

The unique properties of laser light such as its spectral purity and itshigh intensity allow for a wealth of new experiments in the field ofnon-linear optics to which Volume 3 will be devoted. The reason for thename "non-linear optics" is easily explained. When we deal with electro-magnetic fields in matter, we have to supplement Maxwell's equations withmaterial laws. For example, the dielectric displacement D is connectedwith the field strength E by the relation

D = ee0 E (1.23)

where e is the dielectric constant of the material and eo the dielectricconstant of vacuum. Provided e does not depend on E, (1.23) is a linearrelation, which is well fulfilled in the case of light fields produced bythermal sources.

When E becomes very high, the linear relation (1.23) is no longer valid.On the purely phenomenological level of a macroscopic theory we replace(1.23) by relations of the following type

D = eo(E I E + e2 E 2 + e3 E 3 + ...) ( 1.24)

where, for simplicity, we have neglected the vector character of D and E.Using such non-linear relations (1.24) in Maxwell's equations, leads to

entirely new types of solutions. When we let a light wave with frequency coimpinge on a "non-linear crystal", new waves are created in its interior,associated with additional frequencies

6.)2 = 2co (second harmonic generation)

(03 = 3co (third harmonic generation).

ation. To For instance when the incident light is red, in the crystal blue light may be

atomic or generated.

i only by When we shine two waves with frequencies co and co' on a nonlinear

circle. As crystal, new waves with sum or difference frequencies are generated:

but is, as co + = + co'

order on co_ = co —

eir mutualIan in thein of thists of lightntum me-eld and aenir.illa ofIly we willquantiza-

us to treatirption by:ms as their photon,y a shortis leads ushich have

lathemati-systems.

laser and

(1.25)f fof2,7 2 +


The laser can produce such high field strengths E that the non-linearities in(1.24) are substantial. Thus the laser has enabled the realization of a longfelt dream of physicists, namely to pass beyond the linear range of classicaloptics.

Yet what is the physical reason on the microscopic level for the non-linearrelation (1.24)? The answer may be given in two different ways. Anapproach often used generalizes the classical oscillator model, by assuminga non-linear restoring force. That is in eq. (1.1), f is assumed to depend onthe displacement, q, of the oscillator

While such an approach may provide us with an intuitive understanding ofthe underlying processes, a satisfactory theory must treat the atoms (or,more generally, matter) by means of quantum mechanics. On the otherhand, the laser light field can be treated classically, which we will justify inVolume 2. In the framework of this "semi-classical" theory, we maycalculate the polarization of the medium, P, by means of the induceddipole moments of the individual atoms. We will demonstrate in section4.5 by an illuminating example that these dipole moments depend in anon-linear fashion on E, so that P becomes a non-linear function of E. Butsince

D = e0 E + P

the relation (1.24) follows.In conclusion, let us pick another example from non-linear optics. So

far, we have talked about "non-linear dispersion". There are, however, alsonon-linear absorption and emission effects. To explain this, let us adoptEinstein's picture of emission and absorption. In his case, the Einsteincoefficients did not depend on the photon number. For instance, inabsorption the transition rate of an atomic system going from the atomground state 1 to the excited state 2 is proportional to the photon numbern. But at high enough photon numbers, n, again available from the laser,this transition rate may be proportional to n 2, or n3 , etc., and in suchsituations one has the absorption (or emission) of more than one photon ata time by an individual atom. These processes seem to be important forisotope separation etc. We shall give a typical example of such a "multi-photon absorption" process in section 4.4.

These examples by far do not exhaust the field of nonlinear optics. Theyare merely meant to indicate that there exist a whole new world offascinating non-linear phenomena.

2. The E

2.1. Wav.

Let us sU.century Nequationsthe propahe couldlirtrc

Let usamplitudepropagateWe will sequationsfew simpfig. 2.1)

E(x.

k is the NA

k =

The freqtthe periot

=

To writeinclude ti

Note ti-phase so.

It willThis desc

on-linearities ination of a longrige of classical

r the non-linearrent ways. Anel, by assumingd to depend on

(1.25)

tderstanding ofthe atoms (or,• On the othere will justify ineotryt. may)f ucedrate in sections depend in a-.lion of E. But

lear optics. So, however, also;, let us adopt

the Einsteinr instance, inrom the atomhoton numberrom the laser,• and in suchone photon atimportant foruch a "multi-

r optics. Theyiew world of

•

2. The nature of light: Waves or particles?

2.1. Waves

Let us start with the idea that light can be described by waves. In the lastcentury Maxwell (1831-1879) formulated his famous electromagnetic fieldequations. He showed that these equations allowed for solutions describingthe propagation of electromagnetic waves. In particular it turned out thathe could derive the velocity of light from the quite independent constantsof electromagnetism. Thus he concluded that light is an electromagneticwave.

Let us consider a simple example of such a wave. We identify its fieldamplitude with the electric field strength. We assume that the wavepropagates in x-direction and that the field strength points in z-direction.We will show later on that such a wave is indeed a solution of Maxwell'sequations. For the time being, however, we want to remind the reader of afew simple facts about waves. The most simple form of such a wave is (cf.fig. 2.1)

E(x, t) = Eosin(kx — cot — (p). (2.1)

k is the wave number which is connected with the wavelength A by

k = 27r/X. (2.2)

The frequency v and the circular frequency w = 2Try are connected withthe period of an oscillation T by

= 2 7rv = 27/ T. (2.3)

To write the expression for a plane wave in its most general form, weinclude the phase p in (2.1).

Note that a shift of time t by at is equivalent to a shift op = co& of thephase cp. We shall use this relation later. For this section we choose cp = 0.

It will turn out advantageous to use the complex description of wavips.This description is based on the well-known relations between cosine, sine

Fig. 2.2. Examdashed dottedNote both thamplitude Ow

avit, hso

breaks dowLet us, h

superpositicbeats anddifferent wa

Etot(

38 . 2. The nature of light: Waves or particles?

Fig. 2.1. Snapsho of the sine-wave (2.1). The field strength E(x, t) is plotted at a given timeversus the space coordinate x. A is the wave length.

and the exponential function

cos a = -})[ exp(i a) + exp( — i a) ]

sin a = (1)[exp(ia) — exp(—ia)].

Using (2.4) and (2.1) we obtainE(x,t)=-- A exp(ikx — iwt) + A* exp(— ikx + iwt)

where the amplitude A is given byA = E.0/ (2i).

For use in later chapters we will write (2.5) in the formE(x,t)= E"(x,t) + E"(x,t)

whereE" cc exp( — icot)

is called the positive frequency part and its conjugate complex E" thenegative frequency part of the electric field strength. This notation soundssomewhat strange because the + sign in E(+) is connected with the — signin the exponential (2.8). We will understand this notation only later whenwe treat the electromagnetic field quantum mechanically.

Since Maxwell's equations in vacuum are linear it follows that a super-position (fig. 2.2) of two partial waves is again a solution of Maxwell'sequations. This is expressed mathematically by

E(x,t) = Ei (x,t) + E2(x,1). (2.9)

It must be noted, however, that this superposition principle can be violatedwhen waves propagate through matter and this will indeed be aft importantsubject of study in nonlinear optics.

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)Using eleint

Etot( x

The new wa

rc

Ak = (.

The total varepresents aslowly modfunction, II,anniiition

§2.1 Waves 39

given time

Fig. 2.2. Example of the superposition of two sinusoidal waves E l and E2 . El is shown by adashed dotted line. E2 by a dashed line. The resulting superposition is shown by a solid line.Note both the regions of enhancement (positive interference) and of decrease of totalamplitude (negative interference).

40

available, however) one may even expect that the superposition principleIt is also interesting to note that at very intense fields (which are not yet

breaks down for waves propagating in vacuum.(2.5) Let us, however, return to our much simpler considerations. Here the

superposition principle (2.9) allows us to understand the phenomena ofbeats and of interference. Using for E, and E2 two sine waves with

(2.6) different wave numbers ki and frequencies cop j = 1,2, the total wave readsEtot(x,t) = A sin(k l x — w i t + yo 1 ) + A sin(k 2x —w2 1 + P2). (2.10)

E the Etot( t= 2A sin(Tcx — Z5t + sri;)) coS(Akx — Awt + Atp). (2.11)(—) -I sounds

e — sign The new wave numbers and frequencies are given by'er when

= (k, + k2 )/2, =(w1 + 6)2)/2 , = (9, 1 + 9)2)/2

a super- Ak = (k, — k 2 )/2, At4 = co2)/2, Acp = (co l —faxwell's(2.12)

(2.9) The total wave (2.11) is represented in fig. 2.2. The first factor in (2.11), I,represents a rapidly oscillating wave in space and time. Its amplitude is

violated slowly modulated by the second part, II (cf. fig. 2.2). Since the second

vortant function, II, can vanish at certain space—time points we obtain a completeannihilation of wave amplitudes by means of the superposition principle.

•

(2.7) E, E2

Using elementary formulas of trigonometry we cast (2.10) into the form(2.8)

40 2. The nature of light: Waves or particles?

This negative interference can be easily studied by letting light propagatethrough a narrow slit. According to Huygen's principle each spacepoint,which is hit by a wave, becomes the starting point of spherical waves (cf.fig. 1.5). The superposition of these spherical waves gives rise to new wavefronts. A plane wave hitting a slit makes each point of the slit to thestarting center of a wave with equal phase. When the waves propagatefrom the different points phase lags occur. Due to these phase lags it mayhappen that a wave amplitude maximum coincides with the minimum ofanother wave. In this case negative interference occurs and we shallobserve dark stripes on a screen (fig. 23). According to diffraction theorythe dark stripes occur under an angle a according to the relation

sin a = mit/D, m= 1, 2, 3 ... . (2.13)

m is an integer, A is the wavelength of the incident light and D is thediameter of the slit.

In actual experiments using natural (thermal) light sources there are twofundamental difficulties to be taken into consideration. Light sources havea finite dimension and their atoms emit light independently of each other.Since the phases of the wave tracks emitted by the individual atoms areuncorrelated it may seem, at first sight, that the interference patterndescribed above does not occur when emission by many atoms takes place;quite the contrary is true. Since the phases of the individual wave tracksare uncorrelated, interference effects stemming from different wave tracksvanish when we average over the uncorrelated phases and the resultingtotal interference pattern is just the sum of the individual interferencepatterns. We shall explicitly perform such phase averages later in thisbook.

Fig. 2.3. Plane wave front is hitting a slit. Each point of the slit is a startitig center of thewave with equal phases. The figure shows how the different lengths of paths ckuse a phase lagwhich results in negative interference shown by the example.

Anotheimonochrospread (be(Volume 2the diffrac

A furthYoung'swith verythen passeslits 1 and

The exrsynchronoDue to thior other rtthis well-klight is prcTiibit

Fig. 2.4. Thstemming fnreseg intt

intensity

plane slit screenwaves

screen

§2.1 Waves 41

ht propagatespacepoint,

al waves (cf.to new wavee slit to thees propagatee lags it mayminimum ofnd we shalltction theory

(2.13)

tnd D is the

her twosou ave

,f each other.al atoms areence pattern, takes place;wave trackswave tracks

the resultinginterferencelater in this

Another difficulty rests in the fact that a light source does not radiatemonochromatic light but light which always possesses a certain frequencyspread (being equivalent to a spread of wave lengths). As we will see later(Volume 2) the laser has made it possible to excite the field in the plane ofthe diffraction slit with equal phases.

A further experiment to demonstrate the wave character of light isYoung's interference experiment (cf. fig. 2.4). Light emitted from a sourcewith very narrow diameter is made parallel by a collector lens. This lightthen passes through two slits of a screen. We will assume that in the twoslits 1 and 2 the light has the same phase.

The experimental setup can be compared with water waves where thesynchronous dipping of two bars into water causes two interfering waves.Due to this interference there will be regions in which water remains at restor other regions where the water amplitudes are particularly high. To castthis well-known phenomenon into a mathematical form we imagine thatlight is propagating from the two points 1 and 2 in form of spherical waves.To exhibit the most important features we neglect any polarization effects

light source tense double slit

ig center of theause af lag

Fig. 2.4. The basic experimental arrangement of Young's double slit experiment. Lightstemming from a light source is made parallel by a lens and passes through two slits. Tli)eresulting interference pattern is observed on a screen.


and represent the light wave in form of a usual spherical waveE a (1/r)exp[i(kr — cot)]. (2.14)

r is the distance between a point where the field amplitude is measured andthe point where the field amplitude has been created. In our specialexample we have two spherical waves and we consider a point r, hit bythese waves. Using the notation of fig. 2.4 we find

Ef +) = E0(1/r1 )exp[i(kr1 — cot)] .1-1 (1/r1 )exp(—icot 1 ) (2.15)and

E'= E0(1/r2 )exp[i(kr2 — cot)] (1/r2 )exp(—i6.12 ). (2.16)On the right-hand sides we have introduced new times t, by the relations

t, = t — rdc, w = ck. (2.17)These times differ from the original time t by the time T, = r,/c, which isneeded by the light waves to propagate from their original space points tothe point under consideration. By this time lag the field amplitude suffers aphase shift. Again we obtain an interference pattern which is representedin fig. 2.5. We shall discuss this pattern in more detail below. Let us firstdefine, however, intensity. According to electromagnetic theory light inten-sity is defined by

1(60 E2 + —1 B2)tt o

where we use the MKS units. The contribution of the magnetic inductionB to the intensity equals the contribution of the electric field strength, E.

Fig. 2.5. The interference pattern of Young's double slit experiment. Left: The light intensityindicated by dark and light stripes. Right: A photogrammatic plot of the inteqty (abscissa)along the screen (ordinate).

Thus it is sufirst part adecompositicwriting E in

E(x,t)

In later appnot only deassumed, hoexp(icot).

I a EV

In all exp,average oveThis meansa time. T1

(1/

instead of (2(2.18)

(2iwT)

and a simila(2.18) into tEo- we obta

24+)E

We now us,inverse freq

T

or written ii

coT 1

Accordingthe contribtcontributiotintensity

Ii=

whir h

110 where we have dropped all constant factors. Our brackets will indicate a

§2.I Waves 43

Thus it is sufficient for our purposes to use as definition of intensity thefirst part ct E2 . This expression can be further simplified when we use thedecomposition (2.6) of E in positive and negative frequency parts, i.e.writing E in the form

E(x,t) = Er )(x)exp[-icatj + Ar )(x)exp[iwt]. (2.17a)

In later applications we shall even admit that the factors E (+), E(-) maynot only depend on the spatial coordinate x but also on time t. It isassumed, however, that this time dependence is much slower than that ofexp(iwt). Inserting (2.17a) into our definition of the intensity we obtain

/ cc EV' exp[ -2itat] + E4 -)2 exp[2iwt] + 2E4 +)E6 -) . (2.18)

In all experiments measuring light intensity the measurement implies anaverage over times large compared to the period of a single oscillation.This means that we do not measure I but rather its temporal average overa time T. Thus we have to consider

1= (1/T)f 772 I(t)dt (2.19)- T/2

instead of (2.18). Inserting (2.18) into (2.19) we obtain for the first term of(2.18)

(2iwT) i [exp(iwT) - exp(-iwT)].E4 +)2 (2.20)

and a similar one for the second term of (2.18). Inserting the last term in(2.18) into the integral and neglecting the slow time dependence of E0+,Eo- we obtain

24+)E4 -) . (2.21)

We now use the fact that the measuring time T is much bigger than theinverse frequency w i.e.

T>> w 1 (2.22)

or written in a different way

off >> 1. (2.23)

According to (2.23) the contribution (2.20) to (2.19) is much smaller thanthe contribution (2.21). Thus we may safely neglect the rapidly oscillatingcontributions which allows us to introduce the following definition of theintensity

II= <E4 + )E6->> (2.24)$.

c induction>trength, E.

tight intensitysity (abscissa)

(2.14)

leasured andour special

int r, hit by

(2.15)

(2.16)

the relations

(2.17)

/c, which istce po ts toide rs arepr ntedLet us first

• light inten-

time average

<... >= (1/T) f T/2

. . . dt. (2.25)r— T/2

This allows us later on to take care of such cases in which E, E6- aretime dependent.

2.2. Classical coherence functions

The results of the preceding section allow us to define some importantcoherence functions. By means of these coherence functions we want tocast the concept of coherence into rigorous mathematical shape. In orderto find out how such functions look, let us consider again Young'sinterference experiment in more detail. The total field strength at a spacepoint on the screen at time t is given by

44t (t) --= E(+)(t — r1 / c) + E(+)(t — r2/ c) E (+ )(1) + E(+)(2).

(2.26)

By using the intensity formula (2.24) we readily obtain

I = <E(-)0)E(+)(1)> + <E(-)(2)E(+)(2)> + <E(-)0)E(+)(2)>+ <E(-)(2)E(+)(1)> . (2.27)

Using the abbreviation

<E(-)(i)E(+)(j)> G(i,j) (2.28)

we write (2.27) as

I = G(1,1) + G(2,2) + G(1,2) + G(2,1). (2.29)Let us try to understand the meaning of the individual terms. To this

end we assume E(÷)(1) in the form of a spherical wave

E(+)(1) = const(l/r0exp[ (2.30)

just as in Young's experiment. The same form is, of course, adopted forE+ (2). When we insert these expressions into (2.28) for i =j = 1, the twoexponential functions cancel and we obtain an expression proportional1/r1 2 . A corresponding result is obtained for G(2,2).

Let us now consider G(1,2). Here the factor exp(icot) cancels out but thefactor expf —ico(t2 — t,)] still remains Thus we find

G(1,2) = const(1/r1 r2 )exp[ —ito(t2 — t 1 )]. (2.31)

Now recall that according to its definition ti = t — I)/ c. We thus cecognizethat varying the space point on the screen we incidentally vary it and t2.

Since the frecitvaries relative)slowly. To exhand G(2, 2) asturn to the cadepend on titG(2,2) by A a

G(1,2) =

The intensity

/ = 2,

G(1+ G(2

The plidikfactconsidWsobtain the shaBecause an inhave the inecpmathematical

When we ccmum of brightdark interferenand I is. Itby dividing it

"""

Fig. 2.6. How to viphase lag (D. The chis obtained from tEvidently the minit•

§2.2 Classical coherence functions 45

Since the frequency of visible light is rather high, the factor exp[ico(t 2 — ti)]varies relatively quickly, whereas the factors r /- I and r2 I change only veryslowly. To exhibit the essentials it is therefore sufficient to consider G(1, 1)and G(2, 2) as constants and to study only the behavior of G(1, 2). We nowturn to the case where E, E- are not necessarily purely harmonic butdepend on time in a more complicated way. We abbreviate G(1, 1) =G(2,2) by A and put

G(1,2) = Bexp[i(1:1 ]. (2.32)

The intensity I then reads

1= 2A + 2B coscl)

G(1,1) ± G(1,2) . (2.33)+ G(2, 2) + G(2, 1)

The phase factor (I) varies when the point r on the screen is changed. Let usconsider I as a function of the phase factor O. In the case B = A we thenobtain the shape shown in fig. 2.6, for B <A the shape shown in fig. 2.7.Because an intensity can never become negative for physical reasons wehave the inequality B < A (which can actually be proven in a rigorousmathematical way).

When we consider the variation of an intensity on the screen a maxi-mum of brightness follows a minimum. The contrast between bright anddark interference fringes is greater, the bigger the difference between Iand I • is. It is useful to normalize this difference between I and I •by dividing it by /max + I. This leads us to the definition of fringe

Fig. 2.6. How to visnali7e the individual terms of eq. (2.33) for the intensity depending on thephase lag 4). The dashed line shows 2B cos 4) 2A cos 4), i.e. the case B = A. The intensity Iis obtained from the dashed curve by shifting it into the /-direction by the amount 2A.Evidently the minima of the solid curve lie at I = 0.

:le important, we want totpe. In order:ain Young'sth at a space

E(+)(2).626)

+)(2)>(2.27)

(2.28)

(2.29)rms. To this

(2.30)

adopted for= 1, the twoproportional

; out but the

(2.31)

us recognizemy t 1 and t2.

(2.25)

E, E;;" are

Fig. 2.7. We show what the intensity I looks like in the case that B < 2A. The dashed curverepresents 2B cos O. To obtain I we have to shift the dashed curve by an amount 2A into thedirection of the I axis. The resulting total intensity is represented by the solid curve. Note thatthe minima of f are now shifted away from the (I)-axis. A comparison with the result of fig. 2.8shows that complete darkness at certain points is not reached and that the contrast hasbecome smaller.

visibility V by the relationV = ('max Imin )/ ('max +

(2.34)

In the case of our example (2.33) V is given byV= B/A =IG(1,2)1/G(1,1). (2.35)

As we will see in a minute G(1, 2) contains useful information about thecoherence of waves. For this reason

G(1,2) = (E(-)(1)E(÷)(2)> (2.36)is called the mutual coherence function. Since the fringe visibility containsa normalization (division by I + I • ) it is reasonable to do the samewith (2.36), i.e. dividing it by G(1, 1). This is, however, a somewhatoverspecialized procedure because in our present example we had assumedthat the first terms in (2.29) were equal. To take care of the general case wedefine quite generally the complex degree of mutual coherence

7(1,2) —G(1,2)

(2.37)[G(1, OG(2,2)]ii2

As can be shown mathematically

< 1 (2.38)always holds.

It is most useful to compare the coherence function (2.36) [or theexpression (2.37)], which we have introduced by means of a con4deration

of Young's dot1.11. To faciliti:the positive anthe arrangemeidelay, similarhitting the suetwo new wavesadequate geneproduce the delfunctions. In Ifields E1 and EAll we have tostart new wayinstance in a wproperties arepoints x l and;

<41)( xi

where the apprThe expressi

space and timchoose x 1 = x2this case (2.36Et i = t 2 fixed. IIbecomes particand a space-fu:

Ei(x,t)=

where qj may s

(2.36a) =-

In such a casetreat temporaldealing with thto the case in vNow we woulexpression (2.37the coherence (

To study theinto two parts.two slits. Furl"

ashed curve2A into thee. Note thatAt of fig. 2.8:ontrast has

1111(2.34)

(2.35)

about the

(2.36)

y containsthe same

somewhati assumedal case we

(2.37)

(2.38)

i) [or thesideration

•

of Young's double slit experiment, with the one we considered in section1.11. To facilitate this comparison, let us use the definition (1.19), based onthe positive and negative frequency parts of E. We quickly recognize thatthe arrangement of Young's experiment serves as a beam splitter and adelay, similar to that of fig. (1.30). Indeed, consider a plane wave fronthitting the screen with the holes. Then the two holes split this wave intotwo new waves which are generated by means of Huygen's principle or itsadequate generalization. The different path lengths of the two wavesproduce the delay. We may now slightly generalize the concept of coherencefunctions. In principle, we can measure the coherence functions of twofields E i and E2 at space points x i and x2 and times 1 1 and t2 , respectively.All we have to assume is that El and £2 at these space points and timesstart new waves, e.g. spherical waves, which we can let interfere, forinstance in a way known from Young's experiment. Because the coherenceproperties are "faithfully" transmitted (in vacuum) from the initial spacepoints x 1 and x 2 to x, we can measure

(E1-)(x,,t1)E1+)(x2,t2)> (2.36a)

where the appropriate time-average is taken.The expression (2.36a) is a correlation function of E's with respect to

space and time. Two special cases are of particular interest. When wechoose x 1 = x2 , we may consider (2.36a) as function of t i and t2 only. Inthis case (2.36a) describes temporal coherence. Similarly, we may chooset i = t2 fixed. In this case (2.36a) refers to spatial coherence. The situationbecomes particularly simple, when we can split Ej (x, t) into a time-functionand a space-function

Ei (x,t)= q1(t)14;(x)

where qj may still fluctuate. In this case (2.36a) factorizes

(2.36a) = <qY )(t 1 )qr(t 2 )>u l (x 1 )u 2(x2 ). (2.36b)

In such a case the problem of calculating (2.36a) is essentially reduced totreat temporal coherence. We shall meet such a case in Volume 2 whendealing with the laser. Of course, all above expressions can be specializedto the case in which El and E2 refer to the same field: E i(x, t) = E2(x,t).Now we would like to demonstrate by means of examples how theexpression (2.37) for the complex degree of coherence can be used to studythe coherence of light waves.

To study the coherence of a single light wave we have to split the waveinto two parts. This was done in Young's interference experiments by thetwo slits. Furthermore we have to introduce a phase lag between the two

parts and then we have to calculate (2.37) for the two thus constructedwave tracks. Let us consider to this end two examples.

(a) In the case of a pure sine wave the field is described byE(+) = Eoexp[ (2.39)

Let us assume that this wave is duplicated (split) and phase lags t (1) or 1(2)are applied (fig. 2.8). Inserting the resulting field amplitudes into G(1, 2) wefind

1E012f• T/2

G(1,2) = , exptiw(t — t ()) ) — iw(t — 1 (2) )) dt. (2.40)- T/2

The integration is quite elementary and yields

G(1,2) = Eo 1 2 exP[ iw ( t") t(1) ) ] . (2.41)

G(1, 1) and G(2, 2) can be calculated still more simply: = Eo I 2 • Thus the

complex degree of coherence of a purely sinusoidal oscillation reads

y(1,2) = exp[iw(t (2) — / (1) )]. (2.42)

Its modulus

I Y I = 1

(2.43)

has attained its maximum and this maximum value is retained for all phaselags. Thus we see that a sinusoidal wave has a maximal degree ofcoherence.

(b) Let us consider a second example namely that of a finite wave trackof length to (cf. fig. 2.9). In order to determine G(1, 2) we duplicate (split)the wave track and introduce again a phase lag. Inserting the two corre-sponding expressions for the field amplitudes of the wave tracks into

Fig. 2.8. This figure shows two waves, which are to be thought of as infinitely extended, ofequal amplitudes and wavelengths but which were subjected to time lags tO) and t(2)•

Fig. 2.9. This figwave track but 9.

G(1,2) we ohphase lag (meof this simplerepresented irdecreases witsteadily decrewhen we havefunctillintr

2.2.1. Cohere,

The mutualE(-)(1) and Aof higher ord

<E(-)(1

where E(-)(,tion functioncomplicated

Fig. 2.10. The a

•

s constructed Elt)

(2.39)

ags t o) or t (2)

Ito G(1, 2) we

• IA 17A• ,T

It. (2.40)

to •I

Fig. 2.9. This figure shows a wave track of finite length to in time (solid line) and the samewave track but with a time lag (dashed line).

(2.41)

2 . Thus then reads

(2.42)

•(2.43)

for all phaseii degree of

e wave trackplicate (split)le two corre-

tracks into

G(1,2) we obtain only a nonvanishing contribution to G(1, 2) when thephase lag (measured in time units) is smaller than the length to. The resultof this simple calculation, which we leave to the reader as an exercise, isrepresented in fig. 2.10. According to this result the degree of coherencedecreases with increasing time difference. The coherence is thereforesteadily decreasing. We will present more examples of realistic cases laterwhen we have defined the quantum mechanical analogue of the coherencefunctions introduced above.

2.2.1. Coherence functions of higher order

The mutual degree of coherence (2.36) depends on two field amplitudesE(-)(1) and E(+)(2). It is, however, possible to define coherence functionsof higher order in which more than two field amplitudes occur:

<E(-)(1)E(-)(2)...E(-)(n)E(+)(n + 1)...E(+)(N)>

where E(j) stands for E(-)(xj,ti) etc. When we discuss such correla-tion functions later in detail we will see that we have to discuss morecomplicated measurement processes which enable us to measure such

y extended, ofnd t (2)•

111

Fig. 2.10. The decay of the complex degree of coherence (2.37) as a function of the time lag.

•

correlation functions. The statistical properties of light fields, especiallythose of thermal sources and of lasers, can be described by such correla-tion functions. In particular, they will allow us to distinguish uniquelybetween laser light and light from thermal sources.

The interference patterns observed in Young's experiment seemed to bea clearcut proof of the wave-character of light. However, we know that thisis only one aspect. Let us now turn to the other aspect, namely to theparticle-aspect. This new development can be traced back to 1900.

2.3. Planck's radiation law

We can visualize the meaning of Planck's law in a simple way as follows.When we look into a cold furnace it appears black. When we heat up thefurnace its walls first appear to glow red and later, at still higher tempera-tures, white. Planck's law aims at describing this change of colour or, moreprecisely speaking, at determining the intensity distribution of the electro-magnetic field as a function of frequency. Since the atoms of the walls emitand reabsorb light, thermal equilibrium is established between the atomsand the electromagnetic field. When the radiation energy in a frequencyinterval v,v + d v is measured, curves as shown in fig. 2.11 are found.Clearly the maximum of the intensity distribution is shifted to higherfrequencies when the temperature is increased. This shift is expressed byWien's law according to which the frequency shift of the intensity maxi-mum is proportional to the increase in temperature. Let us now calculatethe energy u(v)dv of the radiation in the frequency interval v, v + dv. Tothis end we make use of the wave theory of light and represent u(v)dv in

u(v)

Fig. 2.11. Planck's law. The energy density u( p) plotted versus frequency for differenttemperatures 71< T2 < T3 . The shift of the maximum with increasing tempertture is clearlyvisible.

the formu(v)dv

In order toconsider a nfeatures of tistrength in o

E = Eo:

Since the wMaxwell's tlinfinitely hitx-direction amust vanishspec

These consicbetween the

E = Eo:

The wave rfrequency vi:

ck.

Fig. 2.12. One-conducting wall

4111

§2.3 Planck's radiation law 51

especiallych correla-h uniquely

.!med to be)1,v that thislely to thetoo.

as follows.ieat up the:r tempera-ar or, morethe electro-

•walrthfrequency

are found.to higher

pressed byisity maxi-v calculate

+ dv. Tou(v)clv in

the formu(v) dv = number of waves in the

interval d v • mean thermal energy ofone wave.

In order to determine the number of waves in the frequency interval d v weconsider a model, namely standing waves in a metal box. The essentialfeatures of this problem can be explained when we study the electric fieldstrength in one dimension and represent it in the form

E = Eo sin kx. (2.45)

Since the walls of the box are assumed to be metallic, according toMaxwell's theory E must vanish at the walls (which are assumed ofinfinitely high conductivity). Choosing the walls perpendicular to thex-direction at the coordinate x = 0 and x = L the sine function of (2.45)must vanish at these points. This is achieved by choosing k in (2.45) in thespecial form

k = nv/L, n= 1,2,3.... (2.46)

These considerations imply that only specific forms of E are permitted inbetween the walls, namely (cf. fig. 2.12)

E = Eosin(nv x / L). (2.47)

The wave number of an electromagnetic wave is connected with itsfrequency via the velocity of light

u.) = ck. (2.48)

(2.44)

or differentre is clearly Fig. 2.12. One-dimensional example of a standing electric wave between two perfectly

conducting walls.

•


On the other hand we have, of course,

= 2 rv. (2.49)

From the relations (2.46), (2.48), (2.49) we obtain

v = cn/(2L). (2.50)

Since n is an integer we can enumerate the waves. By means of (2.50) wethus obtain as number of possible waves in the interval dv

dn (2L/c)dv. (2.51)

In this way we have solved our task to determine the number of waves inthe frequency interval dv, at least in one dimension. In reality we have, ofcourse, to deal with three dimensions. As can be shown by means ofMaxwell's equations the field amplitude can be again expressed by sineand cosine functions for instance in the form

Ez CC sinkx x sin ky ycoskz z. (2.52)

Again the wave numbers lc, are given byrn,/ L, n.= l,2,3..., = x,y,z. (2.53)

We define k = lik!+ k y2 + q and n = in 2z n y2 + z2n In entire analogyof (2.50) we obtain the relation

v = cn/(2L). (2.54)

To count the number of possible waves belonging to the frequency rangev,v + dv we consider a coordinate system in which the axes correspond tothe numbers nx , ny , nz , respectively (fig. 2.13). Each point of this spacewith integer coordinates represents a possible state of the electromagneticfield oscillation. When we choose the frequency interval sufficiently large

Fig. 2.13. This figure shows the section of the spherical shell in the n-space (compare text).

we may assu.tion treatingspherical shemust be posi8 and becaucounted indinumber of p(

dN =

Expressing itfrequency int

dN —8:

We now hadetermine tit(formulas weiassillat tthe nan

00W = E

n.•1

where A, isequilibrium.

When denwave was cotHe calculate(identified it veration usinglater that thiIndeed one nwave on accc

Let us retexperimentallhad to makeenergy levelsfollow from (prescription (

W„ hi'

where p is theabsorption frf

•


we may assume that the points can be counted by means of an approxima-tion treating them as continuously distributed. The number of points in aspherical shell of thickness dn is given by 4rn 2 dn. Since all numbers ni

must be positive, this result must be divided by the number of octants, i.e.8 and because we have two directions of polarization which have to becounted individually we must multiply our result by 2. We thus obtain asnumber of possible states in this spherical shell

dN = irn 2 dn. (2.55)Expressing in it n by (2.54) we obtain the number of possible waves in thefrequency interval 1, ,11 dv

8/7E3 dN — p (2.56)

c3

We now have to deal with the second part of our task, namely todetermine the mean thermal energy of a wave. To this end we have to useformulas well known in thermodynamics. Anticipating later results weassume that the system has energy levels Wn , n = 0, 1, 2, ... . According tothermodynamics the mean thermal energy is given by

W = E PnW (2.57)n= o

where A, is the probability to find the state n occupied in thermalequilibrium. We will write down pn below.

When deriving his theory, Planck assumed that each electromagneticwave was coupled to an atom which could be represented as an oscillator.He calculated the mean thermal energy of such an atomic oscillator andidentified it with the mean energy of the specific light wave under consid-eration using the equipartition theorem of thermodynamics. We will seelater that this assumption of material atomic oscillators is unnecessary.Indeed one may directly calculate the mean energy of an electromagneticwave on account of quantum postulates.

Let us return, however, to Planck's ideas. In order to obtain theexperimentally correct distribution function for the light intensity Planckhad to make an assumption which was quite revolutionary at his time. Theenergy levels of an harmonic oscillator must not be continuous as it wouldfollow from classical theory but they must be quantized according to theprescription (fig. 2.14)

Wn = hv n (2.58)

where v is the frequency of the light wave or equivalently the emission andabsorption frequency of the corresponding atomic oscillator, h is Planck's

(2.49)

(2.50)

of (2.50) we

(2.51)•of waves in•we have, of)3, means of•ssed by sine

(2.52)

•.53)

Aire analogy

(2.54)uency rangeJrrespond to

this spacetromagneticciently large

compare text).


W easily calculaiWs

h w z= no EWs

hw By inserting (.W4

w Ft/ = EW

3fiw where we havt

= 1 / (inwThe right-hancof the partitio,

WI

hw

Fig. 2.14. According to Planck's fundamental idea the energy of an harmonic oscillator mustbe quantized. The energy levels being separated by the amount hw h p . Note that co .= 2 VP

and h h/(2v).Due to the sigeomill seri,

constant and n is an integer number 0, 1, 2, ... . The occupation probability Z = [ 1 —p„ was chosen by Planck to be identical with the Boltzmann distributionfunction of classical statistical mechanics. According to this (fig. 2.15) This allows us

= Z -1 exp[ — Wn/ (kr)] (2.59)

where k is Boltzmann's constant, T absolute temperature, Z a normaliza-tion factor which normalizes the sum over the p„ to unity and is called thepartition function. From the condition of normalization of the p, 's Z is

W3

W2

W1

Wo

Fig. 2.15. The energy levels of the harmonic oscillator are indicated on the ordinate. Parallelto the abscissa the corresponding occupation probabilities pn , n =.n 0, 1, 2, 3 are plotted.

W hv /

Now we halenergy in thewe obtain

u(v)dv =

and thus

u( p )clv =

This formula rand in the freqwe obtain the t

P( P) = (81This is Planck'experimental d,learned two tfnumber of fieldthat it is necess,

•

/3 = 1/ (kT). (2.62)


easily calculated and reads00

E exp[ — Wn/ (kT)]. (2.60)

By inserting (2.59) and (2.60) into (2.57) the mean energy is given by

Wn exP(-13K)1/[ E exP(—P)]n= o n=o

where we have used the abbreviation

scillator mustthat ir Due to the simple form of W, (2.58), the partition function is a simple

geometric series and can easily be evaluatedprobability Z =[1 — exp(— hvP)] - I . (2.64)iistribution;. 2.15) This allows us to calculate (2.63) and we obtain

W= hv/[exp(hv/3) — 1]. (2.65)(2.59)Now we have all the results together in order to determine the radiation

normaliza- energy in the interval v,v + dv. By inserting (2.61) and (2.65) into (2.44); called the we obtaine pn 's Z is

u(v)dv = dN• W (2.66)

and thus

(2.61)

The right-hand side of (2.61) can be written as derivative of the logarithmof the partition function Z

W= — To- ln(Z). (2.63)

u(v)dv (8 .771i/c3 )v 2hy/[exp( i3hv) — 1]. (2.67)

This formula represents the energy of the radiation field in the volume Vand in the frequency interval v,v + dv. When dividing (2.67) by V and dvwe obtain the energy density p

p(v) = (87r/c3 )v 2hv/[exp(i3hv) — 1]. (2.68)This is Planck's famous formula which is in excellent agreement with allexperimental data (cf. fig. 2.11). In deriving this crucial formula we havelearned two things. First a mathematical trick how to determine thenumber of field modes in a cavity and, still much more important, the ideathat it is necessary to quantize energy levels. Planck's derivation was based

mate. Parallel'lotted.


on the idea that light is still composed of waves. In section 2.5 we shall seethat there is an even much more elegant access to Planck's law namely bythe assumption that light is composed of particles.

2.4. Particles of light: photons

In the preceding sections we have seen how interference experiments couldconvincingly prove that light consists of waves. Thus it could seem that aunique decision has been made in view of Huygen's hypothesis. There are,however, experiments which can be understood only if we assume thatlight is composed of particles. I mention here two well-known examples.

(a) The photoelectric effect.When ultraviolet light impinges on metal surfaces it can free electrons. Thekinetic energy of the emerging photoelectrons has been measured depend-ing on light intensity and light frequency. According to the wave theory oflight, the energy of a light wave is proportional to its intensity. Thereforeone should expect that on account of energy conservation the kineticenergy of the emitted electrons increases with increasing light intensity.This is, however, not the case. With increasing light intensity more elec-trons are emitted but the energy of an individual electron remains un-changed (cf. fig. 2.16). On the other hand, when we vary the frequency oflight, the kinetic energy of the emitted electrons changes. As Einstein hasshown, this result can be easily understood by the assumption that light iscomposed of particles which are called photons. In this theory it isassumed that the energy of a single photon is given by E = hi', where h isPlanck's constant and v the frequency of the light wave. The kinetic energyof an electron is then connected with the photon energy by (fig. 2.17)

mvZ / 2 -= hi' — A (2.69)

where in is the mass of an electron, Vi the velocity of the emitted electron,

rri 2-2- V

h v

Fig. 2.16. The kinetic energy of the electrons emitted from the metal surface are plottedversus the energy of an incident light quantum. A is the work function, i.e. miiiimum amountof energy which is necessary to free an electron from the metal.

and A the worpay when leavtally verified iconstant. By itmust have also

p hv / c

The experimerand momentur

(b) The CotIn this experirlight field actfollowing ef fecThe oscillatintcome an electsimilarly to th,electron mustdireclif thplace the

ArlirrelyThen a photcenergy and m

hi' + Wel

Fig. 2.17. This fenergy hv intoextension L is nindicated by thea maximum enetbelow the zero omodel will findsection 3.6.

11111

•

$2.4 Particles of light: Photons 57

and A the work function. It is a potential energy which the electron has topay when leaving the crystal (fig. 2.17). The relation (2.69) is experimen-tally verified in an excellent way and can be used to determine Planck'sconstant. By invoking the theory of relativity one can show that a photonmust have also a momentum p according to the relation

p =hv/c. (2.70)

The experimental demonstration that photons are particles, with energy hi'and momentum hv / c, can be done by means of the Compton effect.

(b) The Compton effect.In this experiment a light field hits free or loosely-bound electrons. If thelight field acts only as an electromagnetic wave we should expect thefollowing effect:The oscillating field would make the electron oscillate which would be-come an electric oscillator and thus be capable again of emitting lightsimilarly to the Hertzian dipole. Accordingly the light field emitted by thiselectron must have the same frequency as the incident light. Thus thedirection of the incident light wave can be altered (i.e. light scattering takesplace) but the frequency remains unchanged.

An entirely different effect would occur if light is composed of particles.Then a photon collides with an electron (fig. 2.18). By using the usualenergy and momentum conservation laws the relations

hi' + Wei = hi" + (2.71)

h vX

Fig. 2.17. This figure presents a simple model explaining the decomposition of the photonenergy hv into the work function A and the kinetic energy mv3. The metal of linearextension L is modelled by a box. In it each metal electron can occupy an energy levelindicated by the horizontal bars within the box. The energy levels are filled by electrons up toa maximum energy. Since the electrons are bound within the metal the total energy must liebelow the zero of the potential energy, which is identical in our picture with the x-axis. Thismodel will find its justification by the quantum mechanical approach we shall discuss 4section 3.6.

.5 we shall seelaw namely by

eriments couldid seem that asis. There are,e assume thatvn examples.

electrons. Thesured depend-wave theory ofsity. Therefore3n ttlikkineticugh nsity.;ity m C elec-n remains un-e frequency of.S Einstein hasan that light iss theory it ishi', where h iskinetic energy

2.17)

(2.69)

lined electron,

sface are plottedminimum amount


prhv/c

p'rhy/c

Fig. 2.18. Compton effect. Photon incident from the left hits an electron at rest. The photonis scattered by an angle as shown in figure.

and

hi' /c = hi" /c +/el (2.72)

must be fulfilled. An elementary calculation yields the result that thefrequency of the scattered photon is shifted with respect to that of theincident photon according to the formula

v/ -1 — = (2h/m 0c 2 )sin2(0/2). (2.73)

This result is achieved if the electron is treated relativistically. Experimen-tally it turns out that one finds both an unshifted frequency which wouldpoint to the wave character of light (Rayleigh scattering) but in additionthe Compton effect shows a well defined frequency shift in accord with(2.73).

Exercise on section 2.4

(1). Derive (2.73).

Hint: Use (2.71), (2.72) and the relativistic formula We, m 0c2 (electron atrest), We'1 =--=7 mc 2 = m0c2(l _ v 2/c 2 ) - 1/2.

2.5. Einstein's derivation of Planck's law

In order to explain this approach let us briefly consider ideas about thestructure of atoms known at Einstein's time. According to these ideas,which are mainly due to Bohr, the electrons orbit around their nuclei inatoms in well defined orbits with discrete energy levels. Let us considertwo orbits with energies WI and W2 (fig. 1.14) According to Bohr'shypothesis the emitted light frequency is given by hi, = W2 - W1 . Now letus consider an ensemble of atoms of which a number N1: is in their

groundstateexcited stattions betweemission oflower to thethe numberradiation fi.

(d N/d

In order totwo differer

(a) stimulThe numbe-proportionaDenoting titransition re

(d N/d

4111,ntlBy this procany lightfielN2 and, usir

(d N/d

In thermalconstant thttransitions.

NiBup

It should 134

Fig. 2.19. Left:state 2 into thincident on anlevel and to emof the electron 4

Absorption ofand is now trar

•

§2.5 Einstein's derivation of Planck's law 59

groundstates with energy W1 whereas a number N2 of atoms is in theirexcited states with energy W2 . Following Einstein we now consider transi-tions between these two levels taking place under the absorption andemission of light. The number of transitions of atoms per second from thelower to the upper state, by means of absorption of light, is proportional tothe number of atoms in their groundstates and to the energy density of theradiation field, p. Introducing the proportionality factor B 12 we thus find

(dN/dt)abs = N1B12p.

In order to be able to reproduce Planck's formula Einstein had to postulatetwo different emission processes (fig. 2.19):

(a) stimulated emissionThe number of transitions of atoms from the excited states 2 is assumedproportional to the number of atoms N2 and to the energy density p.

Denoting the proportionality factor by B21 we obtain the correspondingtransition rate

(dN/dt)em.in = N2 B21 p. (2.75)

(b) spontaneous emissionBy this process atoms can emit light spontaneously, i.e. without presence ofany lightfield. The corresponding transition rate is assumed proportional toN2 and, using a proportionality factor, reads

(dN/dt)em,s = AN2 . (2.76)

In thermal equilibrium where the occupation number of atoms remainsconstant the number of up-transitions must equal the number of down-transitions. This yields the equilibrium condition

N1 B 12 p = N2 B21 p AN2 . (2.77)

It should be noted that this condition is necessary but not sufficient for

(2.74)

lectron at 2 2 2

rk..rkft-A.

1 1 1 Fig. 2.19. Left: Spontaneous emission of a photon. An electron may jump from its excitedstate 2 into the ground state emitting a photon. Middle: Stimulated emission. A photonincident on an atom with an excited electron may cause the electron to jump to the groundlevel and to emit an additional photon. Similarly, n incoming photons can cause a transitionof the electron of the upper to the lower level and its emission of an additional photon. Right:Absorption of one photon by the electron. The electron has initially been in its ground stateand is now transferred to its upper state.

tbout the:se ideas,nuclei inconsider

o Bohr'sNow letin their

t. The photon

(2.72)It that thehat of the

"13)xperimen-ich would

additioncord with

thermal equilibrium. Indeed we shall encounter later situations in which arelation (2.77) holds also for situations far from thermal equilibrium. Thecondition of thermal equilibrium will be introduced below [eq. (2.79)]where an explicit assumption about the N's is made. Looking at (2.77) itmight seem to us that all quantities are unknown. We will see, however,that we can determine all of these quantities. Solving (2.77) for p yields

p =[N,B 12 / ( N2 B21 ) - 1] -1. A/B21 . (2.78)

According to statistical mec'hanics the occupation numbers N1 and N2 aregiven in thermal equilibrium by

Ni = const exp[ - Wi/kT]

N2 = const exp[ - W2/ kT]. (2.79)

In particular, we obtain

N1 /N2 = exp[ - (W1 - W2 )/ kT]. (2.80)

In the following we will make use of Bohr's postulate

W2 - W1 = hi' (2.81)

where v is the frequency of the light corresponding to the transition 2 -) 1.In order to determine the relative size of B 12 and B21 we invoke the evidentpostulate that the energy density p must become infinite when temperaturebecomes infinite. This can be achieved only [compare (2.78) and (2.80)] if

T co, p co, i.e. B 12 = B21 (2.82)

holds. Since we need not to distinguish between B 12 and B21 we shall dropthe indices. Thus (2.78) reduces to

p= [exp(h y/kT) - (2.83)

To determine the rate A/ B we specialize formula (2.83) to small frequen-cies so that

hv <kr (2.84)

holds. This case had been treated earlier in classical theory by the Rayleigh-Jeans' law which describes the density p by

p(v) (87r/c 3 )v2kT. (2.85)

Expanding the exponential function in (2.83) for small exponents hv/kTwe can approximate (2.83) by

p kTA/ (hvB). (2.86)

A comparison of this result with (2.85) yields immediately

A / B = (87r / c 3 )hv 3 . (2.87)

Let us atDO,

The intawhere D(volume.we obtai.

a9(

The coefaim of ascopicAccordinof all th,second otime

N2

11111L be(2.90) in

B =

The steptions amourselvetthis poin

We wsomewhf

P(P‘

The firsv,v + disingle ploccupybnon-equannihilatemporaneous erate. By(2.74)-(

dirt,

11111

§2.5 Einstein's derivation of Planck's law 61

ions in which aquilibrium. Theow [eq. (2.79)]King at (2.77) it11 see, however,1 for p yields

(2.78)

N1 and N2 are

(2.79)

(2.80)

•(2.81)

-ansition 2 -* 1.oke the evidenten temperature

and (2.80)] if(2.82)

, we shall drop

(2.83)small frequen-

(2.84),y the Rayleigh

(2.85))onents hv/kT

(2.86)

(2.87)

11110

Let us abbreviate the right-hand side of (2.87) byD(v)hv. (2.88)

The interpretation of D(v) becomes evident when we think of Planck's lawwhere D(v)= 8Trv 2/c3 denotes the number of modes in the cavity of unityvolume. We shall need this quantity in a minute. Inserting (2.87) into (2.83)we obtain indeed again Planck's formula

p(v) =[exp(hv/ kr) - 1] -I 8Trv 2hv/ c3 (2.89)

The coefficients A and B are called Einstein's coefficients. It will be theaim of a later chapter to derive these coefficients by means of a micro-scopic theory. The physical meaning of A can be found very simply.According to (2.76) AN2 is a number of spontaneous transitions per secondof all the atoms. Thus A is the number of spontaneous transitions-per-second of a single atom and therefore has the dimension of the inverse of atime

AN2 = N2/r, A = 1/T. (2.90)

T can be interpreted as the spontaneous lifetime of an atom. By inserting(2.90) into (2.87) we can give B the form

B = (hvD(v)r) -I (2.91)The steps performed above can be generalized to non-equilibrium situa-tions and thus play a basic role in modern laser theory. Here we confineourselves to indicating some simple generalizations. We will come back tothis point in much greater detail in Volume 2 dealing with laser theory.

We write the energy density of the radiation field in the followingsomewhat different form

p(v)dv = (87Tv 2dv / c 3 )- hv • [exp(hv / kT) - 11 - 1

The first factor I is again the number of field modes in the intervalv,v + di, in the unit volume. The second factor II is the energy of thesingle photon. The last factor can be interpreted as the number of photonsoccupying individual waves. We now extend Einsteins considerations to anon-equilibrium case. We assume that each atomic transition creates orannihilates a photon. Then the following rate equation must hold. Thetemporal increase of photon number is given by stimulated and sponta-neous emission. This rate is decreased by the corresponding absorptionrate. By denoting the number. of photons by Fi we obtain by means of(2.74)-(2.76) the prototype of a rate equation for photon production

dri/dt = (N2 - Arl )Bp + AN2 . (213)

(2.92)


Now let us analyze this photon number Ft. To this end we assume that anatom emits photons in a certain frequency range A y or absorbs photonsout of this frequency range. The total number of photons ti is then givenby the number of possible modes in the volume V multiplied by thenumber of photons of an individual mode. The number of modes has beendetermined in the previous section. Again, using the abbreviation (2.88) weobtain

= V D(v)Av- n(v). (2.94)

We now wish to transcribe eq. (2.93) into an equation for the photonnumber n(v). By using the relations (2.91), (2.92), and (2.94) we are led tothe equation

dn/dt = (N2 — N Own + wN2 (2.95)


w = (VD(v)AvT) - I . (2.96)

As we will see later in the frame of laser theory, (2.95) is based on theassumption that the photons of the frequency interval A y are all producedor absorbed at exactly the same rate. This is not quite the case. Neverthe-less (2.95) is a good starting point to develop the basic ideas of a lasertheory.

When we look back at this chapter we are still confronted with a puzzle.On the one hand, there is convincing evidence that light behaves likewaves, on the other hand, there is evidence, not less convincing, that lightconsists of particles. As we have indicated in chapter 1, it was left toquantum theory to reconcile this apparent contradiction and we shall comeback to this question in chapter 5 and subsequent chapters.

3. The natur

3.1. A wave et

According toparticles. To aare atfedthe focarries a mommore convenit

2irv =

and27t/X = A

(co = circularduce an abbrt

h = h/2r

This allows us

W = hi'

and

p = h /A

in the form

W h(A)

andp = hk .

Originally, inleft, i.e. we a

and

p = h/X

in the form

W = hia

and

(3.5a)

(3.4)

3. The nature of matter. Particles or waves?

3.1. A wave equation for matter: The Sehrodinger equation

According to chapter 2, light does not only appear as waves but also asparticles. To a wave with frequency v photons each with an energy W =are attributed. Using further that c = Xi, we can cast formula (2.70) intothe form p = h/X telling us that a photon, attributed to a wavelength X,carries a momentum p. For later purposes we can cast such relations into amore convenient form. Just recall that

2-v = (.4 (3.1)

and27/X = k

(co = circular frequency and k = wave number). Furthermore we intro-duce an abbreviation for Planck's constant divided by 27, namely

h= h/27. (3.3)

This allows us to write the original relations

W = hp (3.4a)

assume that an.bsorbs photonsri is then given

Atiplied by themodes has beenriation (2.88) we

(2.94)

for the photon,4) we are led to

(2.95)

(3.2)

p = hk. (3.5)

Originally, in the theory of light, we read these two formulas from right to

left, i.e. we attributed particle properties to waves. De Broglie had), the 1

(2.96)

is b. on theare all producedcase. Neverthe-ideas of a laser

ed with a puzzle.;ht behaves likeincing, that light1, it was left toad we shall comers.

64 3. The nature of matter. Particles or waves?

ingenious idea to attribute wave character to particles by reading eqs. (3.4)and (3.5) from left to right. So as I represent it here it sounds of coursetrivial, however, note that his idea was to think of particles, for instanceelectrons, as waves when everybody was convinced that these were merelyparticles. According to his hypothesis, a particle with energy W andmomentum p should have the possibility to manifest itself as a wave. Thishypothesis was fully substantiated by electron diffraction at crystal surfaces(fig. 3.1). As we know from diffraction theory, we can expect the diffrac-tion pattern only if the wavelength is of the same order of magnitude asthe dimension of the diffraction grating. In the case of electrons, therequired grating is of the order of the lattice distance of crystals.

Historically, it is worth mentioning that the experimenters Davisson andGermer had done some of these experiments even before De Broglie'shypothesis was established, but they could not interpret their results. Let ustry to cast this hypothesis into a more mathematical form. The simplestexample of a wave is given in the form

exp[ikx — 'Ica] (3.6)

as we have seen before in chapter 2. Invoking the superposition principlefor them we can immediately explain diffraction experiments. As we knowfrom many other examples in physics, expressions describing waves arealways solutions of certain wave equations. This suggests that we shouldlook for an equation to which (3.6) is a solution. Since we deal with freeparticles we have to require that energy and momentum are connected bythe relation

W = P 2/ (2m ) . (3.7)

On the other hand, W and p can be expressed by co and k according to

electron source

targetFig. 3.1. Experimental scheme of the arrangement of electron diffraction lexperiment byDavisson and Genuer. When the detector is put in different positions as igdicated in thisfigure, diffraction minima and maxima are found.

(3.4) and (3.When we wrelation. Aswith respeclwave functiwe obtain

aih —

at ex

and accordi.3ih—at

ex

Of course,proceed in zthe right ha(3.6) twice vevelply pthe owin

—

h21.= "

By equatingby (3.6), wehand, the e;entiation pr(of (3.8) andthe knowndetermined,

h2—

This is indc.freely movitSchrodingerkeep in minderived onlyone has toform. Thenpossible andSchilliger

§3.1 A wave equation for matter: The SchrOclinger equation 65

ading eqs. (3.4)unds of course2s, for instancese were merelyenergy W andIs a wave. Thiscrystal surfacesect the diffrac-f magnitude as

electrons, thetals.s Davisson ande De Broglie'sr results. Let us1. The simplest

(3.6)

ts. As we know3ing waves arethat we shoulddeal with free

c connected by

(3.7)

k according to

on experiment by3 indicated in this

•

(3.4) and (3.5). Thus as a result of (3.7), a relation between k and w follows.When we wish to derive a general wave equation we have to secure such arelation. As we know from wave equations they contain differentiationswith respect to space and time. Therefore we are led to differentiate thewave function (3.6) with respect to time. By multiplying the result with ihwe obtain

aih

Ttexp[ikx – ica] = hwexp[ikx – ica] (3.8)

and according to (3.4)

. aih—

at exp[ikx – iwt] = Wexp[ikx – iwt]. (3.9)

Of course, we have multiplied (3.8) with ih to obtain just the energy W. Weproceed in a similar way to obtain the expression p 2/2m which occurs onthe right hand side of eq. (3.7). Differentiating the exponential function(3.6) twice with respect to x we obtain the factor (–k 2 ). To obtain p 2 andeventually p 2/2m we have still to multiply ( – k 2 ) by – h 2/2m. This yieldsthe following equations

h 2 a 2–

2m ax2 exp[ikx – icot]

h 2k 2 exp[ikx – icot] = —

2P

m

2 exp[ikx – iwt]. (3.10)

2mBy equating (3.9) and the right-hand side of (3.10) and dividing both sidesby (3.6), we obtain the relation (3.7) as we had wished it. On the otherhand, the expressions in (3.10) were obtained in a general way by differ-entiation processes so that we might equally well equate the left-hand sideof (3.8) and the left-hand side of (3.10). Writing more generally instead ofthe known function (3.6) a wave function 11,(x, t), which is still to bedetermined, we find eq.

h2 a2 a ,— t) = in—y

at(x,t). (3.11)

2m ax2

This is indeed the famous Schrodinger equation for the special case of afreely moving particle. Of course, to some readers this derivation of theSchrOdinger equation may look somewhat heuristic. The reader shouldkeep in mind, however, that all fundamental equations of physics can bederived only in an inductive way. On account of more or less evident factsone has to develop a certain hypothesis and to cast it in mathematicalform. Then one has to draw as many conclusions of this formulation aspossible and to compare them with experimental results. In the case of theSchrOdinger equation and the more general Schrodinger equation of a


particle under the influence of external fields these predictions have beensubstantiated very thoroughly. One limitation should be observed, how-ever: the SchrOdinger equation refers to particles moving in the nonrelativ-istic domain. We will use the SchrOdinger equation in our book becausemost of the important effects in lasers and non-linear optics can be treatedby it. For the fully relativistic case, the Dirac equation must be considered.

Within the heuristic spirit of our derivation it is quite simple to derive aSchrodinger equation for a particle moving in a force field. We assumethat this force field can be derived from a potential V. When the force isconservative, in classical physics the energy conservation law holds. Weintroduce the momentum p which is connected with the velocity dx/dt ofa particle by p = m dx/d t. The sum of kinetic and potential energy thenyields the constant energy W

p 2/ (2m) + V = W. (3.12)

The expression on the left-hand side is known in mechanics as theHamiltonian

H = p 2/ (2m) + V. (3.13)

When V is constant, the relation (3.12) can be obtained from a Schrodin-ger equation by adding the expression

VII. (3.14)

to the left-hand side of (3.11). It has turned out that this Schrodingerequation remains valid even when V depends on space x and time t. Thisyields the Schrodinger equation

[ h2 a2— — — V(x) 4.4x, t) = i h —

at4,(x, 1).

2m ax2a

(3.15)

In general we have to deal not only with one-dimensional but three-dimensional motions. To extend the Schrodinger equation (3.15) to thiscase we write (3.7) in the form

p)2 +1,2)/ (2m).

The sum over the squares of the momentum components is to be replacedby a sum over second derivatives with respect to x,y, z in order to obtainthe correct wave equation

a2 a2 a2 a2(3.17)

ax2 ax2ay2az2

The A symbol is called Laplace-operator. Often the resulting Schrodinger

equation is A

HIP = if

In it H is the

H= --

In writing thbe understoowith 4) and ti

ProvidedSchrOdinger1.h.s. of eq. (_the r.h.s. co,mathematicsof a product.

,t)

whellikepeinserting thisequations, orreads ih(af/exercise 4 on

f = exp

The same p,containing ti

44x, t)

Since the dilonly on the eW. Then onexp(—iWt/hindependent

Htly =

Since H hasWe will get rwe shall prediscover wespond to clot

(3.16)

§3.1 A wave equation for matter: The SehrOdinger equation 67

is have beenserved, how-.e nonrelativ-)ook becausein be treated

considered.to derive a

We assumethe force is

w holds. Weity dx/dt ofenergy then

(3.12)

anics as the

03.13)a Schrodin-

(3.14)

Schrodingertime t. This

(3.15)

but three-3.15) to this

(3.16)

be replacedler to obtain

(3.17)

Schrodinger

4110

equation is written in a somewhat formal waya

--= ih—tp. (3.18)at

In it H is the Hamiltonianh2

H= – —2m

+ V(x). (3.19)

In writing the Schrodinger equation in the form of eq. (3.18), (3.19) mustbe understood in such a way that V stemming from H must be multipliedwith tp and that in addition the Laplace-operator acts on 1,1).

Provided V is time independent we can derive the time independentSchrodinger equation. In such a case the expression in brackets on the1.h.s. of eq. (3.15) is an operator containing x and acting only on x, whilethe r.h.s. contains the operator 8/81 acting only on t. As is shown inmathematics in such a case one may find the solutions 4'(x, t) in the formof a product,

tP( x , i ) = f(t) ,P( x) (3.20)

where f depends only on t and tP(x) on the r.h.s. of (3.20) only on x. Afterinserting this ansatz into (3.15), we can split the resulting equation into twoequations, one for f( 1) alone, and one for tP(x) alone. The equation for f(t)reads ih(allat)= Wf, where W is a still undetermined constant (cf.exercise 4 on this section). The solution of this equation reads

f = exp[ — iWt/h].

The same procedure can be applied to the Schrodinger equation (3.18)containing the Hamiltonian (3.19). We therefore try the ansatz

tp(x, t) = exp( – iWt/h)tp( x). (3.20a)

Since the differentiation, with respect to time, acts on the r.h.s. of (3.18)only on the exponential function we obtain on the r.h.s. of (3.18) the factorW. Then on both sides of (3.18) we are left with the exponential factorsexp(– iWt/h). Dividing both sides by this factor leaves us with the timeindependent Schrodinger equation

Htp = wp. (3.21)

Since H has the dimension of energy, W must have this dimension also.We will get acquainted with solutions of equations (3.21) in section 3.3 andwe shall present further examples in sections 3.4 and 3.7. As we willdiscover we find solutions of the Schrodinger equation (3.21) which corre-spond to closed orbits in classical mechanics. The basic new features in the


wave theory of matter will be that for bound states we obtain a discretesequence of energy levels W quite in agreement with Planck's originalhypothesis. We shall distinguish these discrete energy levels by indices andwrite W1 , W2 , ... . To these energy levels there belong wave functionswhich we denote correspondingly by indices tp,, 'P2.....

Since we deal with mechanics which treats the motion of particles andsince this motion is quantized we call this theory "quantum mechanics".For its understanding the correct interpretation of the wave function iscrucial. In contrast to conventional wave theories it has turned out that tpcan be correctly interpreted only in a statistical sense. According to thisinterpretation hp(x)I 2 di/ gives the probability of finding the particle(electron) around the space point x in the volume dV (fig. 3.2).

The reason for this interpretation can easily be seen. Consider anexperiment analogous to Young's double slit experiment (cf. section 1.10),but now done with an electron beam. Or, more realistically, we may thinkof the electron diffraction experiment by Davisson and Germer which wediscussed at the beginning of this section. Let us assume that we try tomeasure the scattered electrons by a set of individual localized electroncounters which we have arranged around the sample (similarly to fig. 3.1).When we lower the intensity of the electron beam so much that only oneelectron is present at a time, the following happens: Only one of thecounters clicks, and we are not able to predict which one. However, whenwe repeat this experiment very often and plot the number of events versusthe position of the individual counters, we obtain a curve which coincideswith the one given by I tp(x)I 2 , except for a constant proportionality factor.4, is the wave-function determined by the Schrodinger equation. Since wecan cover all positions by the counters by shifting them, x is indeed thecontinuous space variable. Clearly, I 0(x)1 2 d V is a measure for the proba-bility to find the electron around the space point x in the volume dV, as wehave just postulated it.

Asof thebelow)partich

Theconditisum (oa certation cc

hencefindices

Oreorthogvector:

Exerci

(I) ENSchroc

whereity int

47F

Show

where

41110

Fig. 3.2. Itgx, 01 2 is plotted versus x. The shaded area shows the fdy•babilitYthee probability of finding the electron within the interval x...x dx.

§3.1 A wave equation for matter: The SehrOdinger equation 69

obtain a discrete As we will find again and again in this book, the statistical interpretationPlanck's original of the wave function (and also of other quantities which we will define

els by indices and below) is the price we have to pay for the reconciliation of the wave andg wave functions particle pictures.

The interpretation of 10 2 dV as a probability implies a normalizationn of particles and condition. According to fundamental postulates of probability theory the-awn mechanics". sum (or the integral) over the individual probabilities to find the particle at'aye function ip is a certain point x must be normalized to unity. We shall use this normaliza-turned out that 4. tion conditionkccording to thisling the particle f IIP(x)1 2 dV= 1 (3.22)3.2).en. Consider an henceforth. Furthermore one may show that wave functions with different(cf. section 1.10), indices are orthogonal to each other, i.e. they obey the relationsily, we may thinkerme which we f 114,(x)1/)„(x)dV = 01, for m = n (3.23)

tet e try to for m n.oca electron Here we have included a case m = n which corresponds to (3.22). Thisilarly to fig. 3.1). orthogonality property (3.23) is quite analogous to the orthogonality of 2ch that only one vectors whose scalar product vanishes.Dnly one of the•However, whenof events versus Exercises on section 3.1which coincides

)rtionality factor. (1) Determine the normalized solutions ci(x) of the time-independentuation. Since we Schri5dinger equation with• x is indeed the:-e for the proba- h2 a2

H= --- (W real!)olume dV, as we 2m ax 2

where ç(x) obeys the periodic boundary condition. L is length of periodic-ity interval.

so(x + L) = p(x).

Show that

roLT:(x)Tm(x) x =j d

where

bility lap x,6 01 2 dx,1 2rrn

cp,,(x) = exp(i kx), k = n = integer.L

(2) Treat the problem of exercise 1 in three dimensions, where

q9(x + L,y,z)= cp(x,y,z)

q9(x,y + L,z)= co(x,y,z)

cp(x,y,z + L) = 4p(x,y,z).

(3) Treat the following one-dimensional problem. A free particle is en-closed in a box whose walls at x = 0 and x = L cannot be penetrated bythe particle. What are the energies and wave functions? Check the ortho-gonality of the wave functions.Hint: Solve the free particle Schrodinger equation for 0 x < L andsubject the wave functions to the condition 0(0) = 4/(L)= 0. Use asuperposition of the form

tP(x) = a exp(ikx) + bexp(—ikx).

(4) Show that the time-dependent Schrodinger equation (3.15) can be splitinto

ihaf/at = Wf and 1-14/(x) = WO(x)

by means of 4,(x, t) = f(t)0(x).Hint: Transform the resulting equation into

ihallat HtP(x) At) tP(x) •

Since the 1.h.s. depends only on I, the r.h.s. only on x, but x and t areindependent variables, (*) implies that each side can be equal only to aconstant, which we call W.

(5) Show thatCO

E exp( iffint/h)On(X)n

where the c,i 's are constant coefficients and the are solutions of thetime-independent Schrodinger equation (3.21), i.e.

=

is a solution of the time dependent Schrodinger equation (3.18).Hint: Insert (*) into (3.18) and use ( * *).

(*)

(*)

Mea

The staticonsequeagain anof a partireminiscerolling dionly withthrow a ciLet us (Itwhen thr(by the sprobabili

1110ave-gravitymasses. Iindividua

Let usmeasurinspace poplays tiltcoordinaThis alio'averagemechanic

=

The stati.space co(ing the a

px =

For its dquantum

§3.2 Measurements in quantum mechanics and expectation values 71

3.2. Measurements in quantum mechanics and expectation values

The statistical interpretation of the wave function has many importantconsequences and the reader is well advised to recall this interpretationagain and again into his memory. When we measure the space coordinateof a particle it can be found at that point or another one. This is stronglyreminiscent of rolling dice. When we wish to make predictions whenrolling dice or on other stochastic processes we can make such predictionsonly with respect to probabilities or to average values. For instance, if wethrow a die very many times, we can determine the mean number of spots.Let us denote the probability of obtaining a certain number of spots n

when throwing a die once by P„. Then the mean number of spots is definedby the sum over the number of spots n = 1, , 6 multiplied by theprobability P,,

= 2 nPn . (3.24)

This average is strongly reminiscent of the calculation of the center ofgravity in mechanics, n corresponding to the coordinate and P„ to themasses. In a similar interpretation (3.24) means nothing but weighting theindividual measured values n.

Let us apply this idea to quantum mechanics and let us start withmeasuring the particle coordinate. The probability of finding a particle atspace point x in the volume dV was given by 14/(x)I 2 dV. This expressionplays the same role as P„ in the case of dice. The measured spacecoordinate x of the particle now corresponds to the number of spots n.

This allows for direct translation of (3.24) and thus for a definition of theaverage value of the space coordinate x of a "particle" in quantummechanics.

i =fx 111,( x )12dv.

The statistical interpretation of quantum mechanics holds not only for thespace coordinate but equally well for the momentum. The rule for calculat-ing the average value of the momentum reads

fix =flP*(x)(h/i)-(x)dv. (3.26)

For its derivation we must refer the reader to the usual textbooks inquantum mechanics.

article is en-enetrated by:lc the ortho-

: x < L and= 0. Use a

caniisplit

3).

(3.25)

(*)

x and t areLai only to a

(*)

'lions of the


Quite generally we can formulate rules for the transition from classical withmechanics to quantum mechanics. In classical mechanics we are dealingwith certain measurable quantities, often called observables, such as coor-dinate or momentum of a particle or its energy, etc. As we have seen insection 3.1 and here again, operators are attributed to these observables. but aThese quantities are called operators because they operate in a welldefined way on the wave functions. For instance, the space operator xmultiples the wave function by it. The momentum operator (h/i) d/d xdifferentiates the subsequent wave function, etc. In order to be able to andmake predictions about the outcome of measurements one has to formaverage (mean) values by the prescription (3.25) and (3.26).

These mean values make a prediction about the experimental average The.values when the experiment is repeated very often. Therefore these meanvalues are also called expectation values. Let us supplement this scheme bythe example of the kinetic or the potential energy. This leads us to table3.1. The

andTable 3.1 siteeOne-dimensional case

Observable

Space coordinate x

Momentum px

Kinetic energy

p2/2m

Potential energy

V(x)

Operator

h a7 ax

h2 a2

— 2m ax2

V(x)

Expectation values

f 0*(x)xsp(x)dV

f 1,P*(x) d V

_ h2 a2fip *(x)Ti—n a,-70( x) d V

f t(x)V(x)*(x)dV

In orhas t(

Anotfromword

_h2 a2Total energy 2m —ax2 + V(x)

p 2 /2m + V(x)

_h2 a2

fo.(x)( 2m ax2 + V(x))0(x)dV

3.2.1. Dirac notation

In the more modern literature, both in texts and original scientific publica-tions, wave functions and expectation values are often represented in stillanother way. As we will see in the next sections we have not only to deal *SU

with expressions of the form

f lii*(x).np(x)dV (3.27a)

but also with expressions of the form

f lp,„(x)x4).(x)dV (3.27)

and similar expressions containing other operators Si instead of x

f 4,:,(x)Shp„(x) ay. (3.28)

The other notation mentioned above reads

11,:,(x)S24.„(x) dV= (3.29)

The English physicist Dirac has split this expression On IS2 In> into Km Iand In>. Making a pun he also split the word "bracket" into "bra" and"ket" which led him to the bra-ket notation

bra and ket(3.30)

< m 1 In>.In order to go over from Schrodinger's notation to Dirac's notation onehas to make the replacements*

tp„(x) In>

dVtp:,(x) (3.31)

from classicalAre are dealingsuch as coor-have seen in

,e observables.ate in a wellce operator xor (h/i)d/dxto be able toc has to form

nental average)re these meanthis scheme bytds us to table

•lues

dV

:) dV

(x) dV

dV

x ))0(x ) dV

Another commonfrom one to Nwords, in the

( 2..) =

•Still, another notation

abbreviation for (3.29) is St„,„•we may arrange the SZ„,„'s in

matrix

'11 11 2 12' • • • • • 2IN

221 1.2n • • • — g2N

2N1 stN2 • - - • aNN

is f cp*(x)12x(x) dV <ciaiX>•

When we let run m and na square array or, in other

(3.32)

.ntific publica-tsented in still)t only to deal

functicled to

62(

whereequatittable 2SZ to eoperat(3.37)(4 Aceigenvmentathe to/.Similauh's

A fSU

recisithatparticlprecis(whethprecis.just ccpostulfuncti

and

We c:soluti,act oton bcmake

'It it

boundedefine°section


mn are therefore called matrix elements. For later use we need the notionof a Hermitian matrix and a Hermitian operator. A matrix (S2„,„) is calledHermitian, if its elements fulfill the relation

= E2tn (3.32a)

i.e. we obtain the conjugate complex of E2 when we exchange the indicesm and n. According to (3.29) an operator S2 can be connected withmn <m1210. An operator S2 whose matrix elements fulfill (3.32b), is

called a Hermitian operator.In the final part of this section we want to complete our presentation of

the formal frame of quantum mechanics. This part is somewhat moredifficult and can be skipped until the reader begins chapter 5 and the laterchapters.

3.2.2. Equations for the determination of the wave function tp. Themeasuring operator

Our considerations of the present section were based on the assumptionthat the wave function p(x) is already given. In section 3.1 we wereacquainted with the Schr8dinger equation as an equation to determineip(x). We now want to show that aside from the Schrodinger equationthere are other equations by which we can determine (p(x). (For simplicitywe again consider the one-dimensional case.) In quantum theory, thespace-dependent part of the wave function describing the motion of a freeparticle with momentum hk is given by

exp[ikx]. (3.33)

On the other hand, we just came across (compare table 3.1) the operatorwhich corresponds to the momentum p x , namely (h/i)d/dx. When weapply this operator to the wave function (3.33) we readily obtain

h d" exp[ikx] = hkexp[ikx] . (3.34)dx

Similarly when we apply the operator of the kinetic energy to the samewave function we obtain

h2 d2 h2k2— — — eXp [ kx] = -

2 exp[ kx]. (3.35)

2m dx 2m

The factor in front of the r.h.s. is just the classical kinetic energyp 2 m (3.36)

From these cases we learn that there are operators SI (for instance S2 =(h/i) d/dx) which, when applied to a wave functionvp, yield the wave

sz(2)p(x) = co(2)4i(x).(3.34) (3.39)


function multiplied by the measured value, for instance hk. Thus we areled to consider equations of the form

4/( x ) = cook(x) (3.37)*

where the index I distinguishes the different functions tp which obeyequation (3.37). The constants co l are called eigenvalues. According to thetable 3.1, we may attribute a quantum mechanical operator which is calledS2 to each classical observable. As is shown in quantum mechanics, theseoperators are Hermitian. According to mathematical theorems, the eq.(3.37) allows for a complete set of eigenfunctions 4,1 with real eigenvaluescol . According to the fundamental postulate of quantum mechanics theeigenvalues col are precisely those values which can be measured experi-mentally. For instance when we identify S2 with the momentum operator,the w i 's are just the values of the momentum we can measure, namely hk.Similarly when we use the Hamiltonian (3.19), i.e. the energy operator, thecol 's are just the quantized energies W1 we can measure.

A fundamental problem in quantum mechanics is whether we canmeasure two different physical quantities simultaneously with absoluteprecision. We have seen in the introductory chapter (namely section 1.9)that in general by measuring a certain quantity, say the location of theparticle, we simultaneously make it impossible to measure the momentumprecisely. We now want to derive a criterium which allows us to decidewhether we can measure two observables simultaneously with absoluteprecision. To this end we introduce the measuring operators S2 (1) and S2(2)just corresponding to these two observables. According to the fundamentalpostulate just introduced above we must require that the same wavefunction 1p fulfills the two "measuring equations"

s-2 (1) ip( x = 6)(1) 1p( x

and

(3.38)

rgy to the same

(3.35)

energy

(3.36)or instance S2 =--, yield the wave•

We can easily find a condition for S-2(1) and SP) so that the simultaneoussolution of (3.38) and (3.39) is possible. To this end we let the operator S2(2)act on both sides from the left on eq. (3.38) and similarly the operator S2(1)on both sides of eq. (3.39). Then in the equation resulting from (3.38) wemake again use of eq. (3.39) and vice versa. Subtracting the two resulting

*It is shown in mathematics that these equations must be supplemented by appropriateboundary conditions on In our following discussion it is assumed that those conditions aredefined, and fulfilled by 4,. For explicit examples consult the exercises at the end of thesection.

need the notion(S2,„„) is called

(3 .32a)

ange the indicesconnected with

(3.32b), is

presentation ofsomewhat morer 5 and the later

i. The

theelimption)n 3.1 we werem to determinedinger equation. (For simplicityurn theory, themotion of a free

(3.33)3.1) the operatori/dx. When weobtain

equations from each other we immediately find the following equation(2(2) 2(1) _ 2(i)s2 (2) )0 = G0(2)(4(1) (.0(1)61(2)) = 0. (3.40)

Since the left-hand side of (3.40) vanishes for a complete set of ip 's, one hasintroduced a shorthand writing

2(02(2) _ 20)2(1) = 0. (3.41)To check such a relation one has always to think that the total expressionon the I.h.s. of (3.41) must be applied to an arbitrary wave function. Weshall study the significance of the relation (3.41) in some exercises on thissection. Let us assume that the relation (3.41) is fulfilled. Then one showsin mathematics that it is always possible to find the wave functions 0(x) sothat the eqs. (3.38) and (3.39) are simultaneously fulfilled. We may sum-marize these results as follows: The relation (3.41) is necessary andsufficient that the observables to which 52(1) and 2(2) are attributed can bemeasured simultaneously with absolute precision.

3.2.3. The relation between equation (3.37) and the time-dependentSchrodinger equation

From our above considerations, it seems to follow that the Hamiltonian(3.19) plays the same role as any other measuring operator, for instance themomentum operator. This is indeed true where the equation (3.37) isconcerned. For H (3.19), eq. (3.37) is just the time-independentSchrodinger equation. We must bear in mind, however, that H appears inthe time-dependent Schrodinger equation also. This equation plays anextra role, because it determines in any case the temporal developmentof /'(x, t). The connection of (3.37), including the time-independent Schro-dinger equation, and the time-dependent SchrOdinger equation can be seenas follows: We first choose an observable we want to measure, and thecorresponding operator O. Then we solve (3.37) (or perform the measure-ment). We choose a certain 44x) (or the wave function belonging to themeasured value col ). This 1,1.4(x) serves as the initial condition for thesolution of the time-dependent Schrodinger equation at the initial time to(= time of measurement) so that

1(x, to) = tpio(x).

Therefore, after the measurement was made, the further time-evolution ofthe wave-function is uniquely determined by the time-dependent Schrodin-ger equation.

Exercises on s

(1) To get use

(ni

-ft-th2

<n

for arbitraryHints: (a) Us

ern)

(fl) Use

(nr —ai 8x

and integrateanalogous to

(2) Verify theLet so = c, vwhich can als

<xls2199;

<plolx;

Let fp

<x121(p;

and

<TAX)

Hint: Use eac<4)1 0 1X> = ft.

•


Exercises on section 3.2

(1) To get used to the bra and ket notation, show

<nlxIm> = <mIxIn>*

I n\

h a—Taxkl ) = m(

h 3I ax

n)* (S)

(2) Verify the following properties of bras and kets.Let (P= clp, where c is a (complex) constant, and I an arbitrary operatorwhich can also be =- 1. Then

<xls-210=- <x1 0 10> = c<x1010>

<c%Pl u lx>= c*<0101x>•

Let (19 = cop i + c 202 , then

<XIS21(1)>=" <X1 E2 i c t4u I + C202> = cAl E2 140+ c2<x1E2102>

and

Op l s2 lx>--= <C 1 01 + c2021 0 1x> = cr<4, 11 2 1x>+ c1<02101x>•

Hint: Use each time the integral defining <so 1E2I x> ' that is

<TI O I x> = .frP*E2x dV (cf. footnote on page 73).

ig equation

(3.40)

of tp 's, one has

(3.41)

otal expression; function. We:ercises on thishen one showsictions tp(x) soWe may sum-necessary andributed can be

relent

Hamiltonianor instance thettion (3.37) ise-independentH appears in

tion plays anI development)endent Schro-xi can be seenisure, and the

the measure-longing to thedition for theinitial time to

e-evolution oflent Schrodin-

(a)

A2 A2 h2 a2

(nM = m

n)*(y)—

2m ax2T—

m ax2

<n117(x)Im> = <mIV(x)In>* (8)

for arbitrary <n I, m> .Hints: (a) Use

<nIxlm> = f 0:(x)xtP,n(x)dV

(f3) Use

(n_h aiax m ) —

_ To * h a1,1dn (x) --Ipm(x)dV

-.0 ax

and integrate by parts. It is assumed that 10„1 2 —> 0 for I x I ± co. (y)analogous to (/3). (8) analogous to (a).

(3) Calculate

f0LIOn(x)12xdx

h a

o(x)dx

i ax

1 L On* (x)(— .1h72n --E-i)tPn(x) dx

for1

On( x ) = exp(ikx), k =VL

(4) In classical mechanics the vector of angular momentum is defined by

ex = YPz zPy

ey = zpx — xpz

e = xpy — ypx.

Translate these quantities into quantum mechanical operators.Hint: Use the rule

p = h h1 8, a , a \•ax ay az)

(5) The following exercise intends to clarify the meaning of the expansioncoefficients c„ in the wave-packet of exercise 5 on section 3.1.(a) Show that (* ) of exercise 5 on section 3.1 is normalized provided

E ic,j2=n=o

(b) Show that

<4, 1 1/10= E wnlen12n=o

(for hints see below.) This relation is extremely important, because itallows us to interpret the meaning of IC'n I 2 . The 1.h.s. is the expectation(.=-- mean) value of the energy operator (F.-- Hamiltonian H). On the r.h.s.is a sum over the energies W„ of the states (wave functions) on . Thus (* *)can be interpreted as

W= E wnlc.12.n=0

Now compare the formation of this expression with (3.24). Clearly, In

Hints: Insdifferentia

(7) Let E2Put 4, =

<OP*San

2 Trnn = 1, 2 ... .

L

correspondsaddition Ic.

E Pn

(Compare (of Ic„I 2 : lc"n" whenHints: (a) 1and use

<0.1

(b) Insert (H4,„ =

(6) Commtcommutatc

Prove

{h aax

where theCalculate

T,

(*)

00

corresponds to the number of spots that can be found in principle. But inaddition I cn 1 2 corresponds to Pn , including the normalization condition

= 1.rz

(Compare (*).) This gives us the clue to the interpretation of the meaningof I cn I 2 : I cn 1 2 is the probability to find the quantum system in the state"n" when we measure the energy.Hints: (a) Insert (5* of exercise 3.1) into <4, tp>, multiply term by termand use

< IPn I 1P,n > = (* * *)

(b) Insert (5* of exercise 3.1) into <4)1 WI >, multiply term by term, useHIP„= Wn tpn and (* * *).

(6) Commutation relations: We use the following abbreviation for thecommutator of two operators 01,

0,22-020.=[E2,,22]•Prove

h a h 2 a2 1 =

I_i ax 2m aX2

, V(X))1 = --n --aVox k Zrn ax 2 i axr h2 a2 v(x) ) ,x 1 = _ Tx- =—ra

h 2 a h

IA — 2m ax2

where the abbreviation px = (h/i)a/ax was used.Calculate

[ h2 a2 h2 a2v(x))1 =?2m ax 2 ' k 2m ax2

V(x)),x1 = Vk 2m m mi

Hints: Insert after [ • • ] an arbitrary wave function q, and perform thedifferentiations.

m is defined by

ors.

f the expansion3.1.d provided

(* *)

ant, because itthe expectation). On the r.h.s.tpn . Thus (* *)

(7) Let 2 be a measuring operator and IN the eigenfunctions of eq. (3.37).Put 4. = I cply, and show that

41 0 14/> = E 1C/12W/.

4). Car, WnHint: Same as for exercise 5.


3.3. The harmonic oscillator

Consider the motion of a particle of mass m bound elastically to itsequilibrium position (fig. 3.3). As particle coordinate q we take the dis-placement from the equilibrium position. Denoting the force constant by fthe classical equation of motion reads

d2qm— = –fq.dt2

The particle momentum p is connected with its velocity by

dq=

or solving for dq/dt the relation reads

–P/m.

q

K(q)

q

Fig. 3.3. The harmonic oscillator. Upper part: Mechanical model of a ball with mass mdisplaced by an amount q from its equilibrium position. Middle part: The elastic force K(q)as a function of the displacement coordinate q (Hooke's law). Lower part: The potentialenergy V(q) as a function of q.

Different

cTP

The equausing the

H

The HamThe potetrelation

F=

and read:.

=Different.equationthe mintnand (3.45

dt'

dtp

This is atthem whetor it is krconstant t

f=With (3.51

H = -

We entertum p by

P =

(3.42)

(3.43)

(3.44)

•

f= nu42. (3.51)

§3.3 The harmonic oscillator 81

Differentiating (3.43) with respect to time and using (3.42) we obtain

cTiP

•

=

The equations (3.44) and (3.45) can be cast in an especially elegant formusing the Hamiltonian:

„2H =

2m + q2 . (3.46) 2

The Hamiltonian can be obtained as sum of a kinetic and potential energy.The potential energy V is obtained from the force F by means of the usualrelation

avF= _. : == —fq

and reads

f 2V = —2

q . (3.48)

Differentiating (3.46) with respect to p we obtain the right hand side ofequation (3.44) and differentiating it with respect to q we obtain except forthe minus sign the right hand side of (3.45). This allows us to write (3.44)and (3.45) in the form

• ax— a =dt' ap

afi—p

•

= (3.50)dt aq

This is an example of the Hamiltonian equations. We will make use ofthem when quantizing the light field. From the classical harmonic oscilla-tor it is known that its oscillation frequency co is connected with the forceconstant by

iastically to its'e take the dis-:e constant by f

(3.42)

(3.43)

I. (3.44)

(3.45)

(3.47)

(3.49)

With (3.51) the expression (3.46) acquires the form

H =2mP

2 -r M 6)

2— — q 2 . (3.52) 2

We enter the realm of quantum mechanics when we replace the momen-turn p by an operator

h dP= dq

. ball with mass me elastic force K(q)part: The potential (3.53)

(3.56)h2 d 2 mw2q2}1p(q) = wtp(q).

2m dq 22


p and q satisfy the commutation relation

pq — qp =

The best way to understand this relation is by thinking of both sides of(3.54) as applied to an arbitrary wave function tp and using the explicitrepresentation (3.53)

h d h d h_=i dq` i dq'

Differentiating the first term of (3.55) by the product rule, we obtain(h / i)tp + ( h / i)q chp / d q . By subtracting the second term as indicated in(3.55), we do, in fact obtain precisely the right hand side of (3.55). As(3.55) holds for any differentiable function, we may treat (3.54) as anidentity. Although the basis of (3.54) may appear rather trivial, commuta-tion relations are of fundamental importance to many problems in quan-tum optics. Substituting (3.53) into (3.52), we obtain the Hamiltonianoperator of the harmonic oscillator. With it the Schrodinger equationbecomes

=MG)

We further put

tp(q) 11.( h

so that the Schniklinger equation (3.56) takes the form

h d2—2— ---dE2 )(PM = Wc9()'

For later use, we write its Hamiltonian in the form12.. h 6) ( 72 + 42)

where r has the meaning of a momentum. If we were now able to treat theoperators and d/cq as ordinary numbers, it might well appear that the

operator on

( d_ 2

could be reg(— a2 +

which could(—a +

Let us thereby

hw

Byand in

(3.62) =

Here the firshould wish.

—d

the second ti

II —

[The commu(3.54) by su(3.62) are al

follows:

1 t_k

11

NriAs (341 an

(3.54)

(3.55)

To treat (3.56) further, let us introduce a dimensionless coordinatedefined by

(3.57)

(3.57a)

(3.58)

(3.59)


operator on the left hand side of (3.58)

(3.54)2

4. e)_CI_

: both sides of could be regarded as an expression of the typerig the explicit

(_ a 2 2) (3.60)

which could then be factorized in the form(3.55)

(—a + /3)(a + /3). (3.61)Ile, we obtainis indicated in

of (3.55). As.t (3.54) as anvial, commuta-Aems in quan-e HiliktonianingeWfuation

(3.56)

Let us therefore, just as an experiment, replace the left-hand side of (3.58)by

d , \ d \ 11ric.) )kc c)—(P.

By multiplying out the brackets, but taking care to keep the operatorsand d/g in the right order, we obtain

no d2(3.62) = —2— — —c 2

de 4- E } C9(n t 16°.

hco ( d d 1

(3.62)

; coordinate

(3.57)

(3.57a)

(3.58)

Here the first expression is the same as the left-hand side of (3.58) as weshould wish. By the commutation relation

—de

— = i (3.63)

the second term reduces to

II = — hto

(p. (3.64)2[The commutation relation (3.63) was derived directly from the earlier one(3.54) by substituting for q using (3.57)1 The expressions in brackets(3.62) are again operators; let us write them at first purely formally asfollows:

1 t_ d + 0 = b+(3.65)

g )1 i d ± i \ = b.

(3.66)Vf k de i

iAs (3.62) and the left-hand side of (3.58) differ by the term (3.64), we

(3.59)

.ble to treat theippeitat the


introduce the displaced energy

W'= W — Ihto.

Then the Schrodinger equation (3.58) can finally be replaced by

hwb + bq) = W'q).

To derive a commutation relation for b and b + , let us construct the termbb + —b + b and substitute the operators according to (3.65) and (3.66).Using (3.63), we then obtain the basic commutation relation

bb + —b + b = 1. (3.69)

For reasons which will emerge later, we will call operators satisfyingcondition (3.69) Bose operators.

We will now show that, with the aid of operators b + , b and commuta-tion relation (3.69), we can construct eigenstates of the harmonic oscillator.Here we start with the assumption (compare exercise 2 at the end of thissection) that the energy W of the quantum mechanical oscillator is alwayspositive, and certainly has a lower bound. Let us denote the state with thelowest energy value W by sco. When multiplying equation

h c4b + b cpo = Kcpo (3.70)

from the left by the operator b, we obtainhoi(bb + )40 = Kbipo. (3.71)

By using the commutation relation (3.69), let us replace bb+ by (1 + b + b)on the left-hand side of (3.71), so that

hcof 1 + 6 + b}bcoo = Kbcpo. (3.72)

Finally, taking hcobcpo to the right hand side, we obtainhwb + b(bcpo) = (ffic; — hca)bcpo. (3.73)

However this equation shows that 40 is a new eigenfunction with theeigenvalue W — hw, contrary to our assumption that coo is the lowest state.This contradiction can be resolved only if

bcpo = 0. (3.74)

This equation will be used from now on to define the ground state To.Let us try to solve (3.68) using nothing but algebra. With this aim in

mind, let us multiply equation (3.68) by b + from the left:

b + (hwb + b)q) = W'b + cp. (3.75)

Using commutation relation (3.69) and the steps just indicated, we obtainhcob + 6(6 + co) = (W' + hca)b + p. (3.76)

Clearly, if cp iand its eigenvan amount hc,excited eigenti

(1). (b+

Since equatioreigenfunctionsnormalize T„dimensionless

f 99:(i—

As we shallpurely algebrathe normalize(

(P.VT1!

What are the cmultiply (3.74'Obviously KW(') n times by

W' =

or, in terms of3.4),

W= hco(

(3.79) may bequanta of makincreased by cis called a crerquantum is ar3.5).

The formali:of fundamentaof finding an ewant to showthis end we str.

bon = 0

(3.67)

(3.68)

•


Clearly, if cp is an eigenfunction of (3.68), 6 + so is also an eigenfunction,and its eigenvalue is larger than the eigenvalue of the former function byan amount hco. By applying the operator b + n times, we obtain the nthexcited eigenfunction

= ( b+ )4%. (3.77)

Since equation (3.68) is homogeneous in cp, there remains, as usual witheigenfunctions, a free constant coefficient. We will choose its value tonormalize son and from now on we will define normalization in thedimensionless coordinate by

9):(09).(n = 1.

As we shall show below, the normalization factor can be determinedpurely algebraically. For the moment, let us anticipate the result, and writethe normalized eigenfunction in the form

1Con = (b+ )flcpo•

What are the corresponding eigenvalues? To obtain the value for n = 0, wemultiply (3.74) by b + and compare this with the general equation (3.68).Obviously W = 0. As n-fold application of b + to cpo increases the energy

n times by hco, the energy value corresponding to (3.78) isW' = nha) (3.79)

or, in terms of the original energy scale on which (3.46) was based (cf. fig.3.4),

W = hw(n + (3.80)

(3.79) may be interpreted such that in the nth state there are n energyquanta of magnitude hw. By applying b + to cpn , the number of quanta isincreased by one, i.e. one additional quantum is created, and therefore b+is called a creation operator. When b is applied to (p [see (3.71) to (3.73)] aquantum is annihilated, so that b is called annihilation operator (see fig.3.5).

The formalism of the creation and annihilation operators b .' ,b will beof fundamental importance when we quantize the light field. In the contextof finding an explicit example for wave functions and energy levels we nowwant to show that we can easily, derive the wave functions explicitly. TO

this end we start from equation (3.74)= 0 (3.8 V

(3.67)

ed by(3.68)

ristruct the term.65) and (3.66).

(3.69).ators satisfying

) and commuta-nonic oscillator.the end of this

:illator is alwayse stillvith the

(3.70)

(3.71)

+ by (1 + b b)

(3.72)

(3.73)

action with thethe lowest state.

(3.74)

ind state cpo.Vith this aim in

(3.75)

sated, we obtain(3.76)

(3.78)VW

•

2h w


7 h w2

2 h w

3 h w2

Fig. 3.4. The energy levels of the quantum mechanical harmonic oscillator.

and insert the explicit expression for b into it

Nri 0C9° = °.

I 1 d

This is a first-order differential equation. As one verifies immediately itssolution reads

soo = 9Lexp[ —e/2]. (3.83)

The factor OZ, serves to normalize this function and is given by

(3.82)

Thus our procedure allows us to derive simply and explicitly the wavefunction of the groundstate. The next excited wave function can be

I I b(p2

Fig. 3.5. Le: The creation operator b+ causes a transition one step up in the ladder of theenergy levels with their corresponding wave functions. Right: The annihilation operator bcauses a transition one step downwards on the ladder of the energy levels with theircorresponding wave functions.

Fig. 3.6.a-d.The first excite,third excited sta

obtained z

P1 —

or after pc

Co i =The normcontinued

T2 = 7

The first 43.6. Quitegeneral for

„4111711:H

= ) — 1 / 4 (3.84)

4),

(P2

(4)0


)nic oscillator.

11110 (3.82)

obtained according to (3.78) by applying b + on coo. We thus obtain

= b ± q 0 = — 9Z, exp [ —C 2/2] (3.85)v 2 'Lc

or after performing the differentiation

cp i = Nrf 9Z,C exp[ —C 2/2]. (3.86)The normalization factor is again given by (3.84). This procedure can becontinued in an explicit way. The next example reads

so2 = -(2e - 1)exp[ —C 2/2]. (3.87)

The first 4 wave functions of the harmonic oscillator are exhibited in fig.3.6. Quite generally, when we continue our procedure, we find con in thegeneral form

son = H „(C) exp[ —C 2/2]. (3.88)H„ is called a Hermitian polynomial.

's immediately its

(4)0 (x) (a) LP, (x) (b)(3.83)

'en by

(3.84)

(plicitly the wavefunction can be

ip in the ladder of thermihilation operator bm-gy levels with their

Fig. 3.6.a-d. Upper left: The oscillator wave function of the ground state n 0. Upper right:The first excited state, n = 1. Lower left: The second excited state, n = 2. Lower right: Thethird excited state, n = 3. Note the increasing number of nodes with increasing n.•

The following exercises are indispensable for an understanding of manyaspects of quantum optics as will transpire in the course of this book.Therefore the reader is advised to pay particular attention to these ex-ercises.


(1) Prove be –b + b = 1.Hint: Write (bb + –b + b)cp() and use the definition of

ab,b + bye,

(2) Prove

ftp*(E){ — --a2 + 21•4)()d > 0._ 00 ae

Hint: Assume that I cp(E)I 0 for E ---* -4- co and integrate

—aa:2 cP dt

by parts, which yields

(d) Choospart of ( *Hint: Re(

(4) In thequationexercisealgebraictranscribtb+ , b.Example:

<T.

can be trmations

(3) Wave packets, an exampleTo solve the time-dependent Schrodinger equation

a2 afp(C t) f(– — + i2 )(P(Ct) ih at

make the ansatz

(a) Show that (* *) fulfills the Schrodinger equation (s).(b) Show that fcgqp i dE = 0. (The integral can be explicitly evaluated.)(c) Show that 'Co l 2 + IC 1 1 2 must be = 1 so that f 41 2 dE = 1.

Fig. 3.7 (rWave patmil of

ing of manyf this book.to these ex-

(d) Choose co = c, = 1/ Nrf and discuss the time-development of the realpart of (* * ).Hint: Re(exp(i cot)) = cos wt; consult figs. 3.7.

(4) In the above section we have seen how to solve the Schrodingerequation of the harmonic oscillator by purely algebraic methods. In thisexercise we want to learn how to calculate matrix elements by purelyalgebraic methods, avoiding any explicit integration. We note that we maytranscribe any matrix element formed of and a/n into those containing

, b.

Example:00

<99m

can be transformed into those containing b + , b by means of the transfor-mations

= — (b 6+ ) a 1 — (b b+ ).

\if i 4 Oil

(*)

—iwt).

(* *)

Fig. 3.7 (a—j). The motion of a wave packet of oscillator wave functions during a full cycle.Wave packets were first introduced by Schrodinger. The sequence of figures shows themotion of the wave packet ( ) of exercise 3 of this section. We have chosen co c l •• 1.

,aluated.)

• •

We now fcvanish for

<c*b+

<9)01

Hints: We

- cog

Fig. 3.7. (continued)The individual figures show the appearance of the total wave packet Re(cp(x, t)) for thefollowing time sequence: t„= n7/(40, n = 0, 1, , 9. Later we will see that wave packets ofthis and a still more general type play a crucial role in quantum optics. As our figures showthe wave packet reaches its original shape after the period T tg 27r/w. This is a uniquefeature of harmonic oscillators. The reader should be warned that in most other quantumsystems wave packets are spreading more and more in space when time is increasing.

Use partia

10 9

and write

f-

With thest

( * ) =

i.e. the r.h(8): Use (

(5) Calcu,oscillatorHint: Rer(3.86),

cc

•

$3.3 The harmonic oscillator 91

We now formulate the basic rules, where co and x are functions of whichvanish for -± 00. Prove the following rules

<Ti b+ X> = < bcPbC> (a)

<512 1 bX> < b+ cip iX> (I6)

< c*b+ bcPiX> = <cplcb + bx> (7)

<9701 b+ TO> = ° (6)

<To = 0, n > 1. (e)

(qv oscillator wave functions, n > 0).Hints: We treat (a) as an example. Write the 1.h.s. as integral

f C° (1)*() (— OX(E)cq. ( * )-co

Use partial integration

49 *(6)(-4)X(0 1:1 = rao (-aa tP*(t))XWozq-cc

and write

f °° 99* X = f (4.*)X- 00

With these tricks it follows

( 4') = f (—Aat + OTI

i.e. the r.h.s. of (a). (3) and (y) can be proven similarly. Perform the steps!(8): Use (a) and bq)0 = 0; (e): use (a), write

--11•

•

11 C9n = (b+ ) n — b+(b+ )fl-1

1/7-!

(5) Calculate <nWm> and <nkrim>, 77 = 070(0) for the first 3oscillator wavefunctions, m,n = 0, 1, 2.Hint: Remember the bra-ket notation of section 3.2, use (3.83), (3.84),(3.86), (3.87). Integrate by parts and use

lc° exp[ = Ni7-77 .

- 00

(p(x,t)) for thewave packets of

..iur figures showThis is a uniqueother quantum

creasing.

•

(6) Assume that <99„, I q9„> = 8„, , ,,, where the cp's are oscillator wave func-tions. Show:

< q)„,i b+ 140 = + 1 8„,,„+1

VT18m,.-1(i.e. especially

<49,ri b+ I(Pn> = <Coni b i cp„> = 0)

without using any integration!Hints: Use

1 q9„ = (b+

and (a) or (13) of exercise 4 above.

(7) This is a somewhat more difficult exercise: Prove

<49,„ (1). > =using

I = m,n, by% = O.

Hints: Use (6.«) if n < m and bcp„= VT/ so n = 0, 1, 2, ... .Use (6.13) if n > m.Use in both cases <To I > = 1.

(8) Oscillating dipole momentsLet the oscillator be charged with charge e. e is the classical dipolemoment. In quantum mechanics its expectation value is given by

coo= (P*(t',1)(e)(1)(i,t)d.

th+(Pr =/V / !

Evaluate(a) (I,q(b) p = 9(c) irp

0 (i.e.

Fig. 3.8. Example of an oscillating dipole moment caused by a wave packet of the harmonicoscillator.

The waythat thenclosed odiscrete (ing statescatteredspectrumacterizedquantumlevels ofmomentt

mall

3.4. The

It is notin all detan underwill tramknow allwill coatgejnucleus,bigger ththe motiCoulomtwhere cc,

distancereads (m

or wave func-

§3.4 The hydrogen atom 93

Evaluate using the following oscillator wave functions:(a) (17 = Toa)(b) p = (PIM exP(—iwt)(c) p = COTOM CICP1()eXP(—it.at), where 1c0 1 2 + 1(. 1 1 2 = 1, co 0 0, c,0 0 (i.e. a wave packet). (See, e.g. fig. 3.8.)

•

3.4. The hydrogen atom

It is not the purpose of this book to develop the quantum theory of matterin all details. We only wish to exhibit those features that are important foran understanding of the processes going on between light and matter. As itwill transpire later, to understand the basic processes it is not necessary toknow all the details about wave functions and energy levels. Therefore wewill confine our discussions of this and the following sections to somegeneral features of quantum systems.

A still rather simple quantum system consists of an electron bound to anucleus, say to a proton. Since the mass of a proton is about 2000 timesbigger than that of an electron, in a good approximation we can neglectthe motion of the proton. We consider a nucleus with charge Ze. ItsCoulomb interaction energy with the electron is given by — Ze 2/(4ireor ),where E0 is the dielectric constant of vacuum in MKS-units, and r is thedistance between the electron and the nucleus. The Schrodinger equationreads (m 0 : electron mass) (cf. also fig. 3.9)

iassical dipolem by

t ofiiii armonic

1_ h 2 a 2 a 2 a 2 Ze2 2m 0 2 ay 2az2 4ire0r 1P(X) = WIP(x). (3.89)

The wave functions depend on the three coordinates x, y, z. It turns outthat there are two types of solutions to (3.89). The first type corresponds toclosed orbits or bound states in classical mechanics and possesses adiscrete energy spectrum. The other type of motion corresponds to scatter-ing states in which the electron comes from infinity and after beingscattered by the nucleus again travels to infinity. The corresponding energyspectrum is continuous. The wave functions of the bound state are char-acterized by three quantum numbers n, 1, m. n is called the principalquantum number and runs from 1, 2, ... till infinity. It labels the energylevels of the electron in the hydrogen atom. 1 is called the angularmomentum quantum number and takes on the values 1= 0, 1, 2, ... , n — 1.

' m is called the magnetic quantum number. It denotes the direction of the


V (r)

Fig. 3.9. The potential energy V(r) of an electron bound to a positively charged nucleus as afunction of its distance r from the nucleus.

1=0,m=0; 1=1,m=-1,0,1: 1=2,m=-2,-1,012

1-0, m=0; 1=1, m=-1,0,1

n=1

Fig. 3.10. The first three energy levels of the hydrogen atom together with their quantumnumbers. The position of the abscissa indicates the zero of the potential energy whereionisation occurs.

angular momentum vector in space and it takes on the values m = —1,

— I + 1, , I. The energy of the hydrogen atom is given by (cf. fig. 3.10)

n=3

n=2

It is remarkable that quite different quantum numbers and thus differentkinds of wave functions can be attached to the same energy level K. Wecall such energy levels degenerate.

M Oe4 18h 24 n2

(3.90)

Fig. 3.11.The dens'quantum1.= 1 p-fu(a) n = 1,


Fig. 3.11. Electron density distributions for the first eigenfunctions of the hydrogen atom.The density of dots is a measure for the probability density Iii(x)1 2 of the electron in itsquantum state n, 1, m (computer plot). Functions with 1= 0 are called s-functions, those with1= 1 p-functions and those with 1 = 2 d-functions.(a) n 1, 1 = 0, m 0 (1s-function). (b) n = 2, 1= 0, m = 0 (2s-function). (Continued).

thus differentlevel W„. We

•

„n•"s . •

•: • ••lues m = —/,fig. 3.10)

(a)

•(b)

•th their quantum • •:)c•..‘ial energy where .•

• 1: ,

• ••:

..• .

(3.90)

arged nucleus as a


Fig. 3.11. (Continued.) (c) Because 4, iswave functions with m ± 1 and m

— i/ Nri )(02,1,1 — # 1, _ 1 ), i.e. nexample of a 2p-function: n = 2, 1 = 1,

complex for m ± 1, often a linear combination of—1 is used so that p becomes real. Here we used

1, / n. 1 are kept as quantum numbers. (d) Anotherm 0.

(c)

•

•

(e)

.. •

•

•(f)

• f•A‘ •r'• ' •

nbination ofere we used

(d) Another Fig. 3.11. (Continued.) (e) n = 3, / 0, m = 0 (3s-function). (f) Linear combination

= ( -i N/2) (03. - 03. -I).•

Fig. 3.11. (Continued.) (g) n = 3, / = 1, m = 0 (3p-function). (h) n = 3, 1 2, m 0 (3d-function). •


Fig. 3.11.

. : .•

• • ..•;. ••• •. .

: . • • • • •••

rt• :-! 1* :.... •...1. -: •• • 41 .. •. ... ..:A.•• 1.:1:-..4.4,„; .. ':'• •• ' .: - .:•:• :1•-%-si.:;;.*::' .. •

' • • • , # i:•;,,.c........r.:,..._ ...,• -.. 1,4. - q-,.**4•1? •• . • ,

. - .I,: t..:14.111% :•• • • t••• t., ''.•;11).,;•,. .. ; .•,.-;.;-•:- :::;r•!..:;",*•'..,i,

. . . ; ' : ••1 drk:?•-•• • .:..‘. • - :, .! • :-.'- -... • • --- •••• • •

Exercises on

(1) Theedr(We introduce

x = r sin

Volume elerris given by

02

A=ar 2

The Hamilto]

H= -

(g)

• •

• Fig. 3.12. Spherical polar coordinates r,

(g)

Fig. 3.11. (Continued.) (i) Linear combination 11, =(—i/V/ )(03.2.1 — 03,2, -1).


(1) The hydrogen atomWe introduce spherical polar coordinates (c.f. fig. 3.12)

x = r sin,' cos co, y = rsinsinq, z = r cos /9.Volume element dV = dx dy dz = r 2 drsintd1dq. Then the Laplacianis given by

A = — + — — +8 2 2 0 1 1 0 2 0 2 a

ar 2 r Or r 2 sin2+ — + cot .

The Hamiltonian of the electron of the hydrogen atom readsh 2 e 2

2m 4ireor

= 2, m = 0 (3d-

a cp2 a42

(i)


4,n,l,m = Wn Ilin,l,m ( * )(a) Insertio11

The solutions of the corresponding time-independent SchrOdinger equation Hints:

are, as indicated, characterized by the quantum numbers n, 1, m. We give a (b n Pe) n Is.

few explicit examples als

1 = a-3/2 fa°1/1,o,o ° exp[ — p] oNITT

a3/242,0,0 = 0 exp[ — -1- p] (l — r)) (2) Calct- \ITT

ao-3/2 I, =

02,1.0 = exp[ . - - (J]p cos 0 (* *)

4N/177 where *

a -3/20

4'2,1, ± 1 = exp[ --1-p]psin0exp[ -±itp] 41(X

where the normalized distance p= r/ao, with oil")

a() = 4ire0 h2 / (me 2) (3) Provt(Bohr radius) has been used. angular -(a) Show that the functions 11/„ , , , ,, given in (* *) solve the Schrodinger

I.equation (* ) with the energies

e 4 m 1 { e.)W„ = 3272h24 n2 • {ez

Show that the tp, are orthogonal and normalized. {e2

(b) Show that the only nonvanishing dipole matrix elementsHint: Ay

8rt,l,m;n',1',m'= f Ort*,1,m[ex]On',1',m'dv e ,wave fuixercise

are given by the following components of 0(4) Provt

n = 1, 1 = 0, m = 0:

n = 2, 1 = 0, m = 0:

n = 2, 1 = 1, m = 0:

n = 2, 1 = 1, m = -±. 1:

3(z) 0.745a e1,0,0;2,1,0 ^•••' 0[3(z)

2,0,0;2,1,0 = 3a0efor

3(x) 4- ilY) ± 1.053a e1,0,0;2,1, ± 1 1,0,0;2,1, ± 1 ^"" 03(x) io(y) -±4.243a e H ="2,0,0;2,1, -± 1 2,0,0;2,1, 0 •

This gives the selection rules for the z-component It follow

Am m — m' = 0, -=- 1 — = -± 1. H,The corresponding selection rules for the x-, y-components read are opei

Am m — m' = ± 1, AI 1 — 1' = ±1.


klinger equation

*I, m. We give a

Hints:(a) Insert the wave functions (* *) into (a) and perform the differentia-tions.(b) In performing the integrations, make use of a representation of factori-als

fo c°Vexp[ —C] d = n!

(2) Calculate

= f tp*extli dV

where i is the wave packet

xP( x , t) = —1 ( 11,1,0,0( x ) exp[ — iwi t/h] + 1P2,1,0( x ) exp [ —iw2t/h])

of the hydrogen wave functions of exercise (1).

the Schrodinger

its

(3) Prove the following commutation relations for the components of theangular momentum operator e:

[ ex, ey] = ihez

[ ey, ez] =

[ ez , exi = ihey{ e2 , ex] =[ e2 9 ey] = [ ez] =0.

Hint: Apply the commutators and ex , ey , ez , respectively, to an arbitrarywave function tP(x,y,z). Use the explicit representation of ex , ey , ex ofexercise 4 on 3.2 and perform the differentiations.

(4) Prove

[ e.,H] =0

for

s read

h2H= — —

2mA + V(r).

It follows that

H, e 2 , ez

are operators which commute pairwise. Discuss what this means forsimultaneous measurability.•

24.77 eV

eV

25

24

l's


Hint: Proceed as in the foregoing exercise and user 2 = x2 +y2 + z2.

3.5. Some other quantum systems

In all other atoms than the hydrogen atoms there are several electrons in asingle atom. This gives rise to a highly complicated problem due to themutual interaction between the different electrons. However, it has proveduseful to lump all the interactions between the electrons together into asingle potential function in which an individual electron moves. In thisprocedure one first assumes a given distribution of charges of the electronsand then calculates the effective potential of an electron in the field of all

23

22

21

20

the other cequation atelectrons. Tand the pr,approach atermined. Tsymmetry pFock procec

To underelectron posadopt onlyPauli's princbers n, 1, mby a single

Since thepotent

of!. Furtheother interactions. TheseTurning theenergy levelsin atoms. Tatomic physi

0 —

Fig. 3.13. Energy levels of the helium atom. In the singlet and triplet states, the spins of theelectrons are anti-parallel and parallel, respectively. The letters S, P, D, F designate the totalorbital angular momentum of the electrons. The left superscript 1 or 3 designates themultiplicity (singlet or triplet). The nonexistence of a triplet IS state is a direct consequence ofthe Pauli exclusion principle: Two electrons with parallel spin cannot occupy the samequantum state. The big numbers 1 and 2 indicate the bigger of the two principal numbersn, n' of the two electrons.

Fig. 3.14. This fitheir centers of Emotion is descrilthe potential enelevels lying hightexample ff.:only the

§3.5 Some other quantum systems 103

al electrons in agem due to the2.r, it has provedtogether into amoves. In thisof the electrons

n the field of all

.77 eV

the other charges and of the nucleus. One then solves the Schrodingerequation and finds in this way a new set of wave functions for theelectrons. These wave functions then determine new charge distributionsand the procedure can be repeated. In this way when continuing thisapproach approximate wave functions and energy levels can be de-termined. This procedure is called the Hartree procedure and when certainsymmetry properties of the total wave function are observed, the Hartree-Fock procedure.

To understand the structure of atoms it must be assumed that eachelectron possesses an internal degree of freedom, called spin. The spin canadopt only two quantum states. With its aid it is possible to formulatePauli's principle. It says that a state characterized by the quantum num-bers n,l,m and the spin quantum number can be occupied, at maximum,by a single electron.

Since the effective Hartree potential in general differs from the Coulombpotential (3.89) the energy levels of the hydrogen atoms are shifted and inparticular the degenerate levels can split into sublevels for different valuesof 1. Furthermore, in addition to the Coulomb interaction a number ofother interactions are effective in atoms, for instance magnetic interac-tions. These give rise to further splittings and shifts of energy levels.Turning the argument around we can state that a careful measurement ofenergy levels can give us important hints on the fundamental forces actingin atoms. This is why spectroscopic investigations are so important foratomic physics. An example of an energy level diagram is given in fig. 3.13.

V(r)

ttes, the spins of theF designate the totalJr 3 designates theirect consequence of>t occupy the sameo principal numbers

•

Fig. 3.14. This figure shows the potential binding energy of two atoms at a distance r betweentheir centers of gravity within a two-atomic molecule. The atoms can oscillate. Their relativemotion is described by the Schrodinger equation of a quantum mechanical oscillator. Sincethe potential energy V(r) is not harmonic (compare the dashed curve), especially the energylevels lying higher are no more equidistant as in the purely harmonic case. We have thus anexample for the energy levels of a quantum mechanical anharmonic oscillator. Indicated areonly the levels belonging to bound states.


As we know from chemistry atoms can be bound together to formmolecules. Their quantum states are not only determined by the electrons,but also by the nuclei, contributing with their own degrees of freedom. Thenuclei (together with their electrons) can perform oscillations as well asrotations. According to quantum theory these oscillations (vibrations) androtations must again be quantized. While very often the low lying oscil-latory levels are determined by an oscillator potential and are thereforeequidistant, at higher excitation levels deviations may occur (cf. fig. 3.14).We shall discuss some explicit examples in the second volume dealing withlaser theory.

3.6. Electrons in crystalline solids

In quite a number of cases it is possible to form crystalline solids byputting many atoms of the same or of few different kinds together. Againit is not the purpose of our introductory text to derive the whole quantumtheory of electrons in solids. But the most important features, which weshall use later on, can be quite easily seen.

Again we have to deal with a very difficult many body problem due tothe many electrons and nuclei present in a solid. In a first approximation itis assumed that the nuclei form a rigid lattice. In the spirit of theHartree—Fock approximation the electron—electron interactions are re-placed by an effective potential field V(x) in which a single electron moves(single electron model). Since we deal with a regular lattice the potential Vis periodic with the lattice constants (cf. fig. 3.15). As a consequence of thisperiodicity it can be shown that quite independently of the explicit form ofV the wave function must have the form (cf. fig. 3.16).

14,n = exp(ikx)uk,n(x). (3.91)

V x

In it the ftThus an el,reminiscenperiodic IT

number. Aessentiallyquantum rare separacrystal thenarrow spdiscrete enPauli's priifield calminsulator.can cause

tethe tithe next Ixenergy garthe valenccomes posthe conducto photocc

The optinfluenced

Fig. 3.15. A one-dimensional plot of the periodic potential which an electron finds in acrystal.

§3.6 Electrons in crystalline solids 105

)gether to formiy the electrons,of freedom. Thedons as well as(vibrations) andlow lying oscil-d are thereforeir (cf. fig. 3.14).ne dealing with

In it the function u has the same periodicity as the potential function V.Thus an electron in a periodic lattice has a wave function which is stronglyreminiscent of that of a free electron in vacuum. The difference lies in theperiodic modulation factor u. k plays, of course, the role of a wavenumber. As is shown in solid state physics, the energy levels depend onessentially two indices namely the continuous wave vector k and a discretequantum number n. The energy levels are now grouped into bands whichare separated by gaps. Examples are shown in fig. 3.17. In any finitecrystal these energy levels are not entirely continuous but have a verynarrow spacing. That is in other words, they are still discrete. Thesediscrete energy levels can then be filled up by electrons which have to obeyPauli's principle. If the upper most band is entirely filled up, an electricfield cannot cause motion of the electrons and we are dealing with aninsulator. When the upper most band is not entirely filled an electric fieldcan cause a current and we have a metal.

An interesting intermediate case is provided by semiconductors. Inthem, in their ground state, the "valence" band is completely filled, andthe next band, known as the conduction band, empty. However, when theenergy gap is small enough, thermal excitation may bring electrons fromthe valence band into the conduction band and electric conduction be-comes possible. Electrons can also be brought from the valence band tothe conduction band by optical excitation (internal photo-effect giving riseto photoconductivity).

The optical and electrical (and other) properties of solids can be stronglyinfluenced by "impurity" atoms. For instance, impurity atoms in an

illine solids bytogether. Againwhilivantumur hich we

problem due to'proximation it

spirit of theictions are re-electron movesthe potential Vequence of thisexplicit form of

(3.91)Ret exp(ikx) uk(x)

:ctron finds in a Fig. 3.16. The real part of the Bloch wave function is plotted versus the space coordinate x.In this example the electron is periodically concentrated around the individual nuclei on thelattice.•

conductionband

valenceband

[W

f ull

empty

F

full

filled


Fig. 3.17. Energy level diagram of an electron in the periodic potential of a lattice. Left part:The energy Wk of the electron is plotted versus the wave vector k occurring in the Bloch wavefunction (3.91). The curve can be periodically extended along the k-axis. a is the latticeconstant of the one-dimensional crystal considered here. Right part: When we project theenergy levels on the energy axis W we find the allowed bands which are interrupted byforbidden zones also called gaps.

insulator introduce new electronic levels below the conduction band (orabove the valence band). From these levels, electrons can be thermallyexcited to the conduction band leading again to a certain type of semi-conductors.

Some of such impurity atoms have an energy spectrum closely resem-bling the hydrogen spectrum, but with a screened Coulomb potential,

Insulator metalFig. 3.18. This figure shows how the energy band model allows for an explanation of thedifference between an insulator (left part) and a metal. In an insulator the valence band iscompletely filled up by electrons, whereas the conduction band is entirely empty. In a metalthe conduction band is partly filled by electrons.

Fig. 3.19. Applicattemperatures the sethe conducti banor byconduction

allowedband

gap

allowedband

Fig. 3.20. An extesthe energy of a siagrano sails due totion of the wholeelectron in the pe'below the conductelectron of the inifthe electron has lxof the conlir


conduction

band

valence

band

semiconductor

ground state afterexcitation

Fig. 3.19. Application of the energy band model to semiconductors. At sufficiently lowtemperatures the semiconductor is in its ground state. The valence band is entirely filled up,the conduction band is empty and no electrical conduction is possible. By thermal excitationor by shining light on a semiconductor electrons can be excited from the valence band to theconduction band which is then partly filled. This then allows for an electrical conduction.

energy of single electron

=conduction band

valence band

Fig. 3.20. An extension of the energy band model when impurity centers are present. We plotthe energy of a single electron versus the x axis. Note that such a plot must be taken cumgrano sails due to Heisenberg's uncertainty relation. Nevertheless it gives us good visualiza-tion of the whole situation. The horizontal bars show the individual energy levels of theelectron in the perfect crystal. Due to impurities localized states are introduced which liebelow the conduction baud or above the volerice -bartd. 1n ,the left part of this picture theelectron of the impurity is sitting in its ground state, the right part shows the situation wherethe electron has been excited by light or thermal excitation from its ground state into a stateof the conduction band.

f a lattice. Left part:

g in the Bloch waveXIS. a is the latticelie1110project the

rrupted by

action band (orin be thermallyn type of semi-

-I closely resem-lomb potential,

explanation of theie valence band isempty. In a metal

•


Fig. 3.21. To describe the ground state or an insulator of semiconductor one may use insteadof the energy band model a model referring to localized electronic states. Here each nucleusindicated by a plus sign is surrounded by negatively charged clouds of electrons.

— ( e 2Z)/(4iree0r) where E > 1 is the dielectric constant of the crystal. Asa consequence, the spacing between the energy levels is narrower than inthe hydrogen atom.

We will come back to all of these energy level diagrams later whendealing with laser action of semiconductors and with nonlinear opticalproperties of solids.

The optical spectrum of semiconductors without or even with impuritiesexhibits a number of lines which cannot be explained in the single electronmodel. To get an understanding of this new kind of states let us consider asemiconductor in its groundstate. The whole lattice is electrically neutralbecause the positive charge of the nuclei is compensated by the negativecharge of the individual electrons. When we now remove an electron froman atom by exciting it to another band it leaves a positive charge behind itby which it is attracted. As may be shown in more detail not only the

0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 00• 00000 0 0

0 00000001 000 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

Fig. 3.22. Model of an exciton. One electron is kicked away of its individual atom leaving apositively charged hole behind it. It now circles in an excited state around the positive hole.

Fig. 3 at t.the totsband staWcwonnlevel shows thecondition band.

electron butfreely in theand m 2 . Fur,constant a ofnew excitedgive rise to ahole pair isoptically, inzform excitonhole-droplets

Exercises on

(1) Subjectdimensionallattice constzShow:

, 2 ITK =

•


total energy of

all electrons

ground state

(full valence band)

Fig. 3.23. This figure represents the energy levels of an exciton at rest. We have plotted herethe total energy of all electrons of a crystal. The state in which all electrons fill up the valenceband states completely is indicated here as the energy of the ground state. The most upperlevel shows the minimum energy required to lift an electron from the valence band into thecondition band. Below this level we find the energy levels of the bound states of the excitons.

electron but also the remaining positive charge (= "hole") can now movefreely in the crystal similar to particles but with certain effective masses m1and m 2 . Furthermore a Coulomb-force, which is screened by the dielectricconstant a of the crystal, acts between these two particles. Thus we see thatnew excited states of the crystal become possible where the two particlesgive rise to a hydrogenlike energy scheme (cf. fig. 3.23). Such an electron—hole pair is called exciton. When a crystal is highly excited, especiallyoptically, many excitons can be produced which in certain crystals canform exciton molecules (biexcitons) or a new kind of state called electron-hole-droplets.

may use insteadere each nucleusrons.

he crystal. Asrower than in

is la er when1 .ptical

ith impuritiesingle electronus consider aically neutralthe negative

electron fromirge behind itnot only the


(1) Subject the Bloch wave function to the periodicity condition (one-dimensional example) tii(x + L) = tp(x), where L = Na, N is integer, a is

lattice constant.Show:

atom leaving ale positive hole.

277m 2Trmk —

Nam integer.

•


Show:

dx111(x)-7h(x) = 0,dx

I-lint: Write

fork *k'.

6

fL N —1

dx(• • ) = E f (1±1)a ( • • • ) dx.1-0

Further hint: Show that•.(1 + 1)a , a

( ) dx = exp[i(—k + k')Ia] f (• • )dx 5Ia 0

and use

N—

E exp[i(k' — k)la] =1=0

Show that

N, for k' = k1 — exp[i(k' — k)Na] ,

1 — exp[i(k' — k)a]

for k k'.

•N —E exp[i(k' — k)Ia] = 0, for k k'1= 0

Hint: Use (*).

(2) Show that the matrix element

d X sli: ,n(X) eXp[iqx] dn(X) = 0,

dx unless k = k' + q.

Generalize the result to 3 dimensions.Hint: Same as for exercise above.Later we will see that this matrix element decides which optical transitionsbetween Bloch states are possible. The result given is called: k-selectionrule.

3.7. Nuclei

We quite briefly touch on other quantum systems, namely nuclei, whichmay be of importance in future laser experiments. Nuclei are composed ofneutrons and protons (nucleons) which both have masses about 2000 ninesthe electron mass. While neutrons carry no charge, protons - .have a zhartr-opposite to that of the electrons. Though the forces within nuclei are the"nuclear forces" (and not only Coulomb forces) the whole formulation of

Fig. 3.24. En,The interactilfor that nudeindicated inindicated. Thnumbers of aoccupied leve

ptical transitionslled: k-selection

•

He s 1/2 (2) [2] $ 1/20I-i tu is

§3.7 Nuclei 111

11/2 (12)

- p1/2 (2) [126(p 3/2(4)

— 13/2(14)f 5/2 (6)f 7/2(8)h 9/2(10)

1 h

___ s 1/2 (2) L821 1/23 s d 3/2 (4) j .----- s

— d 3/2

__ g 7

5//22 (

(86 7/2

))

------- h 11/2d :,--_- g

- d 5/2

- - h 11/2 (12)2 d

1 gSn ,- 9 9/2 (10) [50] g 9/2

2p ç 5/2 ( 5/2

6)p 1/2 -= p 1/2 (2)

,,f--

if p3/2 (4) P3!2f 7/2 ( 8) [2 8]

Ca d 3/2 (4) [20]2hcu 1 d - s 1/2 (2)

C . ) 2 s d 5/2 (6)

— f 7/2

d 3/2s1/2d 5/2k = k' + q.

0 —p 1/2 (2) [8] p 1/2p 3/2 (4) p 3/21 hw 1 p

Fig. 3.24. Energy scheme for protons (left) and neutrons (right) according to the shell model.The interaction of one nucleon with all others is assumed to give rise to an effective potentialfor that nucleon. A simple harmonic oscillator potential would yield the degenerate levels asindicated in the left column. Due to spin-orbit-coupling this degeneracy is removed asindicated. The possible population numbers of the individual levels are given in brackets, thenumbers of all possible particles in the nucleus are given in square brackets (up to a certainoccupied level). (After Klingenberg, P.; Rev. Mod. Phys. 24 (1952), 63).

:ly nuclei, whichare composed oftbout 2000 timesis *aye a .charge-in nuclei are thee formulation of

d 3/2g 7/2d 5/2i 11/2

g 9/2

113/2/2

p 3/2f 5/2

f--1 97/

22

/

611w

for k k'.•

3p

2f

quantum theory seems to be still valid. A good deal of the energy levelstructure can be understood in the frame of the Hartree—Fock approach.A single nucleon is considered which moves in the force field generated byall other nucleons. It turns out, however, that there is a strong couplingbetween the angular momentum and the spin (see section 3.8) of thenucleons, i.e. a strong "LS-coupling" (cf. fig. 3.24).

Furthermore nuclei can perform collective motions similar to oscillationsand rotations of liquid droplets. What is important in the context of ourlater applications is that the basic concepts of laser theory apply equallywell to transitions between quantum states of nuclei. The essential dif-ference lies in the fact that the photons emitted by these transitions are ofvery short wavelength and lie in the y-ray region.

3.8. Quantum theory of electron and proton spin

The electron possesses not only its three translational degrees of freedombut in addition an internal degree of freedom. Since it can be visualized bya spinning of the electron around its own axis (whatever this means) thisnew degree of freedom is called spin. Some other elementary particles,such as protons, also possess a spin. Since the spin has the physicalmeaning of angular momentum it is a vector s with three spatial compo-nents sx , sy , sz . When the electron (or proton) spin is measured in apreferential direction, say parallel to the z-axis, only two values of sz werefound, namely h and — h. This preferential direction is in most casesgenerated by an external constant magnetic field.

In order to develop an adequate formalism we attribute two differentstates and thus two wave functions to the situation "spin up" or "spindown." We call these two functions T T and 419 1 , respectively. Taking theformalism of quantum theory as developed in our previous sections seri-ously we have to describe a measurement quantum theoretically by theapplication of an operator sz on the corresponding wave function. Wechoose the wave function in such a way that the application of themeasuring operator on the wave function yields the corresponding mea-sured value. Since we have only two measured values, namely 1- h and— fh we thus expect the relations

itszT InCPT SzCP4, — (3.92, 93)

We now look for a formalism which yields the relations (3.92) and (3.93) soto speak automatically. It has turned out that this goal can be achieved bymeans of matrices. We write down the result and then will verify it. We

chose sz in tl

h(sz=

and the spin

`PI =

The multipli.rule of matr.two general

( a b)c cl)

Using thisimmediatelyfunction is cb.

= atp

To obtain tiproducts. W

andía

C92 = b

We define tb

<Ti19)2)

The normaliz

< cPi ciD > =

Once we haNleast in princgive the detaone does to IOne first wriinvokes the

§3.8 Quantum theory of electron and proton spin 113

(3.94)

(3.95)

.e energy level'ock approach.i generated byrong couplingin 3.8) of the

to oscillations:ontext of ourapply equallyessential dif-

nsitions are of

!es illezedom?. vis • ed byis means) this

, tary particles,, the physicalpatial compo-neasured in aues of sz werein most cases

two differentup" or "spin

y. Taking thesections seri-

tically by thefunction. Wecation of theNnding mea-mely 1 h and

(3.92, 93)

1 and (3.93) so,e achieved byverifii it. We

chose sz in the form

_ h 1 osz — —

and the spin functions in the form

C9T = 10)' = (7).

The multiplication of the matrix sz and the spin functions co l , cp.i. obeys therule of matrix multiplication. We remind the reader of that definition fortwo general matrices

(a x = (ax + byc d kY ) cx + dy)•

Using this relation one verifies immediately that (3.94) and (3.95) leadimmediately to the relations (3.92) and (3.93). The most general spinfunction is obtained by the superposition of (p t and 99 1 with coefficients a,b.

= apt + b).a

To obtain the normalization condition we have still to introduce scalarproducts. With sc and SO2 given in the general forms

= ab:

and(p2 =

We define the scalar product by

<(p i l 992 > (ar ,br)( b2 = ata2 + brb2 . (3.100)a 2

The normalization condition for a function cp reads

<TIT> = l a l 2 + 1 b 1 2 = 1. (3.101)

Once we have found an explicit representation for sz it is not difficult, atleast in principle, to determine the explicit form of sx and sy . We will notgive the details here because they don't yield much physical insight. Whatone does to find the explicit form of sx and sy is the following:One first writes down two general 2 X 2 matrices for sx and sy . Then oneinvokes the fact that sy , s are components of angular momentum. In x

(3.96)

(3.97)

(3.98)

(3.99)

\o

—1 \o

(3.102)

(3.103)


quantum theory it is shown quite generally that the components of anyangular momentum obey commutation relations (compare exercise 3 onsection 3.4). s„, sy and sz are then subjected to these commutation rela-tions. The resulting equations for the unknown matrices can then besolved. It turns out that the solution is determined within a certainarbitrariness which leaves some freedom of choice of a particular simpleform but has no bearing on any physics as can be shown in detail. For ourpurposes the following choice is favorable

We now wish to formulate a Schrodinger equation for the spin. It isexperimentally known that with the electron or proton spin a magneticmoment is connected according to (MKS-units)

eh

= 2m

where e is the elementary charge, m the mass of the particle. If m is theelectron mass, the magnetic moment (3.104) is also called Bohr's magne-ton. Since the magnetic moment is a vector parallel (or anti-parallel) to thespin we write more generally

= (3.105)

The factor h is now contained in the spin operator s. We now proceedquite in analogy to the derivation of the Schrodinger equation for theelectron, i.e. we start from a classical expression. The energy of the spin ina spatially homogeneous magnetic field with induction B is according toelectrodynamics given by

— LB (3.106)

We now wish to let the expression (3.106) become an operator analogousto the Hamiltonian operator in the conventional Schrodinger equation ofsection 3.1. We know already how this can be achieved, where s is, as wehave seen, now an operator. Inserting (3.105) into (3.106) allows usimmediately to formulate an equation for the electron spin

— B•scp = Wv. (3.107)

When we chose the magnetic field b in preferential direction along thez-axis we have

B (0, 0, Bz ). (3.108)

In this casefactor, withfunctions (3eigenvalues

w= ±

The energy texpression gantiparallelnow also for

e—

The time-demagnetic fiesolutioitea,for in

= 2-1

is a solutionof other spiivalues forare defined]

<sj>

<sj>

where we h(3.103), (3.9exercise yiek

<sz > =

<sx > =

<sy > = -

The three co.motion is exaround theprecession 01(compare ex*differenta spi to

(3.104)

its of anyrcise 3 onIlion rela-I then bea certainlar simple1. For our

(3.102)

(3.103)

,pin. It ismagnetic

004)

f m is the.'s magne-del) to the

(3.105)

v proceedn for thehe spin in'ording to

(3.106)

Inalogousuation ofis, as we

illows us

(3.107)

along the

•8)

In this case, the left-hand side of (3.104) coincides up to a numericalfactor, with the left hand sides of (3.92) and (3.93). In other words, thefunctions (3.95) are just eigenfunctions to (3.107) with the correspondingeigenvalues

eh ,W = ± --D .2m z

The energy of a spin in a constant magnetic field in z-direction is just ourexpression given by a classical theory of the interaction of a parallel orantiparallel spin. Similarly to the ordinary Schrodinger equation we cannow also formulate a time-dependent Schrodinger equation

- —m

B-sco = ihq. (3.110)

The time-dependent equation has to be applied in particular when themagnetic field is time-dependent. Let us here consider a time-dependentsolution in a constant magnetic field. One then immediately verifies that,for instance, the following wave packet

so = 2 -1/2 [exp(iwot/2)92 1 + exp(-icoot/2)cp t ] (3.111)

is a solution of (3.110). The meaning of the wave packet (3.111) as well asof other spin functions can best be explored by determining expectationvalues for the components of the spin operator. These expectation valuesare defined by

<si > =. <99 1 s; IT>, 99 = ( ba ), I = x,y,z

<s)>-.---(a,b)*sf(ab) (3.112)

where we have to represent si by the corresponding matrices (3.102),(3.103), (3.94). A short calculation which we leave to the reader as anexercise yields:

<sz > = 0 (3.113)

<sx > = -21-h cos wot (3.114)

<sy > = h sin coot. (3.115)

The three components of the expectation value of s form again a vector. Itsmotion is exhibited in fig. 3.25. Evidently in this case the spin rotatesaround the z-axis in the x-y plane. We have thus an example of a freeprecession of the spin. When we choose a more general wave packet cp(compare exercise) the spin still rotates around the z-axis but now in adifferent plane (compare fig. 3.26). The spin thus behaves quite similarly toa spinning top.

(3.109)

Fig. 3.25. The motion of the expectation values of s for the wave packet (3.111) (cf.(3.113)— (3.115)).

As we will see later, this analogy can be carried even further namelywhen other components of the magnetic field are non-vanishing.


Show that <

d—dt

<sx

d—dt

<sy '

d-SSd t "

(cf. also fig.

—dt

<s >

Hint: Userepresentatic

(2) Pr,

S

(1) Choose

cp = exp(i wot/2) + c 2cor exp( —i coot/2)

where

I c 11 2+ 1 C 21 2 = 1.

Calculate:

<sx> =- <971s,,,l(P>

<sy>==-<coisy199>

<sz>-=1-<99iszi(19>•

Fig. 3.26. The motion of the expectation value of the spin s calculated by means of the wavefunction of exercise 1 of the section 3.8. •

and

2Sx = S 2

ket (3.111) (cf.

Show that <s> = (<sx >,<sy > , <sz >) fullfills the equations

<sx > = li<sy>B,

--d-7 <sy > = — tt<sx>Bz

—d

<sz > = 0dt

(cf. also fig. 3.26) or, in short

Ti <s> = au<s> x B, B = (0,0,Bz).

Hint: Use the rule for matrix multiplication and the explicit matrixrepresentations of sx , sy , sz given in the text.-ther namely

ng.

•(2) Prove

sxsy — sysx = ihs,

sy sz — sz sy = ihsx

sz sx — sx sz = ihsy

and

Sx = Sy = Sz —2 2 I h 2( 1 0

0 1 ) •2

ns of the wave

•

0

•

•

_•

4. Response of quantum systems to classical electromagneticoscillations

4.1. An example. A two-level atom exposed to an oscillating electric field

To start with an explicit example let us consider an electron of an atom.The motion of this electron is described by a Schrodinger equation of theform

d 4,1-10 ,/, = ih—

dt. (4.1)

We assume that (4.1) allows for a series of solutions describing stationarystates, which in a classical interpretation means that the electron moves incertain orbits. Quantum mechanically speaking, its motion is described bywave functions with time independent probability distributions I 4/(x)1 2.

We have attached an index 0 to the Hamiltonian to indicate that we aredealing with unperturbed motion.

Now let us subject the atom to an external field, for instance a mono-chromatic light wave or an incoherent light field. Then already in classicalphysics we have to expect that the electron is thrown out of its originalorbit and will perform new kinds of motion. We expect a similar behaviorin quantum mechanics. Fortunately there are a number of cases of practi-cal interest in which the new kind of motion, or more precisely speaking,the new kind of wave function can be determined rather simply. Todemonstrate this procedure let us consider a system with only two energylevels. This sounds rather strange because we know that electrons in atomspossess an infinity of energy levels. However, we will see below that in anumber of cases, when an electron interacts with light, only two energylevels are of central importance and all others can be neglected. This is, forinstance, the case when the difference between the two energy levels underconsideration is quite different from the differences between any other

These are ir(4.6) furthe'electron. W

E(x,t,

In cases ofthe wavelernate x bygravity of t

E=E

By a translThe force .

F = (-

where we

enelts pcha

are vectors

V( x)

We write i

x)

which allccharge timEquationenergy offield. Inset

1/8 =

In many c

fcipt(

vanishespressed 13:true in nu

Hf2=

W ow

1

111

120 4. Response of quantum systems to classical electromagnetic oscillations

energy levels. A further condition for this assumption is that the perturba-tion is not too strong.

Now let us turn to our problem. We have to solve the Schrodingerequation

dIPn — = Hip (4.2)

dtwhere the Hamiltonian

H = Ho + H P (4.2a)

consists of the Hamiltonian Ho of the unperturbed motion and the Hamil-tonian caused by the external field, H P. Let us assume that we know thestationary solutions and energies of the time independent unperturbedSchrodinger equation

110cPn = WricPn •

Since we assume that only two levels, i.e. n = 1 and n = 2 are relevant themost general form of the solution of (4.2) must be chosen in the form of asuperposition of the two unperturbed solutions (n = 1,2). This leads us tothe ansatz

tili = ci( t ) ch c 2(t)p2 (4.3)

where we admit that the coefficients co c2 are still time dependent. It willbe now our task to derive equations for the still unknown coefficients

c2 . To this end we insert (4.3) into (4.2). We may use the fact that piobeys equation (4.2b). We then obtain

dc2h

dc,—

cp 1 + ih—dt

cp2 = c i (W1 + HP )cp i + c2(W2 + HP )cp2 . (4.4)d t

The resulting equation has a seeming drawback. On the one hand thefunctions (pi depend on space and also H P still depends on space. On theother hand the coefficients are supposed to depend only on time and noton space coordinate x. Thus we have to get rid of the space dependence of(4.4). In quantum theory it has turned out that this can be easily achievedby multiplying (4.4) by Tr and q and then integrating over space. Whendoing so we use the orthogonality relation (3.23). We further introduce theabbreviations

= f cp(x)H P(pn(x) dV (4.5)

or, in bra-ket notation =- H" n > . One then obtains

ih—dc,

= c 1 W1 + c l iff; + c 21-11'2d t

dc(4.6)ill-

d; =cc P c P *2W2 +H +2 22 1 H 21

(4.2b)

ions

it the perturba-

he SchrOdinger

(4.2)

§4.1 An example. A two-level atom exposed to an oscillating electric field 121

(4.2a)

and the Hamil-it we know theat unperturbed

(4.2b)

are relevant theinhjirm of aT • cis us to

(4.3)

pendent. It willArn coefficients:he fact that (pj

These are indeed equations independent of space. To discuss the equations(4.6) further we consider the case in which an electric wave acts on theelectron. We represent the electric field strength as wave in the form

E(x,t) = Eocos(k•x — cot). (4.7)

In cases of practical interest, the extension of atoms is much smaller thanthe wavelength (of (4.7)) X = 2 ,77/k. This allows us to replace the coordi-nate x by an average coordinate xo which corresponds to the center ofgravity of the atom

E = E0 cos(kx0 — tot) (4.8)

By a translation of the origin of the coordinate system we can put xo = 0.The force F acting on an electron is given by charge times field strength

F = (—e)E (4.9)

where we have explicitly expressed the negative sign of the electroniccharge by the minus sign. For a (spatially) constant force the potentialenergy is proportional to the distance x. Taking into account that E and xare vectors the potential energy is given by

V(x) = ex • Eo cos cot. (4.10)

We write it in the formV(x) = 11- E (4.11)

')cp2 . (4.4)

which allows for the following interpretation. J is a dipole moment =charge times distance, whereas E is again the time dependent field strength.Equation (4.11) is a relation well known in electrostatics expressing theenergy of a constant dipole moment in a spatially homogeneous electricfield. Inserting (4.11) into (4.5) yields for the special case n = m = 1one hand the

n space. On them time and not

dependence ofeasily achieveder space. Whener introduce the

= f q);(x)exq),(x)dV- E. (4.12)

In many cases of practical interest it turns out that

f (49;(x)x(19,(x)dV = 0 (4.13)

(4.5) vanishes (compare exercise at the end of this section). This fact is ex-pressed by saying that the atom has no static dipole moment. The same istrue in many cases for the excited state so that

= icpIexcp2 dV . E = 0. (4.14)

(4.6) We now turn to matrix elements for which n m and we consider for

where

In ordtypicaiory. TtivelySince ;d2 cluamplitinitial

d,

Acconstate. ".we maalk•

ofWs

i h

or afte

d:

We h:coeffic(4.25)wavesminutea modtreatmshowsent fie

Inserti

d:

The in

(4.20)


instance

HIP2 = f tpfexcp2 dV . E.

We assume that this matrix element does not vanish and we abbreviate theintegral in (4.15) by 191 2 , so that

H1P2 = *12 •E (4.16)holds. In the next chapter we shall show how to solve the equations (4.6)under the assumption (4.13), (4.14), (4.16). As is seen, at no instant wemade explicit use of the specific form of the wave functions so that themethod is quite general. On the other hand we must not forget that wehave still to justify why we are allowed to deal only with two levels. Weshall come back to this point in later chapters.


(1) Let cp(x) have the property cp(—x) = cp(x).

Show 1=-7 f fccso*(x)xq)(x)dx = 0. Show that the same is true forx) = —co(x).

Hint: The integral remains invariant (unchanged) if x —› — x. Show that onthe other hand I-+ —1 under the transformation X---÷ -X, by replacingeverywhere x by — x.

4.2. Interaction of a two-level system with incoherent light. The Einsteincoefficients

We start from equations (4.6) assuming

= H =0.

As a first step towards the solution of (4.6) we make the substitutionsc 1 (t) = d l exp[ —iWit/h]c2(t) = d2 exp[ —iW2t/h]

where the coefficients d 1 and d2 are still unknown timetions. Inserting (4.18) we readily verify that the energiesfrom (4.6) and we are left with the equations

. dih —d =d2 HP expEdt 1 12

ih--d2 = d 1 H2P1 exp[ii-ot]a't

(4.15)

(4.17)

(4.18)

dependent func-WI , W2 drop out

(4.19)

§4.2 Interaction of a two-level system with incoherent light. The Einstein coefficients 123

(4.15)

Dbreviate the

(4.16)uations (4.6)) instant weso that the

rget that weo levels. We

(4.21)where we have used the abbreviation

(7) (021 = — /h

•

is true for

how that on)37 replacing

lie Einstein

In order to solve (4.19) and (4.20) we apply a procedure which is quitetypical for quantum theory. It is called time-dependent perturbation the-ory. To apply this procedure we assume that the perturbation is compara-tively small. We shall see somewhat later what "smallness" means exactly.Since the perturbation is small we may assume that the amplitudes di andd2 change only slowly in time. Note that without perturbation theseamplitudes would be entirely time independent. In a first step we adopt theinitial condition

d1 (0) = 1, d2(0) = 0. (4.22, 23)

According to this initial condition at time t = 0 the atom is in its lowerstate. Since the coefficients d i , d2 change only slowly with increasing timewe may assume that it is a good approximation to insert on the right handside of (4.19) and (4.20) these initial values (4.22), (4.23). Thus (4.20) nowreads

(4.17)

itutions

H2P, exp[icT)t] (4.24)ih—dtd2

or after integration over time

d2 = (—i/h) f flifi (T)exp[ii5T] di% (4.25)

We have put the lower limit of the integral equal to 0 so that thecoefficient d2 fulfills the initial condition (4.23). To evaluate the integral in(4.25) further the electric field strength is represented as a superposition ofwaves whose phases (I) are statistically distributed. We will explain in aminute what is meant by "statistically". We note that this approach is onlya model which can be substantiated later by a fully quantum mechanicaltreatment in which also the electromagnetic field is quantized. Our modelshows on the other hand all essential features of the action of an incoher-ent field on an atom. Our hypothesis thus reads

(4.26)

d2(t) = (—i/h) E Ex A i exp[i(Dx] f exp[i(<7.) — wx)T]dT. (4.27)

The integral which we abbreviate by Sx is readily evaluated

Sx f t • • dT =[i(i; — x )] - 1 texp[i(cT) — u)„)t] — 1). (4.28)

(4.18)E(t) = E Ex exp[i(Dx— itaxt] + c.c.

wx>ondent func-4/2 drop out Inserting it into (4.25) and noting the form of 11f, (4.16) we obtain

(4A 9)

1620)

(4.30)


We are now interested in the probability with which the level 2 is occupiedat a later time t. To this end we form I d2 2 = 4d2 . Inserting the corre-sponding expression (4.27) for d2 and its complex conjugate for (11 weobtain

i d21 2= h-2 E exp[i(k — ito x , ]( EA- 1,21)( Eit• 02*t ) Si St- (4.29),A'

We now use our assumption that the phases O A are statistically distributed.In order to evaluate (4.29) properly we therefore have to supplement it bya statistical average over the phases. We indicate the phase average bybrackets. As is shown in the exercises for statistically independent phaseswe obtain the relation

<exp[i4)A — WA ]> = SAA.

where SAA , is the Kronecker symbol O NA, = I for A = A' and = 0 otherwise.As a consequence of the Kronecker symbol all terms of the double sum(4.29) vanish unless A = A'. This yields

<1c1 2 (t)I 2 > = h -2 E I EA l 2 0.7) — 2•4Sif12[(z) WX)I/2] • l e AI I 2

(4.31)


EA = eEA . (4.32)

e is a unit vector in the direction of the light polarization. Equation (4.32)implies that the incident lightwave is polarized in a single direction. Let usdiscuss the result (4.31) first by looking at the individual terms. For

tax, i.e. in the absence of resonance the sine function varies periodi-cally, i.e. we don't get any remaining transition for the atomic state 1 intothe atomic state 2. On the other hand, in case of resonance cT.) = cox wenotice that Id2 1 2 increases proportional to t 2 . This is a result which seemsto contradict our experience: When we let light impinge on a set of atoms,on the average the number of excited atoms increases linearly with timeand not quadratically. For a long time this result was considered asentirely unphysical. Later we will see, however, that this result is veryreasonable when we let an atom interact with entirely coherent light.

By postponing this problem to a later chapter let us come back to thecase considered here, namely incoherent light. In this case neither thenon-resonant nor the resonant case yield the expected result namely alinear increase of occupation number of level 2. This dilemma is resolved,however, when we assume that the incident light field comprises a wholefrequency band. This is realized when we deal with light from thermalsources, i.e. incoherent light.

§4.2 Interacti

With theseintermediated

—dt <1 d2(4

The square oing furthermc

I EXI 2 =By inserting t

(02)1

distrib wSince u.ly

to an in gratall unimportaclose to resor

/( Z5) G°

instead of (4integrals. It 1

expression fo

—dt <I d2(

We are nowhypothesis onside of (4.37)upper level passumption tl

This relaticleft-hand sidein the excitetintensity I,and a constat

P( w ) =

where e0 is th

410

oris §4.2 Interaction of a two-level system with incoherent light. The Einstein coefficients 125

1 2 is occupiedting the cone-ate for 4 we

(4.29)

Ily distributed.Dplement it byse average by

,endent phases

(4.30)

= 0 otherwise.Le double sum

•/2]•le-192112

(4.31)

(4.32)

quation (4.32)-ection. Let usal terms. Foriaries periodi-Ic 1 intoce = cox wet which seemst set of atoms,trly with timeconsidered asresult is verytnt light.e back to the>e neither the;ult namely ala is resolved,'rises a wholefroeermal

instead of (4.35). The integral (4.36) can be easily found in tables ofintegrals. It has the value g . Collecting all factors we obtain the finalexpression for (4.33) namely

--d7 = h-221rie2112/(-7)). (4.37)

We are now in a position to compare this result directly with Einstein'shypothesis on the absorption and stimulated emission of light. The left-handside of (4.37) is just the temporal change of the occupation number of theupper level per atom. This rate is caused by the incident light under theassumption that initially the atom was in its ground state.

This relation can be easily extended to N atoms. In this case on theleft-hand side we then have the temporal change of the numbers of atomsin the excited state. On the right-hand side, we would have the lightintensity I, the number of atoms N, with initially occupied ground statesand a constant coefficient which we may call B 12 . Replacing / by

p(w) = 2e0 1(w) (4.38)

where ao is the dielectric constant of vacuum, we are exactly led to

(dN/dt) = B12NIP(4.38a)

With these ideas in our mind, we continue to treat (4.31). As anintermediate step we differentiate (4.31) with respect to time

—d <Id

2 (t)l>2 = h -2 E lExl 2 (c7.) — (.0x) -1• 2sin[(rj — cox)dle-02112.

d t X

(4.33)

The square of the field strength is connected with the intensity. Introduc-ing furthermore an intensity I per frequency interval we put

1E),1 2 = /(wx)Acox. (4.34)

By inserting this expression into (4.33) we are left with the evaluation of

(1/h 2 )E /(wx),6a.ox sin[( —-Co tax)ti

(4.35)Co' W_ WA

Since usually the frequencies of the incident light wave are continuouslydistributed we proceed from a summation over individual frequencies 6.)),to an integration replacing cox — T:..) by w. Neglecting for the moment beingall unimportant factors and assuming furthermore that EA does not changeclose to resonance cox = 'Co we have to evaluate the expression

1(cT)) 1 clwsin(ot)/w (4.36)-

which is nothing but Einstein's equation which we got to know in section perturbati2.5. Comparing (4.37) with (4.38a), for the special case N1 = I we im- terms aremediately obtain an explicit expression for the Einstein coefficient non-lineal

IT applies toB 12 = e 1/211

2• (4.39) write dow

tO reader nuWhen we repeat the above calculations with the two levels 1 and 2 an electroexchanged we will find exactly the same result. This means that we could perturbatihave considered the emission process in exactly the same way as the that of secabsorption process leading to the same Einstein coefficient. The transitionscan take place only when an external field is present. This is why we have Hi =to call the emission process, just considered, stimulated emission.

The above outlined approach permits us to derive the Einstein coeffi-cients from a quantum theoretical treatment. There are two unsatisfactorypoints, however. First of all, we observe that the formalism does not yieldany spontaneous emission. Indeed if there is no field then there is noperturbation which can cause any transitions. Within quantum theory thisgap can be only surmounted when the electromagnetic field is quantized.We will come back to this point in chapter 7.

Furthermore, the assumption about the statistically distributed phasesseems, to some extent, artificial and one should wish that also this resultcomes out automatically. Indeed we shall see in later chapters that therandom phases are again a consequence of a fully quantum mechanicaltreatment of the light field.

Exercise on section 4.2 to spin systhe wave f

(I) We define the phase average by 92n = 9

<expli(f .277.

d1 f 277

Of — cflk )]) = -27T 0 (Di T77.- 0 d ctok exp[i(tj — (1)k)]. As is shosexactly rep

Prove (4.30) by performing the integrations. q(t) =

43. Higher-order perturbation theory In sectionexponentia

In the preceding paragraph we had assumed that the incident lightwavecan be considered as a small perturbation. This made it possible to exp[ —

calculate d2 or d1 in a simple manner. This app i Lion is indeed Thus we mjustified in most cases when light stems from thermal sources. By means ofthe laser it has become possible, however, to generate light of very high =intensity. Therefore in a number of cases our approximation of section 4.2is no more sufficient. It is necessary to determine higher order terms of the (

whose Ha:

H=.

We assum

11".The correfquantum rways, conrthe case ofthe triple r

The wasbra-ketfollowing

Is §4.3 Higher-order perturbation theory 127

ow in section perturbation in a systematic way. As we shall see later, such higher order

i = 1 we im- terms are of fundamental importance in laser theory itself but also inicient

non-linear optics and in non-linear spectroscopy. Since the procedureapplies to quite general quantum systems and perturbations, we do not

(4.39) write down the corresponding Hamiltonian explicitly. For visualization thereader may, however, identify the unperturbed Hamiltonian with that of

,els 1 and 2 an electron in a potential field, for instance in the hydrogen atom, and the

:hat we could perturbation operator as stemming from an incident lightfield similar to

; way as the that of section 4.2. We therefore start from the Schrodinger equationhe transitions

d

why we haveHIP = ihtp—d7

(4.40)sion.

nistein coeffi- whose Hamiltonian is assumed in the form

unsatisfactory H = Ho + H". (4.41)ioes not yield

imn no

this

We assume that the unperturbed problem has been solved

HocPn = KS% • (4.42). is quantized. The corresponding wave functions and energies are distinguished by the

quantum number n. This quantum number can be interpreted in variousibuted phases ways, corresponding to the problem under consideration. For instance in

lso this result the case of the hydrogen atom, the quantum number n stands in reality for

oters that the the triple n, 1, m.mechanical The wave functions con depend on the electron coordinate. Using the

bra-ket formalism (compare section 3.2) one can reformulate the wholefollowing procedure so that it applies to any quantum systems, for instanceto spin systems. Here, however, to consider a concrete case let us assumethe wave functions in the form

(Pn = q( x). (4.43)

As is shown in mathematics the wanted solution of eq. (4.40) can beexactly represented as a superposition of the complete set of functions cp„:

440 = Edn(t) cPn- (4.44)

In section 4.2 we saw that it is useful to extract from the coefficients c n anexponential function

exp[ —iWnt/h].

Thus we make the ansatz

Ip(t) = Ecn(t)exp[ —iWn t/h]cpn(4.45)

instead of (4.44).

4)k )1

ent lightwavet possible toon is indeed. By means ofof very high

rteof sip 4.2

f the

(4.48)fqH Pcp„ d V <mIH P I n >


The coefficients c still unknown time-dependent functions. Our taskis to solve the equation

(Ho + H P )tp = ih cl(+Pt (4.46)

by means of the ansatz (4.45). To this end we insert (4.45) into (4.46) onboth sides and use equation (4.42). Carrying out the differentiation withrespect to time we find

E c(1 ) exp[ — iWn t/h]( W HP)yon

ih= E exp[ — iWn t/h]( W„c„(t)

dc„(t)

Pdt P"•

Clearly, on both sides of (4.47) those terms cancel which contain the factorWn . In order to get rid of the spatial dependence of the functions in (4.47)we multiply (4.47) from the left with one of the functions 99: and integrateover the whole volume. When doing so we obtain expressions of the form

where the two expressions on the right-hand sides are only differentabbreviations of the integral on the 1.h.s. After these manipulations, thesystem (4.47) reduces to

ih dcm(t)

= E cn(t)exp[iwmn t]Ig ndt


hwmn = — Wn . (4.50)

To specify the problem we have to fix the initial state at initial time to . Weassume that a certain state q) was occupied at initial time to. This meansthat the whole expression (4.45) reduces to con° at that time. This implies aninitial condition for the coefficients cn which reads

c(t0 ) = 3„ , „ o . (4.51)

We define the "O's" approximation for our iteration procedure. Thecoefficients of this approximation will be denoted by an upper index 0, i.e.we define

c„(°) c,r ( to ). (4.52)

The basic idea of perturbation theory is as follows. Since the perturbation

HP is relattion only 1given by (yields

ddt cm

By integrathe c's in

41,)(1

(4.47)

(4.49)

The uppesymbol 01(4.51). Instill be tir

•siarbiinsert anorder to4,2) . Contrelation

(r+i

Using the

-km(

we can 'A

(1+TCm

The relat:cg' onimportan.

c„,(2)(t

For I =

c (1) (1ni•

§4.3 Higher-order perturbation theory 129ions

tions. Our task

(4.46)

into (4.46) onrentiation with

(4.47)

itain the factorctions in (4.47)

)nsaOltegratelie form

(4.48)

only differentUpulations, the

(4.49)

(4.50)

.ial time to. Weto. This meansThis implies an

(4.51)

Tocedure. TheDer index 0, i.e.

(4.52)

ie pilorbation

H P is relatively small, the coefficients cn will differ from the O's approxima-tion only little, at least for sufficiently small times. We therefore use cn(to),given by (4.51), as an approximation for c(t) on the r.h.s. of (4.49). Thisyields

(1,; )(t) = (—i/h)exp[iw „, n o t H,f, n 0 . (4.53)

By integration over time we immediately obtain an explicit expression forthe c's in first approximation

c(t) (—i/ h) f HdT + 8„,. no . (4.54)to

The upper index (1) now indicates first approximation. The Kroneckersymbol on the right hand side of (4.54) takes care of the initial condition(4.51). In the following we have to note that the matrix elements (4.48) maystill be time dependent. An explicit example is that of the time-dependentlight field of section 4.2. So far we have only recovered the same expres-sion as (4.25) of section 4.2, however now generalized to a system witharbitrarily many levels. To proceed further we may imagine that we nowinsert an improved expression for c(t) on the right-hand side of (4.49), inorder to obtain on the left-hand side once again an improved coefficientc (n2) . Continuing this procedure step by step the 1 + 1 step leads us to therelation

c"(t) = ± (-0)E f fc)(T) exp[ia„,,„T]M n (r) dT. (4.55)n to

Using the abbreviation

Iin,„(T) = exp[icomnr]Hn(T)

we can write (4.55) in the form

c(1+1)(T)---= c;7?) + (—i/h )E ft fi,„„(T)70(T)dT.to

The relation (4.55) is a recurrence formula. With its aid we can calculatec„,(I+ I) once cn(1) has been determined We can learn about the mostimportant aspects when looking at 1 = 1. In this case (4.55) reads

c(t) = (—i/h )E f`ii,„,,(T)cg)(,)dT + c. (4.56)s oni

For 1 = 0 we get (4.54) which we write in a slightly generalized form as

c (1)(t)= (—i/h)E H 2(T)th-co) + co)n, no n2 nt

n2 tO

(4.57)

(4.55a)

(4.55b)


where the c" are given constants. It remains our task to express thecoefficients in second order, c„,(2) , by cm(°). To this end we insert (4.57) into(4.56) which yields

tc(t) = cm(o) + ( — i/h)E f H„,„,(T)di-)

„, to

t 7,+ (—i/h) 2 E H„,„ , ( T,) 1-1002(T2 ) dT2c°) . (4.58)

n i , n 2 t o to 2

The expression (4.58) contains quite a number of highly interesting effectswhich will be discussed in the following sections and, in still much greaterdetail, in the volume on non-linear optics. Readers who are not too muchinterested in mere mathematics can end this section here and proceed tothe following sections. For the sake of completeness we now demonstratethe general result for c„(I+ D . Eliminating all intermediate c's we obtain

c 1 (t) = c?+ (—i/h)E (ITC?n i to

▪ ( —i/h )2 E T1 ) d f2

( r2 ) d + • •Ino, toto

▪ —i/h ) 1+' E TO dr, f dr2 .n i , n 2 ... n 1+1 t o to

X f Hn2ni (T1 + 1 ) dT1± 1c(n°).to

The coefficients c(t) which were originally wanted are obtained bypushing this kind of perturbation theory to infinite order, i.e.

c(t) = lim c(t). (4.60)1—)C0

The series expansion we have derived here is called a Born series inphysics. In section 7.7 a formally different approach to time-dependentperturbation theory will be presented.

4.4. Multi-quantum transitions. Two-photon absorption

In this and the following sections we wish to give some explicit exampleswhich will elucidate some of the physical contents of perturbation theory.

We consider a quantum system, e.g. an atom with three energy levels, 1,2, 3 (cf. fig. 4.1). We subject this atom to a light field described by its

(4.59)

Fig. 4.1. 2-rThe electrointermediateelectron malevel 2 whiclbetween thesinglAhtgro teocc‘rameach of whii

electric fi(matic sucl

2w =

We furthe

We now athe atomchange ofgeneral foi

H,r,„ -

We assum

H Pmm

and furthe

Hf3=

butHIP2

Although

§4.4 Multi-quantum transitions. Two-photon absorption 131

o express theert (4.57) into

2

3 3

2

(4.58)

resting effectsmuch greaternot too muchad proceed to

demonstratewe obtain

•

Fig. 4.1. 2-photon absorption illustrated by an example of an atom with 3 levels. Left part:The electron makes a transition from its ground level 1 to its excited level 3 via anintermediate level 2 which lies energetically inbetween the levels 1 and 3. Right part: Theelectron makes a transition from its groundstate 1 to its excited state 3 via an intermediatelevel 2 which lies energetically above the level 3. Note that in both cases the energy differencebetween the intermediate state and the groundstate does not coincide with the energy hco of asingle light quantum. If the energy difference between the intermediate state and thegroundstate is equal to the energy hw of the light quantum a physically different processoccurs, namely an absorption cascade which consists of a sequence of absorption processeseach of which can be treated by first order perturbation theory.

r2 ) dT2

(4.59)

obtained by

(4.60)

electric field strength. First we assume its frequency w purely monochro-matic such that

26.) = (431 = (W3 — W1 )/h. (4.61a)

We further assume that

(A) Wmn, m,n = 1,2,3. (4.61b)

lorn series in-ne-dependent

We now apply the results of section 4.3 to this problem. We assume thatthe atom is initially in its ground state, 1, and we are interested in thechange of the occupation number of level 3. The matrix elements have thegeneral form (compare also (4.15))

= A„,,, exp[ —iwt] + c.c. (4.62)

We assume as usual

HX,,,= 0, m = 1,2,3 (4.63a)

and furthermore(4.63b)HIP3 = fp; = 0

licit examplesbation theory.iergy levrels, 1,scribed by its

butIlf2 0, 1/11 0, Hf2 0, H2''3 0. (4.63c)

Although (4.63b) seems to forbid transitions between levels 1 and 3, we

•

1

2 t ^Ti ,„

62)(0 = ( —i/h) H32( Tt ) drif H21 ( T2 ) d T2to to

(4.64) Fig. 4.2. In tinumber of di(ordinate). Thplotttclin 4th

Exercise on

Perform th,

fi.,„„(T

where (onusChoose to -Show that

43. Non-remoment. Fr

We assumtresonancesystem andimmediatekbut ratheroscillations.

tp(x, t)

as have be

132 4. Response of quantum systems to classical electromagnetic oscillations F §4.5 No

will show, that nevertheless such transitions become possible via a two-stepprocess. To this end we use perturbation theory of second order, i.e. (4.58).We study the change of the occupation number of level 3, i.e. we investi-gate c?)(t). The individual contributions to c?) on the r.h.s. of (4.58) canbe discussed as follows: Due to the initial condition, c? = 0. In thefollowing sum, 6°) = cr = 0, and cr = 1. However, fi31 = 0 due to ourassumption (4.63b). Thus we are left with the study of the last term on ther.h.s. of (4.58). Owing to the initial condition, c„(°) 0 only for n 2 = 1.Furthermore, we have already identified m with 3, m= 3. Due to (4.63a, b,c) the only non-vanishing matrix elements H Ii32 and H3n i, Es n arei l ^219respectively. Consequently, cS2) is given by

so that we have only to evaluate the double integral. As may be shown(compare exercise at the end of this section) the most important contribu-tions to (4.64) stem from the first expression cc exp[ itot] in (4.62). Withits use we obtain after integration, choosing to -= 0

n 21c (i/h)t

W2

- 3 2

. (4.65)

hc —

From this result it follows that the incident field can cause transitions fromthe lower level 1 to the upper level 3. This process can be visualized asfollows: The electron first goes from level 1 to level 2 by absorption of alight quantum with energy hco and then from level 2 to level 3 again byabsorption of a quantum ho.. Note that within this process it is by nomeans necessary that the energy difference 4' 2 — W1 coincides with ha).Such a process in which energy is not conserved is called virtual transition.Note, however, that energy conservation is required for the total process

W3 = 2hw. (4.66)

The total process is that of two-photon absorption. To deal with ann-photon absorption process we shall learn to use perturbation theory ofnth order.

Since in many cases the incident light is not entirely monochromatic onehas to extend the formalism in analogy to section 4.2, taking into accountrandom light phases. One then readily establishes that the occupationnumber of level 3 increases linearly with time and not quadratically. Sincethe corresponding formulas are somewhat lengthy and will be treated inthe volume on nonlinear optics, we will not present them here (see fig. 4.2).

•

§4.5 Non-resonant perturbations. Forced oscillations of the atomic dipole moment 133

ria a two-stepler, i.e. (4.58).e. we investi-of (4.58) can= 0. In the

0 due to ourt term on the1 for n 2 = 1.

to (4.63a, b,in and ii21,

(4.64)

ly be shownint W11(4.

-ith

(4.65)

isitions fromiisualized asorption of a1 3 again byit is by no

les with hco.21 transition.tal process

(4.66)

al with anm theory of

romatic onento accountoccupation

tcally. Sincee treated insee fig. 4.2).

EVV,a2 010

00101

incident intensity (red)

Fig. 4.2. In this experiment done by Kaiser and Garrett (1961) the increase of occupationnumber of the upper level is measured by observing the fluorescent intensity in the blue(ordinate). The abscissa shows the intensity of the incident red light. The double logarithmicplot shows clearly the increase of the absorption rate being a quadratic function of theintensity of the incident red light.


Perform the integrations over time in (4.64) using

ii„,„(r) = exp(ita ,r)(exp[ —iwt] + exp[ica]),

m,n = 3,2 and 2,1where w = co and w32 ± (021 = 2w.Chooseto = 0.Show that for large enough time t, the leading term equals const- t.

4.5. Non-resonant perturbations. Forced oscillations of the atomic dipolemoment. Frequency mixing

We assume that the frequency of the external perturbation is not inresonance with any of the transition frequencies (,)„,,, of the quantumsystem and ca„,,, does not coincide with any multiple of 6). We will seeimmediately that the coefficients c(t) cannot grow in an unlimited waybut rather oscillate. We will first investigate the physical meaning of theseoscillations. We use wave functions of the form

4,(x, t) = cn(t) exp[ — iWn t/h]q i(x) (4.67)

as have been introduced in section (4.3). By means of (4.67), we can

10


calculate the dipole moment

0 = f tp*(x, t)( —ex)0(x, t) d V. (4.68)

We have already met a special case of this expression in section (4.1).Furthermore generalizing (4.68) we introduce the abbreviation

0„„ = f so:(x)excp„(x) dV. (4.69)

Inserting (4.67) into (4.68) we obtain

0= — E c(t)c,,(t)exp[iwn,„t]i nn . (4.70)

In (4.67) and thus in (4.70) the coefficients c(t) are still unknown. To findthem explicitly we evaluate them by means of 1st order time-dependentperturbation theory. We use them in the form

cn(t)c il)(t) = c? + an (4.71)

where the coefficient a proportional to the perturbation

(4.72)

We adopt the usual initial condition

= (4.73)8n,no•

Furthermore we assume that the atom does not possess constant dipolemoments

0„,„, = 0, for all m. (4.74)

Because the perturbation is assumed small we treat the coefficients asmall quantities. We first keep only those terms of (4.70) which are linearin an , which is consistent with first order perturbation theory. Thus (4.70)reads

= - E (4(0 exp[ito„,„ ot]Omn. + c.c.). (4.75)

We now wish to calculate (4.75) explicitly. To this end we use the explicitrelation (4.57) of perturbation theory. The corresponding perturbationHamiltonian is taken in the form

H P = ex•E, E = 2E,:; cos wt (4.76)

so that the matrix elements have the form

HI' n(t) = Amn [exp(—iwT) + c.c.] (4.77)

§4.5

which WI

dipole inperiodicprocesse:the exterthe initiEmathemameans patom an(we let thcan write

The intel

a: =

•where wt

amn

Thus the

0=

After sor

0(0

where wt

E =-From (4.in the satdipole mtheory asmass mfound thEbasis forquantum

3DS §4.5 Non-resonant perturbations. Forced oscillations of the atomic dipole moment 135

which we just encountered before [cf. (4.15), (4.62)]. As shown below, thedipole moment of the atom oscillates under the influence of the appliedperiodic field. In this treatment we are not interested in switching-onprocesses, but rather in the steady state response of atoms to the action ofthe external field. For this reason we let to go to — too. In order to suppressthe initial oscillations which occur in switching-on processes, we use amathematical trick: We add a factor exp[— y(t — TA to the perturbation. Itmeans physically that we slowly switch on the interaction between theatom and the field. After all integrations over time have been performedwe let the constant y go to 0. Using the expressions introduced above wecan write the coefficient a* in the form

(—i/h)f t exp[iwm„r]H2„(T)exp[ —y(t — r)] dT. (4.78)

The integral can be done immediately. We readily find

a:= (i/h)AL,0{(— I exp[ — il2„,„ot]

+ (—i52) l exp[ —al: not]) (4.79)

where we have used the abbreviations

2mrt = Wmn W' Elm+ n = Wmn 6')• (4.80)

Thus the dipole moment 0 acquires the form

= (1/h) E ,47„„.{2;in'o exp[ icat] + exp[ — 1Ca]}Anno ± c.c.

(4.81)After some simple algebraic manipulations we find

0(t) = EWE(l / h)10wmno,„n012 2 (4.82)

( omn o ) —

where we have used as above the relationE = 2E0' cos wt. (4.83)

From (4.82) it is evident that the atomic dipole moment oscillates exactlyin the same manner as the impinging field oscillates. This relation betweendipole moment and electric field strength had been treated in classicaltheory assuming the following model. A harmonically bound particle withmass m and electric charge e is subject to a harmonic electric field. It isfound that the particle oscillates in phase with the field. This relation is thebasis for the classical theory of dispersion. In (4.82) we have found itsquantum mechanical analogue.

(4.68)

section (4.1).

(4.69)

(4.70)

'own. To findne-dependent

41)(4.71)

(4.72)

(4.73)

nstant dipole

(4.74)

ficients aich are linear-. Thus (4.70)

(4.75)

e the explicitperturbation

(4.76)

(4.77)

•

We now study whether higher terms of perturbation theory can givesimilar oscillatory contributions. To this end we consider the second sumin (4.58). To exhibit the new aspects clearly we assume that the incidentlightwave is a superposition of two parts with frequencies (4 1 and (.42 so thatthe perturbation Hamiltonian has the form

H= {E, 2 cos co l t + E2 2 cos w2 t}ex. (4.84)

The corresponding double time integrals in (4.58) can be easily evaluated.We assume as stated in the beginning that

cof(0„,,„ 2(.0)(0„.,„ ( j = 1,2), (02 ±- (0 1(0„,„. (4.85)

To demonstrate the general idea we consider a typical term. After perform-ing the integration it reads

(4.86)

Let us use this expression to evaluate the corresponding dipole moment.We then find

(4.87)

We immediately recognize that the perturbation causes an oscillation ofthe atomic dipole moment with the sum frequency co l + 4)2 . Taking all thesecond order terms of (4.58) one may show that in principle all otherfrequency combinations

26.) 1 , <a l + co2 , 2(.02, co2

0, -W I + ca2, 0, —(0 1 — (02 (4.88)

can occur. To obtain this result we have considered only the second orderterms of the coefficients c„,(2). For the sake of completeness, we mentionthat also products stemming from perturbation theory of first order mustbe taken into account when expressions of the form (4.70) are calculated.We will do this in a systematic manner in Volume 3. In the presentcontext, it has been our main objective to explore the structure and themeaning of the contributions of perturbation theory up to second order.As we know from the theory of electromagnetism, oscillating dipoles canbe the source of electromagnetic radiation. In the present case that meansthat the dipole moments (4.87) can cause electromagnetic waves at fre-quencies of the form (4.88). This is one of the simplest examples of

frequencdes w i adifferencportantgreater c

4.6. Inte

We now4 purely c

adventwill also

We a:atomic tatom ha

rl

Atte,matrix e

H8

where

a =

Under C.h —

d

ih4

For furt

c.(t

which y

ih-

We noitot

)11S

§4.6 Interaction of a two-level system with resonant coherent light 137

eory can giveLe second sumt the incidentand co2 so that

(4.84)

,ily evaluated.

n• (4.85)

kfter perform-

• (4.86)

pole moment.

(4.87)

oscillation ofraking all theiple all other

(4.88)

second orderwe mentionorder must

-e calculated.the present

ture and theecond order.; dipoles cane that meansvaves at fre-exadies of

frequency mixing in which by non-linear interaction two waves of frequen-cies co, and co2 are, so to speak, transformed into new waves with sum anddifference frequencies. This example illustrates the origin of many im-portant effects in non-linear optics and such processes will be discussed ingreater detail in Volume 3.

4.6. Interaction of a two-level system with resonant coherent light

We now treat eqs. (4.6) under the assumption that the incident light field ispurely coherent. Such experiments have become only possible after theadvent of the laser and they have led to entirely new phenomena whichwill also be discussed in Volume 3.

We assume exact resonance between the incident light field and theatomic transition frequency coo = (W2 — WI )/h. Again we assume that theatom has no static dipole moment

= 1/12 = 0. (4.89)

The electric vector is assumed in the form E = E0 coswt. The perturbationmatrix elements (4.15) have the general structure

I-11'2 = a(exp[icoot] + exp[ —icoot]) (4.90)

where

a =1E0 - ITT(x)excp2(x)dV. (4.91)

Under this assumption the eqs. (4.6) simplify to. d

t c 1 = W1 c 1 + c2 a(exp[ico0 t] + exp[ —icoot])

ih—dt

c2 = W2C2 c i a*(exp[iwot] + exp[ —icoot]).

For further simplification we make again the substitution

ci (t)= exp[ —iWt/h] di(t)

which yields

= d2 a(l + exp[ — 2ico0t])

(4.95)

ind

"2 = d 1 a*(exp[2ico0 t] + 1). (4.96)

dtWe now assume that d1 and d2 change very little over times in whichexp(2ico0 t) changes. In other words, we assume that exp(2iw01) oscillates

(4.92)

(4.93)

(4.94)

P.

,b


much more quickly than d 1 or d2 . When we integrate eq. (4.95) over a timeinterval which contains many oscillations of the exponential function butwhich is small enough so that d2 has changed little the integral over theexponential function practically vanishes (cf. fig. 4.3). This justifies neglect-ing the exponential functions exp(2iw0 t) and exp( — 2icoot) compared tounity. This approximation is called the rotating wave approximation. Thename may sound somewhat strange, however, we will see the reason for itin section 4.7 when we treat the response of spins to alternating magneticfields. Under the rotating wave approximation eqs. (4.95), (4.96) acquirethe form

ih —d

dI = ad

2'ih—

dtd

2 = a*d i . (4.97, 98)

dt

To solve these coupled equations we first eliminate d 2 . To this end wedifferentiate the first eq. (4.97) with respect to time and replace theresulting d d2/d t according to the second equation (4.98) by

Using the abbreviation12 = jal/h (4.99)

cl1(t).sin(2wt)

AII aaI 1J "4 1

di(t)

AT —

Fig. 4.3. This figure shows how to visualize the approximations explained in the text after eq.(4.96). The upper part shows the function cl 1(t)sin(2wt) as a function of time. The dashedcl.;.-ve shows CO which changes much more slowly than the sin-function. Due to the rapidoscillation of the sine function the total function d I sine(2wt) changes its sign very rapidly.When we average over a time during which tip) has changed but little, the sine-function hasmade several- oscillations, the positive and negative areas, which we indicate by shaded areas,practically cancel against each other. Lower part: In the absence of . rapid oscillations theaverage over a time interval T leaves di (t) unchanged.

we obta

d2—dt

This isgeneral

d1(

a and 13

To obta

d2(

We noelectorThis cc

WI(

To obicomple

a

By mez

d2

With torigina

c11

C2 '

To discare thetotal wtell ustively.

P1

as a futhe cotlower sis empt

be

§4.6 Interaction of a two-level system with resonant coherent light 139

we obtain

d2d

1 + S2 2d 1 = 0. (4.100)

dt2 This is the well-known equation of a classical harmonic oscillator. Itsgeneral solution has the form

di (t) = a cos 0/ + #sinI2t. (4.101)

a and # are constants which must be fixed by the initial value of d i and d2.To obtain d2 we insert (4.101) in (4.97)

d2 (t) = (ih/ a)-c—i d 1 = i(hS-2/a)[ —a sin Ot + #cos SZt]. (4.102)dt

We now return to physics and assume that at the initial time t = 0 theelectron is in quantum state 1. This means that at t = 0, d1 = 1 and d2 = 0.This condition can be fulfilled by choosing a = 1 and )3 = 0. We thusobtain

di (t) = COS W. (4.103)

To obtain a particularly simple expression for d 2 we decompose thecomplex constant, a, into a modulus and phase factor

a = lalexp[ix]. (4.104)

By means of (4.102) we thus obtain

d2(t) = exp[ —ix] sin Qt. (4.105)

With the results of (4.103) and (4.105) the coefficients c which weoriginally wanted acquire the form

c 1 (t) = exp[ /h]cosOt (4.106)

c2(t) = exp[ —ix] exp[ —iW2 t/h] sin Ot. (4.107)

To discuss the meaning of our result we remind the reader that c i and c2

are the coefficients of the unperturbed wave functions occurring in thetotal wave function (4.3). According to quantum mechanics, c 1 1 2 and IC212

tell us the probability of finding the particle in the states 1 or 2, respec-tively. Plotting the corresponding probability

--.1c 1 (t)1 2 = cos 2Ut (4.108)

as a function of time we learn how the occupation of that state changes inthe course of time. To visualize our result we identify cp 1 with the energeticlower state. At time t = 0 this state is occupied whereas the excited state 92is empty. Under the influence of the coherent resonant field the occupationnumber of the lower state decreases while that of the upper state increases

15) over a time1 function butegral over thestifies neglect-compared to

ximation. Thee reason for it.ting magnetic(4.96) acquire

(4.97, 98)

) this end wei replace thed i mik

(4.99)

the text after eq.me. The dashedaue to the rapidgn very rapidly.ine-function hasby shaded areas,oscillations the•


Fig. 4.4. This figure shows the time dependence of various expectation values of an electronwithin a 2-level atom under the impact of a resonant coherent driving field. The upper partshows the variation of the occupation number of the lower electronic level with time. Themiddle part shows the variation of the occupation number of the upper level with time. Thelower part shows the variation of the dipole moment D(1) (4.111) with time. The dashedenvelope is given by I #12 I sin 2 al.

more and more. This means physically that the electron goes over from thelower state to the upper state on account of the absorption process.However, if the interaction between field and atom goes on, the electrondoes not stay in the upper state but goes down to the lower state with anoccupation number described by (cf. fig. 4.4)

P2 IC2(012 = SiII2 2t. (4.109)

Thus under the influence of a coherent resonant field the electron oscil-lates back and forth between its lower and upper states. It is interesting tostudy the behavior of the dipole moment

D = f 2,1,*(—ex)spdV (4.110)

during this process. By inserting (4.3) with (4.106) and (4.107) in (4.110),we obtain after a short calculation

D.= 012 sin coot sin 2S2t. (4.111)

For simplicity, we have again assumed that the atom does not possess

g4.7 Th‘

permanentbe interprefrequencydescribed 1approximabeen discuterms give

In theis entirelyWe will trt

4.7. The nfields

Since the sinteresting

atoirrise rA good

the followiAn additio4.5). Wephenomen.trons or ntnuclear mmeasuremtand especibiologicaltreatment.

B =

accordingvectors of

Bo =

Since the(4.112) for

cp(t) =

with still u

§4.7 The response of a spin to crossed constant and time dependent magnetic fields 141

permanent dipole moments and we have assumed 0 12 real. This result canbe interpreted as follows: The dipole moment oscillates with the transitionfrequency coo but its size first increases and then decreases in a waydescribed by sin 22t. These results were obtained using the rotating waveapproximation. The effect of the non-resonant terms on the solution hasbeen discussed by a number of authors. It turns out that the non-resonantterms give rise to a small shift of frequency.

In the case of two-level atoms this shift is called Autler—Townes shift. Itis entirely analogous to the socalled Bloch—Siegert shift in spin resonance.We will treat spin resonance in the following chapters.

4.7. The response of a spin to crossed constant and time dependent magneticfields

Since the spin is a two-level system as is the two-level atom, we may expectinteresting analogies between the behaviour of spins and of two-levelatoms under the action of external fields. Since these analogies have givenrise to important new phenomena we will now discuss spins.

A good deal of important spin resonance experiments are done underthe following conditions. We apply a constant magnetic field in z-direction.An additional alternating magnetic field is applied in the x—y-plane (cf. fig.4.5). We shall see that such experiments lead to interesting spin flipphenomena. Depending on whether these experiments are done by elec-trons or nuclei the experiments are called electron-spin resonance (ESR) ornuclear magnetic resonance (NMR). Such experiments allow for exactmeasurements of magnetic moments and are used for structural analysisand especially for the study of relaxation processes in liquids, solids, andbiological material. Now let us turn to the corresponding mathematicaltreatment. We write the magnetic field induction in the form

B Bo+ B P(t) (4.112)

according to the constant part and the alternating part. We choose thevectors of these two fields

B= (0, 0, V), B P(t) = (Bf(t),ByP(t),0). (4.113, 114)

Since the Schrodinger equation (3.110) is now time dependent due to(4.112) for its solution we make the ansatz

p(t) c1(t) (pt + c2(t)T1ci(t)

(4.115)c2(t)]

with still unknown, time-dependent coefficients c 1 , c2 . To derive equations

lea of an electroni. The upper partel with time. Theel with time. Theime. The dashed

over from the)tion process.• the electronstate with an

(4.109)

.lectron oscil-interesting to

(4.110)

7) in (4.110),

(4—.111)

not possess

•

Fig. 4.5. Scheme of the crossed magnetic fields. The field in z-direction is time independent.(a) The time-dependent field oscillates parallel to the x-axis. (b) The time-dependent fieldrotates with frequency co in the x —y plane.

for these coefficients we insert (4.115) into (3.110). By using the relationsfor matrices we readily obtain

y(— h<40) C I it(BxP iBP)C2 = i h deld (4.116)tdc2

— it(Bf + iByP )c, + Ihwoc2 = ih . (4.117)dt

We have introduced the frequency coo by the relation

hwo

To simplify the subsequent calculations we assume that the transversemagnetic field rotates with the frequency o.) around the z-axis, i.e. weassume it in the form

Bf = Fcoswt, B'— —Fsin wt. (4.119)

The specific form of (4.119) allows us to simplify (4.116) and (4.117). Tothis end we use the relation

-± ByP = F(cos wt 7s: i sin wt) Fexp[ :Picot]. (4.120)

We thus obtain instead of (4.116) and (4.117)

(—hco0/2)c 1 — iiFexp[iwt]c2 = ih dc,

(4.121)dtdc2

— jaFexpl —iwtjc, + (hco0/2)c2 = ih .

dt(4.122)

By comparing (4.121) and (4.122) with eqs. (4.92) and (4.93) of section 4.6,a complete analogy is revealed, if the rotating wave approximation is used.Using the following substitutions

c 1 (t) = 411 (0 exp[iwot/2], c2 (t) = d2(t)exp[—iw0t/2] (4.123)

*4.7 Th

and assunsolution (2

T(t)

whereEl =

To discus(fig. 4.6). \the reader.

<sx> =<sy>

<s,>

AccordingposIg a(00 flof s7,111r, saround thethen back

Fig. 4.6. Themagnetic fielccomills a.

§4.7 The response of a spin to crossed constant and time dependent magnetic fields 143

and assuming resonance, i.e. (...) = too, we can immediately write down thesolution (again assuming as initial conditions c 1 (0) = 1, c2(0) = 0).

(p(t) = isin(S2t)exp[ —icoot/2]q) 1 + cos(S2t)exp[ico0t/2]cp 1 (4.124)

whereS2 = (4.125)

To discuss the physical meaning of this result we take expectation values(fig. 4.6). We leave the evaluation of the expectation value as an exercise tothe reader. The result reads

<sx > = (h/2)sin(2S2t)sin(w0 t) (4.126)

<sy > = (h/2)sin(2S2t)cos(w0 t) (4.127)

<sz > = ( h/2) cos(2 ) (4.128)

According to this result, the spin components in the x,y-plane are com-posed of a superposition of a rapid precession of the spin with frequencywo and a modulation of frequency 20. Representing the expectation valuesof sx , sy , sz as a vector we readily obtain fig. 4.7. While the spin precessesaround the z-axis it flips from the +z-direction into the —z-direction andthen back so that it oscillates back and forth between these two directions.

sz>

<sx>

<sy>

Ali, Ai1 " I

Fig. 4.6. The motion of the expectation value of the spin under the action of crossedmagnetic fields according to the arrangement of fig. 4.5. Shown are the individual spincomponents according to equations (4.126)-(4.128).

tions

time independent.me-dependent field

ig the relations

IP (4.116)

(4.117)

(4.118)

the transversez-axis, i.e. we

(4.119)

tnd (4.117). To

(4.120)

(4.121)

(4.122)

) of section 4.6,mation is used.

)t/1110(4.123)

Fig. 4.7. Motion of the expectation value of the spin vector of the same experiment as givenin fig. 4.6.

Thus the spin behaves as a spinning top under the impact of externalforces.

Let us consider this process once again more quantitatively. At timet = 0 we have

<sz> = (4.129)

We now wish to determine the time after which the spin has flipped intothe horizontal plane, i.e.

> = O. (4.130)

This is the case when, according to (4.128) the cosine vanishes, i.e. when

22t = Tr (4.131)

holds. Thus the flipping time is given by

t Tr/ (42) =- rh/ (41.tF). (4.132)

After this time, the spin has been turned with respect to the vertical axis byan angle Tr. A field BP causing such flipping is called a r or 90° pulse.When the field is applied for double the time, the spin is flipped by Tr or180°.

These results form the basis of important spin resonance experiments.By application of an external resonant field we can flip the spin from onedirection to another. In practical experiments, the magnetic field does notrotate with the spin frequency but has a fixed direction. The then resultingequations have exactly the same form as (4.121), (4.122) except for anoscillatory additional term which is neglected in the rotating wave ap-proximation. [Compare the discussion following (4.95), (4.96).] This ex-plains the notation "rotating wave approximation", because in applying it

we passrotating

Exercises

(1) Calcu

(2) Show(4.127), e

c<".

j<5

C-17(5or 001

7<S

Hint: Use

(3) Calcul.(4.131), (4.

h = 1

II= 1

10's

Hint: (4.13duration tisystem witmay vary f

(4) Let BP(4.2 repeatil

4.8. The an

When we c.reaching arcists to pre(pheniinaanal

riment as given

ofAgmal

cly.Vtime

(4.129)

flipped into

(4.130)

es, i.e. when

(4.131)

(4.132)

tical axis byT 90° pulse.ped by it or

experiments.in from oneeld does notien resultingcept for anig wave ap-i).] This ex--I spill" it

§4.8 The analogy between a two-level atom and a spin 145

we pass from a (magnetic field) wave in a constant direction to onerotating with the spin precession.


(1) Calculate <sx >, <sy >, <sz > using (4.124).

(2) Show that <sx >, <sy >, <sz > which are explicitly given by (4.126),(4.127), (4.128), respectively, obey the equations

= It(<sy >B, - <sz >By ) (s)

d t <sy > = 1.4<sz >Bx - <sx >Bz ) (* *)

- c-17 <sz > = p,(<sx >By - <sy >Bx ) (* * *)

or in short

—d

<s> = ti<s> x B.dt

Hint: Use the explicit result of exercise 1, or (4.126)-(4.128).

(3) Calculate numerical values for B to achieve 7/2 or 7-pulses. Use(4.131), (4.125) and

h = 1.055 X 10 -34 W S2

= 1.165 x 10 -29 V sm

10 -6 S < t < 1 S.

Hint: (4.131) and the treatment of section 4.7 are valid only for pulse-duration times t which are much smaller than relaxation times in thesystem which describe incoherent (phase-destroying) effects, such timesmay vary from milliseconds up to seconds for real systems.

(4) Let B P(t) be incoherent. Treat the spin-transition in analogy to section4.2 repeating the individual steps and replacing E by B.

4.8. The analogy between a two-level atom and a spin

When we compare the results of section 4.7 to those of section 4.6, farreaching analogies become evident. These analogies have allowed physi-cists to predict and observe phenomena in the optical region because suchphenomena had been found earlier in spin resonance. We now list someanalogies.

(i) Analogy between time-independent wave functions and energies.The spin of the electron is exposed to a constant magnetic field inz-direction (compare Section 3.8). The atom is not exposed to any externalfields (see table 4.1).

In the preceding sections, we described the two-level atom in terms ofthe complex expansion coefficients c,(t), c2(1). In order to see the fullanalogy between a spin-1- and a two-level atom, a description of thetwo-level atom in terms of the following quantities turns out to be useful

= cfc2+

= i(crc2 — c i c;) (4.133)

= I C21 2 ICII 2'

These quantities J, 372 , :S7'3 are the counterparts of the expectation values(sx >, <sy >, <sz > in the spin-1 case, and we will use them below.can be considered as the components of a vector, which is called thepseudo-spin. We now compare the wave functions in the time-dependentcase, i.e. the solutions of the time-dependent Schrodinger equation.

(ii) Wave packets without external alternating fields

Two-level atom= C1cp1+ C2922

compare section 4.1.

Table 4.1. Analogy between spin and two-level atom

Two-level atomWave Spatial

function representation

These wayena:

C2

These wav<(a) Magnet(b) The ele

• =

Under the

jk =

we ill• =

We decomr

812— v

By using (4

• = 215

Furthermornumbers of

I C2 1 2 —With these

spin

<sx,

<s,, :

<Sz

whereab*

(Compare elthat, e.g. 13z-componeni

Spin= c 1 c9 i + cypc

compare (3.97)

Spin Spindirection function

ions §4.8 The analogy between a two-level atom and a spin 147

= f (4(0 + cIsoI )( — ex)( + c2T2 ) dV.

Under the assumption that O n = 422 = 0, where

19./k = f ( — ex)th d V

we have

= *219 12 + CICI421•We decompose 4, 2 = IV, into its real and imaginary part

412 = 0 12 it512 •

By using (4.137), (4.136), and (4.134) we obtain

= +

(4.134)

(4.135)

(4.136)

(4.137)

(4.138)

id energies.gnetic field into any external

om in terms ofto see the fullcription of the,ut to be useful

(4.133)

)ectation valuesbelow.h is called theti pendentqua n.

)atiaIientation

•

These wave functions are responsible for the following physical phenom-ena:

Free precession of spin Free oscillation of dipole moment

c 1 = a exp[iwot/2] c 1 = a exp[—iWit/h]

c 2 = b exp[— iwot / 2] c2 = b exp[— iW2 t / h].

These wave functions allow us to calculate the expectation values of(a) Magnetic dipole moment of the spin (e / m)<s> , (3.105).(b) The electric dipole moment of the electron of the two-level atom

Furthermore, the difference of the expectation values of the occupationnumbers of levels 2 and 1 can be written as

l c21 2 — 1 c 11 2 = g3* (4.139)

With these expressions in mind, we may establish the following analogies:

spin

<sx >= hiallblcoscoo(t — to)

<Sy > = ha b ( sinw0(t — to)

z> = —2(I b 1 2 la12)

whereab* =

"pseudo"-spin

= 21allbl coscoo(t — to)= 21allb 1 sin (.40(1 — t0)

= b 1 2 1a12)

oh* = i a li b l exPliwo t oi.

(Compare exercise (1) on section 3.8.) Note that this analogy is formal sothat, e.g. s 3 must not be interpreted as a quantity proportional to thez-component of the electric dipole moment. On the other hand, such

analogies will prove useful below when we discuss the response of atwo-level atom to external fields.

(iii) Response of a spin or a two-level atom to a coherent resonant field

spin pseudo-spin of two-level atom

<s x > = (h / 2) sin Mt sin raot , = sin 2S2t sin cot

<sy > = (h / 2) sin 22t cos (Jot, = sin Mt cos wt

<sz > = (h/2)cos 2S2t, = - cos 20t

compare (4.126)-(4.128) compare (4.108),(4.109).

(iv) Equations of motion of expectation values.The equations of motion for both the spin- and the two-level atomsubject to an external field may be written in the following form:

spin pseudo-spin of two-level atom

— <s> = gs> X B, —d t

 = x dtwithB = (Bx , By , Bz ), St = (219E, 0, co)

The transition of the electron from its upper level to its lower level andback (compare section 4.6) can be put in parallel to the up and downmotion of a spinning top via the spin analogy. Therefore, the corre-sponding phenomenon shown by electrons is often called optical nutation.The reader should be warned, however, that the expression "opticalnutation" is sometimes used by some other authors in a somewhat differ-ent sense.

(v) Free induction decayLet us consider an ensemble of spins in a sample in a constant magneticfield. In their lowest states the spins will point downwards and theirindividual spin functions are q. These states have no dipole moment inthe x-y-plane. Now let us apply a resonant alternating magnetic field witha ir-pulse. Then all spins are brought to the x-y-plane in which they startto rotate with frequency coo. In many practical cases, the magnetic field Bois inhomogeneous (for various reasons) so that each individual spin sensesits individual field B.33 , where j distinguishes the different spins. In section3.8 we saw that the magnitude of the constant magnetic field determinesthe precession frequencies of the spins. Thus, for different magnetic fieldsdifferent precession frequencies will result. To the spin of each electron orproton a magnetic moment corresponds which oscillates in the same way

as the spining magnetspin precesspins prece:phase accorthe intensityfree inductic

Owing toprocess mayat time 1 =short light Ielectrons in(compare sedescribed bn

= 2 -

whicl.oxabove.

Such expesolids. Duewhat differetronic oscillieach atom. Eelectromagn,after a certaconsequencefree inductio

(vi) Spin-We first givttreated quana sample wh nfirst apply awave functioinduction de,been applied180° aroundspins precessfigure, are inbetween therestoration oachieved as

tons

response of a

t resonant fieldwo-level atom

;t

,(4.109).

two-level atomform:atom

•

lower level and• up and downore, the cone-ptical nutation.ession "opticalmnewhat differ-

istant magneticards and their>ole moment in;netic field with

•vhich they start•agnetic field Bolual spin sensespins. In sectionield determinesmagnetic fields!ach elearon or

the same way

•


as the spin does. According to the theory of electromagnetism, an oscillat-ing magnetic dipole moment emits electromagnetic radiation. Thus thespin precession leads to electromagnetic radiation. However, because thespins precess with different frequencies, their emitted fields get out ofphase according to the spread of dipole moments (cf. fig. 4.8c). As a resultthe intensity of the emitted radiation decreases. The whole process is calledfree induction decay.

Owing to the analogy between spins and two-level systems, the sameprocess may occur, e.g. for electrons of atoms with two energy levels. Firstat time t = 0 all electrons of the sample are in their lower states. Then ashort light pulse of the type of a 7-pulse is applied which brings theelectrons into a mixed state composed of the upper and lower level(compare section 4.6). The further motion of each individual electron isdescribed by the wave packet

= I/2{exp[ —iW1 t/h]q9 1 — i exp[ — iW2 t/h] 9)2)

which is connected with an oscillating electric dipole moment, as shownabove.

Such experiments have been done, for instance, with atoms embedded insolids. Due to different surroundings the individual atoms possess some-what different electronic energy levels so that the frequency of the elec-tronic oscillation, described by the wave packet is somewhat different foreach atom. Since each electron acts as an oscillating electric dipole, it emitselectromagnetic radiation. Due to the spread of oscillation frequencies,after a certain time the oscillating electric dipoles get out of phase. As aconsequence, the emitted light pulse decays and we are dealing with thefree induction decay.

(vi) Spin- and photon-echoWe first give a qualitative description of this effect, which will then betreated quantitatively. Let us consider an ensemble of spins or electrons ina sample where initially all spins or electrons are in their ground states. Wefirst apply a 7-pulse to the spins (or electrons) by which we prepare theirwave functions in a mixed state and the spins (or electrons) start their freeinduction decay. In the echo-experiment, some time after the 7-pulse hadbeen applied, a 7-pulse is applied. It has the effect of rotating the spins by180° around the 1-axis which is shown in figs. 4.8. After the 7-pulse, thespins precess further with the original speed and, as is evident from thatfigure, are in phase again after a time which approximately equals the timebetween the 7T- and 7-pulse. Thus they can emit radiation as before. Therestoration of the original state of the spins (all spins in phase) can beachieved as long as we can neglect irreversible processes causing an


S 3

(a)

V(b)

(c)Figs. 4.8(a)—(f). Scheme of the spin-echo experiment in the rotating frame. The figures andtheir legends apply both to the spin and the pseudo-spin. (s 0-0 s 2 4-0 sy , s+-s). (a) Attime t = 0 all spins are in their ground states. (b) By applying a Tr-pulse, the spins areflipped into the s 2 -direction. (The final direction of the spin-vector in the horizontal plane

depends onexternal driprecession frotating fra:

• flips the spiiprecess lure

areloe. 11has.


Abi‘,

(d)

3

(e)

S3

depends on the initial phase angle between field and dipole matrix element). (c) Withoutexternal driving field, the spins start to precess around the s 3 -axis. As each spin has its ownprecession frequency, i.e. the sample shows inhomogeneous broadening (compare text), in therotating frame the spins get out of phase, i.e. they show free induction decay. (d) A v-pulseflips the spins by an angle of ir around the sraxis as indicated. (e) After the Tr-pulse, the spinsprecess further with their own frequency. (f) The phase lags have been cancelled, i.e. all spinsare in phase again. Thus the ensemble of (in phase) oscillating spins emits an observablesignal, i.e. the "echo" (of the initially exciting w-pulse).

dons

•

ne. The figures and• sy , s3 4.4 s: ). (a) Atnilse, spins arehe tal plane

irreversible dephasing of spins. Quite a similar experiment can be donewith electrons and we leave it as an exercise to the reader to discuss theeffect of a r-pulse and a subsequent Tr-pulse.

We now treat the whole process quantitatively (see also fig. 4.9). Weconsider an ensemble of spins or of two-level atoms with different transi-tion frequencies w = (W2 — W1 )/ h. Since the spread of transition frequen-cies (or, equivalently, energies W1 , W2 ) is usually caused by a superpositionof many effects (for instance inhomogeneities in solids, if the atomsconsidered here are embedded in a solid), one may often assume aGaussian distribution f(w) of w around a center frequency i7):

( _

f04 ) =

)2exp (4.140)46,4)2

Atowhere

(4.141)

is the total number of electrons in our sample. As above, we write the wavefunction of a single spin (or electron) in the form

= c i (Pi + c2992 (or = c t92 1c29). (4.142)

We shall simplify our analysis by considering only a single spin (orelectron) assumed to be in resonance with the external field. The effect ofdetuning is taken into account by averaging the resulting dipole moments

pulses

radiation

Fig. 4.9. Typical time sequence in an echo-experiment. The spins or atoms are excited with asr-pulse, thus starting free induction decay. After some time T a sr-pulse is applied, thus after

approximately 2T the spins or dipoles are in phase again which produces an observablesignal.

c1(t)

c2(t)

where

to. =

where tthe e

<

Experiment

f

which may

= —

where we u

1

N/Tr At

and the facRememb,

the observesample sho-motion of ctime T aftet = T and(4.157). Thi

over the fr(spins (or el

c/(0)

First we ap

over the frequency distribution f(w). We assume that in the beginning allspins (or electrons) are in their ground states

c 1 (0) = 1, c2(0) -= 0. (4.143)

First we apply a 7-pulse (E = 2E0 cos cot) and end up with

1c 1 (t) = exp[ exp[ —itolt],

_c2 (1) = exp[ — ita2 1- 1 ] exp[ — iw21],

where

= Wi/h,

'12 E0— h12,

ftri = ITT

where t = 0 now corresponds to the end of the 7-pulse. By calculatingthe dipole moment gives

P. m = — 1,12 sinw(t + (4.146)

Experimentally, one observes the averaged macroscopic dipole moment

= f f(co)P,,,d6.) (4.147)

which may be easily evaluated

F= —Nausin W(t -F exp[ — -1(Aw(t + 7 1 )) 2 ] (4.148)

where we used the integral

1 l c° exp[ (`) (7-)2sinwt dco sini-atexp[ —1(02]Aco2V•Tr Aw -00

(4.149)

and the fact that we can approximate f0' dca by ffoo dw provided Ato <Co.Remember, that in (4.148) t starts from zero (end of the .7-pulse). Thus,

the observed polarization decays rapidly due to the exponential, i.e. thesample shows free induction decay. Equations (4.144) describe the freemotion of our spins (or electrons) after the excitation. Let us apply, sometime T after the first pulse, a 7-pulse, i.e. we specialize eqs. (4.144) fort = T and then use the resulting quantities c 1 , c2 as initial conditions in(4.157). This gives us, together with (4.156), which is also derived in the

(4.144)

(4.145)

Lions

it can be doner to discuss the

,o fig. 4.9). Wedifferent transi-nsition frequen-a superposition

if the atomsIten assume a

(4.140)

al (4.141)

e write the wave

(4.142)

single spin (orId. The effect ofdipole moments

ns are excited with ais applied, thus after

duces an observable•

exercise 3 below

c 1 (t) = — 1

exp[ + w2(r 1 + T) + wit)]172

1/2

where, evidently, the time t now runs from the end of the 7-pulse. A singledipole moment would read

P = 1112 sin w(t — to ) (4.151)

(4.152)

as is easily seen by inserting (4.150) into (4.134). Finally, let us consider themacroscopic (averaged) dipole moment P. By using (4.151) and (4.149), Pis given by

= N#12 sint.7.)(t — to) exp[ —i(Aw(t — to)) 2 ] (4.153)

i.e. the "echo" shows up with a delay of to 2T after the first f 7-pulsewhere the 7-pulse was applied at time t = T after the excitation.


(1) Free induction decay:Calculate the decay time of freely precessing spins in an inhomogeneoustime-independent magnetic field Bz with spread ABz.Hint: Calculate

1 rw0+A„,/2<sx > clw, where W CC Bz.

AG) J(40—Aw/2

(2) Discuss what -1 7T- and 7-pulse means for the wave functions ofelectrons.(3) A two-level atom driven by a coherent external field may be describedby the wave function

(4.154)

where the coefficients cj obey the equations

i,dt

= wc +19.1 2 .E(t)c2, 1—dt C2 = W2C2 3'12.E(t)C1•

exp[ — i(w2T2 co / (r i T) + ca2 t)] (4.150)

Comparetions c1(0

c1(t)

in the catfield the

ci(t)

c2(1)

where12 =

(4.155)

11/ = c 1 (t)q) 1 + c2(t)2

4.9. Cohe

An impoi

ewlt ,tcdampingmomentsOn the 01cut electrtbehaviorfields thetransitionand uppe

On thewave pac.establisheupper staelectronsteadily cstates. TIrather sh.experime:supposedelectronicthe questspeaking

as §4.9 Coherent and incoherent processes 155

(4.150)

ulse. A single

(Compare sections 4.6 and 4.7.) Solve (4.155) for arbitrary initial condi-tions c 1 (0), c2(0). Show that the solutions read

c 1 (t) = c,(0)exp[ c2(t) = c2 (0)exp[ —iw2 t] (4.156)

in the case of free motion, i.e. E = 0, whereas in the case of an externalfield the solutions read

c 1 (t) = [c 1 (0) cos12t — ic 2 (0)sin Slt] exp[ — iw1t]

c 2(t) =[c 2 (0) cos SZt — ic 1 (0) sin Ut] exp [ — ico2 t] (4.157)

(4.158)

s consider theind41049),

(4.153)

first r-pulseion.

(4.151)where

(4.152)= Ou•Eo/h, E(t) = 2E0 cos wt, W I = W •

4.9. Coherent and incoherent processes

.homogeneous

functions of

be described

(4.154)

1 . (4.155)

An important comment on our previous discussions must be made withrespect to the impact of incoherent processes. In all the chapters thus far,we have treated the quantum system of electrons or spins neglecting anydamping effects. This is clearly visible from the fact that the dipolemoments of spins or electrons oscillate at a certain frequency ad infinitum.On the other hand, we have distinguished between coherent and incoher-ent electromagnetic fields and we saw that such fields cause quite differentbehavior with respect to electronic transitions. In the case of incoherentfields the electron goes from its lower state to its upper state with a certaintransition probability per second, where any phase relations of the lowerand upper states are ignored.

On the other hand, under the impact of a coherent field an electronwave packet with well defined phases of the lower and upper state could beestablished (more precisely speaking, the relative phase between lower andupper state is well defined all the time). In reality, the motion of anelectron or a spin is subject to various kinds of perturbations whichsteadily cause fluctuations of the relative phase between lower and upperstates. Thus in reality the oscillation of the dipole moment decays in arather short time depending on the individual system. Therefore, all theexperiments which we have discussed so far using coherent excitation, aresupposed to be done in such a short time that the internal dephasing ofelectronic or spin dipole moments can be neglected. We will come back tothe question how to take care of such dephasing effects (more preciselyspeaking to the question of damping and fluctuations) in chapter 9.

•

(1) In exercise (2), section 4.7 we got to know the equations of motion forthe expectation values of the spin operators <sx >, <sy >, <sz > whenmagnetic fields are applied. It is known experimentally that the interactionof the spin with its surrounding causes a damping of the phase of the spinand a relaxation of <se > towards an average value so. To take these effectsphenomenologically into account, Bloch introduced into the equations ( * ),(* *), (* * *) of exercise 2 on section 4.7 the following additional terms:

--Sx l = • • • —\Sx)dt T2

d 1 „= ' • •

dt Y T2

so — <se><sz > = • • +

T1 and T2 are called the longitudinal and transverse relaxation times,respectively.

Exercise: Write down the full Bloch equations replacing the dots aboveby the expressions (*), (* * ), (* * *). Solve the Bloch equations forB = (0, 0, Bz ), where (a) Bz is time independent, = Bz, o, and (b) Bz =B2,0 Bz, I sin w't.

5. Quan

5.1. Exai

In this t

atha&S N

(i) Theinductioi

curl

(ii) An ement cat

curl

(iii) The

div

(iv) Thediv

D andequation:

D =

where ethe magrand A caespecial':scopic tir n I

Ls

)f motion for<si > when

Le interactione of the spinthese effectsluations( * ),tional terms:

5. Quantization of the light field

5.1. Example: A single mode. Maxwell's equations

In this chapter we will deal with the electromagnetic field in classicalphysics which is described by Maxwell's equations. We therefore remindthe reader briefly of those equations. (For their illustration cf. fig. 5.1)(i) The induction equation. According to it a temporal change of magneticinduction causes a curl of the electric field.

aBcurl E = – — . (5.1)

at

(ii) An electric current or/and a temporal change of dielectric displace-ment causes a curl of the magnetic field strength H.

apcurl H =. + . (5.2)

(iii) The magnetic induction B has no sources.

div B = O. (5.3)

(iv) The source of the dielectric displacement is a charge density.

div D = p.

D and E as well as B and H are connected by phenomenologicalequations

D = ee0 E, B = p,p 0 H (5.5, 6)

where e is the dielectric constant, e 0 the dielectric constant in vacuum,the magnetic susceptibility and Ito the magnetic susceptibility in vacuum. eand p. can be determined by experiments. It is a goal of modern theory,especially quantum mechanics, to derive e and p by means of a micro-scopic theory. The relations (5.5) and (5.6) seem to indicate a linearrelation between D and E on the one hand and B and H on the other. It

(ation times,

e dots abovequations fornd (b) B =

•

(5.4)

motion

follows from secwhereas for high.physical results

It is well knoelectromagneticparticles — photoelectromagneticvacuum, i.e. apresent. Thus th

.4nnn•

curl E = —

Fig. 5.1. a— e. These drawings indicate how one may visim lin the meaning of Maxwell'sequations. (a) According to the induction equation, a temporal change of the magnetic fluxwithin the conducting loop causes a curl of the electric field strength which causes a flux ofelectric current in the closed loop (wire). (b) According to eq. (5.2), but without the termap/at, an electric current j causes a circular magnetic field. (c) This figure explains themeaning of (5.2) if the last term ap/at is kept. A condenser charged with positive andnegative charges is discharged. This causes a change of the dielectric displacement D betweenthe condenser plates. The temporal change of D causes a curl of the magnetic field strength.The individual loops of the magnetic field strength along the whole circuitry go over into eachother continuously and are never created or vanish. Thus eq. (5.2) secures that even in spatialregions where there is no material current j there holds H 0, or more precisely speaking, thesecond term on the right hand side of (5.2) takes care of the fact that div.curl H 0. (d) Eq.(5.3) tells us that the lines of magnetic induction have neither sinks nor sources. Thus theymust be closed. (e) According to eq. (5.4) the dielectric displacement D has its sources or sinksat positive or negative charges, respectively.

div E = 0,It can be showna velocity

loco= 1/c:It will turnin vacuum.a special case naits electric vecto

E = (0,0,1

where

Ez = p(1)9i

and where p(t)a mode betweening magnetic iioneself readily t

8— C =

ax z

Since the left-11suggests that we

By = q(1)(

where we havegives p and qyields

dq = copd t

where we have

w = ck.

magnet

(b)

§5.1 Example: A single mode. Maxwell's equations 159

7-

ng of Maxwell'she magnetic fluxcauses a flux of

vithout the termure explains theith positive and'mem D betweenic field strength.;o over into eachit even in spatialely speaking, theH 0. (d) Eq.

Ines. Thus theysources or sinks

(5.16)= wp

follows from section 4.5, however, that this is only true for weak fields,whereas for higher.fields nonlinear relations result. Such relations and theirphysical results will be the central topic of Volumes 2 and 3.

It is well known that Maxwell's equations describe the light field aselectromagnetic waves. We know, however, that light can manifest itself asparticles — photons. This leads us to the question how to quantize theelectromagnetic field. To this end we first specialize the above equations tovacuum, i.e. r = IL = 1, and to the case that no charges or currents arepresent. Thus the above equations reduce to

curl E = — —aBcurl B = e —

aE(5.7, 8)

at 0 0 at

div E = 0, div B = O. (5.9, 10)It can be shown that s 0 e0 has the dimension of the inverse of the square ofa velocity

goeo = 1/c 2 . (5.11)It will turn out (below) that this velocity is identical with the light velocityin vacuum. We shall henceforth use the relation (5.11). Now let us considera special case namely a standing electric wave with wave vector k and withits electric vector in z-direction (cf. fig. 5.2)

E = (0, 0, Ez ) (5.12)where

Ez = p(t)gCsinkx (5.13)and where p(t) is a still unknown function of time. Equation (5.13) definesa mode between two infinitely extended mirrors. To derive the correspond-ing magnetic induction we insert (5.13) into (5.7). One can convinceoneself readily that only they-component of this equation is non-vanishing.

a a B y(5.14)

ax z at

Since the left-hand side of this equation is proportional to cos( kx) itsuggests that we put By proportional to cos(kx). This leads us to the ansatz

By = q(t)(9t/ c) cos kx (5.15)

where we have included the factor 1 / c for later convenience. This factorgives p and q the same physical dimension. Inserting (5.15) into (5.14)yields

dqdt


co = ck . (5.17)

Fig. 5.2. An electromagnetic wave propagating in x-direction.

d t

Now let us consider equations (5.16) and (5.17a) in more detail. First of allwe can differentiate eq. (5.16) again with respect to time and eliminate pfrom it by means of (5.17a). This yields

d2q+ 2 q = O. (5.18)

d t 2This equation is the well-known equation of an harmonic oscillator with acircular frequency co. Equations (5.16) and (5.17a) can be written in a veryelegant form introducing the Hamiltonian

H =-21-co(p2 + q 2 ). (5.19)

With its aid we can write (5.16) and (5.17a) in the formdq al/ dp =dt ap ' dt aq

We have encountered such equations in section 3.3 when we dealt with theclassical harmonic oscillator. Comparing (5.20) and (5.21) with equations(3.49) and (3.50) of chapter 3 we recognize that here again we are dealingwith the Hamiltonian equations of an harmonic oscillator. This then allowsus to definitely identify p with the momentum and q with the coordinate ofan harmonic oscillator. With this identification we have the key in ourhands to quantize the electromagnetic field. This is done by a purely

formal analogy.harmonic oscillzanalogy betweerfirm ground weelectromagneticdensity is define

U(x)

Specializing this

D coE,

and thus obtain

U(x) =1(E

We obtain the t(that volume.

CI= f u(x

In the case of otspatial direction

17= rLuc

By inserting (5.:

=_2191,280

The integration

f sin2 kr(

so that we are lt

= 1L9z,2

We find exact!:However, this knormalization

9t, =eo

Now let us rett

;

Since k is a wave number and c a velocity, co in (5.17) is a circularfrequency. Inserting E (5.13) and B (5.15) into (5.8) yields

dp — wq . (5.17a)

(5.20, 21)

160 5. Quantization of the light field

10 COS2 kx dx =


formal analogy. In section 3.3 we saw how to quantize the motion of theharmonic oscillator. Here we want to do exactly the same. To put thisanalogy between the harmonic oscillator and the electromagnetic field onfirm ground we show that H (5.19) is identical with the energy of theelectromagnetic field mode. According to electrodynamics, the energydensity is defined by

U(x) = f(E . D + B-H). (5.22)

Specializing this expression to vacuum we insert

D = 0E, B = p.oH (5.23)

and thus obtain

U(x)=1(e0E2 + B 2 ). (5.24)o

We obtain the total energy in a given volume V by integrating (5.22) overthat volume

= f U(x) d3x (5.25)

In the case of our one-dimensional example it suffices to integrate over thespatial direction x

= f U(x)dx. (5.26)

By inserting (5.13) and (5.15) into the energy expression (5.26) we obtain

""= IIL(p2 kx q2 cos2Kx2 '0 dx) .

The integration over x can easily be performed using

foL sin2 kx dx = -IL2 ,

so that we are left with= IL ot 2e0( p2 q2).

We find exactly the same function of p and q as occurring in (5.19).However, this identification now allows us to determine the still unknownnormalization factor 9t,. Comparing (5.30) with (5.19) yields

11-2—

to(5.31)

Now let us return to the quantization problem. We wish to utilize the

(5.27)

(5.28, 29)

(5.30)

ion.

7) is a circular

III1(5.17a)!tail. First of allind eliminate p

(5.18)

scillator with aritten in a very

(5.19)

(5.20, 21)

dealt with thevith equationsye are dealingis then allowscoordinate ofLe key in ourby a purely


analogy between the Hamiltonian (5.19) with that of the harmonic oscilla-tor. It is convenient to use its Hamiltonian in the form (3.59). Theequivalence of (5.19) with (3.59) is achieved by putting

p=Vir, q = Vfi (5.32, 33)

so that the Hamiltonian (5.19) acquires exactly the same formhal

H = + (5.34)

Here, however, we know what the quantum version looks like. We have toreplace r by the operator a/ig exactly in analogy to section 3.3. Byexploiting that analogy further we introduce creation and annihilationoperators by

ii a \ h( a)(5.35, 36)

= b

or, solving for p and q

p = (b b), q = (b+ +b). (5.37, 38)

The creation and annihilation operators b + and b obey the commutationrelation [cf. (3.69)]

bb-bb= 1. (5.39)By using (5.37) and (5.38), we can express the free fields E and B by meansof these operators in the form

Ez Kb + - b)11-hi- 9t,sin kx

By = (b + +b) - 9E cos kx

Z,

or

Ez = i(b + -b)alif sinkx

B = (b+ +8 )V— — cos Icx2 L

The normalization factor is given by

— — ,eo L

eotio= 1/ c 2 .

With the transf(expressed by th,3.3 and yields

H = hw(b"

We leave it asHamiltonian cc(c.161

For a numoetrons and the equantity aside fnected with the

B = curl A.

A further relaticpotential namel n

E= Alpwhere V is the s(and B, A and Vthat A can be inchoose the "Cot.

div A = 0.

Choosing B inrelations (5.44) a.

Az - (b+

Therefore, the(5.41a) is given t

A = (0, 0,A

with (5.47).Let us summa

netic field, thevector potentialcreation and antotal energy of thTo complete theSchrodinger equicreation a nen

(5.40)

(5.40a)

(5.41)

(5.41a)

(5.42)


With the transformations (5.35) and (5.36), the Hamiltonian (5.34) can beexpressed by the creation and annihilation operators exactly as in section3.3 and yields

H = hw(b + b +). (5.43)

We leave it as an exercise to the reader to convince yourself that thisHamiltonian could be also derived by inserting (5.41) and (5.41a) into(5.26).

For a number of problems dealing with the interaction between elec-trons and the electromagnetic field it will turn out that we need a thirdquantity aside from E and B, namely the vector potential A. A is con-nected with the magnetic induction by

B = curl A. (5.44)

A further relation holds between the electric field strength and the vectorpotential namely

aAE= – — – grad V (5.45)

at

where V is the scalar potential. As is shown in electrodynamics, for given Eand B, A and V are not uniquely determined. Here all we need to know isthat A can be made unique by an additional requirement. In our book wechoose the "Coulomb gauge"

div A = 0. (5.46)

Choosing B in the form (5.41a) one readily convinces oneself that therelations (5.44) and (5.46) are fulfilled by

Therefore, the vector potential belonging to the mode (5.12), (5.13) or(5.41a) is given by

A = (0, 0,Az ) (5.48)

(5.41) with (5.47).Let us summarize the above results. When we quantize the electromag-

netic field, the electric field strength, the magnetic induction, and thevector potential become operators that can be expressed by the familiarcreation and annihilation operators b + ,b of a harmonic oscillator. Thetotal energy of the field also becomes an operator of the form hu)(b + b +To complete the formalism we will do the following. First, we establish the

(5.42) Schrodinger equation of a single mode and discuss its solution. Since thecreation and annihilation operators b, b + play an eminent role in quantum•

(5.41a)

A z = – (b + +b) —2— - -k- suikx.Vhc4go { 1 .

(5.47)(5.40)

(5.40a)

rmonic oscilla-m (3.59). The

(5.32, 33)

(5.34)

ke. We have toection 3.3. Byd annihilation

(5.35, 36)

•(5.37, 38)

commutation

(5.39)id B by means

optics, we will discuss a number of their properties. Eventually (cf. section5.8) we shall show how we can quantize not only a single mode, but thecomplete light field composed of many modes.

5.2. Sarodinger equation for a single mode

The Schrodinger equation belonging to the Hamiltonian (5.43) readsho.th + b(1) = W(I) (5.49)

where we have shifted the origin of the energy scale so that -1 hco isabsorbed in W. The wave functions will be denoted from now on by O. Aswe have seen in section 3.3 the energy levels are given by

W„ = nhw. (5.50)

This expression permits us to say that the field mode oc sin(kx) is oc-cupied with n photons each of energy hco. The wave function reads

1— (b+)n(1)0 (5.51)

electric fieldObservabl

where the state with no photon present, or in other words, the vacuum strengthstate is defined by

&Do = 0. (5.52) Ez(x,t) Ezi

We now have to apply the general scheme of quantum mechanics to theelectromagnetic field, i.e. we must establish a table corresponding to table3.1 in section 3.2. The observables are the electric field strength and themagnetic induction, and, in a way, the vector potential A. Also the energyis an observable. In quantum theory all these observables become opera-tors. Measured values must be now compared with expectation values. Wequote as an explicit example the following expectation value

<0„jEz1(1)„> i<40„1(b + —b)14)„>F-4 sinkx.2E0 L(5.53)

Since b + ,b have nothing to do with the spatial coordinate x we have beenable to extract sin(kx) and all other constants out of the quantummechanical expectation value. From the exercises of section 3.3, we knowthat the first bracket vanishes, i.e. we obtain

01:0„lEz 1(1)„>= O. (5.54)Similarly we find

< Oni Byi t'n> = O. (5.55)

This result seems surprising because we know from (5.50) that the field bb+

§5

(1) Establish a trfield strength E,(each time for aHint: (examii

mode is occupienon-vanishing apuzzle can be re(5.53) has an arunknown and a(5.54) and (5.55entirely destroy(5.ff)number.

Exercises on sec

(2) Evaluate thedensity for the s(3) Calculate the

{E,b+],[E,b],

where e is the e(0, By , 0), and HNote in partici&and B) implies,strength (or magsured (compare

53. Some useful

In this section V.frequently in thisthe commutation

§5.3 Some useful relations between creation and annihilation operators 165

fly (cf. sectionmode, but the

mode is occupied with a certain number of photons so that the energy isnon-vanishing and we might expect a non-vanishing amplitude. Thispuzzle can be resolved as follows. In classical theory the expectation value(5.53) has an analogue when we imagine that the phases of e and b areunknown and an average is made over them. Then we should also find(5.54) and (5.55). Indeed we will see below that a fixed photon numberentirely destroys the knowledge of phases. However, in eqs. (5.54) and(5.55) we have used just those wave functions with the fixed photonnumber.

0) reads

(5.49)that hco is

w on by O. As

(5.50)

sin(kx) is oc-n reili

(5.51)

;, the vacuum

(5.52)

thanics to theading to tablemgth and theNo the energyecome opera-nt values. We

(5.53)

we have beenthe quantum3.3, we know

(5.54)

(5.55)

thall, field


5.3. Some useful relations between creation and annihilation operators

In this section we derive some relations which we will use later on quitefrequently in this, and the following books. All these relations are based onthe commutation relation

bb+ —b + b = 1. (5.56)

(1) Establish a table corresponding to table 3.1 on page 72 for the electricfield strength E, the magnetic induction B, energy density and total energy(each time for a single mode).Hint: (example)

Observable Operator Expectation valueelectric field

strength

Ez(x,t) Ez(x,t)= i(b + — b) --2—e; sm kx <4)1 Ez(x,t)10>Vi-c7) 1/2— .

(2) Evaluate the expectation values for e 2(x,t), b2(x,t) and the energydensity for the solutions On of (5.49).(3) Calculate the following commutators:

[E,b + ], [B,b], [b+b,b], [H,E],

[E,6], [B,b+], [b+b,b+], [H,B],

where e is the electric field strength (0, 0, Es ), B the magnetic induction(0, By , 0), and H the Hamiltonian (5.43).Note in particular that non-commuting of the operators H and E (or Hand B) implies, that the energy of the field mode and its electric fieldstrength (or magnetic induction) cannot be simultaneously precisely mea-sured (compare section 3.2).


To understand the following property, the reader should recall the mean-ing of commutation relations: We always have to imagine that both sideshave to be applied to an (arbitrary) wave-function. As we know, it ispossible to apply b + (or b) several times to a wave-function (1), i.e. to form(b + )"(1). This leads us to the question of studying the properties of (b+)"within a commutation relation. The following relation can be derived from(5.56):

b(b + ) fl - (b + )"b = n(b + ) n-1 (5.57)

where n is an integer n = 1, 2, ... To indicate how this relation can beproved we choose n = 2. We then rewrite the left hand side in thefollowing way

b(b+ )2 — (b + )2 b = bb+ b + -b+ b + b + b + bb+ -b+ bb+

= (bb+ -b + b)b + +b + (bb+ -b + b).

Making use of the commutation relation (5.56) yields

(bb + -b + b)b + +b + (bb + -b +b)=2b + (5.58)

1 1Relation (5.57) can then be proven by complete induction (compareexercise at the end of this section). In an analogous way we may prove therelation

bb-bb= (5.59)

In sections (5.4) and (5.5) we will encounter wave functions of the form

(1) = E cn(b + )%)0 . (5.60)n.•0

By writing this in the form

(1) = E cn(b + )14)0 (5.61)co

n=0

we are led to study the properties of such a sum over operators:00

f(b) _= E cn(b+)nn-0

For such functions the relation

af(b+) bf(b + ) - f(b + )b -

ab+

§5.3

can be derived. E

b +f(b) -

holds (compareexp(ab + ), (5.63)

bexp[ab+]

b+ expL ajWe now want to

f(a) = exp[

To this end we df( a)

- exta a

where we hosrearranged in the

af(a) — extaa

which allows usagain the definitdifferential equal

af/aa = -j

It is solved by

f(a) = e-aj

where we deducef(0) = b.

Therefore the fin

f(a) = Cat

In putting (5.67)

exp[ab+b]i

In a similar way,exp[ab+b]i

Readers intereste

(5.62)

(5.63)

§5.3 Some useful relations between creation and annihilation operators 167

which allows us to make use of the commutation relation (5.56). Usingagain the definition (5.67) on the right-hand side of (5.69) we obtain thedifferential equation

afiaa = -f. (5.70)

It is solved by

f(a) = e af(0) (5.71)

where we deduce from (5.67) that

f(0) = b. (5.72)

Therefore the final result reads

f(a) = e'b. (5.73)

In putting (5.67) equal to (5.73), the fundamental relation yielded is

(5.62) exp[ab+b]bexp[-ab+b] = e -ab. (5.74)

In a similar way, we obtain the relation

exp[ab + b]b + exp[-ab +b] = e ab +(5.75)

4

.63) Readers interested in more details of the b + ,b calculus are referred to my

all the mean-iat both sidese know, it isI), i.e. to formrties of (b+)"derived from

(5.57)

ation can beside in the

,b+

).

(5.58)

on (comparelay prove the

(5.59)

4 the form

(5.60)

(5.61)

rs:

can be derived. Similarly the relation

af(b)f(b) - f(b)b + - ab (5.64)

holds (compare exercise). When we choose f as exponential functionexp(ab + ), (5.63) reads

bexp[ab + ] - exp[ab + ]b = a exp[ab + ]. (5.65)

Similarly (5.64) is replaced byexp[ab] — exp[ab]b + = -a exp[ab]. (5.66)

We now want to study the expression

f(a) exp[ab + b]bexp[ -ab + b]. (5.67)

To this end we differentiate it with respect to a which yields

af(a) - exp[ab + b](b + bb - bb +b)exp[-ab + b] (5.68)aa

where we have strictly preserved the sequence of operators. (5.68) can berearranged in the form

= exp[ ab+b bb)b

exp[ -ab +b]Ma) (b+ - +aa =_

(5.69)

book: Quantum Field Theory of Solids (North-Holland, Amsterdam,1976).

Exercises on section 53

These exercises are somewhat formal and more intended for the mathe-matically interested readers.

(1)Prove (5.57) by complete induction.Hint: (5.57) is correct for n = 1. Assume that it has been proven up ton = no. Show, that then (5.57) is also correct for n = no + 1.Further hint: Rewrite

b(b + )' 0+ I — (b + )"° + b

as

[b(b + )"° — (b + )"°b]b + + (b + )Nbb + —b + b).

(2) Prove (5.63), (5.64).Hint: Insert on both sides (5.62) and use (5.57) or (5.59).

(3) Derive the Baker— Hausdorff theorem

exp[ab + 136] = exp[ab] exp[ fib + exp[ —1-a13]

exp[ab + fib] = exp[/3b + ] exp[ab] expDa/3].

a, /3 are complex numbers.Hint: To prove ( a ), consider the operator Sl(t) defined by

exp[ t(ab + flb + )] = exp[tab](t), E2(0) = 1.

Differentiate ( a * a) with respect to t and use the definition of OW tosimplify the resulting expressions. In this way a differential equation for12(t) results. Multiply this equation by exp[ — tab] from the left and use(5.63), (5.64) to eliminate the exponentials. Solve this equation for OMunder the appropriate initial condition.To prove ( a a), use the definition

exp [ t( ab + flb +)] = exp [ tflb](r), S2 (0) = 1

and proceed as before.

§5.4 Solutio

5.4. Solution omode. Wave pa

The time depindependent ec

Hoo0

A simrand a siligie-p.

(1:0 = codoo

By inserting (5on the right- E.

superposition (4:0 = codk,

For whattowith respect to

 = <0

and evaluate -inserting (5.77

 = I co

When we use

<00 b cDoand <001b1(1)1expression to

 = ct,c

Differentiating

—d =d t

We leave it a.holds even if(5.76). We als(

—d <6 + >dt

holds.

•

(a)( a a)

(a a)

•

§5.4 Solution of the time-dependent Schrodinger equation for a single field mode. 169

1msterdam, 5.4. Solution of the time dependent Schrodinger equation for a single fieldmode. Wave packets.

The time dependent Schrodinger equation corresponding to the timeindependent equation (5.49) reads

H = ih—d4)

cy dt(5.76)

A simple example of a solution is given by the superposition of a 0-photonand a single-photon function

(1) = c0(1)0 + eel>, exp[ —iwt]. (5.77)

By inserting (5.77) into (5.76) we find that the corresponding terms cancelon the right- and left-hand side. The most general solution is given by asuperposition of functions with all possible photon numbers

(1) = c000 + c 141 1 exp[ — iwt] + • + c„(1)„ exp[ — ituot] +... .(5.78)

For what follows it is interesting to study expectation values of b and b+with respect to the solutions (5.77) or (5.78). We introduce the abbreviation

 = <01b10> (5.79)

and evaluate this expectation value for the explicit example (5.77). Alsoinserting (5.77) into (5.79) yields

 = Icol

+ crc0<c13 1 1b1(1)0 > exp[icot] + 1c11201111b101>.

When we use the relations

<0l b R0> = < <D 1l b 1 40 = <0 1 1b1410 = 0

and <cDolblOi>= 1 (compare exercise 6 of section 3.3) we can reduce thisexpression to

 = ccl exp[ — ica]. (5.82)

Differentiating this expression with respect to time we obtain the equation

(T = — it°(5.83)

7-e--i-e-gr-Fi-ra—s —aire'xercise to the reader to demonstrate that this relationholds even if we use the most general solution of the Schrodingermuation

`(5.76). We also leave it as an exercise to—tNe reader to convince himself that

—dt

 = iw (5.84)

holds.

(* * * )

1 of 12(t) toequation forleft and useon for Sl(t)

2<O0l b l O0> cci c i< cDoi b l O i> exp[ — icor]

the mathe-

oven up to

(5.80)

(5.81)

(1) Prove that (5.77) satisfies (5.76).Hint: Insert (5.77) into (5.76) and compare the individual terms 4).

(2) Show that 4), (5.77) or generally (5.78) is normalized,

< 0 1 0 > = 1, (*)i provided

E lcn 1

2 = 1.n-o

Hint: Insert (5.78) into (*), multiply term by term and use

< 4). (1). > = 8n..

(3) Calculate the expectation value for the electric field strength (singlemode) using (5.77). Compare the result with (5.54).

(4) Show that (5.83) holds for

 013. 1b1 n13> (* * )

where 4) is given by (5.78).Hint: Insert (5.58) into (* * ), multiply the sums term by term and use(compare exercise 6 of section 3.3)

01)„ b (Dm > = VT?? -1-

(5) Derive (5.84).Hint: Same as before, but

<(1)„lb+14)„,>= Vm + 1 5„.,„+1.

(6) Repeat exercise (3), but now with 4) = (5.78) (a repetition of formerexercise).

5.5. Coherent states

In the preceding section we have written down the most general solution ofthe time-dependent Schrodinger equation of a single mode. Choosing timet = 0 this solution acquires the form

= c000 + c 1 (13 1 + • + cnck + • • (5.85)

where the functions (I)„ can be written more explicitly by means of the

photon creatior

On = Oil/In laser theorya fundamental

c„ = (a"/

As is known frthe probabilityequivalently, tprobability ma(

I cni 2 = (kThis expressiorwith the Poissc

Inserting

cI = exp[

The sum overexponential fuoperator still rdefined for nu]containing pov(compare secti

= exp[

When doing sc

A IC

Fig. 5.3. The Poi:

terms On.

se

strength (single

)3, term and use

§5.5 Coherent states 171

photon creation operator in the form

= (1/VT! )(b + )'00 . (5.86)

In laser theory as well as in non-linear optics a special form of (5.86) playsa fundamental role. In it the coefficients have the form

c„ = (a n / VT-z!) exp[ —1a1 2 /2]. (5.87)

As is known from quantum mechanics the absolute square of c gives usthe probability of finding an n-photon state when the photon number (or,equivalently, the field energy) of the single mode is measured. Thisprobability reads explicitly

Ic„I 2 = (Ial 2n /nOexp[—Ice1 2 ]. (5.88)

This expression is well known in probability theory because it is identicalwith the Poisson distribution (fig. 5.3).

Inserting (5.87) and (5.86) into (5.85) we obtain00

(13 = exp[ — I al 2/2] E (an / n!)( b + )" 00 . (5.89)n=1:1

The sum over n in front of 00 is strongly reminiscent of the usualexponential function. The only difference lies in the fact that b+ is anoperator still acting on 00 whereas the exponential function is originallydefined for numbers. However, in quantum mechanics one writes that sumcontaining powers of b + in a formal way again as an exponential function(compare section 5.3). We therefore write

to = exp[—Ia1 2/2]exp[ab + ](130 . (5.90)—

When doing so we have to keep in mind that b ÷ is still an operator acting

lan I 2 exp(-Ia12

)/n!ICIr

,tition of former

-ieral solution of. Choosing time

(5.85)

y means of the Fig. 5.3. The Poisson distribution as a function of n shown for different parameter values a.

n

on the subsequent function 4)0 . We now want to show that (5.90) has quitespecific properties. To this end we apply the annihilation operator b on 4).We make use of the commutation relation (5.65) we have derived earlier inchapter (5.3) namely

b Lab-t. 114-=. 4x.expi ab + ] (5.91)which we now write a little bit differently namely

bexp[ab + ]= a exp{_ 71-,12(PLWWhen we apply both sides of this relation on 4)0 and multiply it byexp( -1a1 2/2) we obtain

b4) = a4). (5.92)Thus we see that the application of b on 4) is equivalent to multiplying 4)by a. In the sense of quantum mechanics 4) is an eigenfunction to theannihilation operator b. This has important consequences. When we haveto calculate any expectation value by means of such a 4) we have simply toreplace the -operator b by a, and correspondingly the creation operator b+by a* provided all annihilation operators b stand on the right-hand sideand all creation operators b + on the left-hand side of the total operation 52.

We illustrate this statement by a few examples and leave their proof tothe reader as an exercise

<401 4:1)>= a, <01b+ b1(1)> =100

(5.93, 94)

= ( a* - a) sin kx. (5.95)

The eigenfunctions Oa belonging to different a's are not orthogonal toeach other. We rather find

4)14) >= ex (- 1 1a1 2 - 1 1 )31 2 + aV). (5.96)

(For a proof and further discussion see exercise 4 hereafter.)The function (5.90) can be also obtained as solution of a Schrodinger

equation. To this end we consider the Schrodinger equation(hwb + b + y*b + +yb)(11 = WO. (5.97)

This equation which describes the "displaced harmonic oscillator" appearsquite frequently in quantum physics (compare exercise 3 hereafter).

We claim that (5.90) with appropriately chosen a is a solution of (5.97).We insert (5.90) into (5.97) and use the abbreviation

= exp[ -1a1 2/2]. (5.98)

Vre-iiike further use of (5.92). This yields

9Lexp(ab + )(hcoab + +y*b+ +7a)4)0 = W9Lexp(ab + )410 . (5.99)

We multiply both sides from the left side by a,- exp(-ab + ). Since the

functions b + (I),can be only fillvanish. This yie

hwa +

andya = W.

From (5.100) vit

= -7*/From (5.100) ac

PV=results. In coatcoherent state.states, that

<4:01b1 711 = <,

is < I. Howevestates.

Exercises on sec

(1) Prove (a) (5.Hints:(a) Multiply (5<0 1 0 > = 1.(b) Use

<T1 b X> = ((exercise 4 of set(c) Use (5.93).

(2) Prove <4)1(1)Hint: Use (s) o

(3) Equation (5.'quantum optics.(a) Quantum inharmonic oscilla+ Ko. (We callqo = 0 for Ko =

§5.5 Coherent states 173

5.90) has quitevrator b on (I).rived earlier in

(5.91)

multiply it by

(5.92), multiplying (I)unction to theWhen we havehave simply tom operator b+right-hand sideal operation I.; their proof to

(5.93, 94)

(5.95)

orthogonal to

(5.96)

.)a Schrodinger

1

(5.97)

llator" appearstreaf ter).ution of (5.97).

(5.98)

)(kr (5-99)b + ). Since the

functions b + (1:10 and Coo are linearly independent, the resulting equationcan be only fulfilled if the coefficients of 6 + (Do and of (Do respectively,vanish. This yields

ham + y* = 0 (5.100)

andya =

From (5.100) we derive

= — Y * / (hw).From (5.100) and (5.101) the eigenvalue

(hw) (5.102)

results. In conclusion we quote an important maximum property of acoherent state. One may show quite generally, i.e. for arbitrary quantum.states, that

<4:011)1(13><O1b+10> t

<01b+blo>,_

is <1. However, the states for which y11 = 1

states.


(1) Prove (a) (5.93), (b) (5.94), (c) (5.95).Hints:(a) Multiply (5.92) with <(1)I from the left and form <4: 0 1b1(1)>. Use<01(1)>= 1.(b) Use

<92 1 bX> = <b+cPIX> (*)(exercise 4 of section 3.3) and (5.92).(c) Use (5.93).

(2) Prove <01(b + )mb"143 > = (a*)"a", where (I) is a coherent state.Hint: Use ( * ) of exercise (lb) m times and use (5.93) n times.

(3) Equation (5.97) occurs at several occasions in quantum mechanics andquantum optics. To see this, treat the following exercises:(a) Quantum mechanics: the displaced harmonic oscillator. Subject theharmonic oscillator of section 3.3 to a constant force Ko, so that = —fq+ Ko. (We call this oscillator displaced, because the equilibrium positionqo = 0 for Ko = 0 is shifted to q; = K0/f).

(5.100a)

(5.101)

(5.103)

are precisely the coherent

Derive the additional term to the Hamiltonian (3.46), stemming from Ko.Transform p and q into b + ,b. Then (5.97) results, where

Y,Y* cc Ko.

(b) Quantum mechanics: the forced harmonic oscillator. Repeat the samesteps as in (a), for a time dependent Ko.(c) Quantum optics (electronics). Start from Maxwell's equations (5.1)-(5.4), put for the electric current j = (0, 0,jz ),j,= jo cos kx, p = 0, D = e0E,B = p.0H.Go through the same procedure as in section 5.1 and show that in H.(5.19), an additional term containing jo arises. Perform the same quantiza-tion procedure following (5.32). Note that Jo can be time dependent.

(4) Prove the "unorthogonality" relation (5.95).Hint: Insert (5.89) for a and # in <0.14)fl >, multiply term by term, use< 11.1 > = 8„,, „ and sum up again.

5.6. Time-dependent operators. The Heisenberg picture

We consider as

<40(t)Ibl(1)(t)

where <Kt) is gSchrodinger equat

ih (IOW

- hd t

Our goal will it bcoperator and 4)(tproblem can be s(quite generally, wt

dcI)ih—

dt = 114)

where H is a get(5.108). If H wereorder differential

In all our considerations we have treated b and b + as given time indepen-dent operators which act on wave functions which in the general case maybe time dependent. The wave functions had to obey the time-dependentSchrodinger equation. This representation is called the Schrodinger repre-sentation or the Schrodinger picture. In quantum optics and laser theory aswell as in other fields of quantum mechanics another representation isfrequently used. To this end remember what we have seen in section 5.4. Itturned out there that the expectation values of b and b + obey theequations (5.83), (5.84), quite independently of the original wave functions.This leads us quite naturally to the question of whether we can findoperators which are time-dependent and which automatically obey theequations (5.83) and (5.84) i.e.

dt b + = icob + , —

db =

dt(5.104, 105)

In this case, the total time dependence of the physical process is containedin the operators whereas the wave functions are now supposed to beentirely time independent. This representation is called the Heisenbergrepresentation. We now show in a systematic way how this Heisenbergpicture can be derived. To make our procedure as transparent as possiblewe treat the explicit example of a single mode. Later on we shall see,however, that the Heisenberg picture has a quite universal meaning inquantum theory.

41(0 = exp( -

However, this rek(1)(0) a wave funcidefine the exponeits power series e:

exp( - iHt/h

By inserting (5.1(term by term wespecial case of H

(I)(t) = exp(

0(0) is an arbitrtposition of the so

b(1)„ =

in the form00

43(0) = E cn-0

Now let us insertthat (I)(t) on th,

§5.6 Time-dependent operators. The Heisenberg picture 175

We consider as an explicit example the expectation value

<0(t)lb14)(t)> (5.106)

where OM is given by a general solution of the time-dependentSchrodinger equation

ih d4)(t) — hwb + b4)(t). (5.107)dt

Our goal will it be to cast (5.106) into a form where b is a time-dependentoperator and OW is replaced by a time-independent function. Since thisproblem can be solved not only for the Schrodinger equation (5.107) butquite generally, we start with the time-dependent Schrodinger equation

d(I)in— = H (5.108)

dt

where H is a general Hamilton operator. We seek a formal solution of(5.108). If H were not an operator but a number, (5.108) is an ordinary firstorder differential equation whose solution reads

= exp(—iHt/h)(1)(0). (5.109)

However, this relation remains valid in the case that H is an operator and10 (0) a wave function given at time t = 0. We have seen above that we candefine the exponential function of an operator (formerly, e.g. b or b + ) byits power series expansion

.0exp( — i Ht/h) = E -n ! ( — i H/ h)"t ". (5.110)

By inserting (5.109) with (5.110) into (5.108) and comparing both sidesterm by term we readily verify that (5.108) is solved by (5.109). In thespecial case of H = hoth + b, 4(t) reads

--= exp(—io.th + bt)43(0), (5.109a)

(KO) is an arbitrary wave function, which can be represented as a super-position of the solutions of the time independent Schrodinger equation

b(1)„=

in the form

4)(0) = E c„(II„

Now let us insert (5.109) into (5.106). Hereby we have to take into accountthat 4(t) on the left-hand side is to be replaced by (I)(0) exp(i Ht/h )

mg from Ko.

eat the same

ations (5.1)—: 0, D = e0E.

w that in H.me quantiza-endent.

by term, use

time indepen-eral case mayne-dependentidinger repre-aser theory asresentation issection 5.4. Itb+ obey theave functions.we can find

ally obey the

(5.104, 105)

s is containedpposed to beLe Heisenbergis Heisenbergnt as possiblewe shall see,.1 meaning in

according to the rules derived in exercise 2 below. Thus we obtain

01)(t)Ibl(1)(t)> = <40 (0) lexp[iHt/h] b exp[ –iHt/h]1,(1)(0)>. (5.111)

b(t)

When we introduce a new operator b(t) as indicated by the bracket ourgoal to rewrite the original expression (5.106) has been reached. We nowturn to the question how to determine b(t). This is best answered by askingwhich equation is obeyed by b(t). To this end we differentiate b(t) withrespect to t which yields

db(t) ddt dt

exp[iHt/h]bexp[ – i Ht/h]

= exp[iHt/h] -h-i (Hb – bH) exp[ –iHt/h].

– hcobSince the commutator between H and b yields –hcob, (5.112) transformsinto

db(t) – i (4) exp [ Ht/h]b exp [ – HO]. (5.113)dt

The right-hand side is identical with b(t) so that the desired operatorequation reads

—d b(t) = –icob(t). (5.114)dt

This is exactly what we had expected from our previous section 5.4[compare (5.83) and (5.84)]. The first-order differential equation (5.114)can be readily solved and yields

b(t) = (5.115)

Since at time t = 0 the commutation relation bb + –b + b = 1 is fulfilled itfollows from (5.115) that b(t) fulfills that relation for all times

b(t)b + (t) – b + (t)b(t) = 1 (5.116)

as can be seen by inserting (5.115) into (5.116).While our above procedure was quite useful in the special case of the

operator b we now formulate a procedure in a way which lends itselfimmediately to general applications. To this end we rewrite the second rowof (5.112) in the form

db(t)=

i tHexp(iHt/h)bexp( –iHt/h)dt h

– exp(iHt/h)bexp(–iHt/h )H) (5.117)

which can be w

db(t) _ idt h

Or introducingdb(t) i

dt hThis is the funHeisenberg picttime independei

There may b,time dependeno

= b(t

When we wish taccount this ad,(5.119) must be

d2(t) idt

In our present eof the sine functof an example(5.121) remain s

Exercises on sec

(1) Heisenberg pAs stated abovharmonic oscillaprovided by spitDerive the Heiseman H = – —e

mg equations wii4.7.

(2) Show<exp( –iHt

where H is the IHints: Use thepower series and

(5.112)

§5.6 Time-dependent operators. The Heisenberg picture 177

which can be written in a shorter way

db(t) idt = h

(Hb(t) — b(t)H). (5.118)

Or introducing the definition of commutators we eventually find

db(t) idt

["(t)].

This is the fundamental relation which is obeyed by operators in theHeisenberg picture. This relation is valid for any operator provided it istime independent in the Schrodinger picture.

There may be cases in which the operator has an additional intrinsictime dependence. For example

= b(t) sin wt. (5.120)

When we wish to derive an equation of motion for S2 we have to take intoaccount this additional time dependence by a partial derivative, so that(5.119) must be replaced by

d52(t) i au(t) —

h[H,S2(t)] + (5.121)

dt atIn our present example the partial derivative refers to the time dependenceof the sine function. Formulas (5.119)—(5.121) have been derived by meansof an example where H is time-independent. We note that (5.119) and(5.121) remain valid even if H depends on time explicitly.


(1) Heisenberg picture and spin.As stated above, the Heisenberg picture does not only apply to theharmonic oscillator, but to arbitrary quantum systems. A nice example isprovided by spin operators. Solve the following problem:Derive the Heisenberg equations of motion for sx , sy , sz using the Hamilto-nian H = — ;13- s (cf. (3.110)), where s = (sz ,sy ,sz ). Compare the result-ing equations with the equations for <sz>,<sy>,<sz>, exercise 2 of section4.7.

(2) Show

<exp( — i Ht / h)4) I x > = <4) I exp(i Ht/h ) x >

(5.119)

where H is the Hamiltonian.Hints: Use the definition of bra and ket. Expand exp(± iHt/h) into a

(5.117) power series and use that H is a Hermitian operator.

e obtain

4)(0)> . (5.111)

the bracket ourached. We nowwered by askingntiate b(t) with

(5.112)

.112) transforms

(5.113)

Iesired operator

(5.114)

ous section 5.4equation (5.114)

(5.115)

= 1 is fulfilled itimes

(5.116)

3cia1 case of the'aich lends itself

the second row

5.7. The forced harmonic oscillator in the Heisenberg picture

The great advantage of the Heisenberg picture lies in the fact that theequations of motion for operators are often strongly reminiscent of classi-cal equations of motion. In particular, this is so for the field operators band b .* . As an example we consider the equations of motion of a forcedharmonic oscillator (compare exercise 3b of section 5.5). The Schrodingerequation of the forced harmonic oscillator can be formulated by means ofb and b + and reads:

(hcob + b + y*b + + yb) 41) = ih d4)

(5.122)

We have encountered the corresponding time-independent equation in(5.97). Now we admit, that y and y* may be time-dependent. While it is arather formidable task to solve this Schrodinger equation with time-dependent y(t)'s the solution of the corresponding problem in the Heisen-berg picture is rather simple. To this end we start from the Heisenbergequation

—dt

b(t) = –h

[H'b(t)] (5.123)

where H is given by the Hamiltonian of eq. (5.122). We leave it as anexercise to the reader to verify by means of the commutation relation(5.116) that the Heisenberg equation of motion now reads

c b(t) = –icab(t) – iy*(t)/h (5.124)

dc-17 1, (1) = iwb + (t) + iy(t)/h. (5.125)

The solution of (5.124) can be found in complete analogy to that ofordinary differential equations and reads

b(t) = – f texp[ –ico(t – T)]y*(T)dr + b(to)exp[ –41(t – to)].

(5.126)b(to) is the operator b in the Schrodinger picture.

Later we will see, in section 8.2, that the driven (or forced) oscillator isthe prototype of a coherent field arising from a classical source.


(1) Derive (5.124) and (5.125) from (5.123).Hint: Use H = hcob + (t)b(t) and (5.116).

(2) Calculate

<4:13(t)1b1(11(t)

<41(t)lb+I(1)(,

<(1)(t)lb+b10

where 4:I) obeys eq(b) 0(0) =Hint: Go over tc(5.126). (Readers8.2.)

(3) Convince youndependent giving req. (5.122)).

5.8. Quantization

In this section wshowed how to qtreader is advised t

We describe thestrength E and thand currents MaxN

curl E = — —ala

div E = 0,

Instead of (5.12) aidescribed as a suprunning waves bucomplicated configa vector, u must d(tion. We distingui:generalize (5.13)

E(x , t) -=

The minus-sign iscorresponding man

B(x , t) = q

§5.8 Quantization of lightfield: The general multimode case 179

(2) Calculate

<4)(t)Ibl4)(t)>

<4)(t)lb+14)(t)>

<4)(t)ib+b14)(t)>

where 4) obeys equation (5.122) and (a) 4)(0) = 4)„ [solution of (5.49)1(b) 4)(0) = 4 [coherent state, (5.90)].Hint: Go over to the Heisenberg representation and use the solution(5.126). (Readers who want to check their results are referred to section8.2.)

(3)Convince yourself that K0 or jz of exercise 3 of section 5.5 may be timedependent giving rise to the same Hamiltonian as before (i.e. to the 1.h.s. ofeq. (5.122)).

5.8. Quantization of lightfield: The general multimode case

In this section we resume our considerations of section 5.1 where weshowed how to quantize a single mode of the electromagnetic field. Thereader is advised to briefly recapitulate that section.

We describe the electromagnetic field by the vector of the electric fieldstrength E and the vector of the magnetic induction B. Without chargesand currents Maxwell's equations in vacuum read

curl E = -aB

curl BaE

= (5.127, 128)at

div E = 0, div B = 0. (5.129, 130)

Instead of (5.12) and (5.13) we now consider the general case in which E isdescribed as a superposition of modes ux(x). u may describe standing orrunning waves but it might describe also spherical waves or still morecomplicated configurations depending on the physical problem. Since E isa vector, u must describe by its vector character the direction of polariza-tion. We distinguish the different modes by an index A. In this way wegeneralize (5.13)

E(x, t) = - E px(t)'3txux(x). (5.131)

The minus-sign is exhibited explicitly for sake of later convenience. In acorresponding manner we generalize (5.15) to the hypothesis

DtxB(x, t) = E qx—vx(x) (5.132)

fact that theent of classi-I operators b-1 of a forced

SchrOdingerby means of

(5.122)

equation in. While it is a

with time-n the Heisen-e Heisenberg

(5.123)

eave it as anition relation

(5.124)

(5.125)

gy to that of

-iw(t-t0)].

(5.126)

I) oscillator isrce.


where we shall determine the connection between the functions vx with uxin a minute. Inserting (5.131) and (5.132) into (5.127) leads us to

a qx— Ep A(t)9Z, ), curl U(X) = – E --vx(x). (5.133)at C

X X

To solve this equation we require that it be fulfilled for each indexindividually. It can be shown that this requirement follows rigorously fromthe fact that u and v are linearly independent functions. The resultingequations can be further decomposed into an equation which must befulfilled by the time-dependent functions qx and px

a—a, = woxat '^

and by the space dependent functions ux and vx

curl ux(x) = —cvx(x). (5.135)

As can be shown rigorously these relations follow uniquely except for anarbitrary constant which we called wx. It wj.11 turn out soon that (..)x is themode frequency. When we insert (5.131) and (5.132) into (5.128) and makethe corresponding steps we obtain

at' A

= cjxqx (5.136).104 a

andcurl vx( x) =

cox—c

ux(x). (5.137)

From (5.129) and (5.130) we readily deduce

div ux(x) = 0, div vx(x) = 0. (5.138, 139)

Thus a solution of Maxwell's equations is reduced to two kinds ofproblems, namely to the solution of the equations for the spatial fieldmodes (5.135), (5.137), (5.138), (5.139) and to those for the time dependentamplitudes qx,px, namely (5.134) and (5.136). As is shown in elec-trodynamics the equations of the field modes can be solved uniquelyprovided adequate boundary conditions are given. We will not discuss thisproblem here but refer the reader to corresponding textbooks.

In analogy to chapter 5.1 we expect that the quantization procedurerefers to qx and px• To this end we repeat the mean steps of section 5.1.The total energy of the electromagnetic field in the volume V underconsideration is given by

U = f dV(ec,E2 + —1 B 2 ). (5.140)o

We assume thto

fux(x)u

fvx(x)v

Again we musit is shown tha(5.132) into (5

U = )2401

(5.143) can be

U= HA

over Hamilton

Hx =

Each of theseered in sectior(5.134) and (5ones (5.16) anexactly the sairelations

9Lx =

(note, that the1qx-A---

PX=Nri

Inserting these

E(x,t) =

B(x,t) =

For later purp

(5.134)


3 with uxto

(5.133)

ch indexrously frome resulting

'eh must be

(5.134)

(5.135)

xcept for anat (...) A is the

8) and make

(5.136)

(5.137)

(5.138, 139)

o kinds ofspatial fielde dependentwn in elec-ed uniquelyt discuss this

n proceduref section 5.1.

e V under

(5.140)

We assume that the field modes are orthogonal and normalized accordingto

fux(x)ux,(x)dV = (SAN (5.141)

fvx(x)vx,(x)dV = SAN. (5.142)

Again we must refer the reader to the theory of electromagnetism in whichit is shown that these relations can be fulfilled indeed. Inserting (5.131) and(5.132) into (5.140) we readily obtain

U = 1E0 E + q09q. (5.143)

(5.143) can be considered as a sum

U = E (5.144)

over Hamiltonians

HA = 1h4.4pi + q0. (5.145)

Each of these Hamiltonians has exactly the same form as the one consid-ered in section 5.1. Furthermore the p's and q's obey equations namely(5.134) and (5.136) which are completely analogous to the correspondingones (5.16) and (5.17a) in section 5.1. This allows us to perform againexactly the same steps as in that section which leads us to the followingrelations

Co

(note, that the spatial modes are now normalized differently)

1 ,qx = — DP ) Dx ) (5.147)

Nrf

PA = (bz — bx). (5.148)

Inserting these quantities into (5.131) and (5.132) we obtain our final result

E(x,t) = Ei(bx — bnIthwa ux(x), (5.149)

B(x,t) = E (bx+ 14- ) V hwo 0 /2 VA(X). (5.150)

For later purposes we present the explicit decomposition of E into its

Otx =4.)A

(5.146)


positive and negative frequency parts:

E( ' )(x,t) = E ibAl/hwa (2,0 ) u(x) (5.149a)

modes accordin

A(x,t) =

f:

] 4

E(-)(x,t) = - ibz i/nw x / (2,0 ) ux(x). (5.149b)

B can be decomposed correspondingly.Now let us anticipate that b -Z , bx have become operators (see below).

Generalizing section 5.1 to the multimode case we obtain the Hamiltonianin the form

H = bx + (5.151)

where we shall drop in the following the zero-point energy

1E hw x- (5.152)

As we know, quantization is achieved when we subject the quantitiesbx, b -Z to commutation relations. Generalizing our former results (5.39) weare readily led to require

bxb;', — bX1- bx= 1. (5.153)

However, the question is open what commutation relation we shall requirewhen the bx's belong to different indices A, A'.

Since the different modes can be interpreted as describing differentquantum systems which are not dynamically coupled, it is suggestive torequire that the operators bx, bZ, commute for different A's. Thus wegeneralize (5.153) to

— b bx= 8xx, (5.154)

and require in additionbxbx, — bx,bx = 0, bbZ — bZ,L, X}- = 0. (5.155, 156)

Of course, our requirement with respect to the commutation relation issomewhat heuristic but we know from numerous calculations and theircomparison with experiments that these commutation relations are thecorrect ones. In section 5.1 we considered in addition to E and B thevector potential A. In the absence of a scalar potential V, E and A areconnected by the equations

B = curl A, E= — —at

A. (5.157)

To find a suitable form for A we decompose it into still unknown field

Inserting this exindividual terms

— p et. xux4

This relation ca_

WA(X) = —

andaa,,PA = _at

By comparing Ct1

ax = —cox

Putting all terms

A

Since bx and b;,'"the magnetic indthe Schrodingeruse the Heisenbt

In conclusionquantization pro*tion. The time-dt

= ihdi

and its time-inde

H(I) = W4).

The Hamiltonian1, 2, ... we mayabbreviation

(n} (ni,r


modes according to

A(x,t) = E ci,wwx(x). (5.158)

Inserting this expression as well as (5.131) into (5.157) and comparing theindividual terms we are led to the equations

aa- paxux(x) = - (x)

at x

This relation can be decomposed into

wx(x) = —9txux(x) (5.160)

andaax

Px = — • (5.161)at

By comparing (5.161) with the relation (5.134) leads us to the choice1

ax = --qx. (5.162)cox

Putting all terms together we obtain for the vector potential A

wxeoA(x, t) = E (bx+ bZ)1/2

ux(x). (5.163)

Since bx and bZ become operators, so do the electric field strength E(x),the magnetic induction B(x), and the vector potential A(x). When we usethe Schrodinger picture, these operators are time independent; when weuse the Heisenberg picture, they will be time dependent.

In conclusion of this section we can now do the last step of ourquantization procedure, namely we can write down the Schrodinger equa-tion. The time-dependent Schrodinger equation reads

= i h—d t

(5.164)

and its time-independent version

= W(I4. (5.165)

The Hamiltonian is given by (5.151). Identifying the index A with numbers1,2,... we may very simply describe the solutions of (5.165). Using theabbreviation

{n} -= {n1,n2,n3...) (5.166)

(5.149a)

(5.149b)

(see below).Hamiltonian

(5 .151)

(5.152)

le quantitiesilts (5.39) we

(5.153)

shall require

ing differentsuggestive to's. Thus we

(5.154)

(5.155, 156)

Dn relation isms and theirtions are theE and B theE and A are

(5.157)

.nknown field

(5.159)

(5.170)<4)(„}10(,,,) > = 8n 1 mt8n 2 m 28n3m 3 • • • 8nNmiv•


we may write the solution as1 (bt)(b2+ )'(b3+ (b n'

Vni!n2!n3!...nx!

(5.167)

(1)0 is the vacuum state which has the property

bx(1)0 = 0. (5.168

The energy W of the field belonging to the function (5.167) is given by

W = E hwx(nx +). (5.169)

Finally we mention that the 4:1:1 's obey an orthonormality relation namely

We refer the reader to the exercises where it is indicated how to prove that(5.167) solves (5.165).

5.8.1. Quantization of the electromagnetic field by means of itsdecomposition into running plane waves

In section 5.1 and in the present section we have treated the followingproblem. We considered the electromagnetic field enclosed in a region, or,more physically speaking, within a cavity. As a consequence the field hasto obey certain boundary conditions. These conditions are essential to fixthe expansion functions ux(x),vx(x). As is evident from (5.13) we treatedstanding waves. We shall make ample use of this type of approach in laserphysics (compare Volume 2). Of course, there are experimental arrange-ments other than that of a given cavity possible. For instance we can makea measurement of the field using a different experimental setup. Then wehave to expand the field into the corresponding new types of modes.

There are other experiments for which standing waves are not theadequate description, but rather running plane waves. Important exampleswill be provided by nonlinear optics where we shall study the propagationof such waves in "nonlinear" crystals. Another example, which will betreated in section 7.6, is spontaneous or stimulated emission. In this caseno boundaries are present and it is "natural" to decompose the field intorunning plane waves. In classical physics, such a decomposition reads

E(x,t)= E exac x{B, exp(—iwxt)exp(ikxx)

+ Bt exp(i x t )exp( — i kxx ) (5.171)

where the individ

X

ex

BAWA

9Z,

kA

Since the normILdifficulties (whicltrick. We subject

exp(ikxx)

to periodic bournWe may quan

procedure we ha.the final result.become operatorread

where k = !old171 = 1.

A choice often n

A =-- — iEe

As it turns out, t(5.153), (5.154),equation, etc. art

In conclusion,choose depends

ty*bxexp[ikxx] + yb;', exp[—ikxx]}B(x) = x eA

-VshwAldo 2V


where the individual expressions have the following meaning:

index numerating the individual waves

ex vector of polarization of wave X

Bx complex amplitude of wave X

cox circular frequency

91,x a normalization factor which we shall specify below

kx wave vector of wave X.

Since the normalization of waves in infinite space provides some formaldifficulties (which one may overcome, however), we shall use a well-knowntrick. We subject the wave-functions

exp(ikxx) (5.172)

to periodic boundary conditions (compare exercise 3 below).We may quantize the fields E and B in a way quite analogous to the

procedure we have outlined above in this chapter. We therefore quote onlythe final result. Again the amplitudes Bx exp( — ico.xt) and Bt exp(icoxt)become operators bx and bZ, respectively. The expansions for E and Bread

E(x) = E exAI— t7*bxexp[ikxx] + ybZ exp[ — ikxx]) (5.173) hwx

(5.174)

where k = k/lk I. y is a complex constant with

IY = 1.

A choice often made is y = — i. The vector potential A reads

V h

A = —iEex 2.4)xeov ty*bx exp[ikx.x] — yb;', exp[—ikxx]).A

(5.176)

As it turns out, the operators bx, bx+ again obey the commutation relations(5.153), (5.154), (5.155), (5.156). The Hamilton operator, the Schr6dingerequation, etc. are identical with our former results.

In conclusion, let us once again stress that the kind of expansion wechoose depends on the physical problem. In free space (without

its

d the followingin a region, or,

ice the field hasessential to fix

5.13) we treatedpproach in lasermental arrange-ice we can makesetup. Then weof modes.

,es are not the,ortant examplesthe propagation;, which will beion. In this casese the field intonsition reads

(5.171)

(5.175)

. (bZ ) nX ... 00.

(5.167)

(5.168)

) is given by

(5.169)

relation namely

(5.170)

”..v to prove that


boundaries), we can use also spherical waves instead of plane waves. Adecomposition of E,B,A into these waves means that we want to treatexperiments (at least "Gedankenexperiments") in which photons belongingto such waves are measured.


The following exercises are arranged as follows. By (1), (2), and (3) thereader will learn what typical solutions ("modes") of the classical Maxwell'sequations look like explicitly and what their orthogonality properties are.

Exercises (4)–(6) deal with important properties of quantized lightfields:(4) deals with the solutions of the time independent SchrOdinger equation,and by (5) the reader will learn about a basic difference between theexpectation values of the fields described by two different types of wavefunctions.

(1) Standing waves of the electromagnetic field in a rectangular cavity withperfectly conducting walls (see fig. 5.4). Its side lengths are L 1 , L2 , L3 inthe x,y,z-directions, respectively. E and B obey Maxwell's equations andthe boundary conditions

Etangential =

Bno = 0 at the walls.

Decompose E and B into modes :ix and vx and show that ux,vx fulfill theequations (5.135), (5.137), (5.138), (5.139) and the boundary conditions

Fig. 5.4. This figure belongs to exercise 1 on section 5.8. Arrangement of the perfectlyconducting walls.

(*), (* *) prov

u = ex,x

tlx ,y =

U X,z = e X,z

Vx ,x = ex,,

Vx ,y = ex,)

V x , z = X,z

ex and ex are c(ex.x,ex,y,eX,z)

ex • k = 0,

k = (4'

Hint: Insert it?,

that ux and V,

y = 0,L2 , Z =

(2) Prove (5.14Hint: Insert ( *

iL nirsin —L

f n7Tcos T

0

(3) Plane waveWe require the

E(x + L.

E(x,y +

E(x,y,z -

and correspon(

( a ), ( a a) provided

ux, = ex" —

2 cos kx x sin ky y sin kzz,' V

Ux o, = exY V

—2 sin k x x cos ky y sin kz Z ,'

U X,z = ex ,z -2i7 sin k x x sin ky y cos kzz,

vx„ = ex ,x T/-2 sin kx x cos ky y cos kzz,

A,y = cos kx x sin ky y cos kzz,V

vx ,z =-2V

cos k,x cos ky y sin kzz.

ex and ex are constant unit vectors with components (ex ,x , ex ,y ,e z ) andrespectively:

ex•k = 0, ex•k = O. ex-ex= 0

L2 L 3 '

k = (-1-n *-- 7--T m —I)

n,m,I integer. V= L1L2L3.

Hint: Insert ux and vx into eqs. (5.135), (5.137)–(5.139). Convince yourselfthat ux and vx fulfill the boundary conditions (*), ( a *) at x = 0, LI,

y = 0, L2 , Z = 0, L3.

(2) Prove (5.141), (5.142) for the solutions given in ( a * a).Hint: Insert ( a a a) into (5.141), (5.142) and use

nir . MIT Lsin—

L xsm—

L xdx = — 8

2 n'm

1 1. nw• MITCOS — XCOS — XUX = — 0 .

L2 n'm

(3) Plane wave solutions:We require the E and B are periodic within the cube with side lengths L:

E(x + L,y,z) = E(x.y,z)

E(x,y + L,z) = E(x.y,z)

E(x,y,z + L)= E(x.y,z)

and correspondingly for B.

)f plane waves. Awe want to treat

photons belonging

), (2), and (3) the.11assical Maxwell'sity properties are.antized lightfields:rodinger equation,.ence between the-ent types of wave

tngular cavity withs are L I , L2 , L3 in.11's equations and

at ux,vx fulfill theundary conditions

meat of the perfectly


Show that the modes ux and vx fulfill their corresponding equations(5.135), (5.137)-(5.139) provided

ux(x) = eA L-3/2 exp[ikx- x]

vx(x) = exL-3/2exp[ikx-x]

where27rn, 27rn 2 27rn3

k = (L L Lni integer.

Precisely speaking, in this case the index A can be identified with (n 1 , n 2 , n3).

(4) Show that (5.167) solves (5.165) with W given by (5.169).Hint: Use the decomposition (5.151) dropping 1E xhtaxbZ bx. Considerbjtobxp(. ) . Use that 1,4, bx. commute with all other b;I: ,bx, A 0 Ao. Nowconsider bjtobx.(b4) flx000 . Consult section 3.3, 5.3•

(5) Calculate

<4)1E(x,t)Ito>

for (a)

(5.167), and (b)

= exp - E ax1 2 1exp[ ax1))+, 00.

Hint: Decompose E according to (5.149) and write (1). in the form of aproduct

cico =illexpi--21-lax1 2 + ax.bZ cloo

While in the case (a), (* ) vanishes, in case (b) we obtain

(*) = E i(ax - al)Vhwx/2e,3 ux(x),

i.e. E takes a form entirely corresponding to the classical expression.

(6) Show that (5.154)-(5.156) are valid also in the Heisenberg picture.Hint: Multiply (5.154)-(5.156) from the left by U -1 = exp(iHt/h) andfrom the right by U = exp(-iHt/h). Write, for instance,

U-Ibxbjt U = U- lb xUU-Uand use

6),(0 = U-lbxUetc.

5.9. Uncertainp

In quantum memeasured simulalways an uncerelations. A JE

observable with

= [ ((1'

where the aver-,

< 0 > =

For hermitian

AS2 1 . AS22

where

[ 0 1, 02] =A famous exar(position). BecE

Ap • Aq

which is the Heprimarily conceand the phase c

Field and phouThe question arthe number of

's, bx's comrin the expan

(5.150). The tin-tors bZo and bxrelation betweerthe other. (Fro(5.179), and

[b + ,b+b]

we find

A(b + +b

b - b+',

*This section is nbe later made in thi

( * )

• This section is rather formal and can be skipped, because no explicit use of its results willbe later made in this book.

In quantum mechanics, operators Up S -2 2 which do not commute cannot bemeasured simultaneously with absolute exactness (cf. section 3.2). There isalways an uncertainty left, the degree of which is expressed by uncertaintyrelations. A measure for the uncertainty in the measurement of theobservable with the operator is the root mean square deviation:

=[(2 - )2)] 1 /2(5.177)

where the average refers to a single wave function:

02 > = <0101T). (5.178)

Foil o p atQ an show quite generally

A0 1' 6n22 >11<[21,221>1 (5.179)

where

[ 0 1, 12 2] = U1S22A famous example is in quantum mechanics S2 1 p'-(momentum), 22 = q(position). Because [p, q] -= —ih one finds:

ft

§5.9 Uncertainty relations and limits of measurability 189

rig equations

h(ni,n2,n3).

bx. ConsiderX X 0. Now

• (.)

5.9. Uncertainty relations and limits of measurability*

(5.180)---

which is the Heisenberg uncertainty relation (cf. section 1.9). We are hereprimarily concerned with the electromagnetic fields, the photon number,and the phase of light.

he form of a Field and photon numberThe question arises how accurately can one measure the field strength andthe number of photons in a certain mode X 0 simultaneously? Since the

brs bx'S—cominute for—thirnfrs, we need 6111Y-consider * same modeX the expansion of E or B which can be obtained from (5.149) and(5.150). The time-dependent mode amplitude is represented by the opera-

! tors N.,. and bx., so that we have merely to determine the commutationrelation between (b + +b) and i(b — b + ) on the one hand and n = bb on

' the other. (From now on we drop the index X or X0 ). On account ofnession. (5.179), and

[b ÷ ,b 4- b] = —6 + , [b bb] = b (5.181, 182)g picture. —3(iHt/h ) and we find

A(b + +b)An b+ +01 (5.183)

6d(b — b + )An (5.184)

Ap•aq >h.

If the averages on the right hand sides are evaluated for states, for which, <±- b + +b> does not vanish, it is evident, that a small uncertainty of n' necessitates a big one of A(b + ±b) and vice versa. Because the electric (or( magnetic) field strength is proportional to b + and b, a precise measure-

ment of the photon number is connected with an uncertainty of the field% amplitudes (This is the case even if no light quanta at all are present).

If we use states for which the right hand sides of (5.183) and (5.184)vanish, at a first glance it would seem possible to determine (b + ±b) and nexactly in a simultaneous measurement. We want to show that thisconclusion is misleading, because in these cases one factor on the left handside of these equations vanishes:

i1) If we use eigenstates of the number operator b + b, both sides vanish,2) The same is true, if we use certain coherent states.

Thus in these two cases the relations (5.183), (5.184) fail to give usinformation about the uncertainty of n, when a coherent state is measuredand vice versa. Therefore one is forced for these two cases, at least, tocalculate An or A( b + +b) directly:

5.9.1. A coherent state (kr (5.90) is given.

Inserting (1) in (An)2 we obtain, according to the definition (5.177)

(An)2 = <0.1b+bb+bl(1),„>— <xl>„Ib+1)illa>2

N./ = < 1%l b+ b + bbitla > + <0.1b+b10.>— 01),,lb+1)102

= <4>a lb + b143„>= lal 2 = h. 51185)

The_meara—square—dexigioLDLA,_cletennined for a coherent g equalsthe average number of photons, It, in that sthre: -Th—e—felative fluctuationsaFelase, however, with ii: An/ii r-T.

5.9.2. An eigenstate of b + b is given: On (5.86).

We obtain

(A(b÷-i-b))2 = <lb +2 + (bb + bb) + b2lOn>

— <cto„lb++b1(1)„>2. (5.186)

Due to (5.54), (5.53) the second expectation value vanishes while the firstreduces to

(5.186) = 2n + 1.

The same i

(Ai(b

The meanstate with r

Phase andHeuristic oamplitude,and a phas

and

b= V

where cp anrepresentat(5.56) that

exp [ ic

This relatic

(Pnand thus o

There ar(1) The de((2) It is netWe demon(5.189) can

exp[ —

In expressmediately rn-represent:vacuum. T1by /n + 1us to replaously deny,Exact treat,

b4=

(5.187) - *The operat

(5.191)qm — ncp = — iinition (5.177)


The same is found for

(Ai(b — b+))2.

The mean square deviation of (b + b +) and i(b — b + ) determined for astate with a definite photon number, goes like the photon number.

Phase and photon numberHeuristic considerations. Since the classical analogue of b,b + is a complexamplitude, one may try to decompose these operators into a real amplitudeand a phase factor

b + = exp[—icp] (5.188)

and

b = VT: exp[icp] (5.189)

where tp and Vn are operators. We assume for the time being that such arepresentation does exist. Then it follows from the commutation relation(5.56) that

exp[isdn — nexp[4] = exp[iq)]. (5.190)

This relation is satisfied if cp and n satisfy the commutation relation

for states, for whichall uncertainty of n:ause the electric (ora precise measure-

ertainty of the fieldall are present).(5.183) and (5.184)

nine (b + -± b) and nto show that this

'tor on the left hand

)oth sides vanish,

84) fail to give usit state is measured) co at least, to

and thus on account of (5.179) the uncertainty relation An Aso follows.

There are, however, two serious difficulties:I (1) The ,decompositions (5.188), (5.121.do not exist,—

(2) It is nre-C-a-sWff-derive (5.191) from (5.190) and not vice versa.We deinoristrite—in`a qii-alifatiVe manner that the decompositions (5.188),(5.189) cannot exist. It is obviously required that

exp[ —isp]exp[ip] = 1. (5.192)

In expressing exp[—icp] by b + /Nrn- and exp[iq)] by b/VTI, we im-mediately run into difficulties: If we calculate these exponentials in then-representation, Vn vanishes in those matrix-elements which contain thevacuum. This difficulty can easily be overcome, however, by replacingby Vn + 1 . One is thus led to a new decomposition of b,b + which allowsus to replace the uncertainty relation (5.191) by one which can be rigor-ously derived. We now turn to an:

(5.186) Exact treatment. We write according to Glowgower and Susskind

11)+610.>2

(5.185)

ierent state, equalselative fluctuations

shes while the first +b 4 = E + (b + b + 1) 1/2 , b (bb + 1)1/2 E . (5.193)*

• (5.187) The operator introduced in (5.193) is not to be mixed up with the electric field.

(bb + 1)1/2 possesses an inverse, E, and E_ are uniquely defined We meby respect to

E,= b + (b ± b + 1) -1/2 , E_-= ( b +k+ 1) 7 1_2..b_ (5.194)*-

where E, are obviously the substitutes for the former exp[ -T-14 The E'shave the property of creating normalized eigenfunctions in the n-representation

(11„ = E ,4)„ (5.195)

nor E_ ,E, are Hermitian operators however, for which anuncertainly -reration of the form (5.179) can be deduced. We thereforedefine

1S = (E_-E,),

and

(— sin so !) (5.196)

C = 1-(E+ +E_), (—cos cp !) . (5.197)

The operators S and C have a continuous eigenvalue spectrum in theinterval from - 1 to +1. Using (5.179) one derives the following uncer-tainty relations:

An•AS >II<C>1

An•AC >-;-I<S>1. (5.198)

From (5.198) the relation

U 7=- (A n)2 (As)2 + (Ac )2 > I (5.199)<S>2 + <C>2 4

follows. In particular one can show for coherent states that -1 > U>The eqs. (5.198) are of no use,if eigensWes offi are involved, and againone has to evaluate AC, AS by the given wave function

(AC )2 = 1<(I)„ I E_2 +E_E++E,E_+E__1(1)„> -= (5.200)r--

andI(AS)2(5.201)

Note for comparison, that

—1 f 2vsin2 so cl =27 0

For a definite value of n, C and Scan take any value between -1 and +1,so that the "phase" is completely undetermined.

IL

<S2>a =

where

(<C>

Here ri

(AS)

From thes10) the olcorder 1 ata great nuelementar:of single 1the exact I

Iniquely defined

(5.194)

[-Tip]. The E'sons in the n-

(5 .195)

er, for which anI. We therefore

(5.196)

We mention some useful...results for expectation values taken withrespect to the coherent states (5.90)

00<S 2 >„ — —ii] — lexp[ —Ft]fi(1 — 2fl E

p![(v + 1)(1) + 2)]"2

(5.202)

where

(na) 2=

li,1a12 (5.203)= ai2

00

<S> cc = exp[ —HRIma) E (5.204)

p—o + 01/2

<C 2 + S 2 >c,= 1 — exp[ —t7] (5.205)

• (5.197)

pectrum in theallowing uncer-

(5.198)

(5.199)

hat > U>Ived, and again

(5.200)

(5.201)

m —aid +1,

?Tr )2

44t1•

1(<C>0)2 (<S>,02 = exp[ —211]( E

p—o v!(v,+ 01/2

(5.206)

\ Here F: always means the mean photon number.

(AS).(AC). >lexp[ (5.207)

From these considerations it is evident that for photon numbers 1 (say10) the old "heuristic" considerations hold, whereas for photon numbers oforder 1 at least quantitative differences occur. In the maser or laser processa great number of photons is involved, so that one may safely use the moreelementary treatment. On the other hand, when the measurement of phasesof single light quanta is discussed, for instance in counting experiments,the exact treatment must be used.

6. Quantization of electronwave field

6.1. Motivation

In the previous chapter we have learned how to quantize the light field. Init we started from field equations (Maxwell's equations). The fields E andB were expanded into modes, for instance the electric field strength wasrepresented in the form

E(x) = E (b;-, -1- bxaxux(x). (6.1)x

In it the functions u(x) obeyed classical field equations. On the otherhand, the amplitudes b, b + became operators subject to certain commuta-tion relations. We then obtained the Schrodinger equation by starting froman expression for the energy density or, more precisely speaking, for thetotal field energy. It has turned out that it is possible and even necessary todo exactly the same for electrons and other elementary particles. Thereason why we did not come across this problem earlier is mainly historicaland partly pedagogical. From the historical development, we know thatelectrons manifested themselves first as particles and only later as waves.This is due to the fact that the wavelength of electrons is so short thatelectron diffraction, etc. could become accessible in experiments per-formed only rather late. However, in principle, we could equally well startfirst from experiments showing the wave character of electrons and thenproceed from there to particles, much the same as we did with the lightfield. In our present case, we know the equation describing the wavecharacter of electrons. It is nothing but the Schrodinger equation

(_ ±_ h2 a + 0 4, = ih dli,

.(6.2)

2m dt

When we take the analogy between the light field and electron fieldseriously we just have to repeat the steps which led us to the quantization

•

•

I

196 6. Quantization of electron wave field

of the light field for the electron wave field. We will do this in the nextsection 6.2.

6.2. Quantization procedure

We note that a somewhat more rigorous way of doing the quantization isbased on Lagrangians. To avoid unnecessary complications we use a morestraightforward method, however, and refer the interested reader to booksdealing with the quantization of the electron wave field, e.g. my own bookQuantum Field Theory of Solids (North-Holland 1976). Now, let us ratherexploit the analogy with the light field. The individual field modes of theelectron obey the time-independent Schrodinger equation

h2— + V)cpj (x) = Wi tpi(x). (6.3)

We expand a general wave function 11.(x) (which is an analogue to theelectric field strength eq. (6.1) into a superposition of eigenfunctions("modes")

11)(x) = E a cpj x (6.4)

Now, however, we have to note a difference between the Schrodinger wavefield and the electromagnetic field. While the electromagnetic field isdirectly measurable and therefore a real quantity, the electron wave field isintrinsically a complex quantity. Therefore we have to consider the com-plex conjugate of eq. (6.4) as well. Having the analogy with the quantiza-tion of the light field in mind, we expect that the functions (T i(x) remainordinary functions in the quantization procedure, whereas the coefficientsaj become operators. For this reason we denote the complex conjugate of99.1 (x) as usual by ce(x), whereas operators ai have as conjugate al , whichare called the hermitian conjugate of a. Since tIi(x) also becomes anoperator, its hermitian conjugate is denoted by tp + (x). We thus obtain inaddition to eq. (6.4)

0 + (x) = E x (6.5)

For what follows, we will assume as usual that the functions cp i obey theorthogonality relations

ice (x)cpj ,(x) d3x =

To derive the appropriate new type of Schrodinger equation which de-scribes the quantized electron wave field we start from an expression for

the total enerenergy U is ti(6.2)

= f

To express eqand (6.5) into(6.6), we obta

H = a

and eventuall

= E

Comparing etstriking analcoperators des(j. It is not oujust give a sh(operators ajt

above-mentiodefinition

a 110 = 0

for all js. Adescribed by

(1)J o•

However, anelectrons occimay occupyaccount). Thican always ai

aj+ + t.

at least in a fSince this tw(However, we

in the next

intization isuse a moreer to books

own booket us ratherodes of the

(6.3)

ogue to theenfunctions

(6.4)

linger wavetic field is

gave field is3r the com-ie

remaincoefficientsonjugate of

.! at whichJ

)ecomes anis obtain in

(6.5)

pi obey the

(6.6)

which de-mession for

§6.2 Quantization procedure 197

the total energy. An expression which plays a role analogous to the fieldenergy U is the expectation value of the Hamiltonian H occurring in eq.(6.2)

= f4/1- Illp d 3 x. (6.7)

To express eq. (6.7) by means of the amplitudes a7, af we insert eqs. (6.4)and (6.5) into (6.7). Making use of eq. (6.3) and the orthogonality relation(6.6), we obtain

h2= E a it ai , ( x ) (— + V(x)}(pf (x)d3x, (6.8)

Wi.(pf(x)

and eventually

= Wiajt ai. (6.9)

Comparing eq. (6.9) with the expression (5.151) of section 5.8 reveals astriking analogy. It makes us think that al" and ai must now becomeoperators describing the creation and annihilation of electrons in the statej. It is not our purpose to describe all details of this quantization. We willjust give a short motivation and then a description of the properties of theoperators a7, aj . Readers interested in more details are referred to myabove-mentioned book. Again we introduce a vacuum state 0 0 by thedefinition

a .0 = 0j 0 (6.10)

for all js. A state in which a single electron occupies the state j is thendescribed by

aj (1 o .) (6.11)

However, an important difference between the operators of photons and ofelectrons occurs. As we have mentioned in section 3.5, no two electronsmay occupy exactly the same state (when the electron spin is taken intoaccount). This is the well-known Pauli exclusion principle. However, wecan always apply twice the creation operator a7 on the vacuum state

a7 a 0 ,40 (6.12)J

at least in a formal way.Since this two-electron state must not exist we require that (6.12) vanishes.However, we could apply a; twice to any other state as well. Since in all

198 6. Quantization of electron wave field

these cases the resulting state must not exist we require that

at4) (6.13)J J

vanishes for any state I. In quantum mechanics this is expressed inoperator form by the requirement

(a.7 )2 = 0 (6.14)

The relation (6.14) is, of course, different from commutation relationsknown from photons. It is nevertheless possible to build up a consistentformalism when the following commutation relations are postulated

a; ak + ak aj÷ = 8jk,

aft a k+ + a, a7 = 0, (6.15)

ajak + ak aj = 0.

In the following we will not make use of the full potentialities of thisformalism but restrict ourselves to applications in quantum optics. Inparticular, we will mainly be concerned with single-electron states only.We just mention that the formalism allows one to cope with many-bodyproblems in a very elegant fashion. A total state with n electrons occupyingthe quantum states j i • • j„ is described by

(1),..i=atat• • • j„at 4:13(6.16)

111 Ji J2 0 •

It is eigenfunction to the Schrodinger equation

(6.17)

With a total energy given by the sum of the individual energies Wj:

Wtot = WJ, + WJ2 + • • • + W,„•

The functions of eq. (6.16) are orthonormal, i.e.

(4)(i) I 4){k)> =6j1 k 1 j2 k 2 • • • 8j.k„•

In particular, we have for the vacuum state

< 4)01 4)0> = 1.

Exercises on section 6.2:

(1) Show that(1) = a 4)0

solves (6.17) with W Wk.Hint: Insert (*) into eq. (6.17) and use eq. (6.15).

(2) Show I

4:1) =

solves (6.1Hint: Sam

(3) ProveHint: Use

<owl

andovoi

These relarelations f(4) It is neformalism.When wewave equaare dealiniinterpret etation is tcto applypresent ea;

Table I

Observl

Charge d(ekgx

Supplemerkinetic ent

1 i2m J

(5) In con

<4:111ev

for the waHint: Use

= Wtot(1).

(.)

§6.2 Quantization procedure 199

(2) Show that(6.13)

expressed in

= clici a i-t;z1k,

solves (6.17) with W = Wktot, Wk.

* * )

Hint: Same as for exercise (1).

(3) Prove eq. (6.19)Hint: Use the auxiliary relations

<4)(1) 1aZ I 2)> = <a k4)(1)I(I)(2)>

and

< 4)(1) 1 a k4)(2) > = <a-k (I)("10(2)>

These relations are introduced by definition in analogy to correspondingrelations for b ÷ , b. Use, in addition eqs. (6.15) and (6.10).(4) It is necessary and possible to formulate expectation values in the newformalism. Here are a few examples:When we consider the usual SchrOdinger equation (cf. section 3.1) as awave equation, we are led to consider 1tp(x)1 2 as an intensity, or, since weare dealing with matter, as a density of matter, and correspondingly, tointerpret ellP(x)1 2 as charge density. Although we know that this interpre-tation is too naive, it is helpful in the present context. Namely, it allows usto apply the general scheme of quantum mechanics table 1, page 72, to ourpresent case:

Table 1Observable Operator Expectation value

Charge density e(x)4i(x) <101e0+(x)0(x)i4>e10(x)1 2 = eEika; akcil (x)(Pk(x).

(6.19)Supplement this scheme for potential energy f Ip*(x)V(x)%p(x)d3x andkinetic energy

-1.1-1/,*(x)( V)21p(x)d 3x!2m

(5) In continuation of exercise (4), calculate

<Oletp + (x)tp(x)14)>, <4)1 f tp+(x)V(x)tp(x)d3x140>

for the wave functions (*), (* *) of exercises (1) and (2).Hint: Use first eqs. (6.4), (6.5) and then exercise (4).

(6.20)

(6.14)

don relations) a consistent,tulated

(6.15)

alities of thisim optics. Inn states only.1 many-body3ns occupying

(6.16)

(6.17)

;ies

(6.18)

•

7. The interaction between light field and matter

7.1. Introduction: Different levels of description

When we wish to treat the interaction between light field and matter wehave to distinguish between different levels of description. These levels arepartly caused by historical development but also by the kind of problemswe have to treat. These levels of description are:(1) Fully classical treatment. The light field is described by Maxwell'sequations, while the constituents of matter such as nucleons and electronsobey Newton's equations. Typically three kinds of problems occur:(a) The electromagnetic field is given and we study the motion of classicalcharges caused by the field.(b) Classical charges or currents are given and the fields created by themhave to be determined.(c) In a third class of problems, the collective motion of particles and fieldsis studied, the dynamics of one system causing the dynamics of the othersystem and vice versa. Important examples are magneto-hydrodynamicsand plasma physics. Some explicit examples will be provided by theexercises at the end of this section.

The next step is usually called:(2) The semiclassical approach. In it the field is treated classically byMaxwell's equations whereas matter is treated by the Schrodinger equa-tion. Typical problems are:(a) The space- and time-dependent, electromagnetic field is given and westudy the temporal change of wave functions of particles under theinfluence of the field. We encountered a number of typical cases in chapter4.(b) Space- and time-dependent wave functions are given. To utilize them inthe classical Maxwell's equation, one first forms expectation values ofcharge distributions and currents, which then act as source terms inMaxwell's equations. We will treat this case in the exercise below and to agreat extent in Volumes 2 and 3.

•

202 7. The interaction between light field and matter

(c) In a number of cases we have to study phenomena in which the fielddetermines the wavefunctions but the latter in turn determine the field.The most important example of this type of approach in quantum optics isthe laser (compare Volume 2), but many phenomena of non-linear opticsare treated by this approach, too.(3) The semiquantum theoretical approach. The light field is treatedquantum mechanically whereas charges and currents are given classicalquantities. This kind of problem leads to the forced harmonic oscillator ofa field mode which we treated in ch. 5, section 5.7. This example showsthat a coherent driving force can cause a coherent field.(4) Fully quantum mechanical approach. Both the light field and motion ofparticles are quantized. This treatment will be the main object of thefollowing part of our book because here the full realm of quantum opticsbecomes accessible.

We close this introductory section with a remark concerning coherence.From the above said it appears that to obtain coherent fields we need acoherent motion of charges. On the other hand to obtain a coherentmotion of charges, e.g. of electrons in atoms, we need coherent drivingfields. So we are in a way confronted with a vicious circle. We will see inthe second volume that the coupled system field-matter is capable ofproducing such coherent motions in a self-organized way so that thevicious circle can be broken.


The following exercises will elucidate our above statements:(la) Fully classical treatment: solve the equation of motion of a particlewith coordinate q, mass m and charge e

d2m q = eE, (*)

d t2

(a) for a constant field E, (fl) for the field E = Eo cos wt.Hints: integrate (*) twice, adding integration constants. Another examplefor (la) (compare section 1.4). A charged particle is elastically bound tothe nucleus: Solve (in one dimension)

d2q dqm— + + jq = eE0 COS cot. (* *)

d t 2dt

Hint: Write cos wt = Yexp(iwt) + exp(— iwt)] and tryq(t) = q 1 exp(iwt) + q2 exp(— iwt) ("forced oscillation").(b) Charge densities p and current densities j are given as functions ofspace and time. In the case of oscillatory charges or currents, the solutions

t,

of Maxwell':solutions ismodel is illuTake a "on(By(x,t) = b

jz(x,t)

Solve Maxvand B depe(c) Classiczmoving in k-

M—Vdt '

where theavcl cl

s-12 =

1

where n isHint: "Musides

E viE AV

E(x) is asthe 1.h.s. cdivergencecontinuity

apat

Identify tidensity p.

This extheory Oh,at sites xjoscillatingare descriread 0

§7.1 Introduction: Different levels of description 203

of Maxwell's equations are electromagnetic waves. The derivation of thesesolutions is beyond the scope of these exercises. However, the followingmodel is illuminating.Take a "one-dimensional" model, in which Ez(x, t) = a(t) sin kx,By(x,t) = b(t) cos kx,

jz(x,t) = jo sin cot sin kx + aEz(x,t),

driving term damping

Solve Maxwell's equations and determine a(t) and b(t)! Discuss how Eand B depend on co and a!(c) Classical model of plasma oscillations. Consider charged particlesmoving in an electric field:

m—d

v = eE(x,),vi = cT x id t

where the index i distinguishes the different particles. Show that theaveraged charge density p may oscillate with frequency

e n

Me0

where n is the particle density.Hint: "Multiply" ( * ) by (e/AV)E,,a, and perform the average on bothsides

E v,—* Arv, E E(x,) --* NE .Jew, teAv

E(x) is assumed to vary slowly over the small volume AV. Then expressthe 1.h.s. of the resulting equation using the current density j, take thedivergence on both sides of that equation and use eq. (5.4) and thecontinuity equation

ap . .+ thy / = 0.

Identify the frequency in the resulting oscillator equation for the chargedensity p.

This exercise may serve as an example of a semiclassical dispersiontheory (theory of the dielectric constant). Consider a set of atoms locatedat sites xi within a sample. A classical, oscillating electric field E can causeoscillating dipole moments of the atoms. In a classical model, the atomsare described by the model of exercise (1), ( * * ), their dipole momentsread * = eq(t).

-:.11 the fielde the field.1m optics isnear optics

is treatedclassical

.scillator ofvie shows

i motion of;ect of theturn optics

coherence.we

c:n t 111,will see in:apable of.) that the

a particle

* )

r examplebound to

ictions of

(s)


In the semiclassical theory, this problem has been dealt with in section4.5, its solution for weak fields is given by eq. (4.82). We denote the dipolemoment of the atom located at xi by Of As shown in classical elec-trodynamics the dipole moments give rise to a macroscopic polarizationP(x,t) = I j8(x — xj),,i , where 6 is Dirac's 8-function. The dielectricdisplacement D is connected with E and P by

D eoE + P.

We can now formulate our exercise: With aid of exercise (1) (* *) or with

eq. (4.81) calculate the dielectric constant, which is defined byD ee0E.

Hint: Replace P(x,t) by a spatial average which contains many atoms, butstill the same expressions of 8, i.e. write P(x,t) as P(x,t) (N/ V)1.1(N:number of atoms in volume V). Use 0 as resulting from (1) (classicaltheory) or 0 as given by eq. (4.82) (semiclassical theory).Discuss the dependence of e on co. Why is this approach only valid forweak fields?(Hint: consult section 4.5).

7.2. Interaction field—matter: Classical Hamiltonian, Hamiltonian operator,Schrodinger equation

In classical physics, the motion of a particle with mass m and charge e isdescribed by the Lorenz equation

m(dv/ dt) = eE + ev X B. (7.1)

In it v is the velocity of the particle, E the electric field strength and b themagnetic induction. We will denote the coordinate of the particle by

x = (x 1 ,x2 ,x3 ) ==- (x,y,z). (7.2)

It is known from electrodynamics that the electric field strength E as wellas the magnetic induction B can be derived from a scalar potential V(x, t)and a vector potential A(x,t) by the following relations

b = curl A, (7.3)

E = — (aA /at) — grad V. (7.4)

We have seen at various occasions that the Hamiltonian of a classicalsystem is a good starting point for quantization. We therefore ask ourselvesthe question whether the equation of motion (7.1) can be derived from aHamiltonian via Hamiltonian equations in much the same way as we havetreated the motion of a harmonic oscillator in section 3.3. In the present

case, findintfore we writ

1H =

We want tcequations tcread as usuz

—X =d t

n. =d t

Inserting eqwith respect

—x • =dt

Sims eq

d t =

Differentiati

m d2x

d t2

In it, we exiyields

m d2x;

dt2

Some care rof Ai with reand time t.the particle

A = A(

However, si7

account thatleads us to t

d Aid t

41111

§7.2 Interaction field-matter: Classical Hamiltonian 205

case, finding a suitable Hamiltonian is not an entirely simple task. There-fore we write down the resulting Hamiltonian right away

1H = fl ( p — eA)2 -F et 7 . (7.5)

We want to convince ourselves that eq. (7.5) leads via the Hamiltonianequations to the equations of motion (7.1). The Hamiltonian equationsread as usual (compare section 3.3)

8H— x • = — , (7.6)dt ap;

aH

dt P" = ax;•

Inserting eq. (7.5) into eq. (7.6) and performing the differentiation of Hwith respect to pj we obtain

1 ,— X • = p•— eAj ). (7.8)dt m

(7.9)

(7.10)

In it, we express dpi/dt by eq. (7.9) and use again the relation (7.8). Thisyields

m d2x; dx aA d Ai ai/-

Je dt — e ax..

Jd 1 2 = dt e ax• —

Some care must be exercised when we wish to perform the differentiationof Ai with respect to time. The vector potential A depends on space point xand time t. It is understood that we have to take that coordinate x wherethe particle is present at time t, i.e. A must be considered as the function

A = A(x(t),t). (7.12)

However, since with varying time x(t) also varies, we have to take intoaccount that variation when differentiating Ai with respect to time. Thisleads us to the relation

dA. aA . dx. aA .J r + J

dt ax dt at •

th in section,te the dipolelassical elec-polarization

ie dielectric

* * ) or with

y atoms, butN / V)6 (N:1) (classical

ily valid for

(7.1)

h and b the.cle by

(7.2)

th E as welltial V(x,t)

(7.3)

(7.4)

a classical3k ourselvesved from aas we havethe present

•

(7.7)

•Similarly eq. (7.7) yields

an operator,

d 1 aA al%)— e-.

-arPi = — „77( p — eA)-(— ea7f)—

ax.J

Differentiating eq. (7.8) with respect to time yields

charge e isd2xJ d dAJ

—mPJ — e dt .d t 2 dt

(7.11)

(7.13)


Using eq. (7.13) we evaluate eq. (7.11) further. As an example we choosej = 1 and write x 1 = x, x2 = y, and x3 = z. We then obtain

= e[0.4

x dx aAy dy aA z dzax dt ax dt ax dt

aA„ dx aA x dy GA,, dz 1 aAx avax dt ay dt az dt —e at — ax (7.14)

eEx.

Using eq. (7.4) we can identify the last two terms in eq. (7.14) with eEx.The terms in square brackets can be rearranged in the form

dy aAx\ dz aA„ aAz3 = dt ax ay ) di k az — ax

and using eq. (7.3) in the formdy B dzBdt z Y •

Equation (7.16) is nothing but the x component of the vector product ofv x B. Thus eq. (7.14) reduces to

m 1.4--2x

= eEx + e(v X B) x . (7.17)dt2

Making the same steps with the other components] = 2, j = 3 we readilyverify eq. (7.1).

So far we have been dealing with a classical particle moving in classicalfields, whose potentials are V, A. When we wish to treat the electromag-netic field as not being given but as a variable with its own degrees offreedom we have to add to the Hamiltonian eq. (7.5) that of the freeelectromagnetic field. We are thus led to a total Hamiltonian

Htot= H + Hf el d • (7.18)

It is possible to derive from eq. (7.18) new Hamiltonian equations includ-ing those of the electromagnetic field. We will not go into the details of thissomewhat lengthy calculation which eventually leads us back to Maxwell'sequations including currents and charges. We would rather go on to showhow to perform the quantization. When quantizing the motion of a particlein section 3.1 we have seen that we have simply to replace the momentump by the operator (h/i) V. As we know from comparison between theoryand experiment, this quantization procedure holds in the presence of a

d2x111

dt 2

(7.15)

(7.16)

magnetic fiel

1H

=

To evaluatesome arbitrasection 5.1, r

V A

which is callparticle coorsequence of(7.19). We th

H = --

which, due tt

We represenderived prey(7.18) in the

Ht,„ = -

where we ha.

H1,1 = -

and

H1,2 = -2-̀

In it a is still

A(x,t)

Therefore we

111,1

§7.2 Interaction field-matter: Classical Hamiltonian 207

magnetic field also. This leads us to the Hamiltonian

1 hH —

= 2n i V — eA) +V,( 2

V = (7.19)

To evaluate eq. (7.19) further we make use of the fact that there is stillsome arbitrariness in choosing A. We adopt again the choice eq. (5.46) ofsection 5.1, namely

V A div A = 0 (7.20)

which is called the Coulomb gauge. Since V is an operator acting on theparticle coordinate x on which A depends, we must carefully conserve thesequence of operators when evaluating the square of the bracket in eq.(7.19). We then obtain

h2 eh eh e2H= — —A V A —

2miA V + —

2mA2 + V, (7.21)

2m 2mi

which, due to eq. (7.20), can be brought into the form

h2 eh , e 2H = — —A — — A V + — A 2 + V. (7.22)

2m mi 2m

We represent the Hamiltonian of the field energy, Hfieid , in the formderived previously in section 5.8. We are now in a position to write eq.(7.18) in the form

h2Iltot = —2m

A + V(x) + E hwxbx+ bx + Hi,l + 111,2

Hel Hfield

where we have used the abbreviationseh , e h

111, 1 = — — A V = — —m

•p, p = V

ml

and

e 2H/,2 = — 2

2m44. (7.25)

In it a is still to be replaced by [cf. eq. (5.163)]

A(x,t) = E(b + bx)Vh/ (2coxE0 ) ux(x). (7.26)

Therefore we find in particular

= E (b;-, + b)( — -)-1/h/ (24)xeo) ux(x) V .MI

(7.27)

we choose

(7.14)

with eEx.

(7.15)

P6)

aroduct of

(7.17)

we readily

n classicaltectromag-degrees off the free

(7.18)

as includ-ails of thisMaxwell's)n to showa particle

aomentumten theory;ence of a

•

(7.23)

(7.24)


The Hamiltonian eq. (7.23) often serves as a starting point from which theinteraction between a particle and the quantized electromagnetic field canbe treated. Many of the underlying processes become still more trans-parent when we use the formalism of the electron wave field quantizationwhich we developed in chapter 6. At a first glance this may appear a littlebit more complicated but the reader will be fully rewarded for his efforts.We will see that this method allows us to describe the interaction processesbetween field and matter in a very transparent way.

7.3. Interaction light field–electron wave field

We proceed exactly as in section 6.2 starting from the expression (6.7) forthe expectation value H of the Hamiltonian H. Remember that H in eq.(6.7) was the Hamiltonian of the "ordinary" Schrodinger equation we gotto know in section 3.1, formula (3.18). Now we use the more generalHamiltonian eq. (7.22) because it describes the electron's motion under theimpact of a general electromagnetic field. Inserting eq. (7.22) into (6.7), wereadily obtain

fie' + fir, (7.28)

where

fie = + (x){-- rm-h2 + V(x)}1P(x)c1 3.r (7.29)

refers to the "unperturbed" motion of the electron.

= 111,1 ± n1,2

describes the interaction of the electron with fields (in addition to V), i.e.in the present context with light field. In correspondence with eqs.(7.24) and (7.25), Hhi and H1,2 read explicitly

= f + (x){ 7:-' -1 21p} ap( x)d3x, (7.31)

111,2 = fo + (x)t kn A 2 )0(x)d3x.( ,2

(7.32)

In complete analogy to chapter 6, section 6.2 we express 1/.(x) and 0+(x)as a superposition of unperturbed wave functions cpi where we choose

h2

He = — —A + qx). (7.33)2m

We thus have

IP(x) = ai T./ ( x ) , 0 + (x) = (pi ( x ) (7.34, 35)

We nowWe havethat we i,our repr,correspor

I1. 1

where the

gA,jk

We leaveanalogow

atioapi

section 7.

111,1

where

g;t,jk

and

tk=

x0 is theconnectedwe havefreedom athe (field-;

(where wetheory,and their I

fitot

where the

(7.30)

$7.3 Interaction light field-electron wave field 209

from which the_gnetic field cantill more trans-

, Ad quantizationy appear a littleI for his efforts.Iction processes

ression (6.7) for3. that H in eq.equation we gote more generalotion under the2) into (6.7), we

• (7.28)

(7.29)

(7.30)

ition to V), i.e.lence with eqs.

(7.31)

(7.32)

(x) and ti, +(x)we choose

(7.33)

(7.34, 35)

We now insert eqs. (7.34) and (7.35) jointly with eq. (7.26) into eq. (7.31).We have encountered such calculations several times, e.g. in section 6.2, sothat we may readily write down the final result. In order not to overloadour representation we represent only H1,1 . In this new formalism thecorresponding Hamiltonians thus read

fiel =E a •

J J .1'

= h E ai+ ak gx jk(bx+ bZ),j , k,

where the coefficients g are explicitly given by

JCT7(x)(ux(x)17)T,c(x)&x. (7.38)

We leave it as an exercise to the reader to express eq. (7.32) in ananalogous fashion by the operators a + , a, b + , b. In a number of practicalapplications the interaction term can still be simplified by means of thedipole approximation. We merely quote the result and will derive it later insection 7.5.

= h E ai+ ak gj ,jk(bx— b );" ), (7.39)j , k,

where

1 k „jk =

hnUXk X 0 jk

eow)

W— W

,

and

tk = (

▪

x ) ( e x ) qp k(x)d3x. (7.39b)

xo is the coordinate of the center of the atom. tk is the dipole momentconnected with the transition k —>j. Now we have to do the last step. Aswe have seen in section 5, the light field possesses its own degrees offreedom and its own field energy which, in quantum theory, gives rise tothe (field-) Hamiltonian

"field = E hcoxbZ bx (7.40)

(where we have dropped the zero point energy). In analogy to classicaltheory, where the Hamiltonian is a sum of those of the electron, the field,and their mutual interactions, we have in quantum theory

"tot =

•

± 1 ./ "field' (7.41)

where the suffix "tot" stands for "total system" (= electron wave field +

(7.36)

(7.37)

1 gX,jk = M 2hWxeo

(7.39a)

•


Hint: Use c3.4 and its

E o = 8

e = 1Use MKSA(2) Show tl-

( cob +

are given b

(I) = —

where a(1)0

= —

Hint: Use t

(3)v tF

E h6,

are given b

=

with

Wtot =

fl >

Hint: Same

(4) Two-ley,Solve:

(ho)b+

g real.Hint: Try

o(n+ I)

light field). In all practical calculations to follow we will use il l. , instead of1/1 . H 1 , H1,1 and H1 are given by eqs. (7.36), (7.37), (7.40), respectively.This implies that we will neglect that term of the Hamiltonian which stemsfrom A'. One may show that this term is small compared to the otherterms of the Hamiltonian provided we deal with atoms (whose electronclouds are rather localized) and with not too high field strengths, i.e. lightfrom conventional light sources. In nonlinear optics, where we often dealwith high light intensities, the term A 2 can become important.

When inspecting the different Hamiltonians we discover that they con-tain two types of operators, namely the Bose operators bZ , bx and the"Fermi" operators ajt , aj . Their commutation relations are given by eqs.(5.154)—(5.156) and (6.15), respectively. However, what about mutualcommutation relations between as and bs? At least in a "gedankenexperiment", we may decouple the electron wave field and the light fieldfrom each other by letting the electric charge go to zero. Since the twofields are then entirely independent of each other it is suggestive to requirethat the as and bs commute. This requirement is part of a theory which isverified up to a very high order (e.g. Lamb shift, see below) so that weadopt the relations

b + — b + a + = 0,J Xaibx— bxai = 0,

b —b a + = 0,j X j ajb7, — b a1 = 0. (7.42)After all this, the reader may guess from his experience from previoussections what will be the end of this section: We write down theSchredinger equation

dcl)=dt

The reader will soon recognize that solving the Schrodinger equation,(7.43), is quite a different thing from writing it down. Indeed, a good dealof the rest of this book is devoted to extracting some of the physicalcontent from eq. (7.43). Before doing so, we advise the reader to payattention to the following exercises which will greatly help him to get usedto the formalism. After having treated some exercises he will most proba-bly enjoy working with operators. He may even experience a "God-likefeeling", by being able to create and destroy photons and particles at willby merely applying the operators!

Exercises on section 7.3.

(1) Calculate gxuk eq. (7.39a) numerically for the two lowest statesj (n = 1,1 = 0, m = 0) and k (n = 2,1 = 1,m = 0) of the hydrogenatom and ux(x0 ) = 1/ -07.(0, 0, 1), V: volume, = 1 m3.

(7.43)

•

§7.3 Interaction light field-electron wave field 211

Hint: Use energies, wave functions, and dipole moment as given in section3.4 and its exercises, respectively. Put cox = — Wk)/h.

Eo = 8.854-10 -12 farads/m,

e = 1.602-10 -19 Coulomb.Use MKSA units.(2) Show that the solutions of

(hwb + b + Wa + (2)43 = Wto,(13

are given by1

4:13 = (b+)(130, n = 0, 1,2, ... with Wto, = nhco,VT-z!

where a4:10 = &Do= 0(*), i.e. 00 = vacuum state, and1

(13 = (b+ra+100, Wtot = W+ nhca.

(3) Show that the solutions of

E hwxbZ bx E Wai+ a j )(1) = kJ/top

are given by1

(1) — (14 ")"'V n ' • • ni•

with

wto, = E n x hto x + E miwp

nx > 0, integer; In • = 0 or 1. (a. 1, (bn° = 1.

Hint: Same as in exercise (2).

(4) Two-level atom interacting with a single mode:Solve:

(hwb + b -4- Wi a l+ a l + W2 ail" a 2 + hgb + a l+ a 2 + hgbaj a l )(1) = Wtot4)

(* )g real.Hint: Try cl>" = a1,0 and

1 1 (1) (n+I) = c,a,+ (b+) c24- (b)' 0,

+ 1)! \rtii

n > 0

(7.42)Dm previous

down the

(7.43)

er equation,a good dealthe physicalader to pay-1 to get usedmost proba-a "God-like-tides at will

owest statesle hydrogen

Vt—i!Hint: Use the commutation relations for the as and bs and (*).

I, I instead ofrespectively.which stemsto the othertose electronths, i.e. light/e often deal

at they con-, bx and the_Oven by eqs.pout mutual

"gedankenie light fieldince the twove to require:ory Ith is) so we


and determine the unknown coefficients by comparing the two sides of (* )when (* *) is inserted into (*). What are the energies Wt.t? Note: for eachn 0 there are two pairs of solution for c 1 , c2 . Plot the two branches of141,7 ) as a function of 4) for fixed W2 — W1 and fixed g. Discuss how Wt(ont)depends on hw, 1412 , W, and g.

(5) Solve the time-dependent Schrodinger equation belonging to (*) underthe initial condition

4)(0) = aP4)0.

Hint: Try a wave packet of the pair of solutions ( *) for n = 0. De-termine the free constants of the wave packet by the requirement4)(0) = 440 . Discuss the time-dependence of the resulting wave function.After which time is the initial state a2I- 4)0 restored?

7.4. The interaction representation

In this section we get to know a procedure which is not restricted to theinteraction between field and matter but which can be applied to manyother cases, too. In it the Schrodinger equation reads

(H0 + H1)4) = ih, (7.44)

where we have evidently split the total Hamiltonian into two HamiltoniansHo and HI . Of course, we may identify this decomposition with the one weencountered in chapter 7, section 7.3. Note that from now on we shall ingeneral omit the "hat" from the Hamiltonian operators to simplify thewriting. There may be no misunderstanding because we will representthe Hamiltonians explicitly whenever it is necessary. Now let us forget forthe moment being that Ho + H1 are operators. Then one may try to applytricks well known from the theory of ordinary differential equations. So letus make the substitution

tI = U, (7.45)

where U and 4; are in general both time dependent. Differentiating eq.(7.45) with respect to time yields

d d U udifdt = dt dt

Since we have in mind to facilitate the solution of eq. (7.44) by means ofeq. (7.45) we subject U to the equation

dUh o (7.47)

We now remind the reader of a result we derived in section 5.6. There we

(7.46)

saw that anIn the preset

U = exr

We now obsonly for nunthe sequence(7.44), we into

HI U41) =

To obtain atequation we

U-111,1

This equatiothe aavia

01.JHIL

We then fine

12/(t).4)

From our pr(7.44) whichbecause it ccwe must notthat our prolthose given iin these case

Exampleusual Bose c

= hc

U readsU=

To evaluateU IbU

They are id

§7.4 The interaction representation 213

sides of (* )ote: for eachbranches of

ss how W;(.7)

(* ) under

saw that an equation of the shape (7.47) can be solved in a formal manner.In the present case one readily verifies that eq. (7.47) is solved by

U= exp[ - iHot/h]. (7.48)

We now observe that all the steps we have done so far can be done notonly for numbers but equally well for operators, provided we always keepthe sequence of these operators. Inserting eqs. (7.46) and (7.47) into eq.(7.44), we immediately realize that 110 drops out so that eq. (7.44) reducesto

n = O. De-requirementtve function.

H = ihU dci) .

dt

rictee:Inieied

(7.44)

familtoniansthe one we

1 we shall insimplify theill representus forget fortry to applyitions. So let

(7.45)

entiating eq.

(7.46)

by means of

(7.49)

To obtain an equation for 45 which has the form of the usual Schrodingerequation we multiply both sides of eq. (7.49) by U-1.

IH/U.1) = ih—dt . (7.50)

This equation can be given a particularly simple form when we introducethe abbreviation

= 1-11 (t). (7.51)

We then find

17, ( t) = ih ddc, . (7.52)

(7.47)

There we•

From our procedure it follows that we have replaced the original equation(7.44) which contains both Ho and H1 by an equation which is simplerbecause it contains only a single Hamiltonian, namely eq. (7.51). However,we must not forget that H1 still contains Ho through U. Thus it may seemthat our problem is not really simplified. However, in many cases, such asthose given in the following examples, H1 can be explicitly calculated, andin these cases, a considerable simplification results.

Example 1 - (displaced harmonic oscillator). We choose b, as theusual Bose operators and 1/0 , H1 in the form

Ho= hcab + b, Hi= y*b + yb + (7.53, 54)

U reads

U= exp[ - io.th + bt] . (7.55)

To evaluate eq. (7.51) we must know the expressions

U -1 bU, U U. (7.56, 57)

They are identical with the expressions we evaluated by formulas (5.74)


and (5.75) of section 5.3, namely when we put a = wt

U -1bU = b exp[ -kat], (7.58)

U -lb +U= exp[iwt]. (7.59)

Thus the Hamiltonian of the interaction representation can be explicitlyevaluated and yields

1711 = y*bexp[- hot] + yb + exp[ kat], (7.60)

and the Schrodinger equation in the interaction representation reads

di)ih-cr

= y*b + iexp[iwt] + yb exp[ -icat]} 4 ).

Example 2 - Two-level atom interacting with a given classical electric field.The operators are now Fermi-Dirac operators obeying the commutationrelations of chapter 6, section 6.2. The unperturbed Hamiltonian 1/ 0 reads

110 = Wi a l+ a, + W2 a2I- a 2 . (7.62)

The interaction with the external electric field is described by

H1 = h(ga2+ a, + g*a,± a 2), (7.63)

where we assume that g contains the external field. The transformationoperator U reads

U = exp[ - al+al + W24- a 2)t/h]. (7.64)

We leave the evaluation of the interaction Hamiltonian if; as an exerciseto the reader and give only the result

111 = htga2' a, exp[i(W2 - W)t/h]

+ ea l+ a2 exp[ -i(W2 - Wi )t/h]). (7.65)

For hints see exercise at the end of this section.Example 3 .- Interaction of a two-level atom with a single quantized field

mode. We specialize the Hamiltonian eq. (7.41) to the case where theelectron can occupy only two levels, i.e. j = 1,2 and where only a singlefield mode is present. Dropping all unnecessary indices in the Hamilto-nian, the Schrodinger equation reads (see fig. 7.1).

d = {Wi a l+ a l + W2 a2I- a 2 + hwb+b}4)

+ h[ga l+ a 2 (b + b + ) + g*a2' a i (b + b + ))(1). (7.66)

We identify 1/0 with the expression

1/0 = W1aa1 + W2 a2+ a2 + hwb + b, (7.67)

and the rest of the Hamiltonian with the interaction Hamiltonian. U now

ti

(7.61)

Op e rat

Fig. 7.1. This figHamiltonian eq. (or the absorptiorright-hand side slower level to itstransition of theannihilation of thThey argicl v

acquires the fi

U = exp[

Since as andthe form

U d

by the decom

U=

and thus

U

We can thenexamples. Usi

(-5 we find as res

(ga l+ a2b

+ gal+

=dt

•

VV2

Operators


T

2 2- -2 2

'111-Ar. eVI.A.s. rtftrUs. WV.Plw fiw hw liw

1 1 1 1

a'a2 b' a'a 1 b a'a 1 b' a' 2a b1 2 2 1

resonant processes inonresont processes

Fig. 7.1. This figure shows how we may visualize the individual terms occurring in theHamiltonian eq. (7.66). The left-hand side represents resonant processes, namely the emissionor the absorption of a photon. Here energy is exactly or nearly exactly conserved. Theright-hand side shows nonresonant processes where the transition of an electron from itslower level to its upper level is accompanied by the creation of a photon, or where thetransition of the electron from its upper level to its lower level is accompanied by theannihilation of the photon. These two processes clearly violate the energy conservation law.They are called virtual processes.

acquires the form

U = exp[ —i( W1aa 1 + W2 4" a2 + hath + b)t /h]. (7.68)

Since as and bs commute with each other we may evaluate expressions ofthe form

U a2bU (7.69)

by the decomposition

U = UatomUfield

and thus

a2 bU = al+ a 2UatomUflehl b Ufield • (7.71)

We can therefore immediately apply the results of the preceding twoexamples. Using the abbreviations

= w2i ( W2 — W1 )/h, (7.72)

we find as resulting Schrodinger equation in the interaction representation

(ga l+ a 2 b + exp[i(co — c-5)t] + g*.(4 a l b exp[— 44) — Co)t]

+ a2bexp[—i(co +(.7.1)t] + g*cg. a l b + exp[i(w + (.7.0t])($

(7.58)

(7.59)

can be explicitly

(7.60)

tation reads

(7.61)

sical electric field.the commutationiiltonian 1/0 reads

(7.62)

:d

(7.63)

le transformation

(7.64)

/11 as an exercise

D. (7.66)

(7.70)

(7.67)

nilto U now(7.73)

(7.65)

;le quantized fieldcase where the

.ere only a singlein the Hatnilto-


By means of eq. (7.73), we can easily explain the rotating wave ap-proximation which we came across on section 4.6, when dealing withclassical fields. When looking at the left-hand side of (7.73) we find twotypes of exponential functions, namely those that depend on (Z5 + (.4)t and((7.) - 6.9t. When is close to co, the exponential functions containing(Z3 + co)t oscillate much more rapidly than those containing (W. - w)t. Asexplained in section 4.6, we may then neglect the rapidly oscillating termsin a very good approximation compared to the slowly varying ones. In theframe of our present description we can easily describe the physicalmeaning of the corresponding processes. The term containing czj I" a lbdescribes the annihilation of a photon where the atom goes from the state1 to the state 2. This is an energy-conserving transition which is expressedby the fact AW = hto. The inverse process in which an electron goes fromthe upper state 2 to the lower state 1, creating a photon, is described bya 1+ a 2 b + . (We have assumed that the energies are such that W2 — W1 > 0.)It is now evident that the other terms a l'a 2b and ift a lb describeprocesses which are physically "unreasonable" because they won't con-serve energy.

There are two problems within our present discussion. First of all, whilethe energy-conserving processes seem to be obvious we must derive theproper meaning of energy conservation in more detail which we will do inthe subsequent sections. Furthermore the reader must be warned againstbelieving that the non-energy conserving processes are unreasonable ornon-existent. We will see, quite on the contrary, that they give rise toimportant effects such as the Lamb shift which we will discuss in section7.8.

Let us briefly return to eq. (7.73) to which we now apply the rotatingwave approximation. In the case of resonance, the Hamiltonian becomestime independent!

Example 4- A (rather) realistic case: interaction of a two-level atom withmany field modes. This example is quite similar to the preceding one. Ourstarting point is again the Hamiltonian eq. (7.41). We keep only thenecessary indices j = 1,2 for the atomic states and A for the field modes.Furthermore, we know from example (3) that terms of the form a l+ a 2 b andal- a l b + can be dropped within the rotating wave approximation. Adopt-ing these simplifications, the Schrodinger equation reads

where, accordi

g—

We identify H,

H0 = Wia

and the rest ofMan. U reads

U = exp{

Owing to the h..6;1; s but also tdecompose Uand U?,

U•tom

From now on,

= hE

+ g,

where

(T' = ( W2 -

Exercises on s(

These two exe:(1) Show:

a l+ a l a

aJJ+ a. aa7 aJ a

d (I)ih—dt = (Wi a l+ a l + W24- a2 }(1) + E

+ hE(go i+ a 2 bZ + 44' a l bx)(1),

bA4)

(7.74)

Ul and U2 , wh

U1 = exp[

U= exp[ - 414

U= U1112

•

•


rotating wave ap-vhen dealing with(7.73) we find twod on (5 + (a)t andactions containingming ((5 – w)t. Asty oscillating termsarying ones. In thezribe the physicalcontaining ajf alb;oes from the statewhich is expressedelectron goes from)n, is described by.hat W2 — > 0.)

a + describeon't con-

1. First of all, whileie must derive therhich we will do inbe warned againste unreasonable or

they give rise todiscuss in section

apply the rotatingmiltonian becomes

two-level atom withveceding one. OurWe keep only the)r. the field modes.le form a l+ a2 b andoximation. Adopt-s

(7.74)

where, according to eq. (7.38)

gA g , I 2 = — —e (2havo) I/2f (pf(x)ux(x) pp2(x)d 3x. (7.75)

We identify 110 with the expression

1/0 = W2a;" a2 + E hca k bk* b k (7.76)

and the rest of the Hamiltonian of eq. (7.74) with the interaction Hamilto-nian. U reads

U = exp –i(Wi a; a l + W24- a 2 + E hcakbZbk )t/h} (7.77)

Owing to the fact that not only the as and a jt s commute with the bxs ands but also the expression b;', bk with all other b, bk., A A', we may

decompose U into the form of a product of commuting operators Uatorn

and Ux

U= tfa tomUA FA 2 • UN (7.78)

From now on, everything goes through as in example (3) and we find

hE {gxcl i± a 2 bZ exp[i(wx – Z5)t]

+ g4" a l bx exp[ – i( cox – ,))t]), (7.79)

where

t7' = ( W2 – W1)/h.


These two exercises are for the more mathematically minded reader:(1) Show:

a l+ a l and 4- a 2commute, i.e. [a l+ a , a 2 ]= 0 ( * )

aft aj and ai,e- ,j k commute,

a7a, and ak ,j k commute.

U1 and U2 , where

U= exp[ ajt/h] commute. x)

U = expj–i(Wi a i+ a l + W2 a 2+ a 2 )t/h] can be written in the form

U= U,U2

(7.79a)


andU -1 = U2-1111-1.

U1 commutes with ajl. and a 2 , U2 commutes with a l+ and al.Hints: Use the commutation relations (6.15) to prove (*). Expand (* *)into a power series of (a; a), insert them into U1 , U2 and use (*). Toprove (* * *) expand both sides of (* * *) into a power series of (a7 a)and rearrange terms.

(2) Show that

U = aP exp[ i(W2 - WI )t /h],

where U is given in exercise (1) (or 7.64).Hints: Use the results of exercise (1) and write the 1.h.s. of (2) (* ) as

U

Write further for example

U -14- U = U2- I laPU/U2 = U2 1aU2.

To determine (* *), proceed in analogy to section 5.3, equations (5.63) to(5.75).

75. The dipole approximation*

In this book, we have described the interaction between an electron andthe light field in two different manners. In chapter 4 we described the fieldby its electric field strength and introduced the term

E(- ex) (7.80)as interaction Hamiltonian. In the present section, on the other hand, weintroduce the interaction between the light field and electron by means ofthe vector potential a, which led to two terms in the classical Hamiltonian,namely

e , e2 42— A p and

2m . (7.81, 82)

(note the change of sign of e, because we are now dealing with the electronwith charge -e). This second formulation, which in classical physicsdescribes the electromagnetic forces on a moving charge properly, formsthe basis for the quantum theoretical formulation presented in this chapter.This leads us to the question, whether the hypothesis (7.80) was only somesort of a model, or if it can be justified properly.

We want to show that eq. (7.80) is a well-defined approximation. We dothis in the frame of the quantum mechanical treatment. First we remark

This section is rather technical and can be skipped.

that for fiewhich reap.the first tentransformedpresent fort

We start

fitot =

where we irpotential vz

wave functiintegral eq.

(7.31)

twolik atAll tio

tions

p( x)

tp+(x)

Inserting ec

f (We will ass

ph = 0

(though owUnder thest

(7.31)

To proceedidentify

nel + -

As we knoN

- e

( * )

1

§7.5 The dipole approximation 219

1-Expand (* *)d use (*). Tories of (a7 a1)

that for fields produced by thermal sources the quadratic term, (7.82),which reappears in H1,2 , eq. (7.32), can usually be neglected compared tothe first term. We will show that the term H1 , eqs. (7.31) or (7.39), can betransformed into an expression entirely corresponding to (7.80) in ourpresent formulation.

We start from the Hamiltonian [cf. eq. (7.41)]

"tot = "el fifield (7.83)

where we inspect eq. (7.31), more closely. We assume that the vector

2) (*) as

( * ) potential varies in its space-coordinate more slowly than the electronicwave functions under consideration. This allows us to put A in front of theintegral eq. (7.31), where we choose A at the atomic center, xo,

(7.31) = — ti,+(x)PIP(x)d3x;

P= (7.84)

411( * *ttions (5.63) to

All essential points of our argument can be seen when we consider atwo-level atom. According to section 6.2, we therefore use the decomposi-tions

electron andribed the field

(7.80))ther hand, wen by means of1 Hamiltonian,

(7.81, 82)

th the electron (730 = m { 21(+) ( xo)P2iaa t A(-)(xo)Pizat+a2)• (7.88)

,ssical physicsroperly, forms To proceed further we introduce the interaction representation where wen this chapter. identifyvas only some

net fifield HO fi/, I HI • (7.89)

As we know from the detailed examples from the foregoing section,

U= exp { iHot/h} (7.90)

ip(x) = a i cp,(x) + a2cp2(x),

1P + (x) = a i+ Ipt(x) + (11(x). (7.85)

Inserting eq. (7.85) into (7.84) we find expressions of the form

f (x) pc; k(x)d3 x (7.86)

We will assume as usual

pil = 0 (7.87)

(though our following approach also works if eq. (7.87) is not fulfilled).Under these assumptions we find

nation. We dorst we remark

•


transforms A (x0 ) into

U -1A(x0 )U = A(xo ,t). (7.91)

We split A(x, t) into its positive and negative frequency parts

A(xo ,t) = ± A(-) (7.92)

where A(+) and A(-) are superpositions of annihilation and creationoperators

A(+) : exp —

exp{icoxt}. (7.93)

Simultaneously, 4- a l is transformed into

(aj a,),= 4 - a l expticT)t), (7.94)

where

==- 6-'21 = (W2 — W1)/h. (7.95)

We will now show that the matrix elements pm of the momentum operator,p, can be transformed into those of the dipole moment, ex. To this end westart from the relation

[ Ha, x] =

(7.96)rn/

whereh 2

Hei = — --A + V(x). (7.97)2m

The relation (7.96) was derived in exercise 6 of section 3.2. Using eq. (7.96)we form

fpcp id3x = Tmi f soltHei x — xHe1 ),:p id3x. (7.98)

The first term on the r.h.s. of this equations can be given another formbecause Hel is a hermitian operator

Tillei x (P1d3x = f X41) 1( /4194 )d3x- (7.99)

Furthermore using

Hei q = W291 HetTi =

we obtain the r.h.s. of eq. (7.98) in the form

mi( W2— — WI ) f TIXT id3x.h

Introducingrelation, atul

P21 =

We insert eqapproximathresonance al

(7.31) =

Now we caras in the int

E(x,t)

where we usto r4Finc,

for

al

This latter aupon whenamplitudes,/3( t) chantexp(—icoAt)over to thesection 5.6)fulfilled, thcpulses it mimade. A catreal or virttinto accoun

nr, I =where we pi

ex21=

When we in

EN)(x.

•

§7.5 The dipole approximation 221

Introducing the abbreviation (7.95), we are eventually led to the importantrelation, announced above:

P21 = mic3-r21- (7.103)We insert eqs. (7.103) and (7.92) into eq. (7.88) and apply the rotating waveapproximation (cf. section 7.4, example 3) i.e. we keep only terms close toresonance and neglect antiresonant terms.

(7.31) = (7.88)

= — e(icTA (+)(xo, t)x2 ,(4" a l ), + Herm. conj.) (7.104)

Now we can do the final steps. Remember that in classical theory as wellas in the interaction representation

E(x, t) =aA(x, t) (7.105)

atwhere we use for A the decomposition eq. (7.92) with eq. (7.93). Since closeto resonance, W A = i, the relevant contributions to eq. (7.104) are those,for which

— -T- it7.)A ( ' ) ( xo, t ). (7.106)at

This latter assumption has a further implication which can best be judgedupon when we interpret the operators 15„ bZ as classical time-dependentamplitudes, flx(t), 13Z(t). Our assumption eq. (7.106) implies that flx(t),13(t) change much more slowly in time than their exponential factorsexp(— iwA t) and exp(iw At), respectively. This conclusion can be carriedover to the operators bx, b;,I- when we use the Heisenberg picture (cf.section 5.6). In general we may assume that this slowness condition isfulfilled, though, for instance in experiments dealing with very short lightpulses it might become necessary to reconsider the approximations justmade. A careful study of these assumptions is also necessary in the case ofreal or virtual multiphoton processes. By taking eqs. (7.106) and (7.105)into account, we can cast eq. (7.104) into the form

H1 — E(x0, 01,21(4- al), E(-)( xo, t)1,12(a1+a2),,where we put, as usual

ex2i =

When we insert the decomposition

E(-)(x,t) = — tE bZ-Vhwa (2e0 ) u(x) (7.109)

(7.91)

(7.92)

creation

(7.93)

(7.94)

opeis end we

(7.96)

(7.97)

eq. (7.96)

(7.98)

:her form

(7.99)

100, 101)

(7.102)

•

1121'

aA(')(xo, t)

(7.107)

(7.108)

•


and a corresponding one for E(+) [cf. eq. (5.149)] into eq. (7.107) andreturn to the Schrodinger picture, we are led to H1.1 , as given in equation(7.39). This completes our derivation. Our derivation shows that the termsAp and xE are approximately equivalent. In processes such as spontaneousand stimulated emission and absorption the assumptions we made aboveare well fulfilled. On the other hand when dealing with multiphotonprocesses the reader is well advised to check these assumptions—or stillbetter, use right away the formulation by means of the vector potential A.We shall base our following chapters on that latter formulation.

7.6. Spontaneous and stimulated emission and absorption

For the treatment of these problems it is convenient to start with theSchrodinger equation in the interaction representation (compare section7.4). We consider a single atom with two levels, 1 and 2, whose upper level,2, is occupied in the beginning (1 = 0). Because spontaneous and stimu-lated emission as well as absorption are processes in which only one lightquantum is created or annihilated, it suffices to take into account only thatpart of the Hamiltonian, which is linear in the light-field operators bZ, bx.

„Ts-= (7.110)dt

Keeping only resonant terms (cf. the discussion following, eq. (7.73)) wefind eq. (7.79) i.e.

111 = hE (gA ct i+ a2 bZ exp[ i(wx — Fo)t]

+ gta*" a l bit exp[ —i(w A — ), (7.111)

where, according to eq. (7.75)

gx ,i2 = — (2hWAE0)-1/2fyvt(x)ux(x)pcp2(x)dV,

hP = -7 v

To solve eq. (7.110), we use perturbation theory. In a first still exact stepwe integrate both sides over time T from T = 0 till T t:

(I) (t) = $(0) (ii )fo lii/ er)43(r)dT. (7.113)

41(0) is the quantum state of the system at the initial time. We shall specify

(7. 11 2)

-4(0) bet(under th,improvec

43(1)1

This appIn this cl

To tresystem: 1

the electr

(1)(0';

(1)0 : vacuInserti,

expressio

a + a2

a

The firstbxcPc

Using th,

a2a:

and the

where us

integratic

(1)(1;

whereActix

From eqby a suma process

elect

is replaceelect

(7.107) andin equation

at the termsspontaneousmade abovemultiphotonms—or stillpotential A.n.

trt with theare sectionupper level,and stimu-

dy lightnt o thattors b, box.

(7.110)

. (7.73)) we

§7.6 Spontaneous and stimulated emission and absorption 223

.40(0) below. In the spirit of perturbation theory, we approximate (i)-(r)under the integral on the r.h.s. of eq. (7.113) by (1)(0). We then obtain animproved function 4) (1)(t) by the formula

(i) (1) ( t ) = -4)(0) +h

f 'Ili( )dr ci)(0). (7.114) o

This approach is called time-dependent perturbation theory in first order.In this chapter we shall base our analysis on eq. (7.114).

To treat (a) spontaneous emission we assume an initial state for the totalsystem: light field + one electron, in which there is no photon present, andthe electron is in its excited state:

41)(0) = a40 , (7.115)

00 : vacuum state.Inserting eqs. (7.115) and (7.111) into (7.114) we are led to evaluate

expressions of the forma+ 1na b a + (1 03 (7.116)2 12

anda + a b + a + cto (7.117)1 2 2 0'

The first expression vanishes, because b A commutes with aP andb A4:00 = 0.

Using the commutation relation (6.15)a2 a1- + a 2 = 1, (7.118)

and the fact that b's and a's commute, we reduce eq. (7.117) to

a + 1) +00 , (7.119)1 X

(7.111) where use was made of a 2400 = 0.After inserting these results into eq. (7.114), we may perform the

integration, and obtain

4-)(t) = 400 + gx-A--1 (1 — exp{i6,6) At])ab;t4)0 , (7.120)x "`')X

where

(7.112) co = co — (7). (7.121)

From eq. (7.120) we see that the original wave function 400 is replaced

exact step by a sum over terms of the form eq. (7.119). Thus our formalism describesa process in which a state

electron in excited state + no photon(7.113)

is replaced by a state

hall specify electron in ground state + one photon.

•

(7.122)Ii— exp[i6,4,/11 2 .


The absolute square of the coefficient connected with the normalizedwave function

b;;I:10,

i.e.

11 — exp[iAtoxt]1 2 ,

gives the probability of finding a photon of the mode A and the electron inthe ground-state. In general, it makes sense to single out a specific modeconnected with spontaneous emission only if the dimensions of the cavity(with closed walls) are of the same order as the wavelength of the emittedelectromagnetic wave. On the other hand, in infinite space or in a cavitywith open sides, there is a continuum of modes, so that we have to sum up(integrate) over a total range of final states:

2

By differentiation of this expression with respect to the time we obtain forthe transition probability per second

sin AwAt= IgAl22

•I

(7.123)LCOx

We encountered an entirely analogous expression in section 4.2, formula(4.33). Having in our mind to replace the sum by an integral, we use therelation (cf. also the mathematical appendix)

in. s cot= 7/15(co). (7.124)

t—)•clo

The meaning of eq. (7.124) is explained in fig. 7.2. 8(w) is Dirac's6-function which is defined by the properties

8(co) = 0 for (.,.) 0 (7.125)

and+e

8(W) dco = 1, e> 0, arbitrary. (7.126)—e

Using eq. (7.124) in eq. (7.123) we obtain

P = 27E1gx1 26(wx — (TO.

We calculate first the sum over A considering only those photons, whichare emitted into a certain group of modes. In order to be specific, we treat

gA

Acox

(7.127)

Fig. 7.2. This Iwhich we obtafrequency. Thtthe other handThe area belov

plailves

u(x) =

e: vector cconsider phiinto an inteV and in thin chapter .Applying sirlie within th

dN -

and therefoi

E • •A

Inserting eq

P(d2)

2

gxAwx


the normalized

the electron inspecific mode

is of the cavityof the emitted

• or in a cavitylave to sum up

(7.122)

welkin for

(7.123)

n 4.2, formularal, we use the

(7.124)

Fig. 7.2. This figure shows us how we may understand the properties of the Dirac 8-functionwhich we obtain of sin tot/co in the limit t —> cc. sin cot/w is plotted as a function of thefrequency. The maximum of this function tends to infinity when t goes to infinity. Onthe other hand the first zero approaches the ordinate more and more when t goes to infinity.The area below the curve, however, is independent of t.

plane waves, so that1

ux(x) = eexp[ikx].VT/

e: vector of polarization, V = volume of the box, with V— co. Weconsider photons emitted into a space angle d52. To transform the sum Exinto an integral we have to count the number of modes dN in the volumeV and in the wave number (or frequency) interval dk (dv). We have seenin chapter 2, section 2.3, formulas (2.54)—(2.56), how this can be done.Applying similar arguments to plane waves eq. (7.128), whose wave vectorslie within the space angle dS2, we obtain

(7.128)

1(.4) is Dirac's dN =V

k 2 dkdS2 (7.129)(2703

and therefore(7.125)

V f • k 2 dkdSZ. (7.130)E =(2703(7.126)

Inserting eq. (7.130) into eq. (7.127) we find

P(d2)(7.127)

thotons, whiche 2

(7)

f 94(x)exp[ikxoxieP(1)2(x)dV2

dS2 (7.131)= 872he0m 2c

3

ecific, we treat

(7.133)(7)3 P (dE2 ) = 872heoP le1/2112",

(7.138)1

X X o: (12ZorbZ4)0,VT!


where

ckx.=

4e in eq. (7.131) is the Einstein coefficient for spontaneous emission of alight quantum of polarization e into (12.

In the dipole approximation [compare eq. (7.39) or section 7.5], eq.(7.131) simplifies to

where

192 , = ef cp1(x)xcp i (x) d V (7.134)

and we have used

P21 = imL3x21. (7.135)

In order to get the total transition probability for the emission of anyphoton, we integrate eq. (7.133) over dS2 and sum up over the twodirections of polarization:

(A13 P 37The0c3 l e 82112' (7.136)

As it will transpire below, P = 1/r, where T is the lifetime of the upperstate.

(b) Stimulated emission. We assume first that there is a definite numberof light quanta in a definite mode A 0 initially present. The normalizedinitial state is then

1 43(0) = aP (b-Zore1)0. (7.137)

N/71-1.-We insert again 4)(0) into the right-hand side of eq. (7.114) and findimmediately that there are two kinds of final states, depending on whetherthe index A in Ex equals A 0 or not:

(7.132)

In case (a), aIn case ( 18), ation. Under tquantum byprobability, w

P = 271.1

The first tern:split the firstconnected viobtain with tiThe remainin

P„ = 2.7

is then the stithe ot r hatthe 8 tioia morthat it is builcorrespondinnormalized v,

(1)(0) =

(in order to treadily obtai

Ps, = 277

With

M = m)

(where L: le

M = —(2

Using furthtity for slim,

Ps,(c1S21 +

A = AO: a t+ (84) n 1 00.

V71!(7.139)

•


In case (a), a quantum of another mode A has been emitted spontaneously.In case (/3), a light quantum has been added to the mode under considera-tion. Under the action of the light field the atom is forced to emit aquantum by "stimulated emission". In order to obtain the transitionprobability, we first naively repeat the above steps, which yields:

P = 27rIgx0 1 2 8(cT) — wx o )(n + 1) + 247. E igx 1 28(z5 — w x ). (7.140)xxo

The first term stems from (fl), while the second one comes from (a). Wesplit the first term (n + 1) into n and 1 and combine the expressionsconnected with "1" with the second expression in eq. (7.140), so that weobtain with this combination just the spontaneous emission into all modes.The remaining expression

P 21rn I gx. 1 28(cT) — cox .) (7.141)

is then the stimulated emission rate. In it no summation over A appears. Onthe other hand, it is necessary to integrate over a continuum in order thatthe 8-function can be evaluated. Thus the formalism forces us to start witha more realistic initial state, which is formed as a wave packet. We assume,that it is built up out of plane waves within a region Alc„, Aky , Aks with acorresponding frequency spread Ace = cAlc. If there are M modes thenormalized wave function reads

1 1 (11 (0) — EaRb,-,')"(1)0 (7.142)

VAJ V7IT Ak

(in order to be quite clear we use k instead of A). With this initial state wereadily obtain

Ps, = 277 E I gkl 28 ( 7) — wk). (7.143)Ak

With27rm

Ak i = L (7.144)

(where L: length of normalization box) we find

V M= L3kkk = k2AlcdS2. (7.145)

(203 x z (27)3

Using further eqs. (7.130) and (7.141) we obtain for the transition probabil-ity for stimulated emission of photons into a space angle dO:

(7.134)

ssioo

. 135)

any)ver the two

(7.138)

Af = rnxtnyinz,

we2n

m2he0caAcoVdS2

2

flpf(x) exp[ ikx .x epp2(x ) dV dS2.

(7.146)

P„(dS2) =

(7.132)

emission of a

tion 7.5], eq.

(7.133)

(7.136)

of the upper

mite numbernormalized

(7.137)

14) and findg on whether

228 7. The interaction between

We write 1 n(dS2) in the

P„(dS2) = pe(Zci,d0)4ed2,

whereire 2

b i =

light field and matter

form

f 44(x) exp[ik ox]epp l (x) dV2

(7.147)

(7.148)2,e h2e0m2(7)2

is the Einstein coefficient for stimulated emission of photons with polariza-tion e into dS2.

nhco pe(cT,, dS2) =

(7.149)A(.0 dS2V

is the total energy of the n photons, divided by the frequency spread, thespace angle, and the volume, or in other words: p is the energy density perunit frequency interval, unit space angle, and unit volume. A comparisonbetween eqs. (7.131) and (7.148) yields one of Einstein's relations:

= hc713(7.150)

bl2,e (203C3

In the dipole approximation the transition probability P„ becomes:

P„(dS2) = pa(ro,d12)4,c1S2, (7.151)

where p is given by eq. (4.38) and

bl.e = I 01211 2(7.152)h2e0

(c) Comparison between spontaneous and stimulated emission rates. Wealready know of one connection, namely the one between Einstein'scoefficients eq. (7.150). This connection can be given another appearance.We determine the spontaneous emission rate, P, per number of modes (notphotons!) in the volume V, the angle dS2 and the frequency range Aco(Fa + Aw) which we had considered just now. By dividing eq. (7.131) bythis number

—

k 26,kV c152 i-O2AcoV dt2 N„,= 87T 3 87T3C3

we find2

f sof(x ) exp[ ikx ox]ep(p2(x) did = Ps„1

(7.154)

so that the ratio of stimulated emission rate to the spontaneous emissionrate is = n(= total number of photons in this range).

(7.153)

P ire 2r- = =

he0m2c7OccoV

(d) Absolp,atom going fquantum outcan be donedifference isof one in its

'dabs = P

where 14,e=that for stimenergy densi,coefficient a.(energy flux

'jabs = I'

The filaseIf the re ,

dn =d t

Introducing

1(co) =

(Ft= n/V: p

dl( co)dt

In writing d.

dI(w) d x

where the at2

a = —C

Since Pab,

generally fo-

d I—dt =

*DO reat


(d) Absorption. The calculation of the transition probability /jabs for theatom going from its ground state to the excited state by absorption of aquantum out of a wave-packet which propagates within a space angle (10can be done in complete analogy to the stimulated emission. The onlydifference is that we start now with an electron in its ground state insteadof one in its excited state:

(7.155)*'jabs = pe (cT.), dS2)b?,e dE2,

where 14 = b e so that the Einstein coefficient for absorption is equal tothat for stimulated emission. The absorption rate is proportional to theenergy density, p, of the incident light. In order to derive the absorptioncoefficient a, we introduce the energy flux density I(w) = pe(Zo- , d2)c c1S2(energy flux per s, per unit area) into eq. (7.155) so that

2b"abs

The decrease of the photon number, n, per s is equal Pabs for a single atom.If there are N atoms, we find

dndt = Pabs N • (7.157)

IntroducingFihwc

I(co) = (7.158)

(Ft = n/ V: photon density) into eq. (7.157) yields

(11(w) be dt

= 1(co)VAco

(7.159)

In writing dx = c dt we obtain the spatial absorption equation

dl(w)= —I(w)a, (7.160)

dxwhere the absorption coefficient

2bNhcoa =

ci/Aw

Since Pabs and Pst play a completely symmetrical role, we find quitegenerally for a system of noninteracting, partially inverted atoms:

d/ = I

NN2 —

dta. (7.162)

*Do not read b squared, but b one-two, because "2" is an index as is "1".•

(7.156)-

(7.161)

(7.147)

(7.148)

with polariza-

(7.149)

:y spread, thegy density perX comparisontions:

41).150)-comes:

(7.151)

(7.152)

ion rates. We:en Einstein'sT appearance.of modes (noticy range Ao.)eq. (7.131) by

(7.153)

21

= — Pn(7.154)

eous emission


We conclude with an important remark about coherence properties oflight created by spontaneous or stimulated emission. The above transitionrates were calculated for an experimental setup, in which the photonnumber is measured. Such a measurement, however, destroys any phaseinformation (compare section 5.9). An appropriate treatment which retainsphase information will be given in section 8.4.

7.7. Perturbation theory and Feynman graphs

In the foregoing section we have discussed processes in which the absorp-tion or emission of a single photon was involved. Both, in the fundamentaltheory of radiation processes as well as in many practical applications ofnonlinear optics, it will be necessary to study effects in which several oreven many photons play a role. For a qualitative and, in many cases, evena quantitative discussion higher-order perturbation theory is useful andnecessary. To get an overview of the possible processes, the Feynmantechnique of graphs is particularly useful. Therefore we present here someof its basic ideas. Let us start with the spontaneous emission of a photon.As shown at the beginning of section 7.6 this process is described by thewave function eq. (7.120). This result can be interpreted as follows. Theinitial state in which an electron in the upper state 2 and no photon arepresent is transformed under the influence of the electron field interactioninto a new state. In it the electron is now in state 1 while a photon withwave vector k(4- A) has been created. This process can be interpreted bymeans of a graph. Here and in the following we shall read such graphsfrom right to left. This may be somewhat inconvenient for the beginner butit is most useful for later applications of the rules which we will developnow. We represent the incoming electron by a solid line with an arrowpointing to the left. The interaction of the electron with the photon field isindicated by a vertex. The outgoing electron after the interaction isrepresented again by a solid line with an arrow. Furthermore the outgoingphoton is indicated by a wavy line with an arrow (compare fig. 7.3).

2

Fig. 7.3. Simple Feynman diagram in which an incoming electron emits a photon ofwavelength A and is hereby scattered into a new state 1.

These and sifurthermore, toprecise rules. Tcform

(5( 1 ) =

It is then the t;prescription for

Tal

For sake co'abbrevi

ej =

so that ei has thour scheme mtintegrate the prprescription yie

C —--

The further diswe do not wistimulated emistructure given

Table 2

Incc

n in

Out

(n -

Ver

•

§7.7. Perturbation theory and Feynman graphs 231

rice properties ofabove transition

hich the photonstroys any phaseent which retains

hich the absorp-the fundamental

Li applications ofwhich several ornany cases, evenTy is useful and

resell"nmansome

s, t y

ion of a photon.lescribed by theas follows. Theno photon are

field interactione a photon withe interpreted bymd such graphsthe beginner butwe will developwith an arrow

e photon field ise interaction isme the outgoingfig. 7.3).

These and similar graphs enable us to describe all processes, andfurthermore, to calculate the function for the final state by means ofprecise rules. To this end we write the function for the final state in theform

cl)( t ) = E cx(t)at b;', 00 + a o. (7.163)

It is then the task of the theory to calculate the coefficients cx(t). Theprescription for this calculation now reads as follows:

Table 12

Incoming electron wave exp(— ie2 1) —.0

Outgoing electron wave exp(ieit)Outgoing photon exp(iwAt)

Vertex — igx 0

For sake of convenience (to get rid of too many hs!) we have used theabbreviation

8.1=

so that e, has the dimension of a frequency. The functions in the middle ofour scheme must be multiplied with each other and finally we have tointegrate the product from an initial time to = 0 until the final time t. Thisprescription yields the coefficient

(exp[ i(e i — E2 co N )t] — 1)cx = gx (7.165)

8 1 — 8 2 ± (A)X

The further discussion can now proceed again as in section 7.6 before andwe do not wish to repeat that here. Instead we now reinterpret thestimulated emission of a photon. It can be represented by a graph of astructure given in fig. 7.4. The table which must now be applied reads

Table 2

1= 1, 2, ... (7.164)

2

Incoming electron wave exp(—ie21) -.0

n incoming photons exp( — in wx.1)A0(n)

Outgoing electron wave exp(ieit) o-n

(n + 1) outgoing photons exp(i(n + 1)wA01)Xo(n+ I)

emits a photon of Vertex — igA0V n + 1 0


X (n)

/n-1.1 2

Fig. 7.4. A diagram showing a process in which an electron and n photons of wavelength Ainteract whereby an additional photon of the same wavelength is emitted and the electron isscattered into the state I.

the coefficients of the corresponding expansion of the wave functions interms of photons and electron states can now be achieved by multiplyingthe functions on the right-hand side of table 2 with each other andeventually integrating the product over time. We leave it as an exercise tothe reader to establish a corresponding diagram and rules for the absorp-tion of a photon.

In our above treatment we have established rules for calculating thetime-dependent coefficients. In this treatment, time is distinguished withrespect to spatial coordinates. We mention that it is also possible to deriverules for Feymnan graphs treating space and time in a symmetric fashionas it is required by the theory of relativity. Indeed, Feynman's originalgraphs were relativistically symmetric. However, since the most importantresults in our context can be visualized by means of time dependentFeynman graphs we will stick to our simpler description.

So far we have been treating processes which occur in first orderperturbation theory. We now discuss some results of second order per-turbation theory and will conclude this chapter by discussing a generalcase. To start with, let us reconsider formula (7.113) of section 7.6. Therewe realized that we could integrate Schrodinger's equation in a formal waywhich led us to the result

430 ( t ) 411 (0) + —1 f iii (T)(r) d r (7.166)M o

The state function '5 and the interaction Hamiltonian are taken in theinteraction picture. Since the solution of the integral equation (7.166) is asdifficult as that of the Schrodinger equation we have tried, in section 7.6,to solve it by an approximation. To this end we had replaced (1)(r) by theinitial state (I)(0). This resulted in an improved state function given by

(i) (1)(t) = (I)(0) + foli1i (r)i)(0) dr. (7.167)

Now we may repeat this procedure replacing 45(r) in eq. (7.166) by the

improved state fustate function 4:0(`

$(2)(t) = ci)(

By introducing tiinto eq. (7.168) w

45(2)( t =

This formulationinitial wave functto evaluate the in

We no ow,side. Agligrwe

Hamiltonian atoequation (7.169) 1

it I-11 must be aprinitial state and t

a +th2 .."'"0 •

When we apply I

b;', a a 2 a

Thus we are deaapply H1 at timedescribes the ernithe photon alreacthe result (7.171)

bxaj a l b;

By using our wtmediately obtainThe subsequentT2 can be represeclosely that partthe emission and

C2(t)a2 too -r

It will bedir tas


improved state function 40(1) . Thus we obtain the following more improvedstate function (I)(2)

1 t_.,$(2)(,)=.1)(0)+Th_foli,(TA(00->d,. (7.168)

By introducing the explicit representation of ci) (1) as given by eq. (7.167)into eq. (7.168) we obtain an explicit representation of (I)(2)

3 (2)(0 = 43(0) + f TI)(1)(0)

+ )21- dr2f T2 dr, iikr2 )iii (1- 1 4(0). (7.169)ih 0 0

This formulation means that on the right-hand side we have to use theinitial wave function and to apply certain operations to it. Finally, we haveto evaluate the integrals.

We now show, by means of an example, how to evaluate the right-handside. Again we use as explicit representation for H1 the interactionHamiltonian atom field. We have encountered the first two terms ofequation (7.169) before so that we focus our attention on the last term. Init H1 must be applied twice in succession to the initial state. We choose asinitial state and treat only virtual processes where e2 — e l wx.

a + o- (7.170)2

When we apply ill first on (7.170), we obtain terms of the form

bZail- a 2 4- 00 = b;:- 440 . (7.171)

Thus we are dealing with the process depicted in fig. 7.3. When we nowapply H1 at time T2 for a second time we obtain two processes. One processdescribes the emission of a second photon, the other one the absorption ofthe photon already present. We treat the reabsorption of a photon. Takingthe result (7.171) into account we must now calculate

6 7,4' a l b 4 cz 1+ 4)0 . (7.172)

By using our well-known rules for field and atomic operators we im-mediately obtain the final state a (Do, i.e. again exactly the initial state.The subsequent action of the two interaction Hamiltonians at times T i andT2 can be represented by the diagram of fig. 7.5. We now consider moreclosely that part of eq. (7.169) which yields the original state on account ofthe emission and absorption processes just discussed

c2 (t)a2l" (Do ± a (7.173)

It will be our task to calculate the coefficient c 2(1) explicitly. To this end

;avelength Xe electron is

ictions inultiplying,ther andxercise toe absorp-

alinehed withto derivec fashion

originalmportantependent

rst orderrder per-

general.6. Therermal way

(7.166)

n in the66) is as

Aion 7.6,-) by then by

(7.167)

) by.

2 1 2

Fig. 7.5. Feynman diagram giving a contribution to the self-energy of the electron. Anincoming electron in state 2 emits a photon which is eventually reabsorbed restoring theinitial state of the electron.

we have only to remember that the vertices stem each time from anapplication of the interaction operator 111 . The factor in front of theoperator of formula (7.171) reads

— igx f r2expti(e, + cox — e 2 )7. 1 ) dr 1 . (7.174)

In it we have made use of the fact that we have to integrate over T1 from 0till T2 . Furthermore we will have to remember that we still have to performthe sum over X. At vertex 2 that factor comes into play which stands in theinteraction Hamiltonian describing the absorption process. By using eq.(7.172) we obtain from the second vertex the expression

— f dr2 expti(e2 — wx — e 1 )T2 ). (7.175)

In it we have to use the same photon wave vector k ( 4=- index X) as informula (7.174). In putting eqs. (7.174) and (7.175) together, the followingexplicit representation of the coefficient c2(t) of formula (7.173) results

T2

c2(t) = 0

dT2 exp[ ia2r2 ] E — f exp[ — i(e i + Wx)(T2 — TinX 0

X (— igx )exp[ — ia2T 1 ] dr' . (7.176)

The result of our little calculation can again be described by rules whichare connected with the graph of fig. 7.5. Eventually we have to integrateover the times T 1 , T2 , where 0 <r1 < T2 < t, and to sum over X. The readerwill recognize that the rules described by table 3 are a systematic continua-tion of the rules given above in connection with perturbation theory of firstorder.

Table 3

Vertex 1Incoming free electron exp[ —ie2T1Emission of a photon —Propagation of electron

and photon from T, to 1.2 exp[ — i( x el X T2 -Vertex 2Absorption of a photon — intOutgoing free electron exp[ie2r2]

Now it is rdeal with stiiexample in ti

After the i

c2 (t) =

The numberfunction itsetimes onlywe may neglform

c2 (t) =

where we ha'

E

Using this re

(t) =

This result I,namely emischange of thtmore and mc

A result rrhigher-ordersection and athat 1 — (AEI

function. Tinform

411 (0 =

We will nowwhat the timback from thusing to the7.4, OW antman 1/0 [cf. (

=

U=


e electron. An•d restoring the

ne from anront of the

(7.174)

er T1 from 0to perform

>tands in thely useeq.

(7.175)

ex A) as inle following3) results

r2 T1) ]

(7.176)

rules whichto integrate• The readeric continua-eory of first

Now it is not very difficult to imagine what such rules look like when wedeal with still more complicated processes. We will get to know a furtherexample in the next paragraph.

After the integration over T 1 we obtain

i t 1 — expti(e 2 — E l — )T2 } C2( t ) = E I gx dr. (7.177)

o

The number 1 under the integral gives rise to a factor t. The exponentialfunction itself leads to a term oscillatory in time. For sufficiently largetimes only the term which increases proportional to t is important so thatwe may neglect the oscillatory term. Thus we obtain an expression of theform

c2(t) = (7.178)


IgAl2 e = E (7.179)

— e l — th)A

Using this result we find the wanted state function (7.173) in the form

$(t) = (1 — iilet)(11- I. (7.180)

This result looks rather strange. It tells us that the process under study,namely emission and reabsorption of a photon, does not lead to anychange of the original state but that its coefficient seems to grow with timemore and more.

A result making sense can be obtained only when we consider also thehigher-order terms of perturbation theory. We will do this in the nextsection and anticipate here its result. According to that analysis it turns outthat 1 — iAet are just the first two terms of an expansion of the exponentialfunction. Therefore we are allowed to write the state function (7.180) in theform

45(0 = exp{ — iAet}a -24" I. (7.181)

We will now assume that the form (7.181) is correct and we want to studywhat the time-dependent factor in eq. (7.181) means. To this end we goback from the interaction picture [state function OW] which we have beenusing to the Schrodinger picture [state function OW]. According to section7.4, OM and OM are connected by means of the unperturbed Hamilto-nian 1/0 [cf. eqs. (7.45) and (7.48)]. Using the relations

= U(t)(7.182)U= expt — iHot/h


and (7.181), and making a small rearrangement of terms we obtain

(I)(t) = exp{ - iAet} Uctif L1-1 U 41)0 . (7.182a)

However, we know expressions I and II from our earlier chapters, namely

I = c4- exp( - ie2 t), II- 'I'.. (7.183)

We thus obtain as a final result

= exp{ - i(e2 Ae)t)00 . (7.184)

Comparing this result with the stationary solution of the Schrodingerequation containing the total Hamiltonian H = Ho+ H1

= exp - i Wt/h )0(0) (7.185)

we readily obtain

W= he2 + he. (7.186)

This relation yields the wanted interpretation of AE. hAE is an energy shiftof the electron in state 2. In the frame of second order perturbation theorythis shift is explicitly given by eq. (7.179). This energy shift stems from theemission and subsequent absorption of a photon. Since energy is notconserved during the intermediate state in which a photon is present thecorresponding processes are called virtual emission of a photon. We arethus led to a result which is also of fundamental importance in the theoryof elementary particles. It tells us that the energy of an electron is shiftedby the virtual emission and reabsorption of quanta, in our case of photons.

So far we have described these processes by dealing explicitly with thebound electron of an atom which can occupy the states 1 or 2. Exactly thesame discussion can be performed when we deal with free electrons withthe momentum p. The only generalization of that case as compared to thecase here consists of the rule that in the intermediate state we have also tosum up over all p vectors of the electron which obey, according tomomentum conservation, the rule

P hk, (7.187)

where hk now denotes the photon momentum. In this more general case,which we will come to in section 7.8, the energy shift depends on". This isusually called self-energy. Now let us briefly determine what the p-dependence of e means. For a free electron its energy depends on p in theform

w0=woo _1„

" k "() 2m ,

where p = Amu.,(compare also St

form

AW =To reveal the sig

8C= 2 m

where 8 is assum(7.188) we are le,

W = Wo + -

where, by comp'.1 1

m* m —

Since 8 4110ssu

m* = m(1 -f

This result putsdence. We reali:electron is chang

That change orole in the modechange of self-ensponding measurwill discuss these

By means of F(study further proin second-order rFigure 7.6 represtSimilarly all highe

Fig. 7.6. An exampleand A', respectively

111)

(7.188)

§7.7 Perturbation theory and Feynman graphs 237

)btain

(7.182a)

)ters, namely(7.183)

(7.184)Schrodinger

(7.185)

(7.186)

)aticlithif t

eoryen

Ins from theiergy is not, present theton. We aren the theoryon is shiftedof photons.

itly with the• Exactly theectrons withpared to thehave also totccording to

(7.187)

general case,mip2 . This isvhat the p-s on p in the

(7.188)

where p = pinitial and m is the electron mass. It can be shown further(compare also section 7.8) that the energy shift can be expressed in theform

= hAe = — Cp 2 (7.189)To reveal the significance of C we write it in the form

8C=— (7.190)

where 8 is assumed to be a small quantity. By now adding eq. (7.189) to eq.(7.188) we are led to formula

P2W = Wco + 2m* (7.191)

where, by comparison, we find1 1 8 (7.192)

m* m m

Since 8 was assumed small, eq. (7.192) is equivalent tom* = m(1 + 8). (7.193)

This result puts the significance of the p-dependent energy shift in evi-dence. We realize that by virtual emission processes the mass of theelectron is changed.

That change of mass due to virtual processes has played an importantrole in the modern development of quantum electrodynamics. Since thechange of self-energy and mass give rise to observable effects, the corre-sponding measurements have been a test of quantum electrodynamics. Wewill discuss these points in the following section 7.8.

By means of Feynman graphs and the rules connected with them we canstudy further processes which lead to a genuine change of the initial statein second-order perturbation theory. Such processes involve two photons.Figure 7.6 represents the subsequent spontaneous emission of two photons.Similarly all higher-order processes can be described by Feyninan graphs.

3

Fig. 7.6. An example in which an electron consecutively emits two photons with wavelengthsA and A', respectively.

2

Fig. 7.7. A Feyprocesses (comp

to determineand subsequ(each time wein the interrintegrations.

E

wher ha"to d inc

iefunctions ma

I ncoi

Outg

Outg

By multiplyiroccurs at ver

exptiAit

where we ha%

= el -

At vertex 1 v(f T2

expt,

Fig. 7.8. A mosthe electron. Shc

•


In concluding this chapter we now deal with perturbation theory inarbitrary order. Again we start from the exact expression (7.166) which wepresented at the beginning of this chapter. As has been seen above we maytry to solve this equation by an iteration procedure. In such a procedurewe determine the nth approximation on the left-hand side of eq. (7.166) byinserting the (n - 1)th approximation on the right-hand side of eq. (7.166)

4)( n )(t) = (0) +Tk' foI t 171 ( T ) d n I ) Tn ) (7.194)

To have a concise formula we put in the 0th approximation

= 4)(0) = 43(0) . (7.195)

It will be our goal to express the nth approximation directly by 40 (0). Tothis end we express consecutively (13(n) by 41) (' 1) , then (I)(n-I) by 4)(n-2),

etc. To recognize the general structure of the resulting expression first lookat formula (7.194) for n = 1 and for n = 2. To obtain 41) (3) let us insert (1)(2)into eq. (7.194). One easily convinces oneself that the resulting statefunction reads

(i)(3)(t ) = 4)(2) ) 3 fo tni ( T3 ) d T3 f 111 ( T2 ) d T2 fT2

Hi ( ) dvD(0).

(7.196)

As can be seen from our examples, each step further yields an additionalterm. Such an additional term is an n-fold integral over the product ofinteraction Hamiltonians taken at subsequent times. Thus we are directlyled to the following explicit expression for the state function in the nthapproximation

3(n)(t) = CO)

•E (ihr9 f t H(T„) dr, f H1(T„_1)(dr„_1...v1 0 0

X f

T2

Hi( TO &re440

Thus the problem of finding (1) (n) is reduced to an evaluation of eq. (7.197).One should not forget, however, that eq. (7.197) contains an enormousnumber of different processes involving the absorption and emission ofphotons. In the present context we confine ourselves to the followingprocess. We consider a sequence of virtual emission and absorptionprocesses as depicted in fig. 7.7. Exactly speaking the diagram of fig. 7.7corresponds to a term with p = 8 in eq. (7.197). Our former results allow us

(7.197)


ation theory in7.166) which wen above we mayich a procedure3f eq. (7.166) byle of eq. (7.166)

(7.194)

(7.195)

tly by 10 (0). To(n-1) by (I)(n-2),ession first looket us insert 4)(2)resteg state

-i1(1-1)dr1(1)(0).

(7.196)

s an additionalthe product ofwe are directlyLion in the nth

(7.197)

of eq. (7.197)., an enormousid emission ofthe following

ad absorptionram of fig. 7.7-esults allow us

•

2 1 2 1 2 1 2 1 2Fig. 7.7. A Feynman diagram of a sequence of virtual photon emission and reabsorptionprocesses (compare text).

to determine the corresponding coefficients in a simple way. Each emissionand subsequent absorption process introduces a factor (- igi). Ateach time we have to sum up over the wave vectors of the virtual photonsin the intermediate states. Finally we have to perform the successiveintegrations. Thus we have to evaluate the following expression

T2E(- 1)4 1gA4 I 2 ...1412fT 8

dT8f dTi exp . ,x • • • x4 0 0 0

(7.198)

where we have indicated the time-dependent factors by dots. We still haveto determine the time-dependent functions under the integrals. Thesefunctions may be obtained by means of the rules:

Incoming electron exp[ - i€211

Outgoing electron

Outgoing photon exp[iwxr].

By multiplying these functions with each other, we obtain the factor whichoccurs at vertex 1

exp{iA l t}, (7.199)


= e l ez- (7.200)

At vertex 1 we identify T with T1 and integrate from 0 till T2

IT'expfiA,T i } dr, = (iA 1 ) -1 (exp(iA 1 r2 ) - 1). (7.201)

. . . .2 1 2 1 2

Fig. 7.8. A more complicated Feynman diagram giving a contribution to the self-energy ofthe electron. Shown is the virtual emission and reabsorption of two photons.


order. Thesediagrams of tlbetween diagr

2 1 2 1 2Fig. 7.9. Similar Feynman diagram as in fig. 7.8, but the sequence of reabsorption processesof the two photons is exchanged.

At vertex point 2 the opposite process occurs introducing a factor

exp( (7.202)

We now have to multiply eq. (7.201) by (7.202) and to integrate over r2.This yields

A(L.11) - 'f ( A ) IN UAexp t —ila 2T2 ikexptiL1 1 7-2.1 — 1 ) T2 = (iA l) I T3 . (7.203)

As usual we neglect oscillatory terms here and in the following becausethey are inessential for large times. At the next vertex we multiply the thusobtained results by a factor of the type (7.199) with T T3 and integrate

r T4(exp{ iA 2T4 ) — 1) If eXptiA T (7.204)

0 2 3}T 3dT

3 i nA 2

By multiplying the result of eq. (7.204) by the corresponding factor ofvertex 4 and again integrating we obtain

j. 75 T4(exp(iA2T4) — 1) T52 1 1exp( —/A 2T4 ) d ,r4= — —

2 iA/ iA2 •(7.205)

iA/iA2

It is now obvious how to continue this procedure. After the last integrationwe obtain

1 1.4f text) —iA T T 3 dT { 4 8) 8 8 nAliA2/A3 (7.206)

4!iA 1 iA 2 iA 3 iA 4 •

The coefficient of the wave function corresponding to diagram fig. 7.7 withfour virtual emission processes can now be formulated by use of eqs.(7.198) and (7.206)

I gx i 2 44it1

— (A)A = —4!(—iAet)4. (7.207)

A 62 81 —

The whole wave function i1)(1) is composed of contributions of different

In each of thethe total proctIn the limit n

i'2( 1 ) = a

Thus we haveshift as discus:

Let us cowdiagrams (graialso occur. Tcthe suilieonbest rfodiagrams of fdiagrams giveprocesses leadobtained by sugraphs.

Exercises on st

(1) Consider ainto modes A).

Draw all Feyniing initial state

= aj 4)C

(2) Determine tat 4)0

in second ordei

•

ss

frPrttl,


order. These different contributions can be represented by a sum ofdiagrams of the type of fig. 7.7. We thus have a one-to-one correspondencebetween diagrams and wave functions:

a2÷ 4011 + (—iAet) + —2

1!

+ • + —n

1!(—iAet)" + • • }.

(7 208)

In each of these diagrams the total wave function remains unaltered afterthe total process except for the time-dependent factor given in eq. (7.208).In the limit n—> oo the summation gives us

(1)2 ( t ) = a24-4:110 expt — itAe) . (7.209)

Thus we have proven that the processes considered give rise to an energyshift as discussed above.

Let us conclude this chapter with an important remark. Besides thediagrams (graphs) considered here quite different kinds of diagrams mayalso occur. To this end we need only to consider processes described bythe succession of interaction operators in eq. (7.197). This discussion maybest be performed again by a graphical representation. For instancediagrams of figs. 7.8 and 7.9 can also occur. As can be shown, suchdiagrams give rise to coefficients proportional to time t so that theseprocesses lead again to an energy shift. The total energy shift can beobtained by summing up all such contributions which are called connectedgraphs.


(1) Consider a three-level atom interacting with the light field (decomposedinto modes A). Let its interaction Hamiltonian be given by

g12 a; a2 bA, exp[i(e t — e2 — wx.)t]

+ g234 a3 b), 2 exp[i(e2 — E3 — 6)t] + h.c.

Draw all Feynman graphs of second order taking into account the follow-ing initial states

40 = aft 00 , j = 1,2,3; =a7 b bcI0, j = 1,2,3.

(2) Determine the coefficients ofat Coo

in second order perturbation theory belonging to exercise (1) and to the

reabsorption processes

g a factor

(7.202)

integrate over T2.

1 ) T3 . (7.203)

ollo becausemultiply the thus

T3 and integrate

(7.204)

)ondimg factor of

. (7.205)jA2

ie last integration

1

(7.206)

;ram fig. 7.7 withby use of eqs.

(7.207)

ions of different

of the p used foIn this we havethat the atom i:more the dipoltthe field amplitfunctions. To scdenote the intel

1 = <n

Furthermore w,fully. To this erk and an indeFurthermore Wt

(41. tok =

While we inith

eventu o o

1

J=

[Compare eq. (may cast (7.211

AW,,/h = (

In it we use tht

1 C —

(2703

For further evaspace angle E2 a

f d3k = f

We then first rover the two oformal we imm.

1 dS2

initial state 4- bZ, b1100 . Discuss these coefficients for the case

8 1 — 62 — (4 X, = e2 83 — (4 ). 2 = 0-

(3) Is the interaction Hamiltonian eq. (7.111) in the rotating wave ap-proximation sufficient to give rise to the diagrams of figs. 7.8 and 7.9 ormust antiresonant terms be included, i.e. the full Hamiltonian H1,1 eq.(7.31) (or its equivalent in the interaction representation) be used?

7.8. Lamb shift

In the preceding section we came across the term self-energy. We now turnto the question of the explicit computation of these self-energies. To comecloser to reality in this section we abandon the model of a two-level atomand take into account all levels of an atom. In particular, we will beinterested in the hydrogen atom. We distinguish the wave functions of thedifferent levels by an index n. Note that in the case of the hydrogen atom ndenotes not only the principle quantum number but the total set of allthree quantum numbers of the electron. We denote the initial state, whichwas formerly denoted by 2, by n, and the intermediate states, formerlydenoted by 1, by n'. The interaction Hamiltonian in the Schrodingerpicture is now taken in the form

H1,1 = h E an+ a, + an), (7.210)n,n'

which generalizes the two-level Hamiltonian of sections 7.6 and 7.7.Taking into account that we have to sum up over the intermediate states ofboth the virtually emitted photon and the electron, the energy shift is givenby

AK/h = (7.211)n' EflWX —

where, as before

he = Wn.In the following we need the explicit form of the optical matrix elements gwhich we therefore write down

= — 7n-e (2hcoxe0V) -1/2f99:(x)(ex fi)cp(x)d 3x. (7.212)*

We have used the symbol p' (h/i) V for the momentum operator instead

*Note that e0 eq. (2.212) is the dielectric constant and is not to be confused with en in eq.(7.211).

•

•

§7.8 Lamb shift 243

case

_ting wave ap-7.8 and 7.9 oranian H1,1 eq.used?

r. We now turn:rgies. To cometwo-level atomar, we will beunctions of theidrogen atom ntota of allial ;tates, formerlytie Schrodinger

(7.210)

; 7.6 and 7.7.ediate states ofg shift is given

(7.211)

itrix elements g

(7.212)*

Terator instead

fused with en in eq.

of the p used formerly to avoid confusion with the classical momentum p.In this we have assumed that the field is expanded into plane waves andthat the atom is situated at the origin of the coordinate system. Further-more the dipole approximation has been made in which we assume thatthe field amplitude is constant over the extension of the electronic wavefunctions. To save space we will use Dirac's notation which means that wedenote the integral occurring in eq. (7.212) by

= <niex (7.213)

Furthermore we have to distinguish the different field modes more care-fully. To this end we replace the general mode index A by the wave vectork and an index j indicating one of the two directions of polarization.Furthermore we use the relation between frequency and wave number

tax tok = ck. (7.214)

While we initially start with waves normalized in a volume V we willeventually go over to an integration which is done by the well-known rule

1 N, f d3k (7.215)V .4 •

1 = 1,2 (2.7)-

[Compare eq. (7.130)]. Using eqs. (7.212), (7.213), (7.214), and (7.215) wemay cast (7.211) into the form

Kniej flin'>12AW„/h = C d3k— 2 (7.216)

k n',J, en — En'

f d3k = f k 2 dk d12. (7.218)

We then first perform the integration over the space angle and sum upover the two directions of polarizations. Since the evaluation is purelyformal we immediately write down the result

f d2 E K n i e; fil re >1 2 = 471K n i 1;1'012. (7.219)

For further evaluation we split the integral over k-space into one over thespace angle 12 and one over the magnitude of k, in k,

In it we use the abbreviation

1 C —

(27)3 2m2he0(7.217)

This leads us to the following result for the self-energy

=1 2e2 cc 1<n' Pln'>12 (7.220)Affin/h

(27)2 3m2he0c3 f0 wdw E

n' e,, en' —

A detailed discussion of the sum over n' reveals that this sum certainlydoes not vanish more strongly than 1/w. We thus immediately recognizethat the integral over 4.) in eq. (7.220) diverges which means that the energyshift is infinitely great. This seemingly absurd result presented a greatdifficulty to theoretical physics. It was overcome by ideas of Bethe,Schwinger and Weisskopf which we will now explain.

When we do similar calculations for free electrons we again find aninfinite result, which can be seen as follows. We repeat the whole calcu-lation above but instead of eigenfunctions T„ of the hydrogen atom we usethe wave functions of free electrons

q9n(x) sop = Dtexp[ ipx/h]. (7.221)

Note that in this formula p is a usual vector whereas j3 occurring forinstance in eqs. (7.212) and (7.220) is the momentum operator h/i V.Instead of matrix elements, which were between the eigenstates of thehydrogen atom, we now have to evaluate matrix elements between planewaves. We immediately obtain

< P'i P> = D72f exP[ — iP'x/h] fiexp[ipx/h] d 3x, (7.222)

and

P'i iI P> = ( h/ i )Pap,p- (7.223)

Furthermore, we have to make the substitution

En — En , — E , (7.224)

but we immediately find

ep — ep, = 0

(7.225)

on account of eq. (7.223). By putting all the results together we obtain theself-energy of a free electron in the form

1 2e2 AW /h = .2 1 dw. (7.226)

(2702 3m2heoc 30

We notice that the self-energy of a free electron of momentum p isproportional to p2 . In adopting our results of section 7.7 we may im-mediately state that eq. (7.226) can be interpreted as giving rise to a shift inthe mass of the electron.

For the sL

the free etcelectron" re:

Wp =In it, mo is I

Wp =

Thus the tot

Wp +

While the mtion is mo, tathat in this 1

weSiriPThis WS

tromagneticnormally wrobserved ma

1— = —

where 2c1 isequation (7.:shift of the nm. This shift

The argunreason that tan infiniteobserved mawords, in facin the presen

H =

Then using e

P2H =

Thusike t


For the sake of clarity, we repeat our former arguments. The energy ofthe free electron without interaction with electromagnetic field "bareelectron" reads

W = P2/ (2mo)-

In it, mo is the "bare" mass. The energy shift just calculated is

1 2e2 coW =

p2 1 dw. (7.228)(20 2 3m2he0c3

Thus the total energy reads

P2Wp + AW = 2M*

(7.229)P

While the mass of the "bare" electron neglecting electromagnetic interac-tion is mo, taking this interaction into account the electron mass is m. Notethat in this type of considerations one uses m and not mo in eq. (7.228).This follows from the "renormalization" procedure we will describe now.

Since we always make observations on free electrons with the elec-tromagnetic interaction present, eq. (7.229) must be just the expression wenormally write down for the energy of a free electron where m is theobserved mass. Thus we can make the identification

=1 1 1 4e2 r dca — (7.230)

M O (2 102 3m 2 he0c3 o mo

where 2â is merely an abbreviation of the last term of the middle part ofequation (7.230). The electromagnetic self-energy can be interpreted as ashift of the mass of an electron from its "bare" value to its observed valuem. This shift is called renormalization of the mass.

The argument used in renormalization theory is now as follows. Thereason that the result eq. (7.220) is infinite lies in the fact that it includesan infinite energy change that is already counted when we use theobserved mass in the Hamiltonian rather than the bare mass. In otherwords, in fact, we should start with the Hamiltonian for the hydrogen atomin the presence of the radiation field given by

P2 e 2

2m 0 4reorH = + Hint . (7.231)

Then using eq. (7.230) we can rewrite H as

P2

H =e 2

+ J Hmt + (7.232)2m 4ireor

Thus if we use the observed free particle mass in the expression for the

(7.220)

urn certainlyely recognizeat the energynted a greatas of Bethe,

gain find anwhole calcu-atom we use

Ati221)>cal" forrator h/iV V.states of the..tween plane

(7.222)

(7.223)

(7.224)

(7.225)

te obtain the

(7.226)

nentum p iswe may im-

to a shift in

(7.227)

kinetic energy (which we always do) we should not count that part of Hir„that produces the mass shift, i.e. we should regard

Hint + a,P2 (7.233)

as the effective interaction of an electron of a renormalized mass m withthe radiation field.

Returning then to the calculation of the Lamb shift we see that to firstorder in e 2/he we must add the expectation value of the second term in(7.233) to (7.220) in order to avoid counting the electromagnetic interac-tion twice, once in m and once in Hint . Thus more correctly the shift of thelevel n is given by

= a f wdo4 filn>12 + < n I i)2in> c'e En , En — En, — CO CO

where we used the abbreviation00 1 2e2

a f do.) = a = dw. (7.235)(270 2 3m2hE0c3

The second term under the integral in eq. (7.234) can be brought into aform similar to the first term under the integral by means of the relation

<nl 132 In> = E <ni *fii n '>< n 'l 13 1 n > - (7.236)n'

In order not to interrupt the main discussion we will postpone the proof ofthis relation to the end of this section. Using eqs. (7.236) and (7.234) wefind after a slight rearrangement of terms

co E — E

(7.237)AK/h = aE Kn'l '1101 2 n n .n' En en'

We note that the integral over to is still divergent, however, only logarith-mically. This divergence is not present in a more sophisticated relativisticcalculation. Such a calculation yields a result quite similar to eq. (7.237),but with an integrand falling off more rapidly at high frequencies &a > me2.We can mimic the result of such a calculation by cutting off the integral at

procedure o'pression

A= En'

which we inthe weight ft.

KO AtNote that inmind, we ma

In w.'MC

Clearly, we n

ohTo simplify e

E Kn'in'

which we vvil;the right-hameasily evaluat

Ho = —

We readily ol

[ fix Ho]

and in a simil

[

(7.234)

= me2 /h. The integral can be immediately performed and yields Using for V t

AK/h = aE Kn'l filn>1 2 (en , — en)inn'

MC2

(7.238)V= —[e 2 / (4(7.246). Usingpoint charge!)

w.'

where we have neglected I — compared to me 2. The further evalua- A Ition of eq. (7.238) must be done numerically. We first cast eq. (7.238) into LI — =.

a more transparent form by interpreting the sum over n' as an averaging IXI

41


procedure over energy differences. More precisely, we introduce the ex-pression

A = E Kn'l fil n > I2(En' en )1n (7.239)n' MC 2

which we interprete as an average of lnl(W„, — W,,)/(mc2)I by means ofthe weight function

K n 'i fii n >1 2 ( en,— n) • (7.240)

Note that in general this "weight" may be also negative. With this idea inmind, we may formulate the following average

Wn' Wn A

MC 2 av

E Kn 'l fil n >1 2 ( e„ , — En)n'

Clearly, we may now rewrite equation (7.238) in the form

AW,:/h = amn E Kn 'l fil n >1 2 ( En , — en).Wn' Wn

mc 2

av n'

To simplify eq. (7.242) further we use the relation

E Kn 'l Wn) = — 12 (nl[ fi,[ 13,Ho]]In), (7.243)n'

which we will prove at the end of this section. The double commutator onthe right-hand side of eq. (7.243) is most useful. Such a commutator can beeasily evaluated. We assume 1/0 in the form

h2H0 = —--+A + V(x). (7.244)

We readily obtain (cf. exercise 6 on section 3.2)

{ H1 h aV(x) 13 -1 i ax

and in a similar fashion

[ fi,[ fi,Ho ]] = — h 2AV(x). (7.246)

Using for V the Coulomb potential of the electron in the hydrogen atomV = — 1 6' 2/(4 /Teo i x IA we can readily evaluate the right-hand side of eq.(7.246). Using a formula well known from electrostatics (potential of apoint charge!) we find

A-1 = —4/78(x),

x I(7.247)

it part of H„„,

(7.233)

mass m with

.e that to firstcond term in

;netic interac-he shift of the

(7.234)

•235)

nought into a)f the relation

(7.236)

ie the proof ofInd (7.234) we

(7.237)

only logarith-ted relativisticto eq. (7.237),

cies hca > mc2.the integral at

i yields

(7.238)

further evalua-eq. (7.238) intos an averaging

Wn' wn

in (7.241)

(7.242)

(7.245)

•


where 8(x) is Dirac's function in three dimensions. Using this result andthe definition of bra and kets (compare section 3.2), we readily obtain

(7.243) = n—e 2

— e2h02

n) il(Pn(x)128(x)d31 (7.248)4Treolx1

and making use of the properties of the 8-function

= e 2h2

2. (7.249)(7.248)I (pn(0)12E0

We are now in a position to write down the final formula for therenormalized self-energy shift by inserting the result (7.246) with (7.243)and (7.249) into (7.242). We then obtain

1 2e2 in

(2s) 2(27) 2 3m2he0c3

To obtain final numerical results we have to calculate numerically theaverage eq. (7.241) as well as 1%(0)1 2. For the hydrogen atom k(0)I 2 iswell known and is nonzero only for s-states. The average value wascalculated by Bethe for the 2S level. Inserting all the numerical values weeventually find AW,:/h = 1040 megacycles. According to these considera-tions a shift between an S and a P level must be expected. Such a shift wasfirst discovered between the 2S112 and 2P112 level of hydrogen by Lamband Retherford. A fully quantum-dynamic calculation gives excellentagreement with the experiments. While Lamb and Retherford used micro-waves to measure the splitting, more recently Hansch was able to measurethe splitting by optical high resolution spectroscopy.

At first sight, it may seem strange that it is possible to obtain reasonableresults by a subtraction procedure in which two infinitely large quantitiesare involved. However, it has turned out that such a subtraction procedurecan be formulated in the frame of a beautiful theory, called renormaliza-tion, and such procedures are now a legitimate part of theoretical physicsgiving excellent agreement between theory and experiment. Unfortunatelyit is beyond the scope of our introductory book to cover the details of theserenormalization techniques.

In conclusion we briefly present the proof of two auxiliary formulaswhich we have used above. To this end we write the bracket symbols againas integrals and start with the right-hand side of equation (7.236).

(7.251)n'I cP:(x ) V cp„,(x) d3xf cp:,(x9— V 'T„(x') d3x'.

Fig. 7.10. Lamesplitting due toing to the 2S11,2r.h.s. of the figurenormalizationThe numbers in

•

Fig. 7.11. Byrnethe 3rd volume),spectrum of thethe spectrum. [AT. W. Hansch, A

MC 2

wn, —

e2h2

ICP

av 2E0 n

(7.250)


2 Pi2

Fig. 7.10. Lamb-shift for the electron states with n = 2 in hydrogen. The left part shows thesplitting due to spin-orbit coupling. According to Dirac's equation, the energy levels belong-ing to the 2S112 and 2 P1/2 states should coincide. In reality, they are split (Lamb-shift). Ther.h.s. of the figure shows the splitting based on calculations which take into account the massrenormalization and some further, though minor effects treated by quantum electrodynamics.The numbers indicate the corresponding frequencies in MHz.

? '500PC°'

BALMERHa I 11101SERIES

x 40 000

H0DOPPLERPROFILE(300°K)

H HLAMB SHIFT

SATURATIONSPECTRUM

10 GHzv ---,-

Fig. 7.11. By modern methods of nonlinear spectroscopy, namely saturation spectroscopy (cf.the 3rd volume), the Lamb-shift can be measured optically. This figure shows the saturationspectrum of the red Balmer line in atomic hydrogen. The splitting of the levels is indicated inthe spectrum. [After T. W. Hansch, I. S. Shahin,and A. L. Schawlow, Nature 235 (1972) 63;T. W. Hiinsch, A. L. Schawlow and P. Toschek, IEEE J. Quant Electr. QE-8 (1972) 802.]

ula for thewith (7.243)

(7.250)

leric henit

kg, 2 isvalue was

11 values weconsidera-a shift was

n by Lamb.ts excellentused micro-to measure

reasonablee quantitiesa procedureenormaliza-ical physics'fortunatelyails of these

ry formulasrnbols again36).

(7.25 1)

result andy obtain

(7.248)

(7.249)

25t211 10'

11040

17

2 5 1 , 2P12 2

2 P122 P1

2 8

Rearranging summation and integration we obtain

f f d3x d3x'cp:(x)—i V

V /(p„(x'). (7.252)

Since the functions (p„ form, in the mathematical sense, a complete set, thefollowing relation is proved in mathematics

E cp„.(x)4(x') = 8(x — x'), (7.253)n'

where 8 is Dirac's function. This relation is often called "completeness"relation. Inserting eq. (7.253) into (7.252) allows us to perform the integra-tion over x' immediately using the 8 function so that (7.252) reduces to

f d3x99:(x)(-hi V) 2 q)„(x)dV = < nl 132 10 . (7.254)

But this is evidently the left-hand side of equation (7.236).We now prove eq. (7.243) and study the expression

E <nl filn'><ni filn>ww. (7.255)n'

We use the fact that (p„. is eigenfunction to the Hamiltonian 1/0

HocP. , = (7.256)

With its aid we can immediately evaluate

<nl = 141):(x)fifloq9„.(x)d3x

= wn'< n l filn'›

(7.257)

which allows us to write (7.255) in the form

(7.258)E < n l filn>Wn' = E <n I Pliol d>02 1 filn> •n' n'

Using the completeness relation again quite similarly as above we obtain

E <nl fiHoln'><n'l ,51n> = <nl 131-10 filn>. (7.259)n'

In the same way the completeness relation yields

E <nl filn>wn = wn< n l fi2 I n > • (7.260)

n'

Thus we can rewrite the left-hand side of eq. (7.243) in the form

< n i Wn)fiin>• (7.261)

Furtherrmtions we c

<nI

Similarly,

Rh

Vir

Writing cwe obtair

Using thcverify the

7.9. Onct

In sectiointeractinus again

(5(0

In the foius now ccremain a,correct!ensembledeal withsary to se

§7.9 Once again spontaneous emission: Damping and line-width 251

Furthermore, by using the definition of brackets and using partial integra-tions we can readily establish the following relations

(7.252)

lete set, the

(7.253)

wleteness"he integra-duces to

(7.254)

65)

h h<nl 13 Wn fil n > =q(x)V W V q)„(x)d3x

= f[(— Icp:(x)1W„cv„(x)d3x

= f{(— Vrco:(x)]Hoq)„(x)d3x

= f (p„( x)fi2Hoson( x) d'x

= <nli,21/010.

Similarly, we obtain

f (H0p:(x))/329).(x)d3x =fc9:(x)H0132(p.(x)d3x

= < n 1 110 132 10 •

(7.262)

(7.263)

Writing out the double commutators on the right-hand side of eq. (7.243)(7.256) we obtain

— /32H0 + Ho fi2 n> + <n filn> (7.264)

Using the results in eqs. (7.259), (7.260), (7.262), and (7.263) we can easilyverify the relation (7.243).

(7.257)

(7.258)

we obtain

(7.259)

(7.260)

tn(7.261)

•

7.9. Once again spontaneous emission: Damping and line—width

In section 7.6 we solved the Schrodinger equation of a single atominteracting with the radiation field by first-order perturbation theory. Letus again inspect our former solution (7.120), which has the form

lf(t) = aj* + E cx(t)a,+ bP:Do. (7.265)

In the foregoing sections we focussed our attention on the sum over X. Letus now consider the first term, 4 00 . According to it, the electron ought toremain all the time in the upper level, 2. Obviously, this result cannot becorrect! We know experimentally that the number of electrons of anensemble of atoms in the upper state, 2, (in quantum mechanics we alwaysdeal with ensembles) decays exponentially with time. This makes it neces-sary to seek a more exact solution of the Schrodinger equation. We choose,

(7) 4421 = 82 — E l. (7.269a)


as in section 7.6, a two-level atom, but work in the Schri5dinger representa-tion. The corresponding Schrodinger equation which we have already metin section 7.5, eq. (7.74), reads

dt = 1311 4 a l + W2 a -2' a 2 + E hc..IxbZ bx

+ hE (go; a 2 bZ + ga a l bx)}(1). (7.266)

We now try the ansatz

= A(t)4 43.0 exp[ -ie2t]

+ E cx(t)bZ (130 exp[ -i(wx + e i )t], ei = Wi /h. A(t) (7.267)

and c(t) are considered as unknown coefficients still to be determined. Byinserting it into eq. (7.266) and comparing the coefficients of the linearlyindependent components aPto and b + a l" (Do on both sides yields

(7.271) the r

1 -lirn —t-o 00

This is a fo'Dirac's fun(defined with

P tco 6

It thus cutspart into eqtransition pr

2y = P

We oinc

the tio

-= P

We shalllevel causes

The princi.e. a level s

Since we expect that the occupation number of the upper level decaysexponentially, we put A = exp[ -yd. By inserting this into eq. (7.269), wecan perform the integration and find

1 - exp[i(wx - 5)t - yt]cx igx (7.270)

y - i(wx -

By inserting this result and A = exp[ - yt] into eq. (7.268) we obtain

exp[ yt - i(cox - FJ)t] -1Y = I gx I 2 (7.271)

Y i(WX t7))

This result seems to imply a contradiction, because y . on the 1.h.s. is aconstant, whereas the r.h.s. still depends on time. This puzzle can be solvedas follows. Since we are only interested in time intervals t >> 1/(7) and sinceit will turn out afterwards that y < Fo, we may use on the r.h.s. of eq.

dcxdt

where

- igiA exp[- i(c) - 4.)x)1], (7.269)

d A= - jE cat exp[ - wx)t],

dt(7.268)

§7.9 Once again spontaneous emission: Damping and line-width 253

(7.271) the mathematical relation (cf. also the mathematical appendix)

Jim—

1 — exp[ i(w — w)t] p 1 i71.607) 6)).(7.272)

--t+co () — (7) —

This is a formal relation making sense only under an integral. 8 is againDirac's function, defined in section 7.6. P means principal value. It isdefined with respect to an integral as follows

— co (.4)

1/(0) dC0 = lirn f 1 ‘7'—e f( 44)) CIW f 6)) d6)

co — (7) — c7.)-F-e CT) — CO

(7.273)

It thus cuts the "point" w = cT) out of the integral. Inserting the 8-functionpart into eq. (7.271) yields an expression identical with that for the totaltransition probability P, eq. (7.127), besides a factor of 2, so that

27 = P.

We thus find that the inverse lifetime T of the upper level is connected withthe transition probability P by

—1 = p

We shall see later, in section 8.3, that the exponential decay of the upperlevel causes a finite width of the spontaneous emission line.

The principal part of eq. (7.272) causes an imaginary contribution to y,i.e. a level shift of the upper atomic level, 2. This shift is proportional to

WoFig. 7.12. Frequency distribution of a Lorentzian line.

r representa-already met

(7.266)

t) (7.267)

ermined. Bythe linearly

;Ids 411)

(7.268)

(7.274)

(7.275)

(7.269)

(7.269a)

evel decays. (7.269), we

(7.270)

)btain

(7.271)

e 1.h.s. is ain be solved(T.) and since.r h.s. of eq.

N 2 (1)

Fig. 7.13. Experimental decay of the excited electronic level as a function of time. The curvesof fig. 7.12 and 7.13 are Fourier transforms of each other.

I gA l 2 and thus stems from the emission and reabsorption of a photon(compare sections 7.7 and 7.8). This shift is a contribution to the Lambshift. It is only a contribution, because in reality transitions to other atomiclevels are also involved.


(1) Verify the statements made in section 1.10, ch. 1 on the line width andphoton energy measurements by means of the results of the presentsection.Hint: Use eq. (7.270), and the probability interpretation of quantumtheory. Let t oo.

(2) Verify the statements made in section 1.13 on the decay of the meanoccupation number N2 = < 4:1) I al: a21(1)>.

(3) Discuss section 1.12 using the results of the present section.

7.10. How to return to the semiclassical approach. Example: A single mode,absorption and emission

In this book we have encountered two ways of dealing with opticaltransitions in atoms. In chapter 4 we treated the atom quantum mechani-cally and the field classically. In that treatment the field was an externallyprescribed quantity. On the other hand, in this chapter, 7, we treated both

the atoms andthe limit of larcoincide. As wecal coherent fieTherefore we exto establish thiinteraction repo

d _ 1 _d t ih

As an exampleand a 2-level at

a'21- al

• 4)21.

We now make

Aik=

where ilpgiveU = exp[ t

When U acts ofield. By insertil

U—d (1) —dt

or after multiphd ldt = ih

In it we have u:

111 = U-11

In section 5.3 v.

exp[ —sb+

Making use of

ii1 = (4-al

+ a;

Here fig is idtHamiltonian 11

•

•

§7.10 How to return to the semiclassical approach 255

the atoms and the field according to quantum theory. We expect that inthe limit of large photon numbers the results of both treatments mustcoincide. As we have seen at several occasions earlier a quantum mechani-cal coherent field has many properties in common with a classical field.Therefore we expect that a formulation using coherent fields will enable usto establish this connection we have in mind. Quite generally, in theinteraction representation we have

d = 1 mi). (7.276)dt ih I

As an example we use the interaction Hamiltonian for a single light modeand a 2-level atom

= a a i bgexp[i(ra – w)t]+ a l+ a 2 b + g* exp[— – co)t],

°21.We now make the substitution for (13

= U$, (7.278)

where U is given by

U = exp[ /3b + – Mb]. (7.279)

When U acts on the vacuum state this new state describes the coherentfield. By inserting eq. (7.278) into eq. (7.276) we obtain

d 1U—(1) = U4), (7.280)

r

or after multiplication of (7.280) on both sides byd 1

Cit (I)= .111(1).

In it we have used the abbreviation

= '12,U. (7.282)

In section 5.3 we derived the relations

exp[ —flb + +13*b]( bb+ )exp[ My+ 13*b] = ( b+ 16*).b + /3

(7.283)

Making use of them in eq. (7.282) we can already evaluate Ili and find

= aPa 1 f3gexp[i(ZZ) – co)t]

+ a l+ a 2 Mg*exp[–i(c7.) – to)t] + fig. (7.784)

Here fig is identical with the original quantum mechanical interactionHamiltonian Hq 171. The transformation eq. (7.279) thus transforms the

(7.277)

(7.281)

t

of time. The curves

m of a photonm till Lambto atomic

line width andof the present

m of quantum

ay of the mean

tion.

: A single mode,

rig with opticalint= mechani-as an externallywe treated both


original interaction Hamiltonian into two parts. Now one contains aclassical field amplitude fi, 13* and the other is still quantum mechanical.When we consider the definition of g etc. one can be convinced after somecalculations that the following relations hold

fig exp[ Ot] =1,21E(+)(t)

/3*g* exp [ —0t] = 4E(-)(t), (7.285)

where /921 is the optical dipole matrix element and E(÷),E(-) are thepositive and negative frequency parts, respectively, of the classical electricfield strength. Thus we may rewrite the first part of eq. (7.284) in the form

= cq- a 1 t921 E(+)(t) + a l+ a 2 1.911 E(-)(t)• (7.286)

We now see that we can do a perturbation theory quite similarly as inchapter 4 where the electrons are treated quantum mechanically but thefield occurs as classical quantity. In this approach it is assumed that thefield amplitude is so high that we can neglect the quantum fluctuationsdescribed by the quantum mechanical part of eq. (7.284), i.e. H q . Acomplete discussion of the impact of transformations such as eq. (7.278) onthe physical content must include not only the Schredinger equation butalso expectation values and matrix elements. Because the field operatorsb + , b commute with the electron operators a 4 ",a, one readily establishesthat all quantities involving only electronic matrix elements are unchangedunder the transformation (7.278). We leave it to the reader as an exerciseto extend these considerations to many field modes.

7.11. The dynamic Stark effect*

The purpose of this chapter is as follows. A two-level atom is exposed to acoherent resonant driving field. What happens with the spectrum of itsspontaneous emission? We shall see that under high enough driving fieldsthe atomic line will split into three parts (cf. fig. 7.14). The splitting ofspectral lines under the influence of static fields is well known. Forinstance a high enough static electric field causes the Stark effect. In thepresent case, the line splitting is caused by an oscillating electromagneticfield, so that the term "dynamic Stark" was coined to describe this

*The topic treated in this section is still under intense experimental and theoretical study.While some authors (see references at the end of this book) use methods similar to thosepresented in this section, others use the density matrix approach (for more details on densitymatrices see section 9.5). While these approaches give the same results concerning the size ofthe line-splitting, the results differ what the ratio of the heights of the intensity peaks isconcerned. Possibly, the assumption made below, after eq. (7.298), might be still too strong.

Fig. 7.14a. Typicand dot-dashed(Theoretical resin

•

•-n ., 1770 MHz

Fig. 7.14b. This(abscissa to theresults were ob,Schuda, C. R. S

•

•4i.1770 MHz

="40-41ww.-.41-1:n•nnnn MirnIiivr;sn

20 wiCC °EN T61 0 10

4 O N 0)Ce ll' it2r0 3ZA K ION 0 n1n4'0Op FA

tgu SO4e i t Ne T 60TA T, , (4, 70

cb.) ' O N sos H I)SO

-200 -100 0 100 200Dee

Tliti143SCATTERED SPECTRUM,MHz

(CENTERED ON EXCITATION FREQUENCY)

§7.11 The dynamic Stark effect 257

1(w)

Fig. 7.14a. Typical spectra of the emitted radiation for the dynamic Stark-effect. The solidand dot-dashed lines correspond to smaller and larger atomic line widths, respectively.(Theoretical results by Gardiner, private communication.)

Fig. 7.14b. This figure shows the intensity (ordinate) of the emitted light versus frequency(abscissa to the right) for various detunings of the exciting laser light. The experimentalresults were obtained for the hyperfine transition F .•• 2— 3 of the sodium D2 line by F.Schuda, C. R. Stroud, Jr. and M. Hercher, J. Phys. B7 (1974) L 198.

ntains a:hanical.ter some

(7.285)

are theelectric

he form

(7.286)

ly as inbut thethat thetuations

Hq4111278)Wion butleratorsiblisheshangedexercise

.ed to a-1 of itsg fields.ting ofn. ForIn the

agneticbe this

;al study.to those

a densityle size ofpeaks is0 strong.


phenomenon. This terminology is sometimes considered not as adequatebecause the mechanisms of the two effects are rather different.

We start with the Schrodinger equation in the interaction representationin agreement with section 7.4. Since the mode of the driving field will behighly excited we write the corresponding Schri5dinger equation a little bitdifferently by treating the creation and annihilation operator of the drivingfield mode, i.e. b4" , bo, separately from the other modes

d -—dt

(I) = a2( exp[ iAxt] + b0+

+ a;" a l (E'bxexp[– iAxt] + b0 )}(1), (7.287)

where

Ax = W — W. (7.288)

For simplicity, but without losing any essentials, we choose g independentof A and real. The prime at the sums indicates that the sums are takenwithout A = 0. As in the foregoing chapter we put

(i) = U(1), (7.289)

with

U = exp[ b 8 – Mbo].

Making use of the relations (cf. section 5.3 including exercise 3)

U+ boU = bo +

and

U+ b U = b(;'. + 11*

we immediately find

d—(1) = – igk a2(bx exp[iAxt] + /3*)dt

+ a bx exp[ –hl At] + /3)}4 . (7.293)

When we compare eq. (7.293) with the original Schro5dinger equation ofsection 7.6 where we treated the spontaneous emission of a two-level atom,we recognize that new additional terms 13 and 13* occur which stem fromthe external field. It is tempting to make a hypothesis for (1) similar to theWeisskopf-Wigner theory which we treated in section 7.9. However, somecare must be exercised here. To see this, let us start with an initial wave

function withal. (Do.

When we ins(the operatorstransformed

+ "Ts

aand of functic

a l+ b;,'" exj

In the next st(7.293). We th

a + b +2 ex

which meansreturned to itway the funct

4110-Here we hawprocedure wephotons. Thu.into one whicof eq. (7.298)be justified wbig and if theconsideration.additional furfollowing hyp

+

In it, the coeprocedure is(7.299) into ("describing th(are orthogon;only by puttii

(7.290)

(7.291)

(7.292)

•


;rcise 3)

function with no photon present and the electron in the excited state

al- 430 . (7.294)

When we insert (7.294) for (i) in the sense of perturbation theory and letthe operators in front of 4) act on it, we immediately realize, that (7.294) istransformed into linear combinations of the function

4- 00 (7.295)

and of functionsab exp[ xt] (Do . (7.296)

In the next step we insert (7.295) and (7.296) into the right-hand side of(7.293). We then see that (7.296) is transformed into

bje- exp[iAxt]4)0 , (7.297)

which means that we now also find states in which the electron hasreturned to its upper state but a photon is present in addition. In the some-way the function (7.297) gives rise to new terms of the form

a t 13 -xl- bZ.exp[iAxt + iA A,t] 00 . (7.298)

(7.289) Here we have to deal with two additional photons. When continuing thisprocedure we will obtain wave functions with an increasing number of

(7.290) photons. Thus the system of equations is not closed. To turn the probleminto one which we can solve explicitly we will neglect the terms of the formof eq. (7.298) and those containing still more photons. This procedure can

(7.291) be justified when we assume that the production rate of photons is not toobig and if the photons can escape quickly enough out of the system underconsideration. As we have seen, an initial function (7.294) generates

(7.292) additional functions (7.295), (7.296), and (7.297). This leads us to make thefollowing hypothesis for the solution of eq. (7.293):

(I) = + E fkAexp[iAAt]bZ at

d not as adequatefferent.:ion representationriving field will be•quation a little bitator of the driving

(7.287)

(7.288)

ose g ' dependent

iiiaie s re taken

+ D2 a -21- + E D2, xexp[iAxt]b;:-}Co. (7.299)

In it, the coefficients D are still unknown functions of time. Our furtherprocedure is now quite analogous to that of section 7.9. We insert eq.(7.299) into (7.293) and let the operators act on 4). Then we collect termsdescribing the same electron-photon state. Since the corresponding statesare orthogonal to each other we can fulfill the corresponding equationsonly by putting the coefficients of each function equal to zero. This yields

(7.293)

inger equation ofa two-level atom,which stem fromr 43 similar to the9. However, someth an initial wave

•


the equations

—dt

D1

= –igD2fi* ,

—dt

D2

= – igDo3 – ig D1,A,

ddi = – igD2 – igD213* ,

ddt =

Since we have an infinite number of modes A, eqs. (7.300)—(7.303) repre-sent a set of ordinary linear differential equations with constant coeffi-cients. The reader, who is not so much interested in the details of thefurther calculations, may skip the following and continue after eq. (7.328).

Here we want to show how the equations (7.300)—(7.303) can be solvedin an elegant fashion. To this end we make use of the Laplace transforma-tion. Since not all readers might be familiar with this method, we brieflyexplain it. Let us consider a time-dependent function x(t). We then definethe Laplace transform es by the relation

es [ x] = fo exp[ –st]x(t) dt. (7.304)

It must be assumed, of course, that x(t) guarantees the existence of theintegral on the right-hand side. When we choose x = 1 we immediatelyfind its Laplace transform to be

e5 [1] (7.305)

Furthermore we use the identity

x(t) = x(0) + I t cle dx(e) (7306)0 dt'

which can be immediately verified by evaluating the integral. Taking theLaplace transform in eq. (7.306) on both sides we find

(7.307)es [x] = —x(0)

+ f dt exp[ –st]i t dx(t')

s 0 dt'

or, after a slight rearrangementdxse-s[ x] x(0) + es{ —dt 1. (7.308)

This formula allows us immediately to determine the Laplace transform ofthe derivative of x by the Laplace transform of x.

To applyoccurring in

es [ DJ]

Taking thenreadily obtai

sdl =

sd2. , =

• =

• =

As an initialexcited state

D2(0) =

Equills (7

=

By insertingand (7.311))

d2 =S

Now (7.315)abbreviation

r(s) =

T(s) can be

r(s) =

When we us4the line widthose of eq.

411

(7.300)

(7.301)

(7.302)

(7.303)


To apply this formalism we have to identify x with any of the Dsoccurring in eqs. (7.300)-(7.303). For abbreviation we further put

es{ = d. (7.309)

Taking then the Laplace transform of the equations (7.300)-(7.303) wereadily obtain

= D 1 (0) - igfl*d2,s , (7.310)

sd2, , = D2(0) - igieck s - igE

(7.311)

= D 1 (0) -ixdlA S - igd2,s - jesd2, (7.312)

sd2, x, = D2, x(0) - iAxd2s - (7.313)

As an initial condition, we choose that at time t = 0, the atom is in anexcited state and no photon is present. This implies

D2 (0) = 1; D 1 (0) = D 1 (0) = D2, x(0) = 0. (7.314)

Equations (7.312) and (7.313) allow us to express d1 andand d2, x ,1 by d2,s

-ig(s + LI?)dldt,s d2,39 (7.315)

(s + iA x )2 + g211312

-ig(- igP)d2,A,s

(s lAX)2 g21/612

By inserting eq. (7.315) into eq. (7.311) and then solving the eqs. (7.310)and (7.311) yields

sD2(0)d2 5 -

S(S TO)) ±g2/3j2(7.317)

Now (7.315) and (7.316) are determined explicitly. r(s) is the followingabbreviation

(7.316)

g 2(s + iAx) r( s ) = E

+ i02+g2ist2

r(s) can be decomposed into

(7.318)

(7.319)r(s)

1 1 2 s + + s + - igl i31) •

g2

When we use the relation (A.19) of the mathematical appendix we see thatthe line width and frequency shift resulting from r(s) are the same asthose of eq. (7.271), where (7.272) is used, provided s 0 and g(I3) is

(7.300)

(7.301)

(7.302)

(7.303)

(7.303) repre-nstant coeffi-ietails of theer eq. (7.328).:an be solved

tran orma-'efly

-e the efine

(7.304)

stence of theimmediately

(7.305)

(7.306)

1. Taking the

(7.307)

(7.308)

transform of

•

For the foloccurring inquantities

s t,2 = —and

53,4 = —

In order toprobability tceasily deduce

p(t) =

Since the cora few import


absorbed in Ax by a change of the summation variable. As will turn outbelow in a self-consistent fashion, the important contributions to the finalresult, are those in which s is small compared to co. Therefore in thefollowing we will use

r(s) = y ill, (7.320)

where y and S2 are the linewidth and frequency shift without the externalfield.

Thus we have determined the Laplace transform of the wanted functionsD(t) explicitly. We now indicate how we can return from the Laplacetransform to the original time dependent functions. To this end let us writeinstead of s the complex quantity z = a + ib; a, b real. Let us assume forexample that es( f) has the form 1/(s — zo) so that

es[f] — 1 • (7.321)S — Z0

One is readily convinced that

f(t) = exp[zot]; Re(z 0 ) < 0 (7.322)

when inserted into eq. (7.321) just yields the right-hand side. One mayshow that this solution is unique and in this way may establish a whole listof rules for evaluating Laplace transforms. For instance one may show thata factor s corresponds to the differentiation

s —- (7.323)dt

or a power of s to n-fold differentiation

cTd (7.324)

Furthermore when a Laplace transform

1

—

1 (7.325)

G(s) a(s — z 1 )(s — z2 ) . -- (s — z,7)

is given, one may decompose this fraction into partial fractions

1 al a 2 ± an (7.326)S — z 1 s — Z 2 S — Z n

For each of the expressions of the right-hand side we know the originalfunction of the Laplace transform.

It is now a straightforward task to decompose eqs. (7.315) and (7.316)with (7.317) into such partial fractions and to evaluate the final result.

G(s)

In the limitstrong enoug,

2p x

2

•g 2

Px = 2

p), appears hto a continthtinuous frequLorentzian c+ S2/2. The

= — 21g,

•

•


For the following analysis we need the roots si of the polynomials in soccurring in eqs. (7.315), (7.316) with (7.317). These roots are the followingquantities

s I,2 = (7.327)

and

-2 ± I' V1- 2

In order to make contact with experiments we wish to determine theprobability to find a photon of mode A at time t. This probability can beeasily deduced from the hypothesis (7.299) and is given by

p(t) = 1130 1, x(t)1 2 + ID2, x(t)1 2 . (7.329)

Since the corresponding expressions become rather lengthy we quote onlya few important final results. For t oo one obtains

S3,4 = (7.328)

•.322)

ill a whole listde. One may

nay show that

(7.330)(7.323)

In the limit in which I g/31 is much bigger than the line width y, i.e. forstrong enough driving fields, px acquires the form

g 2 \ 2 -1(7.324) PA = T

{ y 2

T + 21° 1 + -T)

(7.325)

y 2

+\21-1

T - 2101 +

+ 2[+72 (— -+ )2 ]

-1-4- 6.) —2- (7.331)

(7.326) px appears here as a function of the mode frequency Wx which belongsto a continuous spectrum. We thus can plot p as a function of the con-

v the original tinuous frequency co. Clearly the first term in the curly bracket represents aLorentzian curve with the width y/2 centered around co = + 2101

) and (7.316) + S2/2. The second term describes the same line but now centered aroundnal result. co = (7) - 2101 + S2/2. The last term finally represents a Lorentzian curve

2

Ax lai - 45'3 - lei - is4g

2 1 PA = 2

1 53 - s S4S3

will turn outis to the final;refore in the

(7.320)

t the external

ited functionsthe Laplace

id let us writeus assume for

(7.321)

Ax + 101 — is3 Ax + Ig$1 - is4

S 3 S4


centered around c,.) = + l/2 and has double the intensity than the twoother curves. According to this formula the original emission line of anunperturbed atom splits into three lines under the influence of an externalelectric field (dynamic Stark effect). The theoretical and experimentalresults are shown in figs. 7.14a and b, respectively. The dynamic Starkeffect can also be observed by absorption of light which hits atomssubjected to a strong coherent resonant driving field.

8. Quant

8.1. Quan

In sectionparticular

I

giE(4-In ancstrength 1componenpositive arthe brackeassumed 1(in this bcquantitiesseen, themeasurablexpectatio(8.1) intonamely E'That meaoperators.which wewhich absand the n(is detectesequenceconfine oi

We noend we c,field and

ensity than the twoemission line of antence of an external

and experimentalThe dynamic Starkwhich hits atoms

8. Quantum theory of coherence

8.1. Quantum mechanical coherence functions

In section 2.2 we got acquainted with classical coherence functions. Inparticular we dealt with the following coherence function

<E,(+)(x',e)E1-)(x, t)> (8.1)In it x and t were the space point and time, respectively, at which the fieldstrength E is measured. The indices i and j in eq. (8.1) refer to thecomponents of the field vector, whereas plus and minus indicate thepositive and negative frequency part, respectively. As the reader may recallthe brackets denote the time average over the measuring time which wasassumed long compared to the inverse of the light frequency. Several timesin this book we have had occasion to learn how to translate classicalquantities into the corresponding quantum mechanical ones. As we haveseen, the classical observable "field strength" becomes an operator. Themeasurable quantities are then derived from the operators by certainexpectation values. When we try to apply this scheme to the translation of(8.1) into a quantum theoretical expression we encounter a difficulty,namely E(-) and E(+) don't commute with each other (cf. exercise below).That means we now have to take care of the correct sequence of theseoperators. It turns out that their sequence depends on the experiment bywhich we measure the coherence function. When we use light detectorswhich absorb light then the positive frequency part must stand on the rightand the negative frequency part on the left. On the other hand, when lightis detected by processes involving downward transitions, the oppositesequence holds. Since the upward transitions are the usual ones we willconfine our analysis to this case.

We now wish to derive the coherence function just mentioned. To thisend we consider a realistic system consisting of a light source, the lightfield and a detector. Each of these individual systems is described fully

•

•

266 8. Quantum theory of coherence

Table 1

Source

Field:

Interactionsource-field:

Hf

H,

113_1

Interactionfield-detector:.ks t:.er tor :

Hf_

Hd

quantum mechanically. The corresponding individual Hamiltonians arelisted in table 1. Thus in principle we are confronted with the solution ofthe Schrodinger equation of the total system composed of source, field anddetector.Its Hamiltonian reads

H H3 + II s_f+ Hf + 141f_ d+ Hd.

HF

Of course, in general we will not be able to solve a Schrodinger equationwith such a complicated Hamiltonian. However, we want to show how wecan reduce the problem in a way which allows for direct comparison withexperiments. It is most convenient to transform eq. (8.2) into the interac-tion representation which we have explained in section 7.4. To this end wemake the identifications

HF + Hd —+ Ho (8.3)

and

HF— d --° Hint

so that the total Schrodinger equation to be solved reads

(H0 + Hint )(1) =- in d:t

Making the hypothesis

(I) = exp [ — iHo t/h

we go over to the interaction representation which now reads

ih 7/7 = ;I/ nt(I)

with

Flint = exp[ iHot/h]Hmt exp[ —iHot/h] (8.8)

as we know from section 7.4. Our first task will be to determine (8.8) more

explicitly. Tdetector amexplicitly.procedure vIts Hamilto

Hd = h

(.7.) = 21 sannihilatiorfunctions 0:

(pi( x).

Adopting t:for the inte

HeOur next tzmade the rtin real trandescribes acombinatio

Since theent system:

[Hd,1

This allow5into two fa

[ ii

where weAt a first

with respecof course,measuringabout the rNamely, h:led to cons

exp[

which,sions

H3 + H,_f+ Hf

113+ H,_ J.+ Hf

H1-1- Hf_ d + Ha Hf_a+ Hd

(8.2)

(8.4)

(8.5)

(8.6)

(8.7)

§8.1 Quantum mechanical coherence functions 267

Hf

j± Hd

miltonians arethe solution ofmirce, field and

• (8.2)

linger equation) show how weimparison withto the interac-To this end we

(8.3)

explicitly. To this end we formulate in a first step the Hamiltonian of thedetector and the Hamiltonian of the interaction between field and detectorexplicitly. To avoid too many complications which would obscure theprocedure we assume that the detector consists of a single, two-level atom.Its Hamiltonian reads

Hd = hwaj a 2 . (8.9)

= w21 is the transition frequency of the atom, aj , ajl- are, as usual, theannihilation and creation operator of the electron in level j. The wavefunctions of the corresponding two states j = 1,2 are, as usual, denoted by

cpi (x). (8.10)

Adopting the general result of section 7.2 we may write the Hamiltonianfor the interaction field-detector as

Hf_ d a 2+ a 1 (– —e

)1 cp1(x)A(+)(x)7 v cp i(x)c13x

+ Hermitian conjugate. (8.11)

Our next task will be to evaluate (8.8) using (8.9)—(8.11). We note that wemade the rotating wave approximation in (8.11), i.e. we are interested onlyin real transitions. To verify our statement we recall that the operator 4" aldescribes an upward transition of the detector atom, while A(÷) is a linearcombination of photon annihilation operators.

Since the detector Hamiltonian and the Hamiltonian HF describe differ-ent systems the corresponding Hamiltonian operators commute

[Hd,HF] = 0• (8.12)

(8.4) This allows us to split the exponential functions containing 1/0 in eq. (8.8)into two factors. Treating the factor containing Hd alone we readily obtain

(8.5)where we use results obtained in section 7.4.

exp[iHdt/h]4- a i exp[ – iHdt/h] = exp[ (8.13)

At a first sight it appears rather hopeless to perform a similar procedurewith respect to that part of (8.8) which contains HF. The reason for that is,

(8.6) of course, that we don't know yet anything about the source. Indeed byIs

measuring light absorption by the detector we want to get informationabout the nature of the source. However, a formal trick helps us quite a lot.

(8.7)

Namely, having in our mind again the decomposition of 1/0 (8.3), we areled to consider an expression of the form

exp[ iHFt/h]AN" )(x) exp[ –iHFt/h] (8.14)(8.8) which occurs in eq. (8.8). But now we recall that we encountered expres-

ine more sions of such a type earlier, namely when we considered the change of

(8.18)ci)(o) = o(o)detectoroF(o)

(8.21)(t) = (I)(0) — L f 1-2 (T) dri)(0).h t


operators in the Heisenberg picture (cf. section 5.6). In our case we maythink that the operator of the vector potential A (positive frequency part)changes under the action of the full Hamiltonian HF describing thedynamics of the systems field and source. We will see in a minute that weneed not evaluate (8.14) explicitly to obtain our final result. We wouldrather make use of the fact that (8.14) just defines the operator A ( ) in thecorresponding Heisenberg picture which allows us to put

(8.14) = A (+)(x,t). (8.15)

According to the positive and negative frequency parts of A in (8.11) wesplit the interaction Hamiltonian into the corresponding parts

Ant = ni(nT ). (8.16)

Our considerations just performed allow us to write now explicitly

IiiVt" ) = a rTie exp[ Of] f 94(x)A(+)(x,t)-Th V q) /(x)d3x. (8.17)

We now wish to deal with the interaction between the field or, moreprecisely speaking, between field and source with the detector. To this endwe have to solve the Schrodinger equation (8.7). We treat a specificexperimental situation. In it, at the initial time t = 0, the coupling betweenfield and detector atom is switched on. At this time we still may decom-pose the initial wave function into a product of the wave functions of thedetector atom and the field + source system

field + source

To be more specific we assume that the detector atom is at that time in itsgroundstate

(1)detector(0) = at (bo, atom •

The field acting on the detector atom may now cause transitions and thenwe may be able to verify by other means that the detector atom has madethe transition into its upper state. So our next task will be to calculate theprobability of finding after time t a detector atom in its upper state

P2( t ) = <a) ( t )i ai a24( t )› .(8.20)

To evaluate eq. (8.20) we need 43(0. We determine that function by meansof first-order perturbation theory in the well known fashion (cf. section7.7)

When we inslooking at tiaccount theourselves tha

P2(z)

We now us,..expression innot shock thcontains twcelectron was,part contain:.reads

P2(ii

In it the bra,

< > =

This latter bprobability (tially deterrrnegative freccoherence ftapparent int:the dipole u:have appearsteps our fineverywhere I

<OF(OY

[Another wathe time der

To be quifree fields irwith the detwith the soui

411

(8.19)

§8.1 Quantum mechanical coherence functions 269

ur case we mayfrequency part)describing theminute that wesuit. We wouldator A ( ) in the

(8.15)

A in (8.11) werts

(8.16)

:plicitly

(x)cii (8.17)

fielrigr, moreor. To this endmat a specificupling between11 may decom-inctions of the

(8.18)

that time in its

(8.19)

dons and thentorn has made) calculate theer state

(8.20)

don by meansn (cf. section

(8.21)

When we insert eq. (8.21) into (8.20) we are left with four terms. However,looking at the action of the atomic operators a. a; and taking intoaccount the form of the initial state eq. (8.19) we may readily convinceourselves that eq. (8.20) reduces to

p2(t)=-2-<$(0)iforni(nv(T) d Tifni(riveodT143(0)>.0

We now use the explicit expression (8.17) for H. When we insert thisexpression into eq. (8.22) we obtain a rather lengthy formula which shouldnot shock the reader too much. He will notice that the resulting formulacontains two major parts. One part refers to the atom containing theelectron wave function and the momentum operators, whereas anotherpart contains the vector potential in the Heisenberg picture. This formulareads

e2 P2( r ) = 2t2 E f t dri t dT'exp[i)(7. ' — r)]

M j, k...x,y,z 0

I h \ I h \x f d3x1 d3x'ql(x)k Vi pi(x)41( xA 7 )99 1( x ')< • • • > •

(8.23)

In it the bracket <...> is the abbreviation for

<...> <11F(0)14(1)(x,T)4÷)(x',T)1(1)F(0)> . (8.24)

This latter bracket is most important for our purpose. It shows us that theprobability of finding the detector atom in its upper state is quite essen-tially determined by the correlation function (8.24) between positive andnegative frequency parts of the vector potential. The connection with thecoherence function quoted at the beginning of this section can be madeapparent immediately. We could have performed the whole procedure inthe dipole approximation in which case the electric field strength wouldhave appeared instead of the vector potential. Repeating all the abovesteps our final result then would depend on E, replacing A in eq. (8.24)everywhere by the corresponding E.

<4: F (0)1 E(x , r)Ek+)(x' ,1-')1(13 F (0)> (8.25)

[Another way of going from eq. (8.24) to (8.25) can be achieved by takingthe time derivatives of the As, compare sections 5.1 and 5.81

To be quite clear let us repeat some essential points. The As representfree fields in the interaction representation. Free refers to the interactionwith the detector. On the other hand, when there is an interaction of Awith the source, the As are the full Heisenberg operators with respect to the

(8.22)

source. We will present later on a number of explicit examples how the Asand the correlation function (8.24) can be explicitly determined. Thequantum mechanical averages in eqs. (8.24) and (8.25) refer to the fieldand source variables but no longer to the detector. In our present modelwhich represents the detector by a single atom, x and x' are very closetogether within the atomic dimensions. When we use a system of atomswith macroscopic dimensions, for instance crystals, one may also - encoun-. itr macroscopic differences of x and x' but we shall not consider this case

urther here. In the dipole approximation we can replace the coordi-nates x and x' in the A's by xo. To evaluate P2 for practical cases one hasto take into account an ensemble of atoms and to sum up over theindividual contributions. Furthermore each of these atoms may have morethan 2 states. One has then as usual to average over the initial states and tosum up over the final states of all these atoms. Whether the double timeintegration T can be replaced by a single one depends on the spread ofatomic energies (band width 6,4) of the counter involved in the countingand detecting process). If z> 1/Ato this can be achieved. These averagingprocesses are somewhat involved but do not give us a new insight at thismoment. Therefore, we just state that we find that the detecting rate isdirectly given by functions of the form of eq. (8.24) or (8.25) where theother factors depend on the detector. As it transpires from our discussionjust performed we may put x = x', t = t'. Thus eqs. (8.24) and (8.25) arethe quantum mechanical analogues to eqs. (2.24) with (2.25). In order toobtain correlation functions containing different times and space points wemay proceed in complete analogy to sections 2.2 and 1.11.

We conclude this section by two comments: (1) it is simple to extend ourprocedure to processes in which a successive annihilation of photonsoccurs. In this case we have to apply H1 severalseveral times on OM, togetherwith the corresponding sequence of time integrations. The probability thatn photons are absorbed is then determined by correlation functions of theform

<0F(0)1M )( x l t1) • • t)

x 4+)(x;„ • • 4,+)(xi, ti)10F (0)>. (8.26)

(2) One might also imagine processes by which light is detected not byupward transitions but by downward transitions. In our specific case insuch an experiment the detector atom is initially in the upper state 2 and isthen caused to make a transition towards the lower state 1. Repeating allour above steps one readily verifies that this has the effect of exchangingthe role of the positive and negative frequency parts of the vector poten-tial A.

Exec+

(1) SixHint:furtloer

(2) Dis

<OA(

8.2. E2

In theWe shs

111/

<4

We defield st

E;

In itor whitand ancases otion fatdent arbut notin (8.2Throwwhich(8.27),whichunaffecfunctio•oi

8.2 Examples of the evaluation of quantum mechanical coherence functions 271

)1es how the As;termined. Thefer to the fieldpresent modelare very close

'stem of atomsy also encoun-asider this caseice the coordi-1 cases one hasa up over thenay have moreal states and tole double time

the spread ofn thiguntinghes raginginsight at this

!tecting rate is.25) where theour discussionand (8.25) are5). In order topace points we

! to extend our)11 of photons(I)(0), togetherrobability that'fictions of the

(8.26)

;tected not by3ecific case in- state 2 and isRepeating all

of exchangingvector poten-


(1)Show that E(-)(x, t) and E(+)(x', t') do not commute.Hint: Use eq. (5.149). bx, bZ are taken in the Heisenberg picture. Usefurther eq. (5.154) (in the Heisenberg picture).

(2)Discuss under which conditions (8.26) can be replaced by

<OF(0)1E,y )(x 1 ,1 1 )• • Ei(: )(x„, t n )E1: )(x „, tn )- • Er(x l ,t 1 )14)F(0)> .

8.2. Examples of the evaluation of quantum mechanical coherence functions

In the following we will show how we can evaluate coherence functions.We shall do this for the following coherence functions

<0:1:01E1')(x4)14:1)>, (8.27)

<01 , t')Ei +)(x, t)10>, (8.28)

<4, 1 E,"(x' t')EI-)(x' , t')E1 +)(x, t)E,(x, 010> . (8.29)

We decompose the positive and negative frequency parts of the electricfield strength in the usual way (compare section 5.8)

Eie"(x, t) = E cox , ,(x)bx, (8.30)

t) = E (8.31)

In it A distinguishes the different light modes which may belong to a cavityor which might be modes of free space. b, bx are as usual the creationand annihilation operators. ux , ,(x) are the classical field modes, in mostcases of practical applications running waves. c, cx are certain normaliza-tion factors (cf. 5.149). In the Schr6dinger picture the Os are time depen-dent and bx, bZ time independent. This picture is applicable only to (8.27),but not to (8.28) or (8.29), however. The reason for this lies in the fact thatin (8.28) or (8.29) operators E occur which contain different times.Throughout this chapter we will therefore use the Heisenberg picture inwhich the whole time dependence is inherent in b, bx. The brackets in(8.27), (8.28), and (8.29) again denote the quantum mechanical averagewhich affects bZ, bx, but leaves all other quantities in eqs. (8.30) and (8.31)unaffected. When we insert eqs. (8.30) and (8.31) into one of the coherencefunctions (8.27)-(8.29), we can take out of these coherence functions allparts of eqs. (8.30) and (8.31) except for bj1", bx. Thus the evaluation of eqs.

(8.27)—(8.29) is reduced to the evaluation of expressions of the form

<0 i bx( t )i 0 >, (8.32)

<1n 1/4,;(e)bx(t)14)>, (8.33)

<4)Ibtt4(t4 )b 3(t3 )bx 2(t2 )bx 1(t 1 )1 0:1)>. (8.34)

We now show, by examples, how we can evaluate such expressions. Wefirst start with free fields in which case the field operators obey theequations

dbx dbZ=

d—iwxbx, = zwxbZ , (835,36)t dt

with the solutions

b(t) = bx(0) exp[ — iwxt], b(t) = b(0)exp[iwxt]. (8.37, 38)

b(0) and b(0) are the operators taken at initial time t 0. We assumethat at that time the Heisenberg picture and the Schrodinger picturecoincide. By inserting eqs. (8.37) and (8.38) into (832)—(834) we readilyobtain

<Olbx(t)10> = exp[—iwxt]<dolbx(0)1(1)>,

<0 b;,',(e)bx(t)I = exp [ ioixt] <0 I b it; (0)bx(0)i 0 > ,

(8.34) = exp [ hax 4r4 iCO A3 t3 i(A) 12t2 hOxit

X < 1:13 I 14:4(0) b(0)bx 2(0)bx ,(0) I (1) > . (8.41)

Thus we have only to evaluate expectation values at the initial time whichwe can easily do in the Schrodinger picture.

We first choose for cl) eigenfunctions of the Schrodinger equation, i.e.eigenfunctions in the energy or photon number representation

1 O A b;;P( bZ)nA2 • (8 A2)-1- . 2 • ••. n /

ynx i !nx 2 ! • • • nx.!

Using methods developed in sections 3.3 and 5.3 we may immediatelywrite down the results. We obtain

<431bx(0)10> = 0, (8.43)

< 4:1:1 11)Z, (0)b(0)10> = 8xx,nx , (8.44)

<0I bZ.(0)bZ,(0)bx 2(0)bx ,(0)I >

= - (nx i + 1)nxiaxixpx,xpx4xi• (8.45)

We obtained the result (8.43) earlier, namely (8.27) vanishes when we are

(8.39)

(8.40)

dealing winumbers.expressionsaw we oE

<CE

<41:11E

Coherenchoose ascoherent f

(I) =

The

Using the

<41b;

Again weby using t

<(1)1E

Similarly1

<4:11E1—

= Exv

which we

(833;

Thus, (85:So far

respect to

•

§8.2 Examples of the evaluation of quantum mechanical coherence functions 273

form

(8.32)

(8.33)

(8.34)

!ssions. Wes obey the

(8.35, 36)

(8.37, 38)

We assumegerOurewe dily

(8.39)

> (8.40)

(8.41)

time which

illation, i.e.

(8.42)

nmediately

(8.43)

(8.44)

. (8.45)

leneare

dealing with free fields and the field being in an eigenstate of photonnumbers. These results put us in a position to write down the explicitexpressions for the coherence functions we wanted to calculate. As we justsaw we obtain

<erlE,(—)(x,t)1(1)>= o,<4)1E1-)(x',e)EV(x,t)lo>

= E IcA l 2nxut ,i (x)ux ,k(x)exp[iwx(t' — t)]A

Coherent initial fields. We now do the same steps for free fields butchoose as initial field (which then, of course, persists for all times) acoherent field

= gLexp[ E 13xb;', 4)0 . (8.47)

The normalization factor 91., is given by

9t,= exp{ E 1,3,11

Using the results of section 5.5 we readily obtain

<0 1 bZi 4) > = 11Z

<4)1 1,Z2bx,1 4)>= ' 2 'I'

<4)1b;',4b;',,bx,bA,14)> = St.M3SA2SAI*Again we may determine the coherence functions (8.27), (8.28), and (8.29)by using these intermediate results. Thus we obtain

<411E1 -)(x,t)14)>= E ctu'L(x) exp[ ico A t /1;1. . (8.52)

Similarly we obtain

<cDIE1-)(x/,e)Ek+)(x,t)1(1)>

= Ectcxut. ,j (x')ux,k(x) exp[ icove — icoxt] (8.53)XX'

which we can also cast into the form

(8.53) = <4)IEJ(-)(x',e)14)><4)1Eil+)(x,t)14)>. (8.54)

Thus, (8.53) factorizes into coherence functions of lower order.So far we have evaluated coherence functions for free fields. With

respect to our next chapters it is most illuminating to calculate coherence

(8.46)

(8.48)

(8.49)

(8.50)

(8.51)

functions of fields driven by prescribed classical sources. For simplicity weconsider a single field mode. As we saw in section 5.7, the operators b andb + of the field mode obey equations of the form

db—dt = —iwb + f(t) (8.55)

and

dt — iwb + +f*(t). (8.56)

In it f(t) describes the effect of a classical source on the field mode. f(t)can be an arbitrary classical time dependent function, for instance it candescribe the oscillations or currents of classical charges. The solution of eq.(8.55) reads

b(t) = exp[— — r)] f(r)dr + b(0)exp[ —i(ot],

B(t) (8.57)

where we have chosen as a specific initial time t = 0. But any other initialtime could have been chosen as well. b(0) is the photon operator at theinitial time and is chosen to coincide with the corresponding operator inthe Schrodinger picture. One readily convinces oneself that eq. (8.57) andthe corresponding hermitian conjugate satisfy the commutation relationsfor Bose operators. We now evaluate the coherence functions. We firstchoose 4> as eigenstate of photon number. We then readily obtain

<19(010> = B(1). (8.58)

Of more interest is the correlation function in which two operators areinvolved, e.g.

<01b;',.(e)bx(t)1(1:0>. (8.59)

Here we now have furnished b + and b with mode indices having in mindcoherence functions of the form (8.32)—(8.34). Inserting (8.57) and itshermitian conjugate into (8.59) we readily obtain

(8.59) = B(e)B(t) + Bt(e) exp[ —i(axt]

+ B ( t ) exp[iwx,t1 + <bbx> exp[itax,t' — iwAt].

(8.60)

The final evaluation of this expression depends on the expectation valuesfor the bs in the Schrodinger picture. Using our results we obtained abovein this section we readily find the following results. When (1) is a photon

eigenstate

(8.59)

If (1) is a co

(8.59)

results. Whpresent at tstates of tiphysical prtransferreddence of fthe source tthe free fiel(

Let us nois that 2k

in which ca:

B(t) =

In the case

B(t) =

i.e. the fieldan exercise Ifunctions fcexponentiallfunctions vinWhat cohes

8.3. Cohere'

We wish to(8.28) allow5neously emi.

r(t + T

It is mostcounter. Col

db+

§8.3 Coherence properties of spontaneously emitted light 275

eigenstate we obtain

(8.59) = B(e)B(t) + ASx 2,, exp[ico(1 — (8.61)

If I. is a coherent field

(8.55) (8.59) = (Bt,(e) + ot,exp[iwx,t1)(BA(t) + fl), exp[ — iw xt]) (8.62)

results. While in eq. (8.61) ri ), is the number of photons of the mode Xpresent at time t = 0, /3)„ igt in (8.62) are the amplitudes of the coherentstates of the corresponding modes. It is interesting to study how thephysical properties of the source, i.e. its specific time dependence, istransferred by the field modes to the detector, i.e. how this time depen-dence of f is reflected by coherence functions. From our results we see thatthe source term f enters into the Bs, but in addition certain properties ofthe free fields also appear as is visible from eqs. (8.61) and (8.62).

Let us now study a few explicit examples for f(t). The simplest form of fis that of a harmonic oscillation

f(t) = A exp[ (8.63)

in which case we readily obtain— i

B(t) = texp[— icoot] — exp[ —icot]}. (8.64)— coo

In the case of resonance, eq. (8.64) reduces to

B(t) = texp[ — hot], (8.65)

i.e. the field mode amplitude increases proportional to time. We leave it asan exercise to the reader to calculate similar expressions and the coherencefunctions for the following cases: (1) f(t) is, in addition to eq. (8.63),exponentially damped and (2) f(t) consists of a sum of exponentialfunctions within a frequency interval around co and with random phases.What coherence functions result when the phase average is performed?

83. Coherence properties of spontaneously emitted light

implicity we!Tutors b and

We wish to show explicitly, how the formalism of the coherence functions(8.28) allows us to directly determine the coherence properties of sponta-neously emitted light. To this end we calculate

F(t + t) = 01)(0)1E"( X, t T)E(+)(X, t)10(0)> (8.66)

It is most important that the atom should serve as source and not as acounter. Consequently E" and E" (or A" and A") must be the full

— iwxt].

(8.60)

lation valuesained aboveis aiton

(8.56)

i mode. f(t),tance it can,lution of eq.

0.57)

other initial!tutor at theoperator in

i. (8.57) andon relationsns. We firsttamn

(8.58)

perators are

(8.59)

ing in mind57) and its

Heisenberg operators:

E(-)(x,t) = exp[iHt/h]

ex exp[ - iicx.r]}- exp[ - iHt/h] ,

(8.67)

where H is the complete Hamiltonian (field + source). Since E(-) andE(+) are decomposed into a linear combination of operators b + , 6, we firstinvestigate:

< 0:1)(0)IbZ(t + T)bx,(t)14)(0)>, (8.68)

whereb(t) = exp[iHt /h]bXI- (0) exp[ - iHt /h],

b(t) = exp[ iHt /h] bx(0) exp [ - iHt/h]. (8.69)

4:1)(0) is the initial state (one atom excited, no photon present). We insert(8.69) into (8.68) and observe that

exp[ - iHt /h] 40 (0) -= OW, (8.70)

so that

(8.68) = <0(t + r)lbZ exp[ -iffr/C1(1)(t)>. (8.71)

We now use the explicit form, eq. (7.267) of OM and find after someelementary algebra:

(8.68) = ct(t + T) eXp[ iCJ A(i + 7)] cx,(t)exp[ + ielT]

X <4 0 a 1 exp[ -ifir/h]at 14)0. (8.72)

Since ai1- 4:00 represents a state in which the electron is in its groundstate andno photon is present, H1 can cause only virtual transitions. Also, since theygive rise only to renormalization effects, we need not consider thosetransitions, so that of exp[ - iHT/h] only exp[- ii-1„,r/h] is left.

Therefore we finally find

(8.68) = c(t + ,r)cx(t)exp[i(cax- (ov)t + (8.73)

When we insert eq. (8.73) into eq. (8.66) we obtain a double sum over X, X'of expressions (8.73), still multiplied with coefficients occurring in eq.(8.67). This evaluation is rather tedious and because it sheds no light onthe quantum theoretical treatment, which has been completely done above,

x.[

.07-

Vhcox2e0 In it, the differe

(3: transitio/ fy: opticair: distance: at8: angle betw021 : optical (eIn deriving eq.

y < 15,

The cond4L(

confine otWv,vanish (on acctexplicit derivati

It is most rm.:oscillator leadsorder to find tiobtain immeclia

y(t + T,t)

This clearly . slcoherence time<e(-)(x,t)> doof e(x, t) is uni

8.4. Quantum E

This section isexperiment whienergy levels.quantum theorcoherence funcconsider the eiduration is shoexcited levels 2state which cal

•

we merely quott

§8.4 Quantum beats 277

we merely quote the result:

r(t + T,t) — exp[ —2y(t — L'e )]- sinr22 13 1421 1 2 exp[icTvr — yr].co-4

2 /reoc4

(8.74)

In it, the different quantities are defined as follows:(7.): transition frequency of the atom,y: optical half-width calculated in eq. (7.274),r: distance: atom-detector,0: angle between atomic dipole moment and direction atom-detector,1521 : optical (electric) dipole moment.In deriving eq. (8.74) we have assumed that

, cy < r = Ix ] » —(7) = r < ct . (8.75, 76, 77)

The condition (8.75) is always well fulfilled. Equation (8.76) means that weconfine ourselves to the wave region of a Hertzian dipole, while r wouldvanish (on account of causality) if eq. (8.77) were not fulfilled. For theexplicit derivation of eq. (8.74) the dipole approximation has been used.

It is most remarkable that the classical treatment ;of a damped Hertzianoscillator leads to exactly the same expression. When we normalize I' inorder to find the complex degree of coherence (compare section 2.2) weobtain immediately:

y(t + T, 1) = eXp[ itT)T — yr]. (8.78)

This clearly shows, that the spontaneously emitted wave track has acoherence time 1/y. Although r(t + T, t) does not vanish, <E(+)(x, t)> =-

<e"(x,t)> do. This can be interpreted as meaning, that the initial phaseof e(x, t) is unknown. This phase cancels out, however, in eq. (8.66).

8.4. Quantum beats

This section is of interest in several respects. First of all it describes anexperiment which allows one to measure the splitting of closely spacedenergy levels. Secondly it will allow us to demonstrate basic ideas ofquantum theory and thirdly it is a nice example of how to evaluatecoherence functions. We first briefly describe the experiment. Let usconsider the energy level diagram of fig. 8.1. By a short pulse whoseduration is shorter than the inverse of the frequency splitting of the twoexcited levels 2 and 3, we excite these levels. The excited atom is then in astate which can be described by a coherent superposition of the wave

exp[ — iHt / h] ,

(8.67)

Since E" ands b, b, we first

(8.68)

• (8.69)

sent). We insert

(8.70)

(8.71)

Ind after some

+ kir]

(8.72)

;roundstate andAlso, since theyconsider thoseleft.

(8.73)

sum over A, A'ccurring in eq.ecis no light onely done above,


e, outgoingpulseI/

MINIMSleaatom

Fig. 8.1. Scheme of quantum beat experiment [after Haroche, in: High Resolution LaserSpectroscopy, ed. K. Shimoda, (Springer 1976).] The left part gives a diagram of the differentatomic levels. To compare this diagram with the text identify l and f' with g. The right handside shows the basic idea of the experiment. The atom is excited by a light pulse in thedirection indicated by the outgoing pulse. In a different direction the light emitted from theatom is measured by a photo multiplier and the resulting signal exhibited on a monitor(compare fig. 8.2).

functions of the two excited levels 2 and 3. From there the electron candeexcite to some of the lower states depending on selection rules. Accord-ing to the different decay channels the excited atom can now emit light atthe corresponding frequencies. In the following we assume that we use adetector which is sensitive for photons emitted from levels 2 and 3 to level1. The essential result of such an experiment is shown in fig. 8.2. In it weplot the intensity of the light absorbed by the detector as a function of time

after the pneous emioccurs, thiby (W3 —frequencyment of ttan import.

We nowone hand Ision (cf. seevaluate cc

In our tr3 of a singthe atom aFurthermosimplaiRatatomWsto the predifferent v%directions (coefficientand 3 1.

111=1016.0

5.0

i I +0

3.0"."

• E periment2.0

I0

0 11 oseci I I

2 m sec1 1

20 3 0 4 0 50 60Channel number

Fig. 8.2. This figure shows the number of counts as a function of time. The oscillatorybehaviour of the quantum beat signal is clearly visible. (After Haroche, 1.c.)

70

—)r )—sIGNAL

P M

where we 1

To solve

drin the interstraightfortWe expectlevel 1 whe

•


:h Resolution Laserxam of the different:11 g. The right hand1 light pulse in theht emitted from thebited on a monitor

rule.Accord-he on can

aw emit light ate that we use a2 and 3 to levelig. 8.2. In it we'unction of time

after the pulse had excited the atom. In contrast to conventional sponta-neous emission (compare section 7.6) where simple exponential decayoccurs, this decay is now modulated. The modulation frequency is givenby (W3 — W2 )/h w32 as we will derive below. Because the modulationfrequency can be directly deduced from fig. 8.2 it allows for a measure-ment of the energy level splitting. This kind of measurement has becomean important method to measure the splitting of energy levels.

We now give a quantum theoretical treatment of this effect which on theone hand generalizes the Weisskopf—Wigner theory of spontaneous emis-sion (cf. section 7.9) and on the other hand allows us to show how we canevaluate coherence functions.

In our treatment we shall deal explicitly with the three energy levels 1, 2,3 of a single atom according to fig. 8.1. We treat the interaction betweenthe atom and the field mode as usual in the rotating wave approximation.Furthermore we start right away in the interaction representation. It is asimple matter to generalize the interaction Hamiltonian of the two levelatom whose spontaneous emission we have treated in sections 7.6 and 7.9to the present case. Since the split energy levels are connected withdifferent wave functions, which may give rise for instance to differentdirections of polarizations of the emitted light, we distinguish the couplingcoefficients g by indices 2, 1; 3, 1 corresponding to the transitions 2 —> 1and 3 —› 1. The interaction Hamiltonian then reads

hE { g214 albAexp[

+ g1aa2b exp[iA21,At]

+ g314a1bxexp[

(8.79a)

(8.80)

70

+ 41 4 a3 bZ exp[iZ1 31, xt]}, (8.79)


co x — ( W.; — WO/h -

To solve the Schrodinger equation

ih -dr "in the interaction picture it seems reasonable to make an ansatz which is astraightforward generalization of the Weisskopf—Wigner hypothesis (7.267).We expect (I) to be a superposition of the excited states 3 and 2 and thelevel 1 where in addition a photon is present. Thus our present hypothesis

e. The oscillatory

411

•


reads

= a3(t)4(1:00 -1- a2(t)at1)0 + E ex(t)bat(100 . (8.81)

Inserting eq. (8.81) into eq. (8.80) and comparing coefficients of 0 and 1photon states leaves us with the following set of equations

a = —iE exp[ —i6, 31,At]g3IcA(t), (8.82)dt 3

X

—a2 = exp[ –i21,xt]g2icx(t), (8.83)U'

—d

cx= – ifglia2 exp[ iA21,xt] glia3 exp [ iA 31, xt]). (8.84)dt

We supplement these equations by the initial condition according to whichthe electron is in a coherent superposition state of levels 3 and 2 at theinitial time t = 0

4)(0) = a3(0)400 + a2(0)4400 . (8.85)

Normalization requires

1 03(0)1 2+ 1 02(0)1 2= 1. (8.86)

In the following we will assume that the two levels 2 and 3 are coupled inthe same way to the electromagnetic field except that these states may giverise to different directions of polarization. As a consequence we expect thatthe two levels decay with the same rate constant y. Accordingly we makethe hypothesis

a .(t ) = a .(0) exp[ – yt]J J

(8.87)

Inserting eq. (8.87) into (8.84) and integrating these equations yields

3 elai(0)c(t) = – i E . – (exp[(iAii,A – y)t] – 1). (8.88)

.j=2 Jidt

y

For our following discussion it will be convenient to split the sum overj = 2,3 into its corresponding parts explicitly

c(t) = c2,A(t) + c3,A(t). (8.89)

We now wish to calculate the field intensity measured in the detector atspace point x and time t. For simplicity we assume that the atom is locatedat the origin of the coordinate system. The expectation value for the field

intensity is gi,

K = <11)(

In this descrtime-indepenccare of by (I).that eq. (8.90'take equal tinto the Schrodpose the positplane waves.polarizationmodes with ti

E(+)(x)

E()In it we havebook by thecreation opera(8.92) into (8.!

K= Ek',k

Evidently wethe form

<41(t)lb:

To this end v(8.94). Evaluabook (see forthe final result

(8.94) =

Note that thethe same time

What shouinvolves the pdoes not shedthe final resu.

intensity is given by

K = <40(t)1E( - )(4E(+)(x)14:0(t)> (8.90)

In this description we are in the Schrodinger picture. In it E( ' ) aretime-independent field operators while the time dependence is fully takencare of by (1). It is an interesting exercise for the reader to convince himselfthat eq. (8.90) is a special case of the coherence function (8.28) when wetake equal times and proceed simultaneously from the Heisenberg pictureto the Schr6dinger picture (see exercise). To evaluate eq. (8.90) we decom-pose the positive and negative frequency parts of the field strengths intoplane waves. Because we have an experiment in mind in which a specificpolarization is measured, it suffices to confine the following expansion tomodes with the corresponding polarization vector e.

E(x) = E bka,keexp[ikx], (8.91)

E(-)(x) = E bz- gckeexp[ — ik.r]. (8.92)

In it we have replaced the usual mode index A used-everywhere in ourbook by the wave vector k. bk , b -Ik- are the photon annihilation andcreation operators. 9Z, k is the normalization factor. Inserting eqs. (8.91) and(8.92) into (8.90) yields

K = E 9tkgck,<4,(01b,+, bk ,10(t)> exp[i(k' — k)x]. (8.93)k',k

Evidently we are now left with the evaluation of the expectation values ofthe form

01)(t)114- bk ,10(0> (8.94)

To this end we have to insert the explicit expression for eq. (8.81) into(8.94). Evaluations of this type have been done at several places in thisbook (see for instance section 8.2) so that we can immediately write downthe final result

(8.94) = ( cI k + c k )(c2,k , (8.95)

Note that the coefficients c, c* appearing in eq. (8.95) must all be taken atthe same time t.

What should now follow is a somewhat lengthy calculation whichinvolves the performance of summations or integrations over k. Since itdoes not shed any light on the physics we skip the details and just presentthe final result. In it we make use of the dipole approximation which

(8.81)

of 0 and 1

(8.82)

(8.83)

(8.84)

ng to whichnd Eli the

(8.85)

(8.86)

coupled ines may giveexpect thatly we make

(8.87)

yields

(8.88)

e sum over

(8.89)

detector atri is locatedor the field

•


means that we use the specific form of eq. (7.39a) for the couplingcoefficients g. The final result then has the following form

kK(x, t) = (47E0 ) - 2 2 E (e1,1i )ai(0)ay,(0)(e191 j,)I x 1 2 "

x 0(t — )exP[(—iw11,-- 1')( 1)1,

wherecoif = (Wi — Wy )/ h , j = 2, 3; / = 2,3.

In it we assume that a special direction of polarization e is measured by thedetector. i are the dipole moment matrix elements for the opticaltransitions from levels j = 2 or 3 to level 1. The field intensity measured atspace point x and time t, as given by eq. (8.96), depends on various factorswhich are worth to be discussed in detail. Starting from the right to the leftwe observe the following.

The field intensity decays exponentially with time t and the decayconstant y corresponding to the well known spontaneous emission. How-ever, this decay appears modulated by the Factor exp(— — Ixl/c)).This factor leads really to a modulation as can be seen when we performthe sum over j, j'. We then readily may verify that this exponentialfunction gives rise to a term proportional to cos(icau(t — I xj/c)). Thisfunction causes the modulation of the light signal and occurs with thefrequency splitting. e is the Heaviside function which vanishes when itsargument t — Ixl/c is negative and it equals unity for positive t — Ixl/c. Itjust reflects the fact that a signal cannot proceed faster than the velocity oflight. Of further particular interest is the appearance of a(0)s, which arethe amplitudes of the initial wave functions. Since these amplitudes arecomplex they still contain certain phases. When we prepare an ensemble ofatoms in initial states with random phases we have to average eq. (8.96)over such phases. In such a case the average products oci a, , i j yield 0and exactly those terms will vanish which give rise to the modulationeffect. This shows quite clearly that the modulation effect depends sensi-tively on the way the initial atom, or more precisely speaking, the initialstate of an ensemble of atoms is prepared. For instance, if we excite theatoms incoherently from their groundstate to their excited states by abroad band source such cancellation of contributions in eq. (8.96) willoccur. On the other hand a short pulse with a time-limited frequencyspread only can cause a coherent superposition of the initial states. Wethus see that the outcome of the photon beat experiment depends in a

(8.96)

(8.97)

sensitive way.not allow us t

However,interpretationbeautiful ramYoung's cloutquote Haroch,

8.4.1. Physical

In the experimby the atom tEexcited state 28.1. When thebetween the tst.

scattered throuamplitu ).tude corrand their sum

leads to the intintegration ovtmodulations itmodulated, res

uct does indeetturn beats areYoung's doublproblem, it is 4

order to detemwill result in atto make use ofwith a polariz.,'example. In thsand there are oithe emission aband filter cent

a result of thisrestricted to afrequency. In thare again losi

*The reader is ad,

•


sensitive way on the kind the initial states 2 and 3 are prepared. Space doesnot allow us to enter the mathematical formulation of that problem.

However, in conclusion of this section we should like to discuss theinterpretation of this photon interference experiment which indeed is abeautiful manifestation of typical quantum mechanical processes similar toYoung's double slit experiment. Since we can hardly do any better wequote Haroche's description of this effect:

8.4.1. Physical interpretation of the quantum beat signal*

In the experiment sketched in fig. 8.1, the impinging light pulse is scatteredby the atom through two different channels, corresponding each to a givenexcited state 2, 3. These channels are symbolized by the diagrams of fig.8.1. When the atom has re-emitted a photon there is no way to distinguishbetween the two possible channels and to tell whether the photon has beenscattered through level 2 (with an amplitude a 2 ) or through level 3 (with anamplitude a3 ). As a general postulate of quantum mechanics, the ampli-tude corresponding to these two indistinguishable processes must be addedand their sum must be squared to yield the expression of the signal. Thisleads to the interfering term ai al', in the expression (8.96) of K(x,t). Afterintegration over the photon energy, these interfering terms produce themodulations in the observed signal. As a 2 and a3 are at resonancemodulated, respectively, at frequencies W2 /h and W3 /h their cross prod-uct does indeed exhibits a modulation at frequency (W3 — W2 )/h. Quan-tum beats are thus a typical quantum interference effect quite similar toYoung's double slit experiment. In complete analogy with this latterproblem, it is quite clear that any attempt to perform an experiment inorder to determine through which channel the photon has been scatteredwill result in a disappearance of the beat pattern. For example, we may tryto make use of the polarization selection rules and detect the light emittedwith a polarization ed which can be emitted by only one level 2, forexample. In that case, however, the matrix element (e d0 3 ) is equal to zeroand there are obviously no beats in eq. (8.96). Another way of determiningthe emission channel would be to put in front of the detector a narrowband filter centered, for example, around the (W2 — W1 )/h-frequency. Asa result of this filtering, the summation over k and k' in eq. (8.93) is nowrestricted to a small frequency interval excluding the (W3 — WI )/h-frequency. In that interval the amplitude remains very small, and the beatsare again lost.

'The reader is advised to compare the arguments presented here with those of section 1.10.

coupling

(8.96)

(8.97)

ed by thee optical.asured atus factorsto the left

11 he layon. How-- I xl/ c)).

performponential'c)). Thiswith thewhen its

- ixl/c. Itelocity ofwhich aretudes aresemble ofeq. (8.96)

= j yield 0odulationrids sensi-the initialexcite theates by a8.96) willfrequencytates. Weends in a

•

0

•

;er picture) andum over i =j =

cp(—iHt/h), H:

9. Dissipation and fluctuations in quantum optics

9.1. Damping and fluctuations of classical quantities: Langerin equation andFokker—Planck equation

In this chapter we will deal with some problems which at first sight seem tohave little or nothing to do with quantum optics. Indeed the correspondingideas and methods were introduced into quantum optics rather late butthey have turned out absolutely necessary for the explanation of manyphenomena, especially in laser physics and nonlinear optics.

The phenomenon which we have in mind is the Brownian motion. Whena particle is immersed in a fluid, the velocity of this particle is slowed downby a force proportional to the velocity of this particle. When one studiesthe motion of such a particle under a microscope in more detail onerealizes that this particle undergoes a zig-zag motion (compare fig. 9.1).This effect was first observed by the biologist Brown. The reason for thezig-zag motion is this. The particle under consideration is steadily pushedby the much smaller particles of the liquid in a random way. Let usdescribe the whole process from a somewhat more abstract viewpoint.Then we deal with the behaviour of a system (namely the particle) which iscoupled to a heatbath or reservior (namely the liquid). The heatbath causestwo effects: (1) It decelerates the mean motion of the particle and (2)Simultaneously the heatbath causes statistical fluctuations. As we shall seelater atoms or the light field are by no means isolated systems. Forinstance optically active impurity atoms in solids interact all the time withlattice vibrations. Gas atoms steadily suffer collisions with other atomswhich form the heatbath for the individual atom under consideration. Thelight field in the cavity interacts all the time with the walls forming aheatbath for the light field. From these examples it transpires that thenotion of a heatbath plays an important role for systems considered in ourbook. Its importance will become fully evident when we apply these

•

286 9. Dissipation and fluctuations in quantum optics

propertiesand to thethe force I

Fig. 9.1. Pathway of a particle undergoing Brownian motion.

(Fomotion antSynergetic

(9.1) only someout that N,N

In the present case the force K(t) is composed of two parts. (1) One is the given in thfriction force which is proportional to the velocity of the particle but with In manyreversed sign

stochasticdifferent ti

0((2) The particle is steadily pushed by the particles of the fluid. To develop <Ft)

a model for these pushes we compare this process with a football game For a pro(with a soccer game) where the soccer players kick the ball again and again proportiombut at each instance for a short time interval only. We idealize these events force K itby assuming an infinitely short time for each individual push. The force stochastic texerted on the particle (ball) is then represented by a 8-function. We motionassume that the individual pushes occur at times ti . Furthermore, we shall do

m—admit that the pushes occur in the right or left direction in a random dtsequence. According to these ideas, we write the force representing the For simplifrandom pushes in the form introduce n

Fo(t) = E 8(t — tf )(-± 1)i . (9.3) Y7oThe equatic

An example for Fo is given in fig. 9.2. cp measures the strength of thepushes. The symbol (± 1)) is meant to indicate the direction of the push at d v

each time tf . The times tj form a random sequence whose statistical

methods to laser theory and nonlinear optics. But let us first return toBrownian motion where we consider a one-dimensional example. Let theparticle have the mass m. We denote its velocity at time t by v(t). Underthe impact of external forces the particle is accelerated according toNewton's law

m v = K(t).

Kfriction = YOV • (9.2)

§9.1 Damping and fluctuations of classical quantities 287

t, tts3L t5

3t return topie. Let the

:coill t tov(t der

Fig. 9.2. Example of a random sequence of kicks OW —F0 ( t )

properties we shall not discuss here. We assume that the pushes to the leftand to the right occur with equal frequencies. Therefore when we averagethe force F0(t) over all pushes we obtain

<Fo> = O. (9.4)

(For a more detailed presentation of the classical theory of Brownianmotion and its general formulations I refer the reader to my book (Haken,Synergetics — An Introduction, 2nd ed., Springer, 1978)). Here we presentonly some of the basic features of classical Brownian motion. It will turnout that we need a fully quantum mechanical treatment which will begiven in the subsequent sections of this book.

In many cases it is important to know the correlation functions of thestochastic force eq. (9.3). Forming the product of these forces at twodifferent times and averaging over all pushes we obtain

<F0(t)F0(0> = C8(t — t'). (9.5)

For a proof see, e.g. the above mentioned book. The constant C isproportional to cr,2 and thus a measure for the strength of each push. Theforce K in eq. (9.1) is composed of the friction force (9.2) and thestochastic force (9.3). This yields the fundamental equation of Brownianmotion

dvm—dt = — Toy + Fa t ) . (9.6)

For simplification of eq. (9.6), we divide this equation by the mass m andintroduce new abbreviations

= Yo/m ; F(t) = Fo(t)/ m. (9.7)

The equation which we now want to consider thus reads

—dv —yv + F(t). (9.8)dt

(9.1)

) One is the:le but with

(9.2)

To develop3tball gamen and againthese events. The forcenction. We,re, we shall

a random:senting the

(9.3)

ngth of thethe push at

e

To this end -(9.10) where N

<v(t)v(t

Since perfontintegration w,obain

Itfrexio o

In this expressintegration ofthe case t> t'

We now consi

I + t' »

but that the ti

It —

In this case, e.

<v(t)v(t'


Due to the relation (9.7), the correlation functions (9.4) and (9.5) arereplaced by

<F(t)> = 0

<F(t)F(1) = Q8(t — t'), (9.9)

where Q is a measure for the strength of the fluctuations. Equation (9.8) isa differential equation of first order whose formal solution reads

v(t) = f t exp[ — y(t — '0] F(T) dT + v(0) exp[ yt]. (9.10)

In it v(0) is the initial velocity prescribed at time t 0. When we wait asufficiently long time which is large compared to the inverse dampingconstant y, the last term on the right-hand side of eq. (9.10) can beneglected. When we then average over all pushes, we obtain by means ofeq. (9.9) the relation <v(t)> = 0. Thus the mean velocity is equal to 0. Onthe other hand, we observe under a microscope that the particle is steadilypushed around and thus achieves a finite velocity after each push. Thevanishing of the mean velocity stems only , from the fact that when weperform the average, positive and negative velocities cancel each other.Thus to get a more realistic measure for the average velocity we must forman expression in which the sign of the velocity no longer plays a role. Thesimplest expression of this kind is <v(t) 2 >. Since it is also interesting toknow how long a velocity is preserved, it is advantageous to consider themore general expression

<v(t)v(e)> (9.11)

Taking t t' we obtain a measure of the size of the velocity independentof its sign. In addition, for t t' we obtain an expression which tells ushow long the velocities remain correlated. The latter can best be seen bymeans of the following limiting case. Letting the time difference between tand t' become very large the particle at time t will no more be able to"remember" what its velocity was at time I', because it has suffered manyrandom pushes in between. In such a case the velocities at times t and t'have become statistically independent. According to basic rules of proba-bility theory the expression (9.11) then splits into the product

<v(t)><v(t')>. (9.11a)

On the other hand, we have seen above that <v(t)> vanishes for t oo, sothat (9.11) vanishes for great time differences. This result means that atthese times there is no longer any correlation. Of course, we expect acontinuous transition of the correlation function (9.11) when going fromt = t' to t — t' oo and we wish to consider this transition more closely.

Fig. 9.46 e exunder Brow

(9.9)

.ion (9.8) isIs

(9.10)

we wait a,e damping10) can bey means ofal to O. On

wW;A'reeach other.must forma role. Theteresting toonsider the

(9.11)

ndependent'itch tells usbe seen bye between tbe able to

lered manynes t and t';s of proba-

(9 .11a)

t oo, sotans that atwe expect agoing from

nore


To this end we form the correlation function (9.11) using the solution(9.10) where we may drop the term « v(0) for sufficiently large t, t'

<v(t)v(e)>= ( fti t

exp[ — y(t — T)]F(71dT'

Jo 0x <exp[ — y(e — T')] AT') dT'› . (9.12)

Since performing the average over the pushes has nothing to do with timeintegration we may exchange the brackets with the time integration andobain

ff exp[ —y(t + t' — T — T')]<F(T)F(T')> dr dr'. (9.13)

0 0

In this expression we may use the relation (9.9). Due to the 8-function, theintegration of the double integral can be replaced by a single integral. Forthe case t > t' we obtain especially

<v(t)v(e)> = Q f exp[— y(t + t') + 2yr] dr

= -2y ft exp[ —y(t — t')] — exp[—y(t + t')]). (9.14)

We now consider a situation in which the times t and t' are so big that

t + t' » 1/y, (9.15)

but that the time difference t — t' is still of the order

It — el 1/y. (9.16)

In this case, eq. (9.14) acquires the especially simple form (fig. 9.3)

<v(t)v(0> = exp[ —y(t — t')]. (9.17)2y

t - t'Fig. 9.3. The exponential decay of the correlation function of the velocity of a particleundergoing Brownian motion. The correlation function is plotted versus the time t — t'.

unity. More pifunction f(v) w

f c° v,:)— cc,

holds. f(v,t)dv,the probabilityinterval v, v +distribution furLangevin equat

a—f = aat at)

[

The factor yv iThe stationary

Ot is the ia

Ot

Ai(

=frQ

Using the fluc(9.22) into the

.ist( v ) = 9

This is of cou

We thus have obtained an explicit expression for the correlation functionof the velocities which tells us both the size of the average of the squaredvelocities and the way the correlation of the velocities dies out. Evidentlythe correlations decay according to an exponential law with decay time1/y.

We now want to derive a formula which allows us to derive the factor Q(which is a measure for the size of the pushes) directly. Taking in eq. (9.17)t = t' and multiplying it by m/2 we obtain on the left-hand side of eq.(9.17) the mean kinetic energy of the particle. According to fundamentallaws of thermodynamics we must assume that the particle is in thermalequilibrium with its surrounding (heatbath). According to thermodynamicsthe kinetic energy of a single degree of freedom in thermal equilibrium attemperature T is equal to kT, where k is Boltzmann's constant and T theabsolute temperature. Thus we have quite generally

L'-727-<v(t)2> = kT. (9.18)

A comparison of eqs. (9.18) and (9.17) yields an explicit expression for QQ = 2ykT/m. (9.19)

As mentioned above Q is a measure for the size of the fluctuations. On theother hand y is a measure for the damping or, in other words, for thedissipation (due to the damping of the velocity kinetic energy is dissipated).The relation (9.19) is probably the simplest example of a fluctuation-dissipation-theorem. The size of the fluctuations is proportional to the sizeof the dissipation. The importance of this relation lies in the fact that itallows us to calculate the size of fluctuations by means of the size ofdissipation.

Equation (9.8) is called Langevin equation. Such equations and theirquantum mechanical generalizations play a fundamental role in lasertheory.

Besides the description of Brownian motion by means of Langevinequations, there exists a second approach namely that by the Fokker–Planck equation. The basic idea about the Fokker–Planck equation is this.Let us imagine a large number of identical experiments in which each timea particle of mass m is undergoing a Brownian motion. At a certain time twe can measure the velocities of the particles in each experiment. Takingthe results of all the experiments together we shall find a certain numberN(v) d v of particles in a given velocity interval v, v + d v. We thus obtaina distribution function N(v) which tells us how frequently the particleshave the velocity v. Invoking probability theory we may go over to aprobability distribution by normalizing the distribution function N(v) to

Fig. 9.4. An examv at a given timevelocity range v


v v+dv

Ilion functionif the squaredJut. Evidently.h decay time

e the factor Qg in eq. (9.17)id side of eq.fundamentalis in thermalrmodynamics;quilibrium atant and T the

(9.18)

resseor Q(9.19)

itions. On the-ords, for theis dissipated).. fluctuation-al the sizee fact that itf the size of

as and theirrole in laser

of Langevinthe Fokker-uation is this.ich each timexrtain time tnent. Taking.-tain numbere thus obtainthe particles;o over to aI don N(v) to

unity. More precisely speaking we introduce instead of N(v) a newfunction f(v) which is proportional to N(v) but for which

f(v, t)dv = 1 (9.20)

holds. f(v, t) dv has the following meaning (compare fig. 9.4). f(v, t) dv isthe probability of finding a particle at time t with a velocity v in theinterval v, v + dv. As is shown in statistical physics the probabilitydistribution function f which belongs to Brownian motion, i.e. to theLangevin equation (9.8), obeys the Fokker-Planck equation

—j = — yv + — — f .at ay

a , [ a Q o2

2 ay2 (9.21)

The factor yv is called drift coefficient, the factor Q diffusion coefficient.The stationary solution with (a/at)f = 0 reads

Using the fluctuation-dissipation theorem, eq. (9.19), we can bring eq.(9.22) into the form

J(v) = 9t,exp[ ---2 kTi•(9.24)m v2

This is of course the Maxwell-Boltzmann distribution function for the

Fig. 9.4. An example of the distribution function obeying the Fokker—Planck equation versusv at a given time t. The shaded area gives us the probability of finding the particle in thevelocity range v v + d u .

J(v) = 9Lexp[ - yv2 / Q]. (9.22)

DI, is the normalization factor and is given explicitly by91, = r y 1 1/2

(9.23)TrQ •


velocity of a particle in thermal equilibrium with a heatbath at tempera-ture T.

Later in this book we shall get acquainted with cases in which the systemunder consideration is coupled to several heatbaths kept at differenttemperatures. Our considerations can be generalized in several respects, forinstance we may consider a La.ngevin equation which contains a moregeneral force. Simultaneously we replace the velocity by a general variableq. The corresponding Langevin equation with a general but still determin-istic force K then reads

—d

q K(q) + F(t). (9.25)dt

F(t) is again the fluctuating force with the properties eq. (9.9). We notethat for the derivation of the corresponding Fokker—Planck equation for(9.25) it is necessary that F(t) is, in technical terms, Gaussian. We will notdiscuss this point here in detail but refer the reader to the literature.

Under these assumptions the Fokker—Planck equation, associated withthe Langevin equation (9.25), reads

f(q, t) ={– K(q) + —2 2

(9.26)

dq a qa a Q a2

Its stationary solution reads

f(q) = 9Lexp – -3- f qK(q9dql . (9.27)

Q is assumed as a constant independent of q and time. Unfortunately thetime-dependent, Fokker— PLanck equation (9.26) cannot be solved ex-plicitly, at least in general, as was possible for a linear force K q. Thuscomputer solutions are required.

We now try to translate our considerations to a damped electric orelectromagnetic field. An electric field propagating in a conductor getsdamped because it accelerates electrons of the material giving its energy tothe electrons. The electrons lose their energy by collisions with latticevibrations. The energy dissipation of the electric field is proportional to theconductivity a of the material. This fact can be derived directly fromMaxwell's equations. From Maxwell's equations including damping onemay derive the telegraph equation describing the propagation of an elec-tromagnetic field in a conductor

a2 a a a2

at 2 eo at

+ — — c.2 IE• = o. (9.28)ax2

We have written down the equation in one dimension, the field strength

being trastanding •field strer

E(x,

where dr(inbetweerand perftdividing t

a2

at2

We havea

8o

The oscill

40)By inserti

–

When wefrequency

Thus wethe electr.

b(t)Apart fro.identicaltake intoto suppler

—b(d t

Note, hosoccurringclassical acal operaincorporatanswer thtgeneral lassioquen


ith at tempera-

hich the systemot at different-al respects, for,ntains a moreeneral variablestill determin-

(9.25)

(9.9). We notek equation forn. We will notterature.ssociiii with

(9.26)

(9.27)

)rtunately the>e solved ex-K q. Thus

td electric ormductor gets; its energy to; with lattice•rtional to theErectly fromtamping onen of an elec-

(9.28)

ield silth

being transverse to the direction of propagation. We now consider astanding wave between two mirrors. For this reason we assume the electricfield strength in the form

E(x , t) = b(t) sin kx, (9.29)

where the wave number k is chosen in such a way that the wave fitsinbetween the 2 mirrors (compare fig. 2.12). By inserting eq. (9.29) in (9.28)and performing the differentiation with respect to x we obtain afterdividing the equation by sin kx

a 2 a[ at 2 t

b(t) = o. (9.30)2KT +-

We have used the abbreviationsa— 2K; kc = coo. (9.30a)EO

The oscillator equation (9.30) can be solved as usual by the ansatz

b(t) = a exp[ - iwt]. (9.31)

By inserting eq. (9.31) in (9.30) yields a relation for the frequency

- 4)2 - 2Kwi + = 0. (9.32)

When we assume that the damping constant K is much smaller than thefrequency wo we may write the solution of eq. (9.32) in the form

(.4) iK ()o. (9.33)

Thus we obtain the result that the time-dependent amplitude b(t) and thusthe electrical field strength (9.29) decays exponentially

b(t) = b(0) exp[ - iwot - Kt]. (9.34)

Apart from constant factors which are of no interest the function b(t) isidentical with the amplitude b(t) which we used in section 5.1. In order totake into account damping, i.e. the exponential decay of eq. (9.34), we haveto supplement the former equation (5.105) by a term - Kb

d ,--b(t) = (-iwo - K)b(t). (9.35)d t

Note, however, that in identifying b(t) of equation (5.105) with thatoccurring in (9.34) we made a big jump. While eq. (9.34) describes aclassical amplitude, b occurring in (9.35) is meant as a quantum mechani-cal operator! Note further that eq. (9.35) is only a tentative step toincorporate dissipation into quantum mechanics. Namely, we still have toanswer the question whether eq. (9.35) yields an operator which fulfills thegeneral laws of quantum mechanics. We will deal with this problem in thesubsequent section.

(1) Solve the time-dependent, Fokker—Planck equation (9.21) with theinitial condition

f(v, 0) = 6(v)

8(v) Dirac's delta function.

Hint: Try

f(v, t) = a(t) exp{ — v2 b(t)1, (*)where a(t), b(t) are time-dependent functions. In order to determine them,insert ( * ) into (9.21), divide by f(v, t) and compare coefficients of powersof v. Solve the resulting differential equations for a, b.

(2) Calculate the following average values:

<v(t)> f f(v,t)vdv,

<v2(t)> f f(v,t)v2 dv,

<v(t)v(e)> f f(v , t; v', t')vv' dv dv'

= f f(v tIv' t')f(v/ t')vv' dv dv'

Hints: f(v, t; v', t') is the two-time probability for finding the "particle"around v' at time t' and around v at time t. This probability (— density)may be written as

f(v,t; v',r) f(v, t > t',

where f(v, t v', t'), the "conditional probability density", gives the proba-bility for finding the "particle" around v at t provided it was around v' att'; f(v', t') is the usual single-time probability density. f(v, tIv', t') is asolution to eq. (9.21) with the initial condition f(v, t'l v', t') 8(v — v').Show that a formal solution reads

f(v, t I v' = exp[(t — t')(--a

yv + --)]8(v — v')av 2 av2Q 82

by inserting (* *) into eq. (9.21). Then rewrite (*) using this formalexpression, expand the exponential into a power series and integrate by

§9.2

parts. The

<v(t)t

(3) Derive(5.2), puttirHint: Forn(5.2). Use t

9.2. Dampi

We now withe dampirhow to quato know aiinteractioncorr.and

d:er

lion for suc

b(tdt

with the sob(t) =

The operat(9.36). Novhave seenrelation (5.instance thand b + weThis, of cowhether eqrelation. W

(bb + —

The brackeinitial timethat the bra

(938)

i.e. an exprtionio+ —

(*)

§9.2 Damping and fluctuations of quantum mechanical variables: Field modes 295

.21) with the

(*.ermine them,nts of powers

te "particle"r (– density)

; the proba-iround v' attl v', t') is a= 8(v – v').

(* *)

this formalntegrate by•

parts. The resulting integral may be easily evaluated and gives

<v(t)v(e)>= exp[ –y(t – t')]<v2(e)>

(3) Derive the telegraph equation (9.28) from Maxwell's equations (5.1),(5.2), putting p = 0 and j = crE.Hint: Form on both sides of eq. (5.1) curl and insert (d/dt)curl H from(5.2). Use the relation curl(curl E) = V (V E) – AE and eq. (5.5, 6).

9.2. Damping and fluctuations of quantum mechanical variables: Field modes

We now wish to deal with our main problem, namely how we can describethe damping of quantum mechanical quantities. In section 5 we learnedhow to quantize the light field. At the end of the preceding section we gotto know an example of how a classical light field is damped due to itsinteraction with the medium. It is tempting to translate eq. (9.35) and itscorresponding solution (9.34) into quantum mechanics by interpreting band its Hermitian adjoint b + as operators. We repeat the tentative equa-tion for such an operator b

—d

b(t) = K)b(t) (9.36)dt

with the solution

b(t) = b(0) exp [ –iwot – Kt]. (9.37)

The operator b + has to obey an equation which is conjugate complex to(9.36). Now we recall a fundamental postulate of quantum mechanics. Wehave seen at several occasions that b and b + obey the commutationrelation (5.116) for all times. If they did not obey that relation, and forinstance the commutation relation equalled 0 for t oo the operators band b + would eventually commute and thus become classical quantities.This, of course, contradicts quantum mechanics. Thus we have to checkwhether eq. (9.37) and its Hermitian conjugate obey the commutationrelation. We obtain

(bb + –b + b), = (bb + –b + b)oexp[ –2Kt]. (9.38)

The bracket on the right-hand side is the commutation relation valid atinitial time t = 0. Adopting the Schrodinger picture at that time we knowthat the bracket equals unity. We thus obtain

(9.38) = exp[ –2Kt], (9.39)

i.e. an expression which vanishes for t --0 co. Thus the commutation rela-tion bb + –b + b= 1 is violated and the equation (9.36) cannot be used in

quantum mechanics. Quite astonishingly the example of Brownian motionof classical physics which we encountered in section 9.1 offers a solution tothis problem. As we have seen the particle keeps its velocity by means of afluctuating force representing the effect of the heatbath on the particle. Inthe case of the electromagnetic field we mentioned that its damping iscaused by the coupling of the field to another system, for instance to theelectrons in a conductor. This leads to the question whether these electronscan exert fluctuating forces on the electric field in much the same way asthe liquid caused fluctuating forces acting on the particle. This is indeed soand we shall come back to a detailed treatment of this problem in the nextsection. Here we anticipate the final result. It will turn out that the effectof a heatbath at temperature T on an oscillator, especially a mode of theelectromagnetic field, obeys the equation

—d

b(t) = ( - 440 – 000 + F(1). (9.40)d tIn this equation, the b's are operators. As we will see below also thefluctuating forces F are operators having the following properties

<F(t)> = 0, <F + (t)> = 0,

<F + (OF + (0> = <F(t)F0> = 0:

<F + (t)F(t')> = 2tcii8(t — (9.41)

<FO)F + (0> 2tc(ti + 1)8(t — t').

The bracket means a quantum-statistical average over the fluctuations, orin other words, over the variables of the heatbath. It is evident from thelast line expressions of (9.41) that the correlation functions of F and F +depend on the sequence of F and F + . This reflects the fact that the F'sare operators. Insofar, there is an important difference between classicalLangevin forces and the quantum mechanical Langevin forces treatedhere. As we shall see, in applications we don't need to know how theaverage is defined in detail, although we will give an explicit examplebelow. All we need to know are the properties of eq. (9.41), and perhapsthe Gaussian property which we will explain later.

It is the mean number of photons present at temperature T. We nowwant to show that the fluctuating forces having the properties (9.41) giverise to solutions of eq. (9.40) which obey the quantum mechanical commu-tation relations. The solution of (9.40) has exactly the same form as that ofequation (9.8), namely the form (9.10). Inserting these solutions into thecommutation relation bb + – b + b and averaging over the heatbath varia-bles we obtain after some calculation the result

<bb + – b + b> = 1. (9.42)

In contrastion relaticthat thisheatbath v

<b + (

<b(

Thus the E.complete zb, b+

<b + (t

Again Ft isthe correlaconstant tc.

Exercises o

(1) Prove eHint: Findinto (9.42)

(2) DeriveHint: Use(9.41).

(3) The mcforce f(t),

<b(t);

Hint: Startdd t

and its Heequations a

9.3. Quantmechanical

In the precing forces e

show by mi

This sectic

§9.3 Quantum mechanical Langevin equations 297

1rownian motionfers a solution toy by means of a

the particle. Inits damping is

• instance to ther these electronshe same way ashis is indeed so)1em in the nextt that the effect• a mode of the

(9.40)

below also theDelp

(9.41)

luctuations, ordent from theof F and F+t that the F'sween classical7orces treatednow how the)licit example

and perhaps

T. We nowes (9.41) givencal commu-,rm as that ofions into theatbath varia-

(9.42)

•

In contrast to eq. (9.39), the new operators b and b + obeytion relation, at least when averaged over the heat bath. Itthat this is all we need to require. When we average b+heatbath variables we obtain on account of (9.41).

 = exp[ —Kt + hoot]

<b(t)> = <b(0)> exp[ — Kt — iwot].

Thus the averaged operators behave exactly as the classical quantities. Incomplete analogy to (9.12) we can construct the correlation functions ofb,

 = iiexp[— t — exp[ iwo(t — t')]. (9.45)

Again ii is the number of photons present in thermal equilibrium. Clearlythe correlation between b .+ and b at times t, t' decays with the timeconstant K.


(1) Prove eq. (9.42).Hint: Find the solution of (9.40) in a form analogous to (9.10). Insert itinto (9.42) and use (9.41).

(2) Derive eq. (9.45)Hint: Use the solution b(t) constructed in exercise (1) and the relations(9.41).

(3) The mode operators b, b + are coupled to a heatbath and a drivingforce f(t), P(t), respectively: f(t) = fc, exp(iat). Calculate

<b(t)> , 

Hint: Start from the equation

—d7 ( — ico — K)b + F(t) + f(t)

and its Hermitian conjugate. F(t) is the fluctuating force. Solve theseequations and use (9.41).

93. Quantum mechanical Langevin equations. The origin of quantummechanical fluctuating forces*

In the preceding two sections we got acquainted with the idea of fluctuat-ing forces and we learned about their properties. In this section we want toshow by means of an exactly solvable example how these fluctuating forces

•This section is somewhat more difficult to read and can be skipped.

the commuta-can be shownor b over the

(9.43)

(9.44)


arise explicitly. In the sense of thermodynamics we assume that anyheatbath (reservoir) with some general properties which we will specifybelow, produces fluctuating forces with the same typical properties. Forthis reason it will be sufficient to choose a heatbath which can be treatedin an explicit manner. Such an example is provided by a heatbath com-posed of an infinite set of harmonic oscillators. We couple the harmonicoscillator under consideration which represents the particle of Brownianmotion (provided it is coupled harmonically to the origin), or the fieldmode, to this heatbath. The Hamiltonian of the total quantum system thusconsists of three parts, namely that of the considered oscillator ("system")

Ho =- hcoob + b, (9.46)

the Hamiltonian of the reservoir oscillators with creation and annihilationoperators B: , B.

HB = E + (9.47)

and the interaction. We assume that the interaction is linear in theoperators k, and b + and their hermitian conjugates. We assume furtherthat the creation of a quantum of the harmonic oscillator of the system isconnected with the annihilation of a quantum of the reservoir and viceversa. Thus the interaction Hamiltonian acquires the form

H1 = hE tg..b + B.+ g:B: b). (9.48)

The Heisenberg equations (compare section 5.6) for b + and B: readdb+

—dt

iwob + + i E g:B:

dB:—

dtitaB: + ig.b + (9.50)

Equation (9.50) immediately allows for the solution

B(t) = fo tb + (r)g.exp[iw(t — r)] dr + B:(0) exp[ hot], (9.51)

where B:(0) is the operator at time to = 0. By inserting eq. (9.51) into(9.49) we arrive at

db+ = iwob + — f b + (r)Elg.1 2 exp[iw(t — r)] drdt

+ iE g:B: (0) exp[ iwt]. (9.52)

We want to shevaluated, at lelwe must first diinteraction bet,expressions in edramatically. C

b + (t) e'

where g+(t)Furthermore, w

E 'La I2

-->

where I L.,1 2 dif.tion of to's.

A characteriE(practically infiithe system undtimplies that thethe idealized czshift the integra

— WO =

and the lower Ireplace the lowetional to Dirac':

f I g 1 2 ex

where K = 41-the 8-function h

f c5(t — T )

By means of th—tcb + (t)

db +dt

where the last tproperties of thi

(9.49)

10t 18(t T)dT = (9.56)


ssume that anywe will specify

t properties. Forh can be treated

heatbath corn-1e the harmonicde of Brownianin), or the fieldturn system thusator ("system")

(9.46)

and annihilation

is Alt in theassume further

of the system isservoir and vice

(9.48)

id B: read

(9.49)

(9.50)

We want to show that the integral occurring in eq. (9.52) can be easilyevaluated, at least in a good approximation. Since b + (T) appears under it,we must first discuss the dependence of b + on T. When we assume a weakinteraction between b + , b and the bath, we expect that the last twoexpressions in eq. (9.52) do not change the time dependence of b, b + toodramatically. Consequently we expect b + to have the form

b + (t) = eutg+ (t), (9.53)

where g+ (t) changes much more slowly in time than its exponential factor.Furthermore, we replace the first sum over w in eq. (9.52) by an integral:

E Ig.1 2 — jc.

> I -1 2 exp(ico(t — T) — iwo(t — T))cico (9.54)4)

where rg1 2 differs from I g.1 2 by a factor which stems from the numera-tion of to's.

A characteristic feature of heatbaths is that they provide very many(practically infinitely many) degrees of freedom, which must be coupled tothe system under consideration, in our case to a harmonic oscillator. Thisimplies that the coupling constants go, are of comparable size. We considerthe idealized case in which I gc,1 2 is independent of co. Furthermore, weshift the integration variable co w' so that

to — to =and the lower limit of the integral lies at w' = —coo. Since coo is large wereplace the lower limit by w' = — oo. The thus resulting integral is propor-tional to Dirac's 8-function

igeexp[i(a(t — T)]tho = 2K8(t — t'), (9.55)

where K = Trig1 2. For an evaluation of the integrals it must be noted thatthe 8-function has the property

(9.47)

3[iwt], (9.51)

; eq. (9.51) into= iwo b + —Kb + g:B: (0) exp[ hat]

(9.57)

F+(t)

(9 52where the last term is evidently the fluctuating force. We determine the

• ) properties of this fluctuating force and consider the average of F (t) over

By means of these results the integral in eq. (9.52) can be replaced by—Kb (t)

db+dt

•

(9.58)<F (t)F(e)

(9.64a)Tr§ = E <43„1§10„>,n=0


the heatbath variables

<F(t )>B = exp[iwt]<B:(0)>B,..

The index B at < >13 means "bath average", the index co at < >B,.

means "bath average over the bath variables B, A, alone." The average<B,„+(0)>B, can be explicitly evaluated. We assume that at initial timet = 0 the heatbath and thus each of its oscillators is in thermal equilibrium.According to quantum-statistics this average is defined by

00

<B: (0)> = E exp[ — nhc / (kT )] <Oni B Pi ck> • (9.59)n=o

(Readers not familiar with such quantum statistical averages will find amotivation of this definition in section 9.5.) Z. is the partition functiondefined by

00

Z, = E exp[ —nhca/ (kT)]. (9.60)11=0

k is Boltzmann's constant and T the absolute temperature. <43„1134-(0)140,7>

is the quantum mechanical expectation value of the operator B: in theSchrodinger picture, taken with respect to the oscillator wave function On.Because we know that

<4',1 B: (0)1 ten > = 0, (9.61)

we obtain

<F+(t)>B,„= 0. (9.62)

In a similar way we find

<F(t)>B, „, = 0. (9.63)

For later use it will be convenient to write averages of the form (9.59) moreconcisely. Let E2 be an arbitrary operator (composed of it, and B: for afixed co), then we define

<E2>B, . = z Tr.(2 exp[— / j(kT)]), (9.64)

where H = hcoB: B.. "Tr" means trace. The trace of an operator E.2- isdefined by

00

where in the present case the On 's are, as before, the eigenfunctions of theharmonic oscillator with frequency w. The equivalence of eq. (9.64),

S2 = B: , with (9.5Sevaluate

X Tr(B: B

where HB is giveneigenfunctions belt(9.65) cf. exerciseoscillator with freq

'MT) =

In a similar way w

<F(t)F (t')>

It follows from excommutator has th

<[F(t),F+(t'

One may even sho'

[F(t),F+(e)

(cf. exercise 3). Eqidamping constanttemperatures it mu

<F (t)F(tl>

and

<F(t)F (t)>

always appear unccoo is the frequenctaken into accouirsteps leading to ec

<F (t)F(t);

$9.3 Quantum mechanical Langevin equations 301

(9.58)

at <The averageinitial time

equilibrium.

>. (9.59)

will find aion function

•.60)

I B PI O n>r B: in thefunction On.

(9.61)

(9.62)

(9.63)

(9.59) morerid 13 + for a

(9.64)

verator S-2 is

(9.64a)

ctions of theeq. 664),

S2 = B, with (9.59) will be shown in the exercise. In a similar way, we canevaluate

<F+(t)gt')>B= E Igw i 2 exp[ico(t – t')]

x Tr(132- B,,exp{ —11kTB D Z. = E I g. I 2exPE io (t Oihw(T)

(9.65)

where HB is given by (9.47) and the trace refers to the multi-oscillatoreigenfunctions belonging to HB . For some intermediate steps leading to(9.65) cf. exercise 2 below. is the mean number of quanta of anoscillator with frequency co and at temperature T:

17,„(T) = Z„-lTr(BB,,exp[– HB / (kT)]). (9.66)

In a similar way we obtain

<F(t)F+(/)>B= E —ico(t — t')](h(T) + 1). (9.67)-

It follows from eqs. (9.65) and (9.67) that the bath- average over thecommutator has the form

<[F(t),F÷(/)]>B= 2K6(t – /). (9.68)

One may even show that (9.68) holds without the bath average, i.e.

[F(t),F+(/)] = 2K8(t – e) (9.68a)

(cf. exercise 3). Equations (9.65) and (9.67) can be directly expressed by thedamping constant K [cf. (9.55)] only for the temperature T = 0. For highertemperatures it must be noted that in practical calculations

<F+(t)F(/)>B= E Ig 2 exp[ico(t — (9.69)

and

<F(t)F (e)>B = E Ig„1 2 exp[ — (9.70)

always appear under an integral which contains a factor exp[iwot], wherewo is the frequency of the harmonic oscillator (light field). If this fact istaken into account, eq. (9.69) may be expressed in the form [compare thesteps leading to eq. (9.55)]

<F+(t)F(/)>B= E Ig,j 2 exp[iw(t — t i )Ji1„(T) = 2K17. 0(T)8(t – t'),

(9.71)

and similarly

<F(t)F (0> = 2K(H, 0(T) + 1)8(t — t'). (9.72)

The 8-function in eqs. (9.71) and (9.72) expresses the fact that the heatbathhas a very short memory. As may be seen from eqs. (9.69) and (9.70), it isessential for the derivation that the heatbath frequencies have a spreadwhich covers a whole range around the oscillator frequency coo. It isinteresting to derive equations for the average values B,, Kb + b > B , and qb,b + 1>B where the average over the heatbath varia-bles is taken. By taking the average over equation (9.57) we obtain by useof (9.62)

—d

B = (iwo — K) B . (9.73)dt

Quite similarly it follows that

—d

<b(t)>B = (— iwo — K)<b(t)> B . (9.74)dt

To derive an equation for Kb + b >8 we solve eq. (9.57) explicitly

b + (t) = b + (0)exp[(ico0 — tc)t]

+ f t exp[(iwo — K)(t — 1)] F (r) dr

+ f f<F (r)F(1)> B expkiwo — K)(t — 7)] dT (9.76)<FA(t)FAt (

+ b + (0)expkiwo — K )d< F( t )> B The occurrencereservoirs are as

+ (t)>Bb(0) exp[( — icoo — K) t] section with a ge

The integrals may be evaluated using eqs. (9.71) and (9.72), and the last can be seen fro

two terms in (9.76) vanish on account of (9.62) and (9.63) so that we find individual contriclassical and rec

—dt

B = —2K B + 2Kii(T). (9.77) which states thatity that the class

ti(T) ti(T)) is the mean number of quanta of the system oscillator

with frequency wo at temperature T. In a similar way we show how the P(q) 9I,ex

fluctuating forces restore the commutation relation. In analogy to eq. One may show th

(9.75)d b

—

and form dtThe fluctuating f

d t Bdt= b b+ —

db = —2K B

<FA

+ f <F (t)F(T)> B exp[— (1(40 + K)(t — T)] d,r <FA+ (t)FAt

<FA+ ( t)FA,(

(9.76) we can de

cytqb,b+

Its general soluti

,b + ]>

When we insert

qb,b÷DB

we find

C = 0which means tha

So far we cccoupled to a restparticles (harmoindependent resethan composedheatbaths and oheatbath variabliof quantum meci

(9.74)

,t1

(9.75)

di (9.76)

and the lastthat we find

(9.77)

:an oscillatorlow how thealogy to eq.

(9.76) we can derive

qb,b+DB= — 2tc<[ b,b+]>B+ 2tc. (9.78)

Its general solution reads

qb,b + DB, , = Cexp[ —2Kt] + 1. (9.79)

When we insert the initial condition

qb,b+]>,,o= 1 (9.80)

we find

C = 0 (9.81)which means that the commutation relation is preserved for all times.

So far we considered only a single particle (a harmonic oscillatorcoupled to a reservoir). The whole procedure can be extended to a set ofparticles (harmonic oscillators), which are in contact with statisticallyindependent reservoirs at temperature T. Again we start from a Hamilto-nian composed of the Hamiltonians of these particles, of their individualheatbaths and of the interactions between particles and heatbaths. Theheatbath variables can be eliminated and we end up with the following setof quantum mechanical Langevin equations

dbZ= (iwx — tcx)bjt + Fx+ (9.82)dt

The fluctuating forces have the following properties<Fx(t)> = <Fx+ (t)> = 0, (9.83)

<Fx+ (t)Fx+ (r)> = <Fx(t)Fx.(0> = 0, (9.84)

<Fx+ (t)Fx,(0> = rix(T)2KAt — (9.85)

<Fx(t)Fx+ (0> = (rix(T) + 1)2Kx8(t — (9.86)]

The occurrence of the Kronecker symbol SAA , stems from the fact that thereservoirs are assumed to be statistically independent. We conclude thissection with a general remark on the statistical properties of FA , FA+ . Ascan be seen from eq. (9.57), FA+ (and Fx) is composed of very manyindividual contributions which are statistically independent. If Fx were aclassical and real quantity, q, one may apply the central limit theoremwhich states that q is "Gaussian distributed". This means that the probabil-ity that the classical random variable q has the value q is given by

P(q) = 9Lexp(— aq2).

One may show that such a concept can be generalized to our operators Fx,

(9.72)

t the heatbathnd (9.70), it islave a spreadncy wo . It isues B,

.tatbath varia-obtain by use

(9.73)

F7. In an appropriately generalized sense, the F's are then "Gaussiandistributed" (for more details see the second volume).


(1) Prove the equivalence of eq. (9.64), (with 2 = ) with (9.59).Hints: Use the definition of the trace and show that

e-11,,../(kT)0 = e-hcon/(kT)on.

(2) Derive eq. (9.65)Hint: Insert F + according to eq. (9.57) and its Hermitian conjugate into

< >B= 11 4:1Tr(e- 1-18/(kr)(F + F))

= E e — H,/(kT)F +F 000 >(n)

Show that

Tr(e -14/(knk,+, (0)B,,,,m)ll = o

for co l co2 and00

= Z -1 E ne (9.66)wi

ii(T) for 6) 1 = w2.

Hint: Split HB into a sum and Of n) into a product over co.(3) Prove eq. (9.68a)Hint: Insert F(t) and F + (t) and use the commutation relations of A,Use eq. (9.55).(4) Calculate <bi'," (t)bx,(1')>, X X', where bjl: obeys eq. (9.82).

Langevin equations for atoms and general quantum systems

9.4.1 Example of a two-level atom

In the previous sections of this chapter we have been mainly concernedwith the question of how to incorporate damping and fluctuations into thetreatment of quantum mechanical systems described by Bose operators b,

b.fluctugasesexpecshowtwo nchanic(a) W(i.e. t.fluctu.such rclassicforcesconsis(b) Wcontaiinteracof thewe are

In teven rmore

To cmay oannihia . weJ'tion vt

ni

Now .instancThen ilevels sj = 2.transitithe irn.to the

d.—7

Since I

§9.4 Langevin equations for atoms and general quantum systems 305

Len "Gaussian

(9.59).

conjugate into

•

,ns of B.,

32).

;MS

concerneditons into thee operators b,

• This allows us to deal in a proper way with the damping andfluctuations of field modes. On the other hand also atoms in solids or ingases interact with their surrounding which serves as a heatbath. Thus weexpect that also atoms or, more precisely speaking, the electrons of atomsshow damping and fluctuations. In the previous sections we learned abouttwo methods to incorporate damping and fluctuations into quantum me-chanics. We may formulate these two methods as follows:(a) We start with the Heisenberg equations of motion of the field mode(i.e. the harmonic oscillator) alone. We then incorporate damping andfluctuation terms into these equations. The damping terms are chosen insuch a way that the expectation values of the operators obey the analogousclassical equations with phenomenological damping terms. The fluctuatingforces must be determined in such a way that the quantum mechanicalconsistency (commutation relations!) is preserved.(b) We start from the Heisenberg equations of motion of the total systemcontaining the field mode under consideration, the heatbath and theinteraction between these two subsystems. We then eliminate the variablesof the heatbath giving rise to damping and fluctuations of the subsystemswe are interested in (namely the field mode).

In the literature both inethods have been applied also to atoms and toeven more general quantum systems. Since the method analogous to (a) ismore elegant and simpler, we will present it here.

To explain the basic idea let us consider a two-level atom whose electronmay occupy the quantum states j = 1 or j = 2. Using again creation andannihilation operators for electrons in the corresponding states j, i.e. a.7,a3 , we can define the mean occupation number of level j by the expecta-tion value

ni = <41 1ajtai lto >; j = 1,2. (9.87)

Now let us consider the coupling of this atom to its surrounding, forinstance to lattice vibrations or to an (incoherent) electromagnetic field.Then in general these interactions will cause transitions between the twolevels so that the occupation numbers change. Consider for instance a levelj = 2. Then its occupation number increases in the course of time due totransitions from the lower level to the upper level and it decreases due tothe inverse process. The transition rates will be proportional in each caseto the occupation number of the initial level. We therefore find

d n2= WI2n2*dt

Since the electron must be in either one of the two levels (even in a

(9.88)

•

superposition state where the electron -jumps" in a statistical fashionbetween the two levels, it must be found in any of the two levels when theoccupation number is measured) we must have n 1 + n 2 = 1. Thus we canimmediately write down also the equation for level 1

dndt = W21n1+ W12n1•

(9.89)

As we have seen in section 4.6, the transition of an electron may be oftenconnected with an electric dipole moment. Leaving aside all constantfactors, we can take as the operators for the positive or negative frequencypart of the dipole moment

(9.90)

and

(9.91)

respectively.Let us consider the corresponding exp-ectation values which we ab-

breviate by a and a*

<Olat±a214)>= a, 01)14- a i lt, >= a*. (9.92, 93)

When the atom is not coupled to heatbaths, a oscillates at a frequency—(02I . When we couple the atom to a reservoir, for instance to thecontinuum of field modes (compare sections 7.9, 8.3), the dipole moment isdamped and decays exponentially. Thus we are led to the conclusion thatthe expectation value of the dipole moment obeys an equation of the form

dadt = —hozia Yi2a.

(9.94)

Now let us reconsider equations (9.88), (9.89), (9.94) and its conjugatecomplex equation. Having in mind the definitions (9.87), (9.92) and (9.93)we can put these atomic equations in analogy with the equations (9.73) and(9.74) describing the damping of the expectation value of a field modeoperator b + or b. Now let us do exactly the same as we have done withrespect to the Bose operators b and b + before. There we translated theequations for expectation values into those for operators b, b. We sawthat we could conserve quantum mechanical consistency only by suitablefluctuating forces. Doing exactly the same in our case we immediately find

the followind—a cdt 2

d—a,dtd—a .dt 2

—d

a+

cdt I

In order tomust requin

<rik(t)

The meaninderived theThere we linthe heatbattthe bracket!.addition taloperators.

Now letfunctions b(again that twe may asst

<1;k(t)

This can actforces in aas a hypoth(

The questquite so simiconservationbe noted thannihilationquantum me

ai+ak,

Such a pro&of the form

(I) =

tron may be oftenaside all constantlegative frequency

• (9.91)

es which we ab-

(9.92, 93)

as at a frequencyinstance to the

dipole moment isle conclusion thattation of the form .

Ind its conjugate(9.92) and (9.93)

aations (9.73) andof a field mode

t have done withwe translated thes b, b. We sawonly by suitable

immediately find


the following equations

rt (1 2+ az = Wn ai a t WI2aia2

dTli al al = W21al±a1 ± W12(142 -61 2 + rii(t),

ddt a2 41 1 = ( 1021 — Y21) ai r21(t),

ddt a' a2 = 1021 — 1'12)4-a 2 r12(t)•

In order to go back from eqs. (9.95)—(9.98) to (9.88), (9.89) and (9.94) wemust require

<r,k( t )>= 0. (9.99)

The meaning of the brackets needs some discussion. In section 9.3 we havederived the corresponding fluctuating forces for Bose operators b,There we have seen that the fluctuating forces depend on the variables ofthe heatbath. This leads us to the conclusion that also in our present casethe brackets mean averaging over the variables of the heatbath and inaddition taking the expectation value with respect to 4:0 of the atomicoperators.

Now let us ask the question how we can determine the correlationfunctions between the fluctuating forces l'. First of all we shall assumeagain that the heatbaths contain a broad spectrum of frequencies so thatwe may assume that the r 's are 8-correlated.

<rik(or,„,(0> = Gth, ,„,s(t - (9.100)

This can actually be justified by an explicit derivation of such fluctuatingforces in a way analogous to section 9.3. Here, however, we introduce thisas a hypothesis in the manner of a model.

The question how to formulate quantum mechanical consistency is notquite so simple. At a first sight one might think that one should require theconservation of the Fermi commutation relations (6.15). However, it shouldbe noted that in eqs. (9.95)—(9.98) only products of a creation and anannihilation operator occur. This leads us to the idea to seek appropriatequantum mechanical relations for such products

ai ak , atam. (9.101)

Such a product has the following property. When we apply ai±ak to a stateof the form

(I) = (I) (9.102)o

statistical fashionwo levels when the= 1. Thus we can (9.95)

(9.96)

(9.97)

(9.98)

(too: vacuum state) it transforms the state j into a new state i providedj = k and it gives 0 otherwise. This property reoccurs when we take theproduct of expressions of the form (9.101). We then readily establish

ak a7 am = a i+ am8kr, (9.103)

using the Fermi commutation relations and the fact that we deal exclu-sively with states containing only one electron. It turns out that therelation (9.103) is exactly the requirement of quantum mechanical con-sistency we need. We will assume in the following that this relation (9.103)holds at least when it is averaged over the heatbath variables. We shallshow below that this requirement allows us to calculate the coefficientsG•kam uniquely and explicitly. Since the corresponding derivation is some-what lengthy we first write down the final result for a two-level atom. Forthe special case of equations (9.95)–(9.98) we obtain the following relations

G 11,11 = W12 n 2 W2In1, (9.104)

G 11,22 = W21n1 WI2n2 = G22,11, (9.105)

G22,22 = W21 n 1 WI2n2, (9.106)

G 12,12 = G21,21 = 0, (9.107)

G 12,21 = WI2 n 2 W21n1 (712 + Y21)n1, (9.108)

G2I,21 = W21 n 1 0'12n2 + (Y12 1- Y21)n2' (9.109)

These relations must be understood as follows. We have to insert into themthe solutions of the averaged equations (9.88), (9.89) and (9.94) so that, ingeneral, n 1 and n 2 are time-dependent functions. In many cases of practi-cal interest we deal with steady states. In such a case

d n i dn 2 0

G21,I2 = (712 1- 27n2, (9.112)

where we have introduced the abbreviation

712 = Y21 = Y .(9.112a)

Similarly we could also simplify eqs. (9.104)–(9.106) by expressing, forinstance, n 1 by n 2 or vice versa. We will illustrate the application of thisformalism by some examples treated in the exercises. Readers not so much

interested in rthis section hethe other hatderivation =-for Gik,hn (9.1.

9.4.2. General

Let us now tt(9.100). We rk = 1,2,...,/reason for thi

ak=

However, theics. When it istate i. Oneequations quinot give hereoperators becelectron state(9.95)–(9.98).

Pik =

[compare (9.1,quantum systereservoirs

<Pair.>and take the tof the systemthe present pr

— <P.dt

The correspoforces are

dt '02

We subject th

(9.110)dt dt

As a consequence, by use of (9.88), (9.89) the relations (9.108) and (9.109)can be simplified

G 12,21 = (712 + y21 )n 1 a- 2yn 1 , (9.111)

state i providedlen we take they establish

(9.103)

we deal exclu-s out that theiechanical con-relation (9.103)ables. We shallthe coefficientsivation is some--level atom. Forlowing relations

(9.104)

• (9.105)

(9.106)

(9.107)

(9.108)

(9.109)

insert into them4.94) so that, incases of practi-

(9.110)

08) and (9.109)

(9.111)

(9.112)

(9.112a)

expressing, for)lication of this.trs not so much

•

interested in mathematical details or in the general case can stop readingthis section here, but they are advised to have a look at the exercises. Onthe other hand, readers interested in the general result but not in itsderivation may proceed as follows: go on until eq. (9.118). The final resultfor Gikam (9.100) is represented in eq. (9.141).

9.4.2. General case of a quantum system

Let us now turn to the explicit determination of the coefficients Gikam in(9.100). We may do this not only for a single electron occupying statesk 1, 2, ... , N in an atom but for an arbitrary quantum system. Thereason for this is as follows. First we may introduce as an abbreviation

ak = Pik . (9.113)

However, the operator Pik has a very general meaning in quantum mechan-ics. When it is applied to a quantum state k it projects this state k onto astate i. One may show that one can formulate the quantum mechanicalequations quite generally by means of such projection operators. We shallnot give here a derivation of the corresponding equations for projectionoperators because the reader may always take the example of a singleelectron state as described by the relation (9.102) and the former equations(9.95)–(9.98). Projection operators obey the relations

Pik Plm = Pim8kl• (9.114)

[compare (9.103)]. We require that these relations remain valid when thequantum system is coupled to heat reservoirs provided we average over thereservoirs

< Pik Plm> = 8kI<1:l,m> (9.115)

and take the expectation value with respect to an arbitrary quantum stateof the system itself. Generalizing the equations (9.88), (9.89) and (9.94) tothe present problem we start from equations of the form

dt D

,'12d

= EJI .J2

(9.116)

The corresponding quantum mechanical equations containing fluctuatingforces are

—P. = E M.. + - .dt '02 ; 11-2u1J2 ./1./2 I-2

(9.117)

We subject the fluctuating forces r to the conditions (9.99) and (9.100).

Furthermore we take into account that the quantum system must be in oneof the states i = 1 ...N (compare the discussion leading to (9.89)). Onemay show that this implies

E F 1.= 1. (9.118)

For the following it is convenient to introduce new abbreviations. Weconsider 13;02 as components of a vector A

(Fiji) = A. (9.119)

Similarly we consider the M's as the components of a matrix

( Mili2ju2) =

Especially we have for t = 0

A(0) =( Piti2(0))• (9.121)

This allows us to represent equation (9.116) as

—d <A> = <MA> . (9.122)dt

In it, M contains the transition probabilities (especially the losses, thenondiagonal phase-destroying terms produced by the heatbaths and"coherent" driving fields). Since these driving fields, if treated quantummechanically, may still depend in an implicit way on the heatbath, theaverage over the heatbath has to comprise both M and A. We now go onestep back and consider the motion of the unaveraged operators A byadding fluctuating forces, r.

—dt

A = MA + F. (9.123)

The driving forces can have any form and in particular can still depend ona7ak , for instance, in the following form

r(t) = L(t)A + N(t), (9.124)

where L and N do not depend on A. In order to come back from (9.123) to(9.122) we must assume

<F> = O. (9.125)

The formal solution of eq. (9.123) can be written as an integral representa-tion

A = f K(t, T)Fo-) cur + Ai, (9.126)

where A I, is a sias usual that ti

K(t,t) =

The initial con

A(0) = (I

where the Pkik

struct

<ABA>=

where - denott

B = (Bk,i

having only or

Bk,x

Using thWordifferentiation

d z8i2h dt '

By differentiatproperties of ('

<f(t)BA)

<2 .1- ICI BA >,

For the evaluaproperty, i.e.

and

<1,112(r)>

We now havethat the flucturesponse of thtout of the averindirectly over

(9.120)

•

nust be in one3 (9.89)). One

(9.118)

eviations. We

(9.119)

(9.120)

49.121)

(9.122)

ie losses, theeatbaths andLted quantumheatbath, thee now go oneerators A by

(9.123)

ill depend on

(9.124)

om (9.123) to

(9.125)

ii representa-

(9.126)

•

where Ah is a solution of the homogeneous part of eq. (9.123). We postulateas usual that the kernel K has the property

K(t,t) = E(= unity matrix). (9.127)

The initial condition requires

A(0) = (Pkik2(0)), (9.128)

where the Pkik :s are operators in the Schrodinger picture. We now con-struct

<ABA> = ((f Ifer)ic(t, T)CIT )B(f tIC(1, T)r(T)dT Ah ))

(9.129)

where - denotes the adjoint matrix. B is assumed as a constant matrix

B = (Bh,i)

having only one nonvanishing element (i,j) = (i 1 i2 , jtj2):

Bk ik 2, 1 1 1 2 = 8018k2i28018/2/2*

Using the property (9.115) we find for the left-hand side of eq. (9.129) bydifferentiation

d / \ 2 v2.11dt 'II/2/ = ui2j, •nn \` '102 ,mn 1mn> •

m,n

By differentiation of the right-hand side of eq. (9.129) and by using theproperties of (9.123) we obtain the following terms

<i;(t)BA>ii, <ABT(T)>ij , (9.133, 134)

>ij , < ABMA >ij (9.135,136)

For the evaluation of expressions (9.133) and (9.134) we use the Markovianproperty, i.e.

<ri,i,(01),,20> = Giii2u1J28(t t') (9.137)

and

<r„i2W > =

(9.138)

We now have to perform the average over the heatbaths where we knowthat the fluctuations have only a very short memory. We assume that theresponse of the particle (or spin) is slow enough for K and Ak to be takenout of the average. (Note that M contains at most the heatbath coordinatesindirectly over the quantum mechanical ("coherent") fields, but in all the

(9.130)

(9.131)

(9.132)

(9.141)- 8,,,„m.,,,2„2„)><P„,„> •


loss terms and phase memory destroying parts they do not appear.) (9.133)then gives the contribution

(9.139)

and the same quantity follows from (9.134) in a similar manner. In order toevaluate (9.135) we contract the product of the A's by means of the rule(9.115), where we make use of the commutativity of the operators P withthe operators of the driving fields within M, which holds in the Heisenbergpicture. We thus obtain

( = E <Pmhmiiivmdm,rt

from eq. (9.135) and a similar one from eq. (9.136). We now compare theresults of both sides of (9.129), thus obtaining the following equation forthe G's which occur in eqs. (9.137) and (9.140).

E6nLI2Miti2tmit

This equation represents the required result. It allows us to calculate allcorrelation functions of the type (9.137) if the coefficients M are given andthe solutions <P,k > of the averaged equations are known. We now showthat in (9.141) all terms containing external fields cancel each other, so thatonly transition rates and damping constants need to be used for the M's.We came across such terms in eqs. (9.95)—(9.98). In these equations thetransition rates Wik and damping constants yik stemmed from the couplingof the system to heatbaths. However, we may also imagine that the systemis coupled in addition to external fields, for instance the classical lightfield, or to other quantum systems. In such a case additional terms occuron the right-hand side of eqs. (9.95)—(9.98) which stem from that coupling.The explicit form of the corresponding terms can be found quite easily. Tothis end we have only to consider the left-hand sides of eqs. (9.95)—(9.98).The left-hand sides are just the time derivatives of operators. However, weknow that such time derivatives can be calculated by means of theHeisenberg equations. Thus the effect of external driving forces or thecoupling to other quantum systems (but not heatbaths!) can be incorpo-rated into (9.95)—(9.98) by adding the corresponding commutators of a7 akwith the interaction Hamiltonian to the right-hand sides of (9.95)—(9.98).In the present context we have, of course, more general quantum systemsin mind but still the coherent part stems from a commutator with a

Hamiltoni(9.141) or.(9.141). Npart of Mof the fori

H=

The evalu;

• H• L

yields imn

E. . hJ1J2

A coliu-show that

Myr,'

When thisterms cancatomic eqi:

dn,—r--•dt "

ddt

By speciali

Gli&

• it =

• =

For many

1110

(9.140)

Hamiltonian. We want to show that such terms do not affect the relations(9.141) or, in other words, that the coherent part of M drops out from(9.141). Now let us formulate this problem more explicitly. The coherentpart of M stems from the commutation of P11 a Hamiltonian which isof the form

H= E c„,„P„,„• (9.142)MII

The evaluation of the commutatori D v \

[ iii2 = I Cmnku" ni,`jatin

m,n

i1i2 (9.144)

•irlzhjlj2

A comparison of this expression with the right-hand side of eq. (9.116)

(9.141) show that

n calculate all AChri I j2 = (9.145)are given and

Ve now show When this explicit form is inserted into the right-hand side of (9.141) these

other, so that terms cancel each other. In the case of the laser, the incoherent part of the

for the M's. atomic equations often reads

equations the d

the coupling dt = E wik Pkk- E at the system

dlassical light dt ijk = Yjk Pjk + rik(t), j k.terms occur

hat coupling. By specializing formula (9.141) we obtainLite easily. To

9.95)—(9.98). G11j1= 8i; ( E ( Wki< Pkk> Wik<Pii>)}

However, we

Leans of the Wfi<Pii> (9.147)'orces or the

be incorpo- Gu,ii= 0. i j, (9.148)

.tors of a1+ ak

,9.95)—(9.98). G1j,j1= E Wik< Pkk> Wki<Pii>) (Yij Yji)<Pii> i j. (9.149)[turn systems

tator with a For many applications, e.g. laser theory, a knowledge of the second

4111

yields immediately

E h ,2-2 -2,2 ,I-1 I ,2.

(9.143)

(9.146)

ppear.) (9.133)

(9.139)

er. In order tons of the ruleerators P withhe Heisenberg

(9.140)

r compare the; equation for

moments of the fluctuating forces, i.e. (9.149) is completely sufficient,because the fields interact with many independent atoms. Thus the results,e.g. on the linewidth, depend only on a sum over very many independentlyfluctuating forces, which possesses a Gaussian distribution (the fluctuatingforces of a single atom are not Gaussian, however).

So far we have shown that eq. (9.141) follows from condition (9.115). Itis rather simple to show that the opposite also holds.


(1) From eqs. (9.95) and (9.96) derive an equation for $ = aa2 — a l" al.Solve that equation as well as those for ajf- a l , a l+ a 2 and calculate

<s> , <(a' al)t(al+a2),,> ,

<(4" .2 1 ),(4" a l ),,> , <s „sr > .

The indices t, t' are the arguments of a7, a k in the Heisenberg picture.Hint: Solve the equations in the same way as ordinary differential equa-tions treating al. ak as classical quantities. (This is justified because theequations are linear in a7ak .) When forming correlation functions, becareful in keeping the sequence of operators.(2) In section 4.9, exercise (1), we encountered the Bloch equations used inspin resonance. These equations contained both the applied magnetic fieldand incoherent damping terms. As we know from section 4.8, an analogyexists between a two-level atom and spin 1. This leads us to the followingexercise. Put in the Bloch equation B= (0,0, Be ). Compare the Blochequations of section 4.9, exercise (1) with eqs. (9.88), (9.89) and (9.94) andits conjugate complex.Hint: Try the following analogies:

a(9.92)+4 <sx >— i<sy>,

a*(9.93) 4-* <sx > + i<sy>,

f(n 2 — n 1 )(9.87) 4-> <sx>.

(3) Consider the same situation as in exercise (2). Derive quantum mechan-ical Langevin equations for the spin operators

s + = sz + isy , s_= sz — isy , sz.

Hint: Use the correspondence

s al+a2, s =-1 (a + 2a — a -I- a 11\

1z 2 2

and tra:correlati(4) A t'electricequatioiAssumeHint: W

d t

In it (d(9.98), v

H =

What isHints:g,

ex(5)classicalequatioi

ddr'

—dt

d t

d t I

Solve I

approx

<a

<(

Hint: T

4111,

•

1112 = f Vi(x)ex(p2(x)d3x,g = E0012 cos tot,


and transform the eqs. (9.95)—(9.99) correspondingly. In this way thecorrelation functions of the corresponding F's can be determined also.(4) A two-level atom is coupled to heatbaths and a coherent classicalelectric field, E = E0 coswt. Derive the quantum mechanical Langevinequations for aj+ ak , j, k = 1,2 and the correlation functions (9.100).

Assume that the dipole moments On = = 0, but *12 0.Hint: Write

d d—dt '

(a ak ) = —dt

(a. a tk ). + (a ak )J incoherent dt coherent

In it (d/dt)(aj+ ilk) incoherent is defined by the r.h.s. of equations (9.95)—(9.98), whereas (d/dt)(ai+ a k coherent is defined by (i /h)[H, aft ad, where

H= E heiajt aj + hgal" a l + hg*ai'a2.

What is the explicit form of g? Show that g a cos wt.Hints: Use the results (9.141) and the discussion thereafter. To determine.g, use sections 7.3, 7.5. (To check your results, consult the followingexercise.)(5) A two-level atom is coupled to heatbaths and a coherent, resonantclassical electric field E = Eo costal. Its quantum-mechanical Langevinequations are given by

—dt

(ai+ a 2 ) = – i(e2 – e 1 )a l+a 2 + ig(a21- a 2 – a a 1 1t+ , .iza t+a z riz(t),

—d (a +a ) =i(e – e 1 )a I+ a – ig*(a22+ a – a a ) — .2 1 2 2 1+ 'Y21 a , a t + 1.21(t),dt

— (a + a ) = ig*a + 2 – r22a – iga2* a l Wzi a t+a t2 2 1 Wi24a2 + (t) ,dt

—d

(a 1+t

a 1 ) = iga2i- a 1 –d

sufficient,the results,ependentlyfluctuating

(9.115). It

az – a t at.Ate

•picture.ntial equa-ecause theactions, be

mas used inpetic fieldan analogy

followingthe Bloch(9.94) and

az – Wzi a t+at Wtz aiaz rn(t),

pi( x) atomic wave function.

Solve the equations for aj÷aj , aj".'a k ; j, k = 1,2 in the rotating waveapproximation (e2 – et 6-21 = w) and calculate

<a l+ a 2 > , <(a 1+ a 2 )>„<(a,1 a 2 ),,> ,

<(cti+a2),(4a1),,>.

Hint: Take the Fourier transform of the equations.

in mechan-

(9.150)

The empty box in the lower right corner immediately leads to the questionwhether there exists in quantum mechanics an equation which correspondsto the Fokker-Planck equation. As we saw above the solution of theFokker-Planck equation has certain advantages. Knowing its solutions weeasily can calculate average values of q and of its powers at any time t bymeans of mere integrations

< gn> = igni(g91)dg.

- 9.5. The density matrix

In section 9.1 we saw that we may mathematically treat Brownian motioneither by means of the Langevin equation (9.8) or in entirely equivalentfashion by the Fokker-Planck equation (9.21). Then in section 9.2 werecognized that the classical Langevin equation (9.8) possesses an analoguein quantum mechanics. Thus our results can be summarized by means ofthe following table 1:

Table 1.

Langevin

Classical 5-1L1 —nil+ F(1)d1dbQuantum mechanical dt — Kb + F(t)

There exists indeed a quantity in quantum mechanics which directlycorresponds to the distribution function f and which allows us to calculateaverage values or, in the sense of quantum mechanics, expectation values.The quantity corresponding to f is called density matrix and is oftendenoted by p. As we know, in quantum mechanics we have to attributeoperators S2 to the observables, say coordinate or momentum. Let 52correspond to q" of eq. (9.150). The quantum mechanical analogue of(9.150) then reads

0.2> = Tr(2p) (9.151)where -Tr" means trace which we will define below [(9.158), cf. also(9.64a)]. We first want to motivate the definition (9.151), then define p, andeventually derive an equation for p. Equation (9.151) can be considered asa generalization of the quantum mechanical average or expectation valuewell known to us:

= f 99:(x)14n(x)d3x. (9.152)

In the fo

=

This notabecause vby mean:functionsNow westatistics -instance,at temperall in a dcwith enericalculateto weightthe definii

As usual \

PnIt

We adoptfor oscillalthermal e,Boltzmanr

PIT =

where the

Z =

and takes ((9.154) carmatrix p

IT

P

The readeiinsert eq. (


Fokker-Planck

af [ aK_+al aq

Q a2 it

di 2 aq2

ck

82 it

84,2

$9.5 The density matrix 317

nian motionequivalent

tion 9.2 wean analogue)y means of

he-:orrAtIl•ndstion of theolutions we3/ time I by

(9.150)

:11 directlyo calculateion values.d is ofteno attributeun. Letialogue of

(9.151)

), cf. alsop, and

sidered astion value

(9.152)

In the following we shall use Dirac's bra and ket notation:

= <niStin>. (9.153)

This notation means a considerable generalization compared to eq. (9.152)because we can use quite general wave functions such as those constructedby means of creation operators. But still, the I n>'s are the usual wavefunctions of quantum mechanics and describe in this sense pure states.Now we wish to incorporate thermodynamics or, more precisely speaking,statistics into the formulation of average values. To this end consider, forinstance, an ensemble of oscillators in thermal equilibrium with a heatbathat temperature T. In this case the quantum mechanical oscillators are notall in a definite pure quantum state. Rather we can only say that a state nwith energy wn is occupied with a certain probability pn . When we want tocalculate the average value of an observable of such an ensemble, we haveto weight the individual expectation values (9.153) by pn . This leads us tothe definition of the quantum statistical average

<2 > = EPn<n I n> • (9.154)

As usual we assume

E = 1. (9.155)

We adopt the definition (9.154) for general quantum systems and not onlyfor oscillators. In classical physics, the pa 's are well known for a system inthermal equilibrium with a heatbath at temperature T. According toBoltzmann we have

p„ = exp(— W„/ (kT)), (9.156)

where the partition function Z is given by

Z = E exp( — W„/ (kT))

and takes care of the normalization of p„ [cf. eq. (9.155)]. We show that eq.(9.154) can be considered as a special case of (9.151) with the densitymatrix p given by

P =EPnin><nl. (9.157)

The reader should note the sequence of In> and <n I, i.e. ket-bra! Weinsert eq. (9.157) into (9.151) and use the definition of the trace of an

410

•


din'

We obtain (with it = Stp) or, extracting

<52> .=E <n'ISZEpnIn><nIn'> dp =n' n d t

= E Epn<n15.210<nin'>. (9.159) Using on then' n obtain the fui

Owing to the orthogonality of wave functions, dp =<n n'> = 8„ n„ dt

eq. (9.159) reduces to or, in short

< S2 > = EPn<n121n>, dp =dt

i.e. eq. (9.154). We can generalize our considerations. The density matrix So far we ha(9.157) is still

I and not of the form In > <ml, i.e. with two quantum numbers n and allow usitil

insofar too special, as it contains only terms of the form formalis

m. Since in practical cases such combinations may also occur, we define S will become sthe density matrix by spirit and an

We start fronP = Ep„,„In><mi (9.160) system comp(

This form may look to some reader unfamiliar, but we may introduce thewhere the braequation (9.162) just as equation defining <ml! Since In> and <m I are time

dependent, p is time dependent too. We are now in the position to derive [A, B] =an evolution equation for p. To this end we differentiate p with respect to For later purr

timedp

c±e = ,,mn i din> z ml + In\ d<ml \.dt k dt dt ) (9.163) d t

mn

arbitrary operator By use of eqsdp

Tri-2 E <nlf2In'> (9.158) ,

mnthe heatbaths

where the pmn 's are constants, i.e. classical numbers. Furthermore, we have the reduced cseen in section 3.2 that we may form expectation values not only for the bath variatime-independent but also for time dependent wave functions. Conse- the "proper"quently we now consider the case that the wave function I n> is time To elucidatdependent and obeys the Schrodinger equation tions and the

ih— l n > • (9.161) harmonic osodt considered in

Similarly, <m I obeys the "adjoint" SchrOdinger equation equation read

(9.162)dp

<mIH = – ihdt— dt<ml. –

§9.5'lledensitymura 319

(9.164)

(9.165)

By use of eqs. (9.161) and (9.162) we obtain

p„, „ ( H I n> < m I — In><mIH)dt =

dt =dP HE Pmnin><ml EPmnin><"1111 ) .mn MW

Using on the r.h.s. of eq. (9.165) the original definition of p, (9.160), weobtain the fundamental density matrix equation

or, extracting H to the left and right of the sum, respectively

(9.158)

(9.159)

dp=

7-(Hp — pH)dt h

or, in shortdp = id t

— —h

H' p]. (9.166)

So far we have established a general frame. We now want to apply thisformalism to treat the dynamics of systems coupled to reservoirs. This willallow us to fill the empty box in table 1. Since the mathematical treatmentwill become somewhat involved we will first give a short description of itsspirit and an explicit example. The basic idea is this (cf. also section 9.5.2):We start from the density matrix pto, and its equation (9.166) for the totalsystem composed of the "proper" system (for instance the field oscillator),the heatbaths and their mutual interaction. Then we derive an equation forthe reduced density matrix p of the "proper" system alone by eliminatingthe bath variables. Such an equation describes damping and fluctuations ofthe "proper" system. We will present the general reduction scheme below.

To elucidate the general features of such reduced density matrix equa-tions and their solutions we first write down the result for the dampedharmonic oscillator whose quantum mechanical Langevin equation we hadconsidered in section 9.2 and derived in section 9.3. This density matrixequation reads

dp = —iw[b

+b,p] + 8t[b +p,b] +[b + ,pb])

d t+ Wbp,b + ] +[b,pb + ]}, (9.167)

where the bracket [A, B] is defined as usual by

[A,B] = AB — BA.

For later purposes we write (9.167) more explicitly in the formdp .dt =

zw(b bp — pb + b) + 8(2b + pb — bb + p — pbb+)

+ t2bpb + — b + bp — pb + b). (9.168)

milkie formCs n and.! define

(9.160)

ve have)nly forConse-is time

(9 .161)

(9.162)

uce theire time) derive;pect to

(9411

•


To get an insight into the meaning of the constants 8 and we derive bymeans of (9.167) or (9.168) an equation for the average values of b + andb + b, respectively. To this end we multiply eq. (9.168) by b + from the leftand take the average value which we define according to (9.151). Sincetraces have the cyclic property

Tr(0 1 12 2 ) Tr(22 2 1 ) (9.169)one easily finds for the average light field amplitude the equation

d — {ico —(— 8)} . (9.170)dtIn an analogous manner we find equations for the photon number

d — 28 — 2(C — 8) , (9.171)

dtand for the commutator

—d

<(bb+—b+b)> = 2(C — 8) + 2(8 — E)<(bb +b)> . (9.171a)dt

Eq. (9.170) allows for the solution

 = 0 expa — (C — a)]t), (9.172)

whereas eq. (9.171) is solved by

 = oexp[ —2(C — 8)t] + 8 8

(1 exp[ —2(C — 8)t]).—

(9.173)

Under the initial condition <(bb + — b 4 b)> = 1 at t = 0, the solution of(9.171a) reads

<(bb + —b +b)> = 1, for all t, (9.173a)

i.e. the commutation relation is preserved for all times, at least in the senseof an average. We can compare the solution (9.172) with (9.43) of section9.2. This allows us to identify — 8 with the former decay constant K. Onthe other hand letting t cc in eq. (9.173) we find the stationary solutionwhen the heatbath is at a temperature T. The average photon numberresulting from (9.173) must be that in thermal equilibrium: FL This allowsus to make the identifications

— 8 kT8 =[exp[ —61h — 1] =— = (9.174, 175)IC,

—1

9.5.1. An example of the solution of the density matrix equation

We will not try to solve the general time-dependent density matrix equa-tion (9.167). However, it is of some interest to consider at least a typical

In this edetermine

example.make the

p(t)

where I nexample"expansicfrom thecovers, ahowever,discuss tlmatrix etrelations

b + Ir

<n I

The Alp

the relaticevaluate

b + pi

By inserti

b + In

In an ana

b+br

A typical

bi

Proceedinevaluate (

nO

co c

§9.5 The density matrix 321

trix equa-a typical

411

example. Here it is convenient to use Dirac's bra and ket notation. Wemake the following ansatz for the density matrix

p(t) = E In>p(t)<ItI (9.176)

where I n> and <n I are time independent. (Incidentally, we learn from thisexample that we may shift the time dependence of the 1 n >'s to one of the"expansion" coefficients, p(t). This is quite analogous to the transitionfrom the Schrodinger picture to the Heisenberg picture. Eq. (9.176) whichcovers, as we shall see below, at least the stationary state. It covers,however, also a certain class of time-dependent states but we shall notdiscuss this question here. When we insert eq. (9.176) into the densitymatrix equation (9.167) or (9.168) it is useful to have the followingrelations in mind

1, 1- In> = V n + 1 In + 1>, bin> = VTz In - 1>, (9.177, 178)

<nib = n + 1 <n + 11, <nib + = <n - 11. (9.179, 180)

The relations (9.177) and (9.178) are well known to us (compare section3.3). The relations (9.179) and (9.180) are just the Hermitian conjugates ofthe relations (9.177) and (9.178). By using (9.177) and (9.179) we can easilyevaluate

b + pb (9.181)

By inserting an individual term of (9.176) into (9.181) we thus obtain

bln> <nlb = Vn + 1 In + 1><n + 11\4: + 1 . (9.182)

In an analogous way we can evaluate

b + bp. (9.183)

A typical term then reads

b + bin> <nl = b +In - 1><nI

= Vi In> <nl. (9.184)

Proceeding with all the other terms in a similar way we may immediatelyevaluate (9.168) with (9.176) which yields

dp„E --dT 1n><nl = 28E P(n + 1 )(I n + 1 > + II - I n > <ni)

00

+ 2

p„n{ln - 1> <n - 11 - In> <nl} . (9.185)o

In this equation, the p„'s are still unknown functions of time to bedetermined. Now we must remember that the bra and kets, or more

derive byof bcf andom the left151). Since

(9.169)

(9.170)

ber

(9.171)

(9.171a)

•(9.172)

(9.173)

olution of

(9.173a)

the senseof sectiontant K. Ony solutionn numberhis allows

.174, 175)00

For n 0

6Po —

or

Pt =

For n = 1

[ 8Po -which dueprocedure

8Pn —

or

Pn +1=

This is a rreadily finc

1110A

By putting

8/C =

eq. (9.197)

P„ = 13(

As we hal,expressed IK and theallows us tc

a =

However, N

distributionPo coincidesthe quantuization condform

Tr p

Inserting (9

Po = (n


precisely speaking,

In><nI

are linearly independent. That means any equation of the form of (9.185)can be solved only when the coefficients of each term (9.186) vanish. Toderive such equations for the coefficients we must transform terms of theform

In + 1> <n + 1 I (9.186a)

into those of the form (9.186). Take as an example00

E p„(n + 1)In + <n + 11, (9.187)n=0

then make the replacement

n + 1 = n' (9.188)

which transforms (9.187) intoco

(9.189)n'

Since the term with n' = 0 vanishes we may let the sum start from n' = 0.Dropping eventually the prime we then find instead of eq. (9.187)

pn _ I nin><nj. (9.190)n=0

Making the same procedure with terms containing n + 1 we eventuallyfind

dpE <ni 2 E In><nj (8np„_ i — 8(n + Op„+ C(n + 1)p„."n=0 n=0

— Cnp„). (9.191)

Due to the above-mentioned linear independence of the expressions (9.186)we are then left with a set of equations

dp„= 2t8npn _ 1 + C(n + Op„ 4.1 — [8(n + 1) + n] p). (9.192)dt

We shall be concerned with its stationary solution only where we putdp„ = 0

(9.193)dt

so that (after a rearrangement of terms) we have to solve the equationsn[Sp,,_ — ip,,] — (n + 1)[SP„ — CP.-1-1] = 0, n = 0, 1,2,...

(9.194)

(9.186)

This is a recursion formula. Expressing p„ by n- I , Pn- I by pn-2 etc. wereadily find

Pn = P0( 8 / O n •

(9.197)

By putting

SA= e -a, (9.198)

eq. (9.197) acquires the form

= Poe -an. (9.199)

As we have seen above, cf. eqs. (9.174) and (9.175), 8 and can beexpressed by directly observable quantities, namely the damping constantK and the thermal photon number Ft, or, equivalently, by ha)/(kT). Thisallows us to determine a. After some algebra we obtain

a = G.)/ (kT). (9.200)

However, with eq. (9.200), the expression (9.199) is just the Boltzmanndistribution we introduced in section 2.3 (cf. eqs. (2.59) and (2.58)). EvenPo coincides with Z -1 as can be seen as follows. Since the density matrix isthe quantum mechanical analogue of a distribution function the normali-zation condition of such a distribution fu. ction also applies, namely in theform

Tr p = 1. (9.201)

Inserting (9.176) with (9.199) into (9.201) brings us to the relation

Po exp [ - an]

(9.202)n=0

$9.5 The density matrix 323

For n = 0 we find

(9.186)

8Po - 01 = 0

'orm of (9.185) or

86) vanish. To Pi = (8/)Po.

rn terms of the For n = 1 we obtain

(9.186a)

{ 8P0 - 01] - 2 [41 - 02] =

which due to eq. (9.195) reduces to Sp, - () 2 = 0 orprocedure can be continued leading to

(9.187) 8Pn EPn+1 = 0

or

(9.195)

P2 = (8/)Pi . This

pn +1 = (9.196)(9.188)

410(9.189)

rt from n' = 0.(9.187)

(9.190)

we eventually

(n + 1)Pn+1

(9.191)

ressions (9.186)

(9.192)

;re we put

(9.193)

ie equations1,2,...

0(9.194)

it is t

'

The i

•where

The e;

We h.(yields

At(

where

f •

Sinceelimineheatbal

=

Therefct

or, after performing the sum:

pc, = 1 — exp[ —a]. (9.203)

Inserting the result (9.202)—(9.203) into (9.176), we find the explicit form ofthe density matrix

00

p (1 — exp[ —a]) E exp[—an]ln><nj. (9.204)n—o

We hope that the reader now has a certain feeling of how to deal with atypical density matrix equation and what some typical properties of adensity matrix are. We now turn to the final step of our treatment, namelywe wish to show how one may derive a density matrix equation for anarbitrary quantum system.

9.5.2. General derivation

We consider the interaction of a free field (e.g. the light mode or theelectron of an atom) with a heatbath or a set of heatbaths. The densitymatrix of the total system obeys the equation

it

r „Ptot = —h L Ptotd

whereH = + Hp_ 8 (9.206)

and1/0 = Hp+ j8 (9.207)

Hp is the Hamiltonian of the free field and reads explicitly

hwb + b light-mode

E he1a7a1 atom

E hEiPii arbitrary quantumsystem described byprojection operators.

The Hamiltonian of the heatbath H8 may consist of a sum of severalHamiltonians corresponding to different heatbaths

H8 = E Hk). (9.209)

Hp_ 8 represents the interaction between the two systems. For our analysis

(9.205)

HF = (9.208)


(9.203)

it form of

(9.204)

:eal with a:rties of aat, namelyion for an

(9.210)it is convenient to proceed to the interaction representation

13tot = exp[iHot/h] oto, exp[ —iHot/h],

fi F_ B -= exp[ iHot/h] HF_ B exp[ —iHot/h]•

The interaction Hamiltonian has the form

liF_ B(t) = hEVk(t)Bk(t)

(9.211)

)de hehe density

(9.205)

(9.206)

bexp[ —iwt] (t) + b + exp[iwt] B(t)

V1(t) hBi V2( t) hB2 (9.212)

E Pu exp[iwut] Bu(t) (9.213)

Vk(t) Bk

exp[iHot/h]Pik exp[ —iHot/h]

= Pi k eXp[hOik t],

where

(9.214)

to.k = e ek. (9.215)J J

The explicit form of the V's suggests that we write

Vk(t) V k exp[i Ac.) k r]. (9.216)

(9.207)

We treat the interaction by perturbation theory up to second order whichyields

Atot = Po+h 1

i )f t dT[IiF_ Ber),po] + (-=1 )2 ft dT2wo

dr'

o h .10

(9.217)

(9.208)

of several

(9.209)(9.219)

where po = ik„(0) and

( • • • ) = ( 17 F - B( r2) 171 F- r 1)P0 F- B( T2)P0 1. F- B( T I)

11;-B( T1)130 11 F-B( T2) Po ll' F-8( T1) 11.F-B( T2 )) • (9.218)

Since we are ultimately interested only in the variables of the field weeliminate the bath variables from (9.217) by taking the trace over theheatbaths

=Tr 0 .

urTherefore, 13 depends only on the field operators. It is essential that at timet = 0 the total density matrix p to, factorizes in that of the free field, p(0)

yields foci

We assu.nonvanis

Tz. -

In the f,correlaticinfinity,:

Jo d

If

the

tic"In the fowise ther,expressiothe interadensity Itment

13(t)

If we furi

TrB(

TrB.k(Bk(0)heatbath.

Zi" I"

x exT

Since the tr.

Zk- IT


and that of the heatbath, pB:

Po =-- Atot(0) = P(OPB• (9.220)

pB may itself factorize into the density matrices of the individual baths

PB = PB, I PB,2 • • • • (9.221)

We allow that these baths are kept at different temperatures 7:1 , so that

pBd = I eXp — (kTin, (9.222)

where

= TrBj(exp{ —HP/ (kT1 )]). (9.223)

For simplification of the further analysis, we may assume

TrB(Bkp) = 0 (9.224)

because the corresponding terms in (9.217) would lead to mere energyshifts, but no damping effects. After taking the trace over eq. (9.217) wefind

ii(t) = p(0) —J dT2i dT i t ...),

t i-2

o o

.• • } = E( Vk ( T2 ) Vie(T i )p(0)TrB(B k( T2 )B kt(r,)pB(0))kk'

- krk( T2 )P(0) Vie( ) TrB( Bk( T2 )PB(0)Bk , ( TI))

— Vk( T I )P (0) Vie( T2 ) TrB( Bk( 7 1 )PB(0) B k'( T2))

P(0) Vk( ) ( T2 ) Trs( PB(0) Bk( TOBre( T2 )) } • (9.225)

The second term on the right hand side of eq. (9.217) has now beendropped because of eq. (9.224). In order to further simplify eq. (9.225), wediscuss the expressions

ft

0 d T2 exp [ hlwkr2 ]j 2 dr1 exp [ T1 Kkie( T2 — T1), (9.226)

where K is an abbreviation for

Kkk'( r2 ,TI) = Kkk'( T2 — T1) TrB(Bk(T2 )Bk ,(T i )pB(0)). (9.227)

The first part of eq. (9.227) expresses the fact that the correlation functionK depends only on the time difference, provided, that the interaction isstationary.* A simple coordinate transformation

*The proof runs as follows: Stationary means that in the Schriidinger picture Bk , Bk . andH, are time independent. We distinguish between two cases: (a) Bk and Bk . belong todifferent heatbaths. The trace (9.227) then factorizes into K kK le where e.g. Kk=TrB.k exp(iHr)72 /h1B(0)exp(—iHr )72 /h]expf — Hik)/(kTk)].Z; I where (9.222) was used.

Because the trace is cyclic: Tr(AB) Tr(BA) we immediately find Kk= whichijer

. (9.225)

as now been;q. (9.225), we

(9.226)

(9.227)

ition functioninteraction is

zture Bk , Bk. andid Bk. belong tovhere e.g. Kk.=(9.222) was used.tly find Kkm•


(9.220)

ival baths(9.221)

7-:„ so that

(9.222)

(9.223)

(9.224)

mere energyq. (9.217) we

•

T2 — T 1 = T

yields for (9.226)

fd r2 exp [ ( 614 + Aco 01'2 ] f exp[ — iild.o k/r] Kkk ,(T) dr.

0 0

We assume that the heatbath has a short memory so that (9.227) is onlynonvanishing for

I T2 — Td < T0 . (9.230)

In the following we consider times t which are large compared to thecorrelation time ro. Then we may simply replace the upper limit r2 byinfinity, so that we finally find for (9.226)

dr2 exp [ i(Acok + Aco )1-2 f exp[ — iCtw k ,r] Kkk ,( T dT • (9.231)

If

k k' = 0 (9.232)

the expression (9.231) becomes

f exp[ K kk .(T) dr. (9.233)

In the following we may simply assume that (9.232) holds because other-wise there appear rapidly oscillating terms which cancel the correspondingexpressions. We now make the second essential assumption, namely, thatthe interaction with the heatbath is so small that during the time T > ro thedensity matrix has changed very little. This allows us to make the replace-ment

ii(t) — j5(0) d t • (9.234)

If we further use the cyclic property of traces

TrB( B k( r1 )13 B (0) B le( T2 )) TrB( B k.( T2 ) B k( )pB(0)) (9.235)

TrB,k(B k(0)exp(— Hik) / ( kTk )D = 0 due to (9.224). (b) Bk and Bk. belong to the sameheatbath. Using (9.222) we write (9.227) in the form

Zk—i TrAexp[ iHr )T2 / h]B k(0) exp[ — iHr2 /h]

X expi iHr)rt / h1B k.(0) exp[ — iHr )T /h] exp[ — HY' )/ ( k Tk )1).

Since the trace is cyclic (9.227) transforms into

4-1 TrAexpEiHr)(1-2 — Ti )/ h]B k(0) exp[ — (T2 — ) / h]

X B k,(0) exp[ — / (kTk)1)

which depends only on T2 — T1.

(9.228)

(9.229)


we may rewrite (9.225) in the form

d(t)dt —

E VkVk,p(0)exp[ i(Awk + Ot]kk'

x j exp[ — iLakr]TrB(Bk(T)Bk,(0)PB(0))dr

— Vk p(0) Vie exp[i(Lok + Awk,)1]

x0

exp[ — iAwkr] TrB(B le( T )Bk(C)PB(0)) dr

+ P(0) Vk Vie exp [ i( Acok + Awk,)t]

X f exp [ — iAwkr]TrB(Bk(0)Bk,(T)PB(0))dr

— Vkp(0) expr i(Acok + tIoJk,)t]

X f exp[ iCtcokr] Tr B I( B k,(0)B k (T)p B(0)) d T . (9.236)

We now assume that the iteration step from time 1 = 0 to the time t maybe repeated at consecutive times so that we may replace the initial timet = 0 by the arbitrary time. The physical meaning of this assumption,which is quite essential for the whole procedure, can be justified by thefollowing consideration: Let us assume that the heatbaths are themselvescoupled to still larger heatbaths and so on. Then these "other" heatbathswill again and again bring back the original heatbaths to a truly thermody-namic, i.e. random state with the originally introduced temperatures Ti. Thereader should be warned that there is a whole literature on statisticalmechanics dealing with this and related problems. The idea of heatbathhierarchies, to my knowledge, is not employed there, but it seems essentialthat the heatbaths are themselves open systems. Otherwise the followingargument cannot be disproved: Let us consider the total system: field +heatbaths as a closed one. Then according to fundamental laws of thermo-dynamics the total system eventually tends to a state with a uniquetemperature, T, in contrast to our assumption that the heatbaths retaintheir individual temperatures 7). Since the integral expressions of the firstand second sum agree provided k is exchanged with k', and the same istrue for the third and fourth term, eq. (9.236) takes the very simple form

d (3dt E f[ Vvii(t), VdA kk,+[Vk,, 15(t)VdA' kk,), (9.237)

kk'

where the (

A kk'

=

A and A' zcase that

Awkholds. ThisHamiltoniafor inst.

ai+aj

It follows t

=Under this

=

It is therefevaluate th(heatbath al(

A kk =

where

Pm

where the coefficients A and A' are defined by

A kk' exP[ i( Awk Awk')t]

X fco

exp[ — iAco k ,T] TrB ( Bk( T )Bk , (0)PB(0)) dT (9.238)

irkk , = exp[ Atok + &)k,)1]

•

(9.236)

:ime t maynitial timessumption,ied by thethemselvesheatbaths

thermody-res T The

statisticalf heatbathis essential

following-n: field +of thermo-a unique

tths retain3f the firstle same ismple form

(9.237)

•

x fexp[ — i6a4kT] TrB ( B k (0) B k T )pB(0)) d T.0

A and A' are in a close internal connection: We consider the particularcase that

AWk k' = 0;

holds. This is suggested by the detailed consideration of the interactionHamiltonian (9.212), (9.213) since it must be Hermitian. In the atomic casefor instance, we always find the following combination

ai+ aj exp[icoif t]Bij + ajtai exp[—hout]Bi-it . (9.241)

It follows that k and k' are connected by

(i,j) = k and (j,i) = k'. (9.242)

Under this assumption we obtain

A'k ,4 ,= f exp[ iCicokT ] Tr B( p B (0)B 1-ct; (0)B le (T )) dT 0

exp[ — Ct(A) T ]T: r B ( B 1,4; (T) B k ,(0)p B(0)) dT) = Atk,•

(9.243)

It is therefore sufficient to discuss the properties of the A's only. Weevaluate the trace over the heatbath in the energy representation of theheatbath alone. (We denote the energy of state n by Mtn .) This yields

A kk = f exp[ — iAwk,T] E <nlBk lm><m B,; In>n , m

x exp[i(Sln — OJT — h1I,1 (kT)]

= E KnIB k lM> 1 2 exp[ —hOn / (kT)] 1 .778(St n — — tIcok,)nm

+

} Z -1 (9.244)*iP 2. — 2n — Awe,

where Z is the normalization of the heatbath trace. In the same way we

*I' means principal value (cf. eq. (7.273)].

(9.239)

= Bk. (9.240)

•


obtain

Ak , ,k = E KniBk lm>1 2 exp[ —h9 m / (kT)] {r8(0„,, — +nm

}ZiP

S2„, — 12,, — Atok

where use was made of eq. (9.222). A comparison of eqs. (9.245) and(9.244) leads to the important relation

ReAkk , = ReA k ,k exp[hcok./ (kT)] (9.246)

which we discuss both for the light field and the atom: (1) light field: witheq. (9.212) we have e.g. k = 1, k' = 2, Awk . =

ReAn= ReAuexp[hw/ (kT)]. (9.247)

(2) atom: with (9.213), (9.241) and (9.242) we have

Re A ,2L, • 4= Re A 2, , exp[ hwii / ( kr)] •

Specialization of eq. (9.237). (a) Light mode. We make the followingidentifications

= b, V2= b + ,

Aw l = —co, = (9.249)

B 1 = B , B2= B.

Equation (9.237) then takes the form

— =[0,b+]A21+[b+i5,b],412+[b,ijb+JAII+[b+,b]At2.dt(9.250)

Since the imaginary parts of A give rise to mere frequency shifts which canbe absorbed into the frequency of the actual oscillator we keep only thereal parts and put

A 21 = C, Al2= 8 = Eexp[—hw/(kT)]. (9.251, 252)

The final equation for the density matrix of the light field alone thus takesthe form

= 8([b +?i,b] +[b + + Cabf'5,b + ] +[b 03b + ]). (9.253)d t

Finally, we may transfrom 13 (interaction representation) into p (Schro-dinger representation), cf. eq. (9.210), and obtain eq. (9.167) above.

(b) Atom. H

Vk—*

B k = B

The density

dji = .dt

We remindinteractionSchrodingerequation (9.the substitut

T.= U-

where U.

U =Mg]

and obtain

(i/h)(1

We multiplyFor the first

(iin)P

The secondreadily convi(9.255a) now

Ua7aL

must be evalitransformed

a7aiexp

where wi j is

(9.245)

(9.248)


;„+ Awk,)

(9.245)

. (9.245) and

(9.246)

ht field: with

(9.247)

40.248)

the following

(9.249)

(b) Atom. Here we make the following identifications

Vk --> a i+ ai , Ato CO), (9.254)

Bk = B";.

The density matrix equation takes the simple form

—d = E [ ajAji,ij +[a i+ iipit adA1, ,u ). (9.255)dt

We remind the reader that is the density matrix of the "free field" in theinteraction representation. According to (9.214) a7 pi are operators in theSchrodinger picture. As a final step of our analysis we transform the wholeequation (9.255) back into the Schrodinger picture. To this end we makethe substitution

= U -113U,

where U is given by

U= exp[ - iHFt/h],

and obtain

(i/h)(HFA - HF) + U U

ii

dt

= E { [ a i+ ay - '17, adAii,ii +[a7 af ,U - Uajt adAyi,u).

(9 .255a)

We multiply this equation from the left by U and from the right by U -1.For the first bracket on the left-hand side of (9.255a) we obtain

(i/h)[HFip].

The second term on the left-hand side transforms into d p /dt. One mayreadily convince oneself that in the commutators on the right-hand side of(9.255a) now expressions of the form

Ua,' ay - I

must be evaluated. As we know [cf. eq. (9.214)] these latter expressions aretransformed into

piexp[iwyt]

where co, j is given by eq. (9.215). Collecting all the transformed terms we

Ab]At.

(9.250)

ts which caneep only the

(9.251, 252)

le thus takes

1). (9.253)

to p (Schro-lbove.

1111

may rewrite eq. (9.255a) in the formdp

= – (i/h)[HF,p] + E [ail" ajpiai+ a dAji,iidt

+[ai+ , pai+ (9.255b)

The time evolution of p is now explicitly determined by two differentterms. The commutator between HF and p describes the coherent evolutionof the system whereas the other terms under the sum describe the incoher-ent processes.We derive the equation for the average of the operator a: a„ by multiply-ing both sides of eq. (9.255b) by a: a„ and taking the trace:

d<a„,* a„ >

+ (a: an a7 ad3a7 a,– a: aaft ajai+

X (i/h)a: a n [ HF , p]). (9.256)

For the further evaluation of the trace we use its cyclic property andarrange the operators in such a way that the density matrix A stands on theright-hand side. We then use the theorem

+ + + _ + (9.257)ai.

which holds if only one electron is present. The relation (9.257) can beproved as follows: consider within the product the terms which bear theindices it and A + 1. Exchanging a a 1 and using the commutationrelations we find

a1Aa, 1 a1A+1 . ah.

= a

(9.258)

The term which still contains aiL iai, contains two subsequent annihilationoperators. These operators are applied to a state which contains only oneelectron. The corresponding wave function to vanishes. If (9.258) is per-formed for all indices, (9.257) results. Using (9.257) we transform (9.256)into

—d

<a + an > = <a: a„>(iwn,„ + 'mn y„,„), for m n, (9.259)dt

cii<a: an,› = E f<af+ ai >Wim – <a: a m>Wm;} (9.260)

where we hay(Wj„,= A

Y. = 2

Awn,. = –

Win, is evidentbe seen by ccsecond part ofthe sum of thconsidered. Ecfrom phase fittions, i.e. Wn„,however, in thinterpreted aswritten in.

P( t ) = 2m,

and consider

Tr(a:ani

where 00 is thtwe find immec

Tr(a:ant

so that we car<a„,+an>=

Exercises on s

(1) A quantuiscribed by optWhat is its de:Hint: Write

dp = ( ddt 171

(dp/d t )incoherer

dpk dt icoher,

dt– Tr E [(a: an al' aiiia7 a,– a: an a7 ajai+ afi3)Afi,u

(9410) dp i rH, p],

k dt /coherent


(9 .255b)

o differenta evolutionhe incoher-

y multiply-

13)Aii,u

,u].

(9.256)

perty andads on the

(9.257)

7) can bebear the

mutation

where we have used the abbreviationsW.= Aj,,, , ,,,j + (9.261)

= E Re ( A mi, +

in = I E ( Wni + Win), (9.262)

a

Acomn = — E + (9.263)a

is evidently the transition rate from the state] to the state m, as maybe seen by considering eq. (9.260). ymn is the phase halfwidth from thesecond part of eq. (9.259). We see that the halfwidth is determined by halfthe sum of the transition rates out of the two states n and m which areconsidered. Equation (9.262) contains also an additional term which stemsfrom phase fluctuations alone which are not accompanied by real transi-tions, i.e. w, eq. (9.261), contains frequency shifts which we shall neglect,however, in the following. We now show that (9.259), (9.260) can also beinterpreted as an equation for the density matrix itself if the latter one iswritten in the occupation number representation. We put

p(t) = E p„,„(t)a,:an (9.264)m,n

and consider

Tr(a:a„p(t)) = E <00a la: a n E 400> , (9.265)

where 00 is the vacuum state and al' 00 a one-electron state. Using (9.257)we find immediately

Tr(a,n+ an p(t)) = pn ,n(t), (9.266)

so that we can identify<a:an > = (9.267)

(9.258) Exercises on section 9.5.

nihilation (1) A quantum mechanical harmonic oscillator (e.g. a field mode) de-

only one scribed by operators b ÷ , b is coupled to a reservoir and a coherent source.

8) is per- What is its density matrix equation?

m (9.256) Hint: Writedp dp dp

, (9.259) dt di /incoherent dt,i coherent

(dP/ d )incoherent is given by the r.h.s. of (9.156) and


where

H hwob +b + yo exp( /a) t )b + yo exp( — t )b .

(2) Calculate Kb>, (b + b> using p = (9.204).Hint: Remember that 02> = Tr(S2p)(3) A two-level atom is coupled to reservoirs and a coherent driving field.Derive its density matrix equation.Hint: Same as for exercise (I).

Mathematk

In this bookfunctions redefinitiikathese funder anintegral it is8-function clis

8(w) =

The limit t —we multiplyan interval

(028(C

—

which by me

—1

limITt --o 00

The effect off(0) while thTherefore (A

f(0) f

which can ahexists and is

•

-iving field.

Mathematical appendix. Dirac's (5-function and related functions

In this book we encountered, at several occasions, Dirac's 8-function andfunctions related to it. Here we give a list of these functions, theirdefinitions and their most important properties. It should be noted thatthese functions are quantities which are meaningful only when they occurunder an integral. When these quantities occur in equations withoutintegral it is understood that eventually an integration is performed. The8-function can be defined in various ways. The one occurring in this bookis

8(co) = lim = li msin wt

LIT j-.00 t

The limit t —> cc of the function sin cat/co does not exist in itself. But whenwe multiply (A.1) by a continuous function f(w) and integrate over w overan interval which contains w = 0 we obtain the following relation

&MAO dw = lim (.2 sin wt f(w)(16)

IT I-

which by means of the transformation wt = a is transformed into

f—1

lim 432 r sin a

f(a / Oda .t--> 00 - W it co'

The effect of the limit t —> co can be visualized as follows. f(cd / t) tends tof(0) while the limits of the integral tend to — co and + co, respectively.Therefore (A.3) reduces to

f(0)1 f c'c' sin a

da (A.4)ca'

which can also be proven mathematically rigorously. The integral over aexists and is equal to Tr. Therefore equating the left-hand side of (A.2) to

•(A.1)

t-soo

(A.2)

(A.3)

O

336 Mathematical appendix. Dirac's 8-function and related functions

(A.4) we obtain the fundamental property of the 6-function

f 8(w)f(w)dco = f(0). (A.5)

It holds if f is continuous and the integration comprises the point co = 0.As can be seen directly from the definition (A.1), 8 is an even function

8( - co) = 8(4 (A.6)

Due to this fact, one may readily deduce from (A.5) the relation

(A.7)

The 8-function can also be defined in three dimensions by

1 8(x) = f exp( kx) d3 k. (A.8)

(2/03

Provided f(x) is a continuous function and the volume of integrationcomprises the origin x = 0 we obtain in analogy to (A5)

18(x)f(x) d 3 x = 1(0). (A.9)

A function closely related to the 8-function is the 8 ± -function in which theintegration over r, which occurred in (A.1), runs only over positive values.Thus the 8 ÷ -function is defined by

1 8+ (w) = —,,

1 lim e -1"T dr = urn (A.10)

Lir t-yeo 0 LIT 1--,00 hit)

When we split the exponential function into cosine and sine we obtain1,. 1 - cos cot 1 ,. sin cot

8+ (4)) = — 11M + —, lilll .

2 71' z 1._, 00 4) Z. IT t--).00 W

While the second part on the right hand side is immediately recognized asthe 8-function, the first represents a new function which we abbreviate byP/ co.

• 1 -costalim

(-1.00

(A.12)

To get an insight into its properties we multiply (A.12) by a continuousfunction f(w) and integrate over an interval -fa, co2

fW2 P dw. (A.13)

Let us consider first the effect of cos wt. When t is very large, cos cot

becomes a vthe positivefig. 4.3 oncos cot is muvalid. On th.goes to 0. Aeffect of P/tOn the othel

-

P has, thenintegral. Kn(write (A.11)

8+(w)

In aciOn

8_(co)

The functiocomplex co-pof integratio

Let f(co)tperform anobtain accor

2 1ri 96

Comparingfunction asintegrationout the into

To define1/(2rico) al1, i.e. just hus now con- cc till + c

in the loweisame as tha

u p2ri

(A.11)

f: 8(w) dw =

Mathematical appendix. Dirac's 8-function and related functions 337

(A.5)

= 0.-;tion

(A.6)

(A.7)

(A.8)

!grit

(A.9)

lich thevalues.

(A.10)

tamn

(A.11)

lized asiate by

(A.12)

tinuous

(A.13)-

cos cot

111

becomes a very rapidly oscillating function so that when integrated over cothe positive and negative contributions cancel each other (compare alsofig. 4.3 on page 138 where a quite similar discussion is made). Even ifcos cot is multiplied by a continuous function f(co) this statement remainsvalid. On the other hand (1 - cos wt)/w vanishes for any finite t when wgoes to 0. As a consequence of these considerations we recognize that theeffect of PR is as follows. Outside of w = 0, PR can be replaced by 1/w.On the other hand the point co = 0 is cut out. Therefore we may write

Pi f(co)- do.) = limo (f : f(: ) (Ito + r2-4(4) dw). (A.14)

— col 4.1

(..) 2 1

P has, therefore, the property of denoting the principal value of theintegral. Knowing the meaning of PR and the 6-function we may finallywrite (A.11) in short as

1 ,. 1 - 1 P

8+ ( 4)) Illn 7T1 t---> co 0,) 277i

In addition to 8 + one may define 8_ which is related to 8 + by

8-( w ) =The functions 8((o), 8+ , 8_ and P/(2vico) can also be defined in thecomplex w-plane. Each time these functions equal 1/(2Triw) but the pathsof integration are different.

Let f(w) be a function which contains no singularity at co = 0. When weperform an integration on a path which is a small circle around co= 0 weobtain according to Cauchy's theorem

1 96f( W ) A

— WA) = f(0). (A.17)2711

Comparing (A.17) with (A.5) we recognize that we may define the 6-function as 1/(2771w) with the additional prescription to perform theintegration over a small circle around co= 0. P is the prescription to leaveout the interval -e to + E when we integrate along the real axis.

To define 8+ we proceed as follows. The clockwise integration over1/(2riw) along a semicircle in the lower half of the complex plane yields1, i.e. just half of the value we would obtain by use of the 8-function. Letus now consider an integration over 1/(27ico) along the real axis from- oo till + oo where we leave out the point to = 0 using a small semicirclein the lower half plane around co = 0. The result of this operation is thesame as that of integrating along the real axis over

1 —P

+-,8(co). (A.18)

2iri co 2

O

338 Mathematical appendix. Dirac's 8-function and related functions

(A.18) is identical with the definition of the 8,-function. On the otherhand we may replace the path of integration containing the circle by apath along the whole real axis when we shift w by a small amount —is.This leads us to the final result

1 ,. 1 1 P 1

8+ = — nm± —28(4 (A.19)2 .771 — is

Similarly we obtain

= — iim 1 ,. 1

= — 1 —P

+ —104 (A.20)2771 s-0 + is 27i (4 2

Refereno

The topicpublications.refersu 0 wt

thefield. Finallycontain relat

There arereview artici,modern (mil(

Wolf, E.Springer S

TextbooksDavydov, ADirac, P. /Oxford 19:Gasiorowic.Landau, LTheory. AA

MerzbacheMessiah, ASakurai, J.Schiff, L. A

TextbooksAbragam,1961

Textbooks on SPIN RESONANCE:Abragam, A.: The Principles of Nuclear Magnetism. Clarendon, Oxford1961•

the othercircle by anount —is.

(A.19)

(A.20)

References and further reading

The topics treated in this volume comprise a vast field with an enormous number ofpublications. Since this book is meant as a textbook for students, I am giving here thosereferences which are of the greatest interest to them: namely, first textbooks on relatedsubjects so that students can learn more about details, in particular on atomic physics andquantum theory. Second, I have listed references to the original work of the pioneers of thisfield. Finally, I included those references which I directly made use of in my book or whichcontain related material.

There are a number of scientific journals dealing with modern optics. Quite excellentreview articles may be found in particular in the following series dedicated particularly tomodern optics:

Wolf, E. Ed: Progress in Optics. North-Holland, AmsterdamSpringer Series in Optical Sciences. Springer, Berlin

Textbooks on QUANTUM MECHANICS:Davydov, A. S.: Quantum Mechanics. Addison Wesley, 1965Dirac, P. A. M.: The Principles of Quantum Mechanics. 4th ed. Clarendon,Oxford 1958Gasiorowicz, S.: Quantum Physics. Wiley, 1974Landau, L. D. and Lifshitz, E. M.: Quantum Mechanics. NonrelativisticTheory. Addison Wesley, 1965Merzbacher, E.: Quantum Mechanics. 2nd ed. Wiley, 1970Messiah, A.: Quantum Mechanics. Vol I, II. Wiley, 1968Sakurai, J. J.: Advanced Quantum Mechanics. Addison Wesley, 1967Schiff, L. I.: Quantum Mechanics. 3rd ed. McGraw Hill, 1968

•

340 References and further reading

(a) Pertu(i) Time-Schrodim(ii) TimeDirac, P

cf. textbc

§1.8. Qu:Fermi, EHeisenbe

cf. textbcFIELD

§1.9. ProB

Born, M.Clauser,.Einstein,d'EspagtHeisenbe.

cf. textbc

§1.11. CcBeran,Cliffs, NtBorn, M.Francon,Glauber,Cohen-TaGlauber,VarennaHanbtayKlauder,.BenjaminMandel,Mandel, 1of Light.

Abragam, A. and Bleaney, B.: Electron Paramagnetic Resonance ofTransition Ions. Clarendon, Oxford 1970Poole, jr., C. P. and Farach, H. A.: The Theory of Magnetic Resonance.Wiley, 1972Slichter, C. P.: Principles of Magnetic Resonance. Springer, New York,1978

Textbooks on QUANTIZATION OF LIGHT FIELD:Bogoliubov, N. N. and Shirkov, D. V.: An Introduction to the Theory ofQuantized Fields. 3rd ed. Wiley, New York 1980Fain, V. M. and Khanin, Ya. I.: Quantum Electronics. Vol I, II. Pergamon,Oxford 1969Feynman, R. P.: Quantum Electronics. Benjamin, New York 1962Haken, H.: Quantum Field Theory of Solids. North-Holland, Amsterdam1976Heitler, W. H.: The Quantum Theory of Radiation. 3rd ed. Clarendon,Oxford 1954Jauch, J. M. and Rohrlich, F.: The Theory of Photons and Electrons. 2nded. Springer 1976Klauder, J. R. and Sudarshan, E. C. G.: Fundamentals of Quantum Optics.Benjamin, New York 1968Louise 1/ W. H.: Radiation and Noise in Quantum Electronics. McGrawHill, 1964Louise11 W. H.: Quantum Statistical Properties of Radiation. Wiley, NewYork 1973Schweber S. S.: An Introduction to Relativistic Quantum Field Theory.Harper & Row, New York 1961

For Chapter I:§1.5. The early quantum theory of matter and lightBohr, N.: Phil. Mag. 26, 476, 857 (1913)Einstein, A.: Ann. d. Phys. 17, 132 (1905); 20, 199 (1906)Einstein, A.: Phys. Z. 18, 121 (1917)Planck, M.: Verh. d. deut. phys. Gesellsch. 2, 237 (1900); Ann. d. Phys. 4,553 (1901)

cf. te7:'books: QUANTUM MECHANICS

§1.6. id 1.7. Quantum mechanicsDeBroglie, L.: Nature 112, 540 (1923)Dirac, P. A. M.: Proc. Roy. Soc. A117, 610 (1928)Heisenberg, W.: Z. Phys. 33, 879 (1925)Schrbdinger, E.: Ann. d. Phys. 79, 361, 489, 734 (1926)

References and further reading 341

nance of

esonance.

ew York,

(a) Perturbation theory:(i) Time-independentSchrodinger, E.: Ann. d. Phys. 80, 437 (1926)(ii) Time-dependentDirac, P. A. M.: Proc. Roy. Soc. A112, 661 (1926); A114, 243 (1927)

cf. textbooks: QUANTUM MECHANICS

Iheory of

'ergamon,

)2msterdam

larendon,

m Optics.

McGraw

iley, New

I Theory.

§1.8. Quantum electrodynamicsFermi, E.: Rev. Mod. Phys. 4, 87 (1932)Heisenberg, W. and Pauli, W.: Z. Phys. 56, 1 (1926); 59, 168 (1930)

cf. textbooks: QUANTUM MECHANICS; QUANTIZATION OF LIGHTFIELD

§1.9. Probabilistic interpretation of quantum theoryBell, J. S.: Physics 1, 195 (1964)Bohr, N.: Nature 121, 580 (1928)Born, M.: Z. Phys. 37, 863 (1926); Nature 119, 354 (1927)Clauser, J F. and Shimoni, A.: Rep. Progr. i. Phys. 41, 1881 (1978)Einstein, A., and Podolsky, B. and Rosen, N.: Phys. Rev. 47, 777 (1935)d'Espagnat, B.: Sci. Am. Nov. 79, 128 (1979)Heisenberg, W.: Z. Phys. 43, 172 (1927)


1. Phys. 4,

§1.11. CoherenceBeran, M. and Parrent, G. B.: Theory of Partial Coherence. EnglewoodCliffs, New York 1964Born, M. and Wolf, E.: Principles of Optics. Pergamon, Oxford 1964Francon, M.: Optical Interferometry. Academic Press, New York 1966Glauber, R. J.: In: Quantum Optics. Eds. DeWitt, C., Blandin, A., andCohen-Tannoudji, C.; Gordon & Breach, New York 1965Glauber, R. J.: In: Proc. of the Int. School of Physics "Enrico Fermi".Varenna 1967. Academic Press, New YorkHanbury Brown, R. and Twiss, R. Q.: Nature 117, 27 (1956)Klauder, J. R. and Sudarshan, E. C. G.: Fundamentals of Quantum Optics.Benjamin, New York 1968Mandel, L. and Wolf, E.: Rev. Mod. Phys. 37, 231 (1965)Mandel, L. and Wolf, E.: Selected Papers on Coherence and Fluctuationsof Light. Vols I, II. Dover Publ., New York 1970•

•


§1 15 Lamb shiftLamb, W. E. and Retherford, R. C.: Phys. Rev. 72, 241 (1947); 79, 549(1950)

§1.17. Nonlinear OpticsBloembergen, N.: Nonlinear optics. Benjamim, New York 1965

For Chapter 2:§2.1. WavesYoung, T.: Phil. Trans. Roy. Soc. 12, 387 (1802)

§2.2. CoherenceSame as for §1.11

§2.3. Planck's radiation lawKubo, R.: Statistical Mechanics. North-Holland, Amsterdam 1965Kubo, R.: Thermodynamics. North-Holland, Amsterdam 1968Planck, M.: Verh. d. deut. phys. Gesellsch. 2, 237 (1900); Ann. d. Phys. 4,553 (1901)Tien, C. J. and Lienhard, J. H.: Statistical Thermodynamics. Holt, Rine-hart and Winston, New York 1971

§2.4. and 2.5. PhotonsSame as for §1.5

For Chapter 3§3.1. Wave equation for matter: Schrodinger equationDavisson, C. and Germer, L. H.: Phys. Rev. 30, 705 (1927)


§3.2. Measurements and expectation values


§3.3. Harmonic oscillatorBorn, M., Heisenberg, W. and Jordan, P.: Z. Phys. 35, 557 (1925)Ludwig, G.: Z. Phys. 130, 468 (1951)


§3.4. HydBethe, H.Electron

cf. textboc

§3.5. AtorBarrow, CNew Yorl,Ha8k0en,19

Herzberg,van NostrKuhn, H.Landau,Mechanic:Sonaiw-eiBr hvWhite, H.1934

§3.6. SolicHaken, H1976Haug, A.:1972Jones, W.Wiley, NeKittel, C.:Kittel, C.:Peierls, R.1955Pines D.:Ziman, J.1965Ziman, J.1967

§3.7. NuclBohr, A. a1969

•


§3.4. Hydrogen atomBethe, H. and Salpeter, E. E.: Quantum Mechanics of one and twoElectron Atoms. Springer, Berlin; Academic, New York 1957


§3.5. AtomsBarrow, G. M.: Introduction to Molecular Spectroscopy. McGraw Hill,New York 1962Haken, H. and Wolf, H. C.: Atom- and Quantenphysik. Springer, Berlin1980Herzberg, G.: Infrared and Raman Spectra of Polyatomic Molecules. D.van Nostrand, Princeton 1964Kuhn, H. G.: Atomic Spectra. Longmans, London 1964Landau, L. D., Akhiezer, A. I. and Lifshitz, E. M.: General Physics,Mechanics and Molecular Physics. Pergamon, Oxford 1967Sommerfeld, A.: Atombau und Spektrallinien. Vol I, II. Vieweg,Braunschweig 1960White, H. E.: Introduction to Atomic Spectra. McGraw Hill, New York1934

§3.6. SolidsHaken, H.: Quantum Field Theory of Solids. North-Holland, Amsterdam1976Haug, A.: Theoretical Solid State Physics. Vol I, II. Pergamon, Oxford1972Jones, W. and March, N. H.: Theoretical Solid State Physics. Vol I, II.Wiley, New York 1973Kittel, C.: Quantum Theory of Solids. Wiley, New York 1964Kittel, C.: Introduction to Solid State Physics. Wiley, New York 1966Peierls, R. E.: Quantum Theory of Solids. Oxford at the Clarendon Press,1955Pines D.: Elementary Excitations in Solids. Benjamin, New York 1963Ziman, J. M.: Principles of the Theory of Solids. Cambridge Univ. Press,1965Ziman, J. M.: Electrons and Phonons. Oxford at the Clarendon Press,1967

(1947); 79, 549

1965

1925)§3.7. NucleiBohr, A. and Mottelson, B. R.: Nuclear Structure. Benjamin, New York1969

n 1965

)6Apphys. 4,

cs. Holt, Rine-'


(b) Photo,Kurnit, N(1964); PI

(c) Free i)Brewer, EArecchi, I

cf. textbo

For Chap§5.1.-5.4.Fermi, E.Jackson,

RAW

§5.5. CohGlauber, I

131, 2766SchrlidingSchwinger

cf. text)),MECHAi

5.6. Heise

cf. textbo.

5.7. DriveLudwig, (

cf. textbo

5.8. QuanSame as I

§3.8. SpinMcWeeny, R.: Spins in Chemistry. Academic Press, New York 1970Pauli, W.: Z. Phys. 43, 601 (1927)Uhlenbeck, G. E. and Goudsmith, S.: Naturwiss. 13, 953 (1925); Nature117, 264 (1926)

cf. textbooks: SPIN RESONANCE

For Chapter 4:§4.1.-4.3. Two-level atom in external fields

cf. textbooks: QUANTUM MECHANICS; QUANTIZATION OF LIGHTFIELD

§4.4. Two-photon absorptionBiraben, F., Cagnac, B. and Gruenberg, G.: Phys. Rev. Lett. 32, 643 (1974)Goeppert-Mayer, M.: Ann. Phys. 9, 273 (1931)Hansch, T. W., Harvey, K. C., Meisel, G. and Schawlow, A. L.: Opt.Comm. 11, 50 (1974)Hopfield, J. J., Worlock, J. M. and Park, K.: Phys. Rev. Lett. 11, 414(1963)Hopfield, J. J. and Worlock, J. M.: Phys. Rev. A137, 1455 (1%5)Kaiser, W. and Garrett, C. G. B.: Phys. Rev. Lett. 7, 229 (1961)Levenson, M. D. and Bloembergen, N.: Phys. Rev. Lett. 32, 645 (1974)Mahr, H.: Two Photon Absorption Spectroscopy. p. 285. In: QuantumElectronics, Vol. 1A. Eds. Rabin, H. and Tang, C. L., Academic Press,New York 1975

§4.5. Non-resonant perturbations


§4.6.4.9. Analogy two-level atom and spin -21-Feynman, R. P., Vernon, F. L. and Hellwarth, R. W.: J. Appl. Phys. 28, 49(1957)Haken, H.: Laser Theory. Encycl. of Phys. XXV/2c. Springer, Berlin 1970

(a) Rotating wave approximation:Bloch, F. and Siegert, A. J.: Phys. Rev. 57, 522 (1940)Rabi, J. J., Ramey, N. F. and Schwinger, J.: Rev. Mod. Phys. 26, 107 (1954)

•


(1-;) exerir.,Kurnit, N. A., Abeita, I. L). art' flarimanr, . S.: i 3, 567(1964); Phys. Rev. 141, 391 (1966)

(c) Free :etrck.iuc ti(-n de. y,Brewer, R. G.: In: CoheIence in Spectrc s.

Arecchi, F T. Bonifacio, R. and Scully, M. 0. Plenum, New York 1978

)rk 1970

1925); Nature

cf. teAtbooh. .V )N-!. :VC

'IV OF LIGHT

32, 61i(1974)

A.,: Opt.

Lett. 11, 414

965)61)45 (1974)In: Quantumidemic Press,

6, 107 (1954)

Fc,r Chapte • 5`. 5. 1.-5.4 QL ‘antLat :c rFermi, E.. R_ev. 1Vod. r1 s 4 ,i'f14:19?2,

Jackson, J. D.: Classical Electrodynamics. Wiley, NeA .i9o7

cf. Qt;./V." l'.. • Y.\* !../* /;_ .

FIELD

§5.5. Cohetert st tef;

Glauber, : PLy.. .;\ 1,, ( ; (196.3',

131, 2 .766 (1963)Schrodinger, E.: Ann. Phys. 87, 570 (1928)Schw:nge, , : Phi. 91. 7?1? (1°53)

cf. textbooks: Q :LI'Z.i 7' • _MECHANICS

5.5. Heisent-Jerg : -e


5.7. Driven aarmcni. osc atLudwig, G.: Z. Phys. 130, 468 (i95i)

cf. tzxtbooxs: 12U us •, M

5.8. Quantization of light field: Multimode caseSame as '(;1* § 5.1. 5

. Phys. 28, 49

Berlin 1970

352 Subject index

- beats, observed signal 278, 282- electrodynamics 16- mechanical coherence functions 271-- consistency 296, 305, 307- fluctuations 303- mechanics 12- noise 2- statistical average 317

Rabi frequency 138rate equation, for photons 61- equations for occupation numbers 305Rayleigh-Jeans' law 60reduced density matrix 319, 324-- matrix, equations of motion 324reflection 1refraction 3relaxation times T1 , T2 156renormalization 33resonant processes 215rotating wave approximation 138, 144, 216,

267, 315Rutherford 8

scalar potential 12, 182, 204Schrodinger 12Schriidinger equation 63, 66, 127- equation, formal solution 232- equation for single field mode 164, 169- equation, interaction representation 213- , two-level atom and single mode 211

equation, two-level atom in multimoclefield 216

- picture 174selection rules 100self-energy 234, 236, 241, 242, 244semiconductor 107single electron state 198solids 104Sommerfeld 12spectrum, dynamic Stark effect 263spherical wave 44spin 112, 147spin echo 149, 150, 152- flipping 144

functions 113, 141- Hamiltonian 114

matrices 104, 113, 114- operators 314

precession 143, 147- resonance 141spontaneous emission 11. 16. 17, 28, 31, 59.

222, 251- emission, coherence properties 275standing wave 293stationary solution, Brownian motion 291- solution of Fokker-Planck eq. 292statistical operator 316stimulated emission 11, 15, 59, 125, 222, 226.

231superposition principle 38

telegraph equation 292temperature 290, 317thermal equilibrium 60, 290, 297, 300, 317- photon number 301, 320three-level atom in multimode field 241time average 265trace 300, 316trace, cyclic property 320transition probability 155, 224, 310- probability for spontaneous emission 253- rate 305two-level atom and single field mode 214- atom, compared to spin 1/2 145- atom coupled to heatbath 305- in classical electric field 214- multimode field 216- atom, resonant coherent external field 137two-photon absorption 130, 132two-time probability density 294two-level atom, in external field 119---- incoherent external field 123

uncertainty relations 18, 32, 75, 189

vacuum state, bosons 84, 86, 184- , for fermions 197vector potential 163, 182, 185, 204, 218, 268velocity of light 5vertex 230virtual processes 33, 215, 233- transitions 276visibility of fringes, definition 44

wave-particle-dualism 1, 18, 21wave function 13- function, normalization condition 69

- , probabilisti- number 293- packet 14, 90,- packet of spin- track 48Weisskopf - Wign,

Subject index 353

Weisskopf-Wigner theory 251Wiens's law 50

17, 28, 31, 59,

-ties 275

motion 291eq. 292

• 125, 222, 226,

Young's experiment 3, 21, 41, 47, 283

zero point energy 182

— , probabilistic interpretation 7, 20, 68— number 293

packet 14, 90, 146, 149, 212— packet of spin functions 115— track 48Weisskopf–Wigner 280

97, 300, 317

field 241

•31•emission 253

mode 214! 145

14

:.rnal field 1372)41 119eld 123

. 189

!el

204, 218, 268

4

ition 69

•

Light +Volume+I

Documents

Transcript of Light +Volume+I