658 J. Opt. Soc. Am. B/Vol. 1, No. 4/August 1984

Statistical mechanics of light-induced forces

Stig Stenholm

Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20 C, SF-00170 Helsinki17, Finland

Received December 27, 1983; accepted March 2, 1984

We consider the laser-induced deflection of a free two-level atom. When Doppler detuning is neglected, we obtaina Fokker-Planck equation as a result of lifting a continuous degeneracy. The problem is offered as an example ofhow an elimination of microscopic, fast degrees of freedom can provide a microscopic statistical-mechanics descrip-tion in classical phase space.

1. INTRODUCTION

Originally the theoretical problems connected with photonrecoil were regarded as interesting mainly because of theirpotential applications in laser spectroscopy. The increasingexperimental activity in the field has enhanced its relevance,but on the other hand there has emerged a realization that thetheory is of fundamental interest on its own merits. Thephoton momentum couples the internal atomic transitionsto the translational degrees of freedom. Thus even the simpletwo-level atom displays all the features of continuum quantummechanics.

The first indication of this was the result by Gordon andAshkin' and Cook2 that light-induced diffusion is not totallyincoherent but that there appear anomalous terms that maydecrease the diffusion. New interest arose when Mandel 3 andCook4 showed that these anomalous contributions are relatedto non-Poissonian statistics of the spontaneously emittedphotons. Some physical understanding of this situation wasattempted in Ref. 5, which led to the conclusion6 that theproblem was related to singular perturbation theory and de-generate-state perturbation theory. The corresponding co-herence effects for the cooling of a trapped ion have beendiscussed elsewhere. 7

A central point in the discussion of Ref. 6 is the fact that,in the absence of spontaneous emission, the atomic momen-tum is a constant of the motion, and any initial distributionis preserved unchanged. This degeneracy of the solution islifted by the perturbation. This procedure correspondsclosely to the one used by Dohm 8 and discussed by Haken 9 fora system with Fokker-Planck dynamics. In this paper theadiabatic elimination procedure outlined in Ref. 6 is recastin a form that essentially follows the idea of Ref. 9 and thusprovides a statistical-mechanics point of view.

The problem is defined in Section 2 along the lines of ourearlier work (see, e.g., Ref. 5). Section 3 carries out the adi-abatic elimination in a way that brings out new features of thederivation. The adiabatic elimination procedure was alsodiscussed recently by Haake and Lewenstein.1 0 In Section4 we apply the formal equations of our derivation to the spe-cific problem set up in Section 2. The result agrees with theone derived earlier, but it follows in an automatic way fromthe formalism instead of the rather delicate approach neededin Ref. 5. The reason is that our present work eliminates the

singular feature of the perturbation from the beginning. Onthe other hand, some physical insight is lost; the resultemerges in its totality, and no physically interpretable con-tributions can be discerned. In Section 5 we briefly summa-rize our conclusions.

2. SETTING UP THE PROBLEM

We consider the problem of one two-level system in a travel-ing-wave laser field. We use the formulation and notation ofRef. 5. The density-matrix equation of motion is given in Eqs.(1) of Ref. 5. The parameters are the following: A is thedetuning, 2K is the laser-field-induced Rabi flipping rate, Y2

and r are the transverse and longitudinal decay rates, and ais the geometric spreading factor of spontaneous decay (usu-ally a = 2/5). For spontaneous decay only between thelevels,

72 1/2r. (1)

As in Ref. 5, we introduce the optical Bloch vector as afunction of atomic momentum hk in the form

Ri(k) = P21 (k + '/q) - P 12(k + /2q),

R2 (k) = i[p 21 (k + /2q) - P12(k + /2q)],

R3(k) = P22(k + q) -pi(k),

(2a)

(2b)

(2c)

where hq is the photon momentum. In addition, we need thetotal population at momentum hk:

F(k) = P22(k + q) + pi(k). (3)

The arguments in Eqs. (2) and (3) are chosen so that withoutspontaneous decay (r = 0) the Bloch vector and F form aclosed system for each choice of k.

We introduce the four-component vector

A(k) = [R(k)]

and find from Ref. 5 its equation of motion,

d A = MA - FNOA + 1/2r(1 + a)Na2A,dt

(4)

(5)

where the one-photon-momentum shift operator is givenby

0740-3224/84/040658-04$02.00 © 1984 Optical Society of America

Stig Stenholm

Vol. 1, No. 4/August 1984/J. Opt. Soc. Am. B 659

ad q a- ,(6)

The linear operators are

-72 -A 0 0

M= L \ -72 2 2K 0] (7)0 -2K -r -r

0 0 0 0io o 0 0

N =- 0 0 0 0 (8)2 0 0 -1 -1

0 0 1 1_

In this paper we neglect the detuning caused by the Dopplershift. The rate of change of k is measured by the Dopplershift qv = qhk/M against the line width of spontaneousemission F, and hence we can estimate that

a9 hq 2E= q -k - qr - ' (9)

ak mr rwhere e is the recoil energy in frequency units. For opticalradiation, the quantity [Eq. (9)] is small (-10-3) and can beused as the basis for a perturbation expansion.

When we neglect spontaneous emission, r = 0, the timeevolution of A is given entirely by the M of Eq. (7). As thisdoes not depend on the atomic momentum hk, all values ofk occur on an equal footing. In particular, the eigenvalues ofthe matrix M determine uniquely this time evolution. Thismeans that, if A(k) is a solution, any other vector of theform

A(k + (q) = eV0A(k) (10)

is also a solution. Hence to the lowest order there is a con-tinuous symmetry in the problem, which gives a continuumof degenerate states. This symmetry has to be broken by theperturbation terms.

By inspection it is seen from Eq. (7) that one eigenvalue ofM is zero. The corresponding right-hand eigenvector is thesteady-state solution of the two-level problem, viz.,

2KA

aO = - (~ ~2j (11)hD e(2 + 722)D

where

D = A2 + 722 + 4K2 Y2/r. (12)

The solution is normalized by taking F = 1, which is a con-served quantity, F = 0. The conservation of the trace of thedensity matrix to this order implies the existence of the left-hand eigenvector e such that

eTM = 0. (13)

In the present representation eT = [0, 0, 0, 1] simply.When the perturbation is introduced, the different states

[Eq. (10)] are coupled. The conservation of F is affected, andits time derivative acquires terms of order and 2. This liftsthe degeneracy of the eigenvalues, and a slow rate of changeappears for the quantity F; the light-pressure force prevents

the occurrence of a steady state. The slowness of the evolu-tion implies that the perturbation problem is a singular one,and caution is required to obtain the right result; c.f. the dis-cussion in Ref. 7.

3. THE SINGULAR PERTURBATIONCALCULATION

We define a projection operator P by setting

(14)PA aO(eT, A).

Because of Eqs. (11) and (13) we have

PM = MP =0, (15)

and P denotes the subspace of slowly varying quantities. Wealso define the complement operator

Q =1-P (16)

and proceed to partition Eq. (5) in the two subspaces. Wewrite the component [Eq. (14)] as

?q= (e T,A)

and obtain from Eq. (5) the equation

t= aj = q

(17)

(18)

which is of the form of a differential conservation law in kspace. If any change of the probability density - occurs, itmust be due to a flow of the current density.

i = [-(eT, PNPaO)77 + /2(1 + cx)(eT, PNPaO)On- (eT, PNQA ) /2(1 + a)a(eT, PNQA). (19)

To zeroth order in 9 any function, F(k) qualifies as asteady-state solution of Eq. (5) and hence can be regarded asa probability distribution for the conserved quantity k. Thesituation is exactly like the one described in Ref. 9. Anychange in F(k) has to be slow and is compensated for by a flowof probability that is due to the current density [Eq. (19)].Our task now is to calculate j as a consistent expansion in theparameter a up to first order, because this determines the rateof change to second order in Eq. (18), which is as far as theoriginal expansion in Eq. (5) has been carried out. Thisprocedure was discussed earlier in Ref. 6, but here we want toapproach the problem from a different point of view.

We write the equation of motion for the component QA inthe form

(d -rQP +Ia)dt -QMQ + QNQdQA =-F'QNPA + Q(d2). (20)

We need QA only to order , and hence we can solve Eq. (20)in the form

QA(t) = exp(QMQt)QA(t = 0)

At- rd | exp[QMQ(t - t')]QNPPA(t')dt'. (21)

This is now exact to first order in . The submatrix of Moperating in the Q space is seen from Eq. (7) to contain diag-onal elements of order 72, r that are larger than by the factore-1 from Eq. (9). Thus the slow drift of PA(t) at the rate is,in the integral of Eq. (21), sampled by the rapidly decaying

Stig Stenholm

660 J. Opt. Soc. Am. B/Vol. 1, No. 4/August 1984

exponential. Hence we can use the usual argument to replacePA(t') by PA(t) inside the integral and carry out the inte-gration

C exp[QMQ(t - t')]dt' = - QMQ [1 - exp(QMQt)].

(22)

This causes no problem because there are no zero eigenvaluesin the subspace Q, where consequently M can be inverted.After such a time that rt >> 1, we can neglect the exponentialin Eq. (22). Then the present argument is equivalent to theadiabatic elimination argument of Ref. 6. After such a longtime, the system has also forgotten about the initial-value termin Eq. (21), which can be safely ignored. The componentQA(t = 0) is, however, not usually equal to zero and affectsthe evolution of PA during an initial period. For the laterstages of a laser-deflection experiment its influence can beneglected. The modification of the initial value in an adia-batic elimination calculation is called an "initial slip" by theauthors of Ref. 10.

Introducing Eq. (21) with our approximations into Eq. (19),we find the current density

j = {-(eT, PNPa0) + 1/2(1 + )(eT, pNpa°)

-F (eT, PNQ QMQ QNPaO)J d} i. (23)

The probability current thus consists of a drift term, a Fick'slaw term, and a connection from the subspace Q, which is ofthe character of an induced backflow of the probability.

In the present case we may freely move the derivatives aacross the coefficients; then formula (18) with Eq. (23) is aFokker-Planck equation for the distribution function ,. It,in fact, coincides with the one derived in Ref. 6 by using adifferent argument. There it was also shown how the deri-vation relates to a singular perturbation approach of theChapman-Enskog type, which is similar to the earlier deri-vation in Ref. 5. We can thus see that the whole adiabaticelimination procedure together with the perturbation ex-pansion is equivalent to a problem in Fokker-Planck statis-tical mechanics. To go to higher order in a becomes senselessaccording to a theorem by Pawula,11 which says that any finite,higher-order truncation leads to negative probabilities.

From Eqs. (11), (13), and (14), we find that

0 0 0 2KA 1P=aOoeT= 1 0 0 -2K2 , (24)

D 0 0 0 (A2 + 2 2)

LO 0 0 D J

which directly gives Q from Eq. (16).The lowest-order term in a in Eq. (23) contains the oper-

ator

0 0 0 2KA 1PNP = 272 K 2 0 0 0 -2KY 2

FD2 0 0 0 -(A 2 + 72 )

Lo oo D I

(25)

The element of this that is needed in Eq. (23) is just the onein the lower right-hand corner, which we write as

112SO - 2K2Y2 /rD. (26)

Inserted into Eq. (18), this leads to the equation for the av-erage momentum

dh (k) = /2hqrso,dt

(27)

in agreement with Eq. (35) of Ref. 5.For the diffusion term we need to evaluate the term of order

a in Eq. (23). The elements PNQ and QNP are easily calcu-lated, but the inverse of QMQ is not so simply evaluated owingto the zero eigenvalue of M. In fact, because of Eq. (15), QMQappears identical with M and hence just as singular.

In order to be able to carry out the inversion, we choose toinvert the matrix:

Mr = is + QMQ

and consider its limit when s goes to zero.s the determinant of Eq. (28) is given by

D = -srD + O(s2),

(28)

To lowest order in

(29)

which clearly displays the zero eigenvalue. A straightforwardinversion gives

4. EVALUATION OF THE FOKKER-PLANCKOPERATOR

The problem that we treat here is identical with the one solvedin Ref. 5. From our general argument we expect the ensuingFokker-Planck operators to be identical, but it is both reas-suring and intrunctive to carry out the calculations impliedby the formal expression (23).

-2KAs -2KIA 12KY 2 S 2KY2 F . (30)

(A2 + Y22 ) r(A 2 + ,22) i

0 -rD Ji

Here only the lowest-order terms, i.e., terms of order s or s,are written out in the matrix. The higher-order ones play norole.

The 3 X 3 submatrix in the upper left-hand corner isnonsingular; it is the desired part of (QMQ)-l. The lowerright-hand corner contains the element s-l, as we expect fromEq. (28). Also, all the elements in the last column are sin-gular.

[S(72r + 4K2) -SAr(M = 1 sAr SY2r

srD -s2KA -2Ki<2S

L 0 0

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Vol. 1, No. 4/August 1984/J. Opt. Soc. Am. B 661

The diffusion term contains the combination

jeT [1/2r (l + )PNP - r2PNQ QMQ QNP a}

K27 2 [( + a) + 4K[2(r-2) 1722( 1 72)PD [YE [2 (- 2- 72 (+ Y)j

(31)

A comparison with Eq. (40) of Ref. 5 shows that this is thecorrect result that also contains the anomalous diffusioncontributions discovered by Gordon and Ashkin and Cook.In the derivation of Ref. 5 we could follow how the correctionterms emerged from the drift of the center of mass and thedipole vector R. Here no such physical interpretation ispossible, but the result follows directly from the formalism,after the initial physical assumptions have been introducedinto the derivation.

5. DISCUSSION

The formal mathematical operations of inverting the matrixQMQ was also needed in Ref. 5 [cf. its Eq. (30)] because theBloch vector R was separated there too. In evaluating thefirst-order terms in i we had also to consider R and F in firstorder to achieve consistency. This corresponds to drift of thedipole moment and the center of mass, respectively. Herethese features emerge directly because of the use of the pro-jected coupling operator QNP. Working fully in the 4 X 4subspace explicitly shows how the formalism of Section 3eliminates all the problems of singularity.

The infinitely degenerate zero eigenvalue is replaced by aseries of small eigenvalues of the Fokker-Planck operator.These determine the time evolution after an initial period andcarry information about the initial state. For the free atomno asymptotic steady state emerges. In contrast, the trappedion approaches a final cooled state, but the formal procedureremains valid and has been carried out quite explicitly.7

The equations that follow from our formalism can be re-garded as a statistical-mechanics flow in the phase space ofthe free-particle momentum hk. In the case of a trappedparticle the distribution function also acquires a dependence

on position. This theory has been worked out by Java-nainen.1 2

In conclusion, I stress that the simple system of an atompushed by a laser field offers an example of a highly nontrivialphysical system that suggests possibilities for investigatingthe basic question of quantum mechanics. In addition, it isan example of how the fast, internal degrees of freedom canbe eliminated to give a statistical-mechanical behavior inphase space. When the memory of the initial stage has faded,all time evolution can be treated by probability flows in thisspace. The initially high degeneracy of the zero eigenvalueis lifted to provide a series of small eigenvalues determiningthe long time evolution of the distribution. These are ob-viously the conditions that prevail toward the later stage ofa laser-deflection process.

ACKNOWLEDGMENT

I want to thank Juha Javanainen and Markus Lindberg, withwhom I have discussed all the ideas expressed in this work;they have also read and commented on the present paper.

REFERENCES

1. J. P. Gordon and A. Ashkin, Phys. Rev. 131, 2766 (1980).2. R. J. Cook, Phys. Rev. A 22, 1078 (1980).3. L. Mandel, Opt. Lett. 4, 205 (1979).4. R. J. Cook, Opt. Commun. 35, 347 (1980).5. S. Stenholm, Phys. Rev. A 27, 2513 (1983).6. S. Stenholm, presented at the Advanced Study Institute of

Electrodynamics and Quantum Optics, Boulder, Colorado, May27-June 8, 1983. The references to other types of singular per-turbation theory can be found in these lectures.

7. J. Javanainen, M. Lindberg, and S. Stenhohm, J. Opt. Soc. Am.B 1, 111 (1984). References to our earlier work can be foundhere.

8. V. DohmPhys. Rev. A 14, 393 (1976).9. H. Haken, Rev. Mod. Phys. 47, 67 (1975), Sec. XI.C.

10. F. Haake and M. Lewenstein, Phys. Rev. A 27,1013 (1983); Phys.Rev. A 28, 3606 (1983); presented at the Fifth Rochester Con-ference on Coherence and Quantum Optics, Rochester, New York,June 13-15, 1983. References to the earlier work can be foundhere.

11. R. F. Pawula, Phys. Rev. 162, 186 (1967).12. J. Javanainen (submitted to J. Phys. B).

Stig Stenholm