136 OPTICS LETTERS / Vol. 6, No. 3 / March 1981

Resonant laser excitation in a two-level model: a nonlinearSchrodinger equation?

Jay R. Ackerhalt

Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

Received November 17,1980

A nonlinear Schrodinger equation for the dynamics of a resonantly'driven two-level model including both T1 andT2 processes is presented. This equation is novel in that it can account both for the rate of population flow intoan energy level and for the destruction of phase information, i.e., it can account for the system's evolution into amixed quantum state.

It is well known that population losses from a quan-tum system can be included within the framework of aSchr6dinger equation by adding negative imaginaryterms to the diagonal matrix elements of the Hamilto-nian.1 The procedure works because the system alwaysremains in a pure quantum state. It is also well knownthat T, and T2 processes cannot be included within theframework of a Schrodinger equation because then thesystem would evolve into a mixed quantum state.

In this Letter it will be shown that at least for onequantum system, a single two-level model driven by aresonant laser field, the standard Schrbdinger equationdescribing this model2 can be modified to include T,and T2 processes by adding nonlinear terms. To provethat this Schrbdinger equation indeed describes thequantum system's dynamics, we will explicitly show itsequivalence to the correct Bloch equations. 2 Sinceinformation in a Schrddinger equation is contained inthe magnitude and phase of complex-state amplitudes,new insights into some physical, but simple, modelproblems can be gained. For example, with respect toT1 processes, we will study spontaneous decay from theexcited state to the ground state. With respect to T2processes, we will consider the randomization of thephase of the excited-state wave function that is due tothe effect of dephasing collisions.

The Bloch equations that describe both these pro-cesses are

Q I,1Pl = 2 v + P22 (la)

Q1P22- = V- P22 ' (lb)

2 T

Q(P22 - P) _ I 1 1 (1c)

2 - T + i-) u, (2)

where we have assumed resonant laser excitation, whichdecouples the variable u from the remaining importantdynamic variables. Since it now plays no role in thephysical process, we will consider the Bloch equationsfor the system to be only Eqs. (1). The transition Rabi

frequency, spontaneous emission rate, and collisionaldephasing rate are given by Q, 1/T1, and l/T 2 , respec-tively. By defining the wave function to be

1h) = C11Cl) + C2 %f2 ), (3)

where 11) and 1b2) are the unperturbed energyeigenstates for the system, we obtain the usual defini-tions for the Bloch variables:

P11= 1C112 ,

P22 = IC 2 12 ,

V = i(Cl*C 2 - CIC2 *)

= 21 C I I C 2 1sin(01 - 02),

(4a)

(4b)

(4c)

where

Cj = ICjIe i. (5)

The nonlinear Schrodinger equation, which describesthe same system as in Eqs. (1), written for conveniencein the form of two equations for the state amplitudes,is

= -2 C + 2 C22 C1 C2

2 2T, 1C112 +C,(a

2T - C 1Cc2* -C 1*C 2C\ (6a)*C _ 0 ~i i 'CIC2* - Cj*C2i =--C1 - _ C9+ C2

2 2T, - 2T2 C1 C2* + C1*C2 j

(6b)

If 1/T 2 = 0 and the spontaneous-emission process di-rects population into a state outside the two-levelmodel, then Eq. (6) reduces into a more standardSchr6dinger equation, since all the nonlinear terms areabsent. As we will see, the first nonlinear term in Eq.(6a) accounts for population being transferred into thelower state, whereas the second term accounts for phaserandomization. Since the collision process dephasesthe excited state, a nonlinear dephasing term that is dueto 1T 2 is introduced in Eq. (6b).

Before showing that Eqs. (6) are equivalent to Eqs.(1), we can rewrite them using Eqs. (4) and (5) in a form

0146-9592/81/030136-03$0.50/0 © 1981, Optical Society of America

March 1981 / Vol. 6, No. 3 / OPTICS LETTERS 137

convenient for separating magnitude and phase infor-mation:

i|Cl|ei~l - IClIeiqli

= -2 I C21eiO2 + 2 ICi ei '2 2T1 JC1 12

+ 1 IC2 12 sin(0l-k02) 1Clil+2T1 I Ci12 C0s(& - 0k2)Iie@, (7a)

i|C 2Iei02 - IC2Iei02k2 = -. ICeiel - T IC2 Iei b2

1 sin(&1 -02) l C21 e) O2

2T2 cosi1 - t02)(7b)

By multiplying Eq. (7a) by e -i and Eq. (7b) by e-i2and equating the real and imaginary parts, Eqs. (7)become

|lll = 2 I C21 sin(1 -I2) + JI (8a)

1C21 = - Q ICIsin(ol - 02)- I 1C2L, (8b)

* I1C21 1 1(72102)2 sin(ki - 0k2)1=2 |C os (-4 I2 ) Cl 1 2 cos(--02)

(9a)

k2 = Q lC COS(01 - 02) + 2TI co(X - 02) (9b)2 1 C212TCS0 02

This last step shows explicitly which terms in Eqs. (6)refer to amplitude information and which terms referto phase information. Since the variable v dependsonly on sin(l -02), we can use Eqs. (9) to create itsequation of motion:

sin-(1 (- 02) = Q jC212 -(Cl 11l COS2 (t1 - 02)2 j C1 1 I C2

II+ I -I C1 sin(ki - 0'2). (10)2 \T2 T 1 IClI 2

Equations (8a) and (8b) reduce exactly to Eqs. (la) and(lb) after we multiply Eq. (8a) by 21C11, Eq. (8b) by21 C21, and use Eqs. (4). Equation (ic) can be obtainedfrom Eqs. (8) and (10) after we substitute these equa-tions into the identity

v = 2C1fl IC2 Isin(4l - 02) + 21(C>1 IC2Isin(ol - '2)

+ 2lCiIIC21sintk0I-02). (11)

Since the system evolves into a mixed state, care mustbe taken with respect to a precise interpretation of thephases for the individual state amplitudes. For ex-ample, a value of 01 - '2 = 0 implies that the phases arerandom with respect to each other, not that they are welldefined and equal. Equation (10) does, however, de-scribe phase information in the quantum system.

The next important question to ask now that thenonlinear Schrddinger equation (6) has been shown tobe equivalent to the Bloch equations (1) is, Is thereanything new to be learned? Let us first consider thespontaneous-emission problem by setting 1T9 = 0.From the derivation of the Bloch equations we now

know that the contribution of the two nonlinear termsin Eq. (6a), when forming the equation for 6, Eq. (II),exactly cancel each other. The decay in Eq. (le) istotally due to the decay term in Eq. (6b), i.e., to the re-moval of population from the upper state. Physicallythis means that the growth of ) that is due to populationinput into the ground state is exactly canceled by therate of randomization of the phase of the ground-statewave function. This balance is a fundamental aspectof 1/Tl-type population dynamics, which has not pre-viously been noted. A second interesting question onemight ask is, How important is the phase randomizationof the ground-state wave function, assuming that weaccount for the population dynamics correctly? Toanswer this question we can compare Eq. (ic) with theequation for j generated by neglecting the phase ran-domization term in Eq. (7a) or Eq. (10):

6 = Q(P22-P11)- 2T1 (I1C1 11Lc 2I v. (12)

Equation (12) will be in good agreement with Eq. (le)when the population is nearly always in the groundstate, which can be accomplished by assuming that allthe population is in the ground state initially and thatlIT1 >> Q. Although it may seem inconsistent to in-crease 1/T, and reduce the importance of phase ran-domization simultaneously, one must remember thatthe actual dynamics depend on the product (1/T,)p 22,which does not increase, explaining this apparent con-tradiction.

Let us now set lIT1 = 0 and consider the effect ofcollisional dephasing of the upper-state wave function.The question one might ask here is, Since the Wilcox-Lamb reduction of Bloch equations3 to rate equationsdepends on the rate of decay of the off-diagonal densitymatrix elements, can we learn something new about thisapproximation from Eq. (10)? The first step in an-swering this question is to state the Wilcox-Lamb ap-proxiaxation in the context of Eqs. (1).

The Wilcox-Lamb approximations assumes that QT2<< 1 such that u reaches its steady-state value veryrapidly with respect to the population dynamics, i = 0.The resulting solution of Eq. (ic) for v in terms of P22- poII is now substituted into Eqs. (1), reducing Eqs. (1)to a system of two rate equations describing the popu-lation dynamics.

Before discussing the Wilcox-Lamb approximationin the context of Eq. (10), let us consider a variation ofthis approximation by setting sint(0i - 02) = 0. Thesteady-state solution from Eq. (10) is

sin(01 - 02) = - A [1 - (1 + R2)1/2],

where

R- 2QT2 ( CiI I cd

In the limit R << 1, Eq. (13) reduces to

sin(k - sb2) R/2 << 1.

(13)

(14)

(15)

Substituting Eq. (15) into Eqs. (1) and using Eq. (4c)gives exactly the Wilcox-Lamb rate equations. In orderfor Eq. (15) to be small, Q2T2 << 1 and the population

138 OPTICS LETTERS / Vol. 6, No. 3 / March 1981

cannot be localized in either the upper or the lower state.The Wilcox-Lamb approximation is therefore a specialcase of this more general steady-state limit. WhereasEq. (13) is a much more complicated function of thepopulations than its Wilcox-Lamb counterpart, Eq.(15), only modest differences in the solutions for thepopulations over a wide range of parameters have beenobserved in computer calculations. In the future weplan to study the Wilcox-Lamb approximation furtherin a three-level system, in which other interesting effectscan occur (see Ref. 3) and in which this more generalsteady-state condition may be important.

For this simple two-level model a novel nonlinearSchrodinger equation that describes both T1 and T2processes has been presented. Since a Schrodingerequation gives both magnitude and phase informationfor the state amplitudes separately, some new insightinto the physics of T1 processes and into the physics ofthe Wilcox-Lamb approximation for T2 processes hasbeen achieved.

It is also possible to create a nonlinear Schrodingerequation describing a two-level model including anonzero detuning (and in which, therefore, the variableu plays a role in the dynamics) and a three-level model

in which all detunings are zero. The complication in-volved in these calculations is that they require a non-standard definition of some of the dynamical variables.A completely general nonlinear Schrbdinger equationhas yet to be found.

I would like to acknowledge conversations with KenKay, Dick Cook, Bruce Shore, Everett Thiele, and JimStone. In addition, I would like to thank Carl Moserand the Centre Nationale de la Recherche Scientifiquefor supporting my tenure at the Workshop on LaserExcitation and Dynamics of Highly Excited Poly-atomics held in Orsay, France, during July 1980.

References

1. M. P. Silverman and F. M. Pipkin, J. Phys. B 5, 1844(1972).

2. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

3. L. R. Wilcox and W. E. Lamb, Jr., Phys. Rev. 119, 1915(1960); J. R. Ackerhalt and B. W. Shore, Phys. Rev. A 16,277 (1977); J. Stone and M. F. Goodman, Phys. Rev. A 18,2642 (1978).