ECEG398 Quantum Optics Course Notes Part 1: Introduction
description
Transcript of ECEG398 Quantum Optics Course Notes Part 1: Introduction
![Page 1: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/1.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-1
ECEG398 Quantum Optics Course Notes
Part 1: Introduction
Prof. Charles A. DiMarzio
and Prof. Anthony J. Devaney
Northeastern University
Spring 2006
![Page 2: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/2.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-2
Lecture Overview• Motivation
– Optical Spectrum and Sources
– Coherence, Bandwidth, and Fluctuations
– Motivation: Photon Counting Experiments
– Classical Optical Noise
– Back-Door Quantum Optics
• Background– Survival Quantum Mechanics
![Page 3: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/3.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-3
Classical Maxwellian EM Waves
E E
E
x
y
z H
HH
λ
v=c
λ=c/υ
c=3x108 m/s (free space)
υ = frequency (Hz)
Thanks to Prof. S. W.McKnight
![Page 4: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/4.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-4
Electromagnetic Spectrum (by λ)
1 μ 10 μ 100 μ = 0.1mm
0.1 μ10 nm =100Å
VIS=
0.40-0.75μ
1 mm 1 cm 0.1 m
IR=
Near: 0.75-2.5μ
Mid: 2.5-30μ
Far: 30-1000μ
UV=
Near-UV: 0.3-.4 μ
Vacuum-UV: 100-300 nm
Extreme-UV: 1-100 nm
MicrowavesX-Ray Mm-waves
10 Å1 Å0.1 Å
Soft X-Ray RFγ-Ray
(300 THz)
Thanks to Prof. S. W.McKnight
![Page 5: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/5.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-5
Coherence of Light
• Assume I know the amplitude and phase of the wave at some time t (or position r).
• Can I predict the amplitude and phase of the wave at some later time t+(or at r+)?
![Page 6: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/6.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-6
Coherence and Bandwidth
Pure Cosinef=1
Pure Cosinef=1.05
3 CosinesAveragedf=0.93, 1, 1.05
Same as at left, and a delayed copy. Note Loss of coherence.
0 5 10-1
-0.5
0
0.5
1
0 5 10-1
-0.5
0
0.5
1
0 5 10-1
-0.5
0
0.5
1
0 5 10-1
-0.5
0
0.5
1
![Page 7: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/7.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-7
Realistic Example
50 Random Sine Waves with Center Frequency 1 and Bandwidth 0.8.
f
0 1 2 3 4 5 6 7 8-0.4
-0.2
0
0.2
0.4
0 1 2 3 4 5 6 7 8-0.4
-0.2
0
0.2
0.4
Long Delay: Decorrelation
Short Delay
![Page 8: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/8.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-8
Correlation Function
I1+I2
21II
![Page 9: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/9.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-9
Controlling CoherenceMaking Light Coherent Making Light Incoherent
Spatial Filter forSpatial Coherence
Wavelength Filterfor Temporal Coherence
Ground Glass toDestroy Spatial Coherence
Move it toDestroy Temporal Coherence
![Page 10: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/10.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-10
A Thought Experiment
• Consider the most coherent source I can imagine.
• Suppose I believe that light comes in quanta called photons.
• What are the implications of that assumption for fluctuations?
![Page 11: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/11.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-11
Photon Counting Experiment
0 5
Clock
GateCounter
tClock Signal
t
Photon Arrival
t
Photon Count3 1 2
Probability Density
n
Experimental Setup to measure the probability distribution of photon number.
![Page 12: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/12.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-12
The Mean Number
• Photon Energy is h• Power on Detector is P
• Photon Arrival Rate is =P/h – Photon “Headway” is 1/
• Energy During Gate is PT
• Mean Photon Count is n=PT/h• But what is the Standard Deviation?
![Page 13: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/13.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-13
What do you expect?
• Photons arrive equally spaced in time.– One photon per time 1/– Count is T +/- 1 maybe?
• Photons are like the Number 39 Bus.– If the headway is 1/5 min...– Sometimes you wait 15 minutes and get three
of them.
![Page 14: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/14.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-14
Back-Door Quantum Optics (Power)
• Suppose I detect some photons in time, t
• Consider a short time, dt, after that– The probability of a photon is P(1,dt)=dt– dt is so small that P(2,dt) is almost zero– Assume this is independent of previous history– P(n,t+dt)=P(n,t)P(0,dt)+P(n-1,t)P(1,dt)
• Poisson Distribution: P(n,t)=exp(-at)(at)n/n!
• The proof is an exercise for the student
![Page 15: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/15.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-15
Quantum CoherenceHere are some results: Later we will prove them.
![Page 16: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/16.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-16
Question for Later: Can We Do Better?
• Poisson Distribution– – Fundamental Limit on Noise
• Amplitude and• Phase
– Limit is On the Product of Uncertainties
• Squeezed Light– Amplitude Squeezed (Subpoisson Statistics) but larger
phase noise– Phase Squeezed (Just the Opposite)
Stopped here 9 Jan 06
n2
![Page 17: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/17.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-17
Back-Door Quantum Optics (Field)
• Assume a classical (constant) field, Usig
• Add a random noise field Unoise
– Complex Zero-Mean Gaussian
• Compute as function of <| Unoise|2>
• Compare to Poisson distribution
• Fix <| Unoise|2> to Determine Noise Source Equivalent to Quantum Fluctuations
![Page 18: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/18.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-18
Classical Noise Model
Add Field Amplitudes
Re U
Im U
Us
Un
10842-1.tex:2
![Page 19: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/19.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-19
Photon Noise
10842-1.tex:3 10842-1.tex:5=10842-1-5.tif
![Page 20: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/20.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-20
Noise Power
• One Photon per Reciprocal Bandwidth
• Amplitude Fluctuation– Set by Matching Poisson Distribution
• Phase Fluctuation– Set by Assuming
• Equal Noise in Real and Imaginary Part
• Real and Imaginary Part Uncorrelated
![Page 21: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/21.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-21
The Real Thing! Survival Guide
• The Postulates of Quantum Mechanics
• States and Wave Functions
• Probability Densities
• Representations
• Dirac Notation: Vectors, Bras, and Kets
• Commutators and Uncertainty
• Harmonic Oscillator
![Page 22: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/22.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-22
Five Postulates• 1. The physical state of a system is described by a
wavefunction.
• 2. Every physical observable corresponds to a Hermitian operator.
• 3. The result of a measurement is an eigenvalue of the corresponding operator.
• 4. If we obtain the result ai in measuring A, then the system is in the corresponding eigenstate, i after making the measurement.
• 5. The time dependence of a state is given by
Ht
it
hi
2
![Page 23: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/23.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-23
State of a System
• State Defined by a Wave Function, – Depends on, eg. position or momentum– Equivalent information in different
representations. (x) and (p), a Fourier Pair
• Interpretation of Wavefunction– Probability Density: P(x)=|(x)|2
– Probability: P(x)dx=|(x)|2dx
![Page 24: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/24.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-24
Wave Function as a Vector
• List (x) for all x (Infinite Dimensionality)
• Write as superposition of vectors in a basis set. (x)
(x)
x
x
(x)=a11(x)+a22(x)+...
...2
1
a
a
...2
1
x
x
![Page 25: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/25.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-25
More on Probability
• Where is the particle?
• Matrix Notation
dxxxxdxxPx )()()( *
Xx †
2
1
21 ** x
x
Xxxx
![Page 26: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/26.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-26
Pop Quiz! (Just kidding)
• Suppose that the particle is in a superposition of these two states.
• Suppose that the temporal behaviors of the states are exp(i1t) and exp(i2t)
• Describe the particle motion.(x) (x)
x xStopped Wed 11 Jan 06
![Page 27: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/27.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-27
Dirac Notation
• Simple Way to Write Vectors– Kets– and Bras
• Scalar Products– Brackets
• Operators
2
1
|
*2
*1|
2
1*2
*1|
2
1
2
1*2
*1
0
0
00
||
x
x
xx
![Page 28: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/28.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-28
Commutators and Uncertainty
• Some operators commute and some don’t.
• We define the commutator as
[a b] = a b - b a
• Examples
[x p] = x p - p x = ih
xp h [x H] = x H - H x = 0
![Page 29: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/29.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-29
Recall the Five Postulates• 1. The physical state of a system is described by a
wavefunction.
• 2. Every physical observable corresponds to a Hermitian operator.
• 3. The result of a measurement is an eigenvalue of the corresponding operator.
• 4. If we obtain the result ai in measuring A, then the system is in the corresponding eigenstate, i after making the measurement.
• 5. The time dependence of a state is given by
H
ti
![Page 30: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/30.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-30
Shrödinger Equation
• Temporal Behavior of the Wave Function
– H is the Hamiltonian, or Energy Operator.
• The First Steps to Solve Any Problem:– Find the Hamiltonian– Solve the Schrödinger Equation– Find Eigenvalues of H
*http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html
Born: 12 Aug 1887 in Erdberg, Vienna, AustriaDied: 4 Jan 1961 in Vienna, Austria*
H
ti
*
![Page 31: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/31.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-31
Particle in a Box
• Before we begin the harmonic oscillator, let’s take a look at a simpler problem. We won’t do this rigorously, but let’s see if we can understand the results.
2
22
2 xmH
xip
m
pmv
22
1 22 Momentum
Operator:
![Page 32: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/32.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-32
Some Wavefunctions
Eigenvalue Problem
H=ESolution
2
22
8mL
hnEn
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1Shrödinger Equation
H
ti
2
22
2 xmti
Temporal Behavior 2
22
8mL
hn
ti
![Page 33: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/33.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-33
Pop Quiz 2 (Still Kidding)
• What are the energies associated with different values of n and L?
• Think about these in terms of energies of photons.
• What are the corresponding frequencies?
• What are the frequency differences between adjacent values of n?
![Page 34: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/34.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-34
Harmonic Oscillator
• Hamiltonian
• Frequency
22
2
1
2
1kxmvH
222
2
1
2
1mx
m
pH
m
k2
PotentialEnergy
x
![Page 35: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/35.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-35
Harmonic Oscillator Energy
• Solve the Shrödinger Equation
• Solve the Eigenvalue Problem
• Energy
– Recall that...
nnn EH ||
nnn EEEH ||
hnnEn
2
1
2
1
2
h 2
![Page 36: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/36.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-36
Louisell’s Approach
• Harmonic Oscillator– Unit Mass
• New Operators
222
2
1qpH
ipqa 2
1 ipqa 2
1†
†
1, † aa
![Page 37: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/37.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-37
The Hamiltonian
• In terms of a, a †
• Equations of Motion
2
1
2††† aaaaaaH
pp
H
dt
dq
H
dt
dp 2
aiHaidt
da ,1
††
†
,1
aiHaidt
da
![Page 38: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/38.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-38
Energy Eigenvalues
• Number Operator
• Eigenvalues of the Hamiltonian
aaN †2
11 HN
EEEH || '|''| nnnN
2
1nEn
![Page 39: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/39.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-39
Creation and Anihilation (1)
• Note the Following Commutators
• Then
1, † aa aaaa †, †††, aaaa
1†† NaNa 1 NaNa
'|)1'('| nannaN '|)1'('| †† nannaN
![Page 40: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/40.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-40
Creation and Anihilation (2)
'|)1'('| nannaN
'|)1'('| †† nannaN
'|''| nnnN
Eigenvalue Equations States Energy Eigenvalues
'|† na
'| n
'| na
2
1'nh
2
1'nh
2
3'nh
![Page 41: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/41.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-41
Creation and Anihilation (3) '|)1('| nannaN
'|)1('| †† nannaN
'|''| nnnN
1'|'|† nna
1'|'| nna
1'|)1'(1'| nnnN
1'|)1'(1'| nnnN
![Page 42: ECEG398 Quantum Optics Course Notes Part 1: Introduction](https://reader036.fdocuments.us/reader036/viewer/2022062301/56814fd6550346895dbd9a2a/html5/thumbnails/42.jpg)
January 2006 Chuck DiMarzio, Northeastern University 10842-1c-42
Reminder!
• All Observables are Represented by Hermitian Operators.
• Their Eigenvalues must be Real