Monte Carlo wave-function method in quantum optics
Klaus M0lmer
Institute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
Yvan Castin and Jean Dalibard
Laboratoire de Spectroscopie Hertzienne de lEcole Normale Sup6rieure, 24 rue Lhomond,F-75231 Paris Cedex 05, France
Received April 7, 1992; revised manuscript received July 8, 1992
We present a wave-function approach to the study of the evolution of a small system when it is coupled to a largereservoir. Fluctuations and dissipation originate in this approach from quantum jumps that occur randomlyduring the time evolution of the system. This approach can be applied to a wide class of relaxation operators inthe Markovian regime, and it is equivalent to the standard master-equation approach. For systems with anumber of states N much larger than unity this Monte Carlo wave-function approach can be less expensive interms of calculation time than the master-equation treatment. Indeed, a wave function involves only N compo-nents, whereas a density matrix is described by N2 terms. We evaluate the gain in computing time that may beexpected from such a formalism, and we discuss its applicability to several examples, with particular emphasison a quantum description of laser cooling.
1. INTRODUCTION
The problem of dissipation plays a central role in quantumoptics. The simplest example is the phenomenon of spon-taneous emission, in which the coupling between an atomand the ensemble of modes of the quantized electromag-netic field gives a finite lifetime to all excited atomiclevels. Usually the dissipative coupling between a smallsystem and a large reservoir can be treated by a master-equation approach'-; one writes a linear equation for thetime evolution of the reduced system density matrix,ps = Trm8(p), trace over the reservoir variables of thetotal density matrix. If we denote the Hamiltonian forthe isolated system Hs, this equation can be written as
pS = (i/Ii)[PsHs] + relax(PS)- (1)
In Eq. (1), -Erelax is the relaxation superoperator, acting onthe density operator Ps. It is assumed here to be local intime, which means that fs(t) depends only on ps at thesame time (Markov approximation). All the system dy-namics can be deduced from Eq. (1). One can calculateone-time average values of a system operator A: a(t) =(A)(t) = Tr[ps(t)A] and also, by using the quantum regres-sion theorem,5 multitime correlation functions, such as(A(t + r)B(t)).
Recently a novel treatment of dissipation of energy froma quantum system (two-level atom) coupled to a zero-temperature reservoir was presented.6 This treatment isbased on the evolution of a Monte Carlo wave function(MCWF) of the small system, which consists of two ele-ments: evolution with a non-Hermitian Hamiltonian andrandomly decided quantum jumps, followed by wave-function renormalization. This approach, which is equiva-lent to the master-equation treatment, is interesting fortwo reasons. First, if the relevant Hilbert space of thequantum system has a dimension N that is large com-pared with 1, the number of variables involved in a wave-
function treatment (-N) is much smaller than the onerequired for calculations with density matrices (-N 2 ).Second, new physical insight may be gained, in particularin the studies of the behavior of a single-quantum system.
The purpose of the present paper is to give a generalpresentation of this method for a wide class of system-reservoir couplings (Section 2). In particular, the methodpresented here is not restricted to a zero-temperaturereservoir. The physical content of the method and its re-lation to previous wave-function treatments in dissipativequantum optics are discussed in Section 3. We then indi-cate how the MCWF formalism can be used for calculatingtwo-time correlation functions, and we give two illustra-tions of this (Section 4). Section 5 is devoted to the pre-sentation of a series of examples to which the MCWFtreatment can be applied. In Section 6 we discuss theexistence of several MCWF descriptions for a given re-laxation operator 2relax, and we illustrate this with an ex-ample of population trapping. Finally, in Section 7 wegive a few indications concerning the convergence of theMCWF method and the gain in computing time that onemight expect.
Soon after the completion of the study by Dalibard et al.6
it was brought to our attention that other approacheshave recently been developed involving a stochastic evolu-tion of wave functions. In the context of nonclassicalfield generation, Carmichael7 proposed an approachnamed quantum trajectories, inspired by the theory ofphotoelectron-counting sequences8 and quite similar tothe spirit of Dalibard et al.6 On the basis of the continu-ous quantum theory of measurement,9 Dum et al."0 devel-oped a Monte Carlo simulation of the atomic masterequation for spontaneous emission. In the framework ofquantum jump theory, Hegerfeldt and Wilser " considereda quantum-mechanical model for describing a single radi-ating atom, which could also be the starting point for aneffective Monte Carlo evolution with atomic wave func-
tions." Finally, the relation of Ref. 6 to a general sto-chastic formulation of quantum mechanics has beenpointed out to us by Gisin.12 Throughout this paper wewill connect our results with the ones obtained in theseparallel approaches.
2. GENERAL PRESENTATION OF THEMETHOD
In this section we present a general description of theMCWF method. We start by presenting the class of re-laxation operators that can be studied by this method.We then present the procedure itself, and finally we showits equivalence with the master-equation treatment.
Since H is not Hermitian, this new wave function clearlyis not normalized. The square of its norm is
(o'M(t + At) 4 '1(t + at))
( iH8t iHt )(8)= 1 - p,
where p reads as
8p = at ((t) H - Ht I(t)) = E aP.
aPr = 500(t) CMI(P(t)) 0.
(9)
(10)
A. Relaxation OperatorThe class of relaxation operators that we consider in thispaper is the following:
2relax(PS) = (CMCMps + PSCMtC) + E CmPSCM.2m m
(2)
This type of relaxation operator is quite general and isfound in most of the quantum optics problems involvingdissipation. In Eq. (2) the Cm operators act in the space ofthe small system. Depending on the nature of the problemthere can be one, a few, or an infinity of these operators.
A series of examples with the dissipation operator in theform of Eq. (2) will be given in Section 5. Here we indi-cate the expression of -Trelax(PS) for the case of spontaneousemission by a two-level system or by a harmonic oscillator,where there is just a single operator C, = \/i7 a- - in therelaxation operator [Eq. (2)]:
For the two-level system formed with a stable ground stateIg) and an excited state e) with a lifetime r7', we have
o = + = e) (g, (- = S- = g)(eI. (4)
For a harmonic oscillator, a - and C- + are related to lower-ing and raising operators:
o,- = b, a+ = bt. (5)
B. Monte Carlo Wave-Function ProcedureWe now present the procedure for evolving wave functionsof the small system. Consider at time t that the system isin a state with the normalized wave function l+(t)). Inorder to get the wave function at time t + at, we proceedin two steps:
1. We calculate the wave function 10(l)(t + at)) obtainedby evolving |+(t)) with the non-Hermitian Hamiltonian:
H = Hs - 2 ECmCrn-
This gives the following for sufficiently small at:
1,0°'(t + at)) = 1- iH8t 1(t)).
(6)
(7)
The magnitude of the step 5t is adjusted so that this calcu-lation at first order is valid; in particular, it requiresap << 1.
2. The second step of the evolution of 10) between t andt + at consists in a possible quantum jump. We leave thisterm intentionally vague here; for particular examples wesee below that this quantum jump can correspond to theprojection of the wave function associated with a gedankenmeasurement process. In order to decide whether thisjump happens, we choose a quasi-random number e, uni-formly distributed between 0 and 1, and we compare itwith 8p. If 3p is smaller than E, which occurs in mostcases since p << 1, no quantum jump occurs, and wetake the following for the new normalized wave functionat t + At:
If E < 8p, a quantum jump occurs, and we choose the newnormalized wave function among the different statesCmlo(t)), according to the probability law Hn = aPm/8P[note that JmJrn = 1 because of Eq. (9)]:
1k(t + at)) = Cm|I(t))/(apm/8t)/ 2
with a probability In = 5pm/8p, Sp > E. (12)
For the particular case of a two-level atom coupled tothe vacuum electromagnetic field, these two steps coincidewith the ones given in Refs. 6 and 7.
C. Equivalence with the Master EquationWith this set of rules we can propagate a wave function1(t)) in time, and we now show that this procedure isequivalent to the master equation (1). More precisely, weconsider the quantity _U(t) obtained by averaging o(t) =10(t))(4(t)l over the various possible outcomes at time t ofthe MCWF evolutions all starting in 10(0)), and we provethat c(t) coincides with ps(t) at all times t, provided thatthey coincide at t = 0.
Consider a MCWF |+(t)) at time t. At time t + t theaverage value of o-(t + t) over the evolution caused bydifferent values of the random number E is
We now average this equation over the possible values ofa-(t), and we obtain
dt hdt= h, H8g] + Sreiax(Th). (15)
This equation is identical to the master equation (1). Ifwe assume that ps(O) = l0(0))(k(0), a(t) and ps(t) coin-cide at any time, which demonstrates the equivalence be-tween the two points of view. In the case where s(O)does not correspond to a pure state, one first has to de-compose it as a statistical mixture of pure states,p(O) = lpilXi)(Xil, and then randomly choose the initialMCWF's among the Jxi) with the probability law pi.
As is mentioned in Section 1, the master-equationapproach and the reduced density matrix give access toone-time average values a(t) = (A)(t) = Tr[ps(t)A], whichcan now also be obtained with the MCWF method. Forseveral outcomes 1k0"(t)) of the MCWF evolution, one cal-culates the quantum average (0"(t)lAJ0"(t)), and one takesthe mean value of this quantity over the various outcomes
(A) () ( - (¢>(i(t) IAlziP,'i(t)) . (16)
For n values that are sufficiently large, Eq. (15) impliesthat (A)(n)(t) (A)(t). We will see in Sections 5 and 7that this MCWF procedure based on the use of Eq. (16) fordetermining average values of operators may be more effi-cient than the master-equation approach.
It appears clearly in this proof that the equivalence ofthe master-equation and MCWF approaches does not de-pend on the particular value of the time step t. From apractical point of view, the largest possible at is preferable,and one might benefit from using a generalization ofEq. (7) to a higher order in 8t, for example, a fourth-orderRunge-Kutta-type calculation. The only requirement onSt is that the various -ri8t, where the hlqi are the eigenval-ues of H, should be small compared with 1. Of course, weassume here that those eigenvalues have been simplifiedas much as possible in order to eliminate the bare energiesof the eigenstates of Hs. For instance, for a two-levelatom with a transition frequency (OA coupled to a laserfield with frequency WOL, one makes the rotating-wave ap-proximation in the rotating frame so that the Imqil valuesare of the order of the natural width F, the Rabi frequencyQ. or the detuning 8 = OL - OA; they are consequentlymuch smaller than coA.
One might wonder whether there is a minimal size forthe time step 8t. In the derivation presented above, it canbe chosen to be arbitrarily small. However, one shouldremember that the derivation of Eq. (1) involves a coarse-grain average of the real density operator evolution. Thetime step of this coarse-grain average has to be muchlarger than the correlation time T, of the reservoir, whichis typically an optical period for the problem of sponta-neous emission. Therefore one should be cautious whenconsidering any result derived from this MCWF approachinvolving details with a time scale of the order of or
shorter than i,, and only t larger than T, should be ap-plied. This appears clearly if one starts directly from theinteraction Hamiltonian between the system and thereservoir in order to generate the stochastic evolution forthe system wave function.6 The condition t >> T isthen necessary to prevent quantum Zeno-type ef-fects. 3 This restriction is discussed in detail in Ref. 11in connection with quantum measurement theory.
3. PHYSICAL INTERPRETATION OFTHIS PROCEDUREWe now discuss the physical content of this procedure. Tothis purpose, we consider at time t = 0 a harmonic oscilla-tor (the same formalism applies to the case of a two-levelsystem by replacement of 0) and 1) by g) and le)) in asuperposition of the two lowest-lying states:
1k(0)) = aoIO) + p3ol1). (17)
We suppose that the oscillator relaxes toward its groundstate 0) with the relaxation operator defined by Eqs. (3)and (5). Therefore we know that at time t = + the os-cillator will be in state 0); it may have reached this statewithout emitting any photon (probability Ia01 2) or with theemission of one single photon (probability 1,6O1 2).
In the MCWF formalism, following the first step of Sub-section 2.B and using Hs = hcoobtb, we have at time 8t
The probability p defined in Eq. (9) for making a quan-tum jump is
8p = rlBoJ28t, (19)
and it corresponds to the probability for emitting a photonbetween 0 and 8t. The choice of the random number E
therefore simulates the result of the measurement of thenumber of photons emitted between 0 and t. The caseap > e corresponds to the detection of a photon, and thequantum jump described in Eq. (12) is simply the projec-tion of the wave function onto the ground state 10), associ-ated with this detection. If such a quantum jump occurs,the wave function 1k(8t)) is simply 10), and it does notevolve anymore.
We now investigate the other part of the wave-functionevolution, i.e., the case p < e, corresponding to the no-detection result. First, we treat the example of the har-monic oscillator. Then we consider the general result andconnect it with previous wave-function treatments inquantum optics, mainly on the basis of the delay function(or waiting time function).
If no quantum jump occurs, the normalized wave func-tion 10(8t)) is proportional to kk(')(at)). Using the fact that8t is small, we get
1(at)) = ao( 1 + rt lpoJ2 10)
+ Po(1 - t lal2)exp(-ioo0t)l1), 8p < .
(20)
We note that in addition to the free evolution at frequency
Molmer et al.
Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. B 527
coo, there has been a slight rotation of the wave function:the probability amplitude of being in the ground state hasincreased, and the probability of being in the first excitedstate has decreased. From the photon measurementpoint of view, the non-Hermitian part of the evolution de-scribed in Subsection 2.B corresponds to the modificationof the state of the system associated with a no-detectionresult of the number of emitted photons. The informationgained in a zero-result experiment and its consequencesfor the evolution of the system has been emphasized byDicke,'4 and, in the context of quantum jumps, by Peggand Knight,'5 by Cook,'6 and by Porrati and Putterman.' 7
For the present problem this rotation is essential. If itdid not occur, i.e., if we were to take
1f(8t)) = aoIO) + /3o exp(-icso8t)1), 8p < e, (21)
the probability of having a quantum jump (i.e., detecting aphoton) between 8t and 28t would be strictly equal to theprobability between 0 and 8t [Eq. (19)], and this would re-peat over and over until a quantum jump would finallyoccur. One would then always find that a photon is emit-ted between t = 0 and t = A, provided that 1,3012 # 0, and
this conclusion would clearly be wrong. Owing to theslight rotation in Eq. (20), however, the probability formaking a quantum jump between 8t and 28t is smallerthan 8p, and it will be reduced over and over, as no quan-tum jump occurs in successive time steps. Assuming thatno quantum jump occurred between 0 and t, we can write1(t)) as
l+(t)) = a(t)10) + 6(t) exp(-ioot)I1), (22)
where a(t) and 13(t) are solutions of the nonlinear set ofequations deduced from Eq. (20):
The probability P(t) for having no quantum jumps be-tween 0 and t is found to be
P(t) = laol 2 + 1,1012 exp(-Ft). (25)
This confirms the statement made at the beginning of thissection: there is a probability aol2 that no jump willoccur between t = 0 and t = o, and there is a probability113N2 = 1 - laol2 for a jump to occur. In this particularcase we have therefore been able to determine completelythe stochastic evolution of |+(t)) (see also Ref. 7):
With a probability P(t),
i+(t)) = a(t)l0) + 63(t)exp(-o0t)I1); (26a)
With a probability 1 - P(t),
l(t) = 10), (26b)
where a(t) and 13(t) are given in Eqs. (24). We note that ifthe initial state of the oscillator is taken as an eigenstate
of b, a coherent state, the wave function is not changed inquantum jumps, and all changes take place during thenonunitary evolution, which takes the wave functionthrough a progression of coherent states (see also Ref. 12).
We now extend this treatment of the zero quantumjump periods to the general case. Suppose that we knowthat no quantum jump has occurred between 0 andt. During this period we find that the wave functionobeys a nonlinear differential equation deduced fromEqs. (7) and (11):
dlq$) ( + (IH - H \)1),dt 2 / (27)
which generalizes Eqs. (23). The solution of this equationis, for a time-independent Hamiltonian,
I+(t)) = ((O)Iexp(iH't/)exp(-iHt/A)1(0))-1 2
X exp(-iHt/i)I(0)), (28)
which generalizes Eqs. (24). This corresponds to an evo-lution with the non-Hermitian Hamiltonian between 0and t:
itdI) = Hodt
(29)
and a normalization of the result at the end of theevolution.
This result allows us to connect our approach to thestandard treatments of resonance fluorescence. In thosetreatments, following the work of Mollow,'5 the atomicdynamics is interpreted as phases of evolution with a non-Hermitian Hamiltonian H, separated by spontaneous-emission processes. This non-Hermitian Hamiltoniancoincides with the one derived in this paper for the par-ticular case of spontaneous emission. If one defines a re-duced atomic density operator a(-)(t) in the subspacecontaining n = 0,1,2,..., fluorescence photons, one getsan infinite hierarchy of equations'8 :
d = ± [ (n) , H] + feeding term(of(nf-1),dt hi
(30)
which corresponds to the atom cascading along the ladderlabeled by the number of fluorescence photons n =
0,1,2,....This picture has been used successfully to interpret
many quantum-optics phenomena, such as resonance fluo-rescence spectra, photon statistics, and quantum jumps.In this formalism an important quantity is the probabilitydistribution of the time interval between the emissions oftwo successive photons, i.e., the so-called delay function orwaiting time function.19-2' When this function is knownanalytically, it can generate an efficient Monte Carloanalysis of the process: just after the emission of the nthfluorescence photon at time tn, the atom is in its groundstate and the choice of a single random number is suffi-cient to determine the time tn+1 of emission of the(n + 1)th photon. This type of Monte Carlo analysis wasused in Ref. 22 to simulate an atomic-beam cooling ex-periment and in Ref. 20 to prove numerically the exis-tence of dark periods in the fluorescence of a three-levelatom (quantum jumps). Recently laser cooling of atoms
Molmer et al.
528 J. Opt. Soc. Am. B/Vol. 10, No. 3/March 1993
by velocity-selective coherent population trapping2'3 andlasing without inversion have been analyzed by this typeof Monte Carlo method.
Unfortunately the delay function cannot be calculatedanalytically for complex systems that involve a large num-ber of levels. Nevertheless, it is possible to generate aMonte Carlo solution of Eq. (30) in which a single randomnumber determines the time of emission of each fluores-cence photon.'" The non-Hermitian evolution given inEq. (29) has to be integrated step by step numerically, sothat the amount of calculation involved is similar to thatrequired by the method presented in this paper. Thephysical interpretations of the two approaches are alsosimilar; we note in particular that for both approachesphysical quantities such as ((t)lAlck(t)) have to be evalu-ated with normalized wave functions. This normaliza-tion is systematically done at every step for the approachpresented in this paper and has to be added in the ap-proach based on the numerical simulation of Eq. (30).
Finally, we note that, as was shown in Gisin,'2 it is alsopossible to simulate a relaxation equation of the type ofEqs. (1) and (2) by a continuous stochastic equation (seealso Ref. 25). In this approach, no quantum jump occurs,but at each step at a small random element is added to thewave function. This approach has been used mainly in adiscussion of the foundations of quantum mechanics, butit can also be the starting point for an explicit solution toquantum optics problems.26 For the particular case ofhomodyne detection of fluorescence light, Carmichael7
has transformed the evolution involving quantum jumps(Subsection 2.B) into a continuous stochastic evolutionsimilar to the one obtained by Gisin.
4. TWO-TIME CORRELATION FUNCTION
Often in quantum optics problems one needs to calculatetwo-time correlation functions of atomic operators A andB, such as
C(t,T) = (A(t + )B(t)).
For T = 0 the Cij(t, 0) are one-time averages and are calcu-lated directly from the master-equation result for the den-sity matrix. The r evolution of the Cij(t,T) is shown to begiven by
i j (t T) = Y Ei jkl Ckl (t, T),aT k1
(33)
where the coefficients Tijkl, which include the evolutionthat is due to the system Hamiltonian as well as the con-tribution of 2Erelax, are the same as the ones giving the evo-lution of the one-time averages (quantum regressiontheorem):
d(Xj(t)) =dt k1
(34)
These one-time averages (Xij(t)) are identical to the densi-ty-matrix elements pji(t); the coefficients 2$ijkl are accord-ingly known from the master equation (1), and one readilysolves Eq. (33).
B. Monte-Carlo ApproachWe now present the procedure that one can use in theMCWF formalism. We first let 14) evolve from 0 to t, asexplained in Section 2. For a given outcome l+(t)) of thisevolution we form the four new states:
Ix+(0)) = (1 B)I0(t)),
Ix±'(0)) = 1 (1 ± iB)4(t)),
(35)
(36)
where ,u±, A _ are normalization coefficients. Now evolv-ing IX±(T)) and Ix_(T)) according to the MCWF procedure,we calculate
C () = (X ± (T) I Al ± ()),
C+(T) = (+(7)lAlX+(7)),(31)
(37)
(38)
For instance, the fluorescence spectrum of a laser-drivenatomic system is obtained by the Fourier transform of thetwo-time correlation function of the dipole. Another ex-ample in the semiclassical theory of radiative forces is themomentum diffusion coefficient, which describes theheating of the atom caused by the fluctuations that aredue to the randomness of spontaneous-emission processesand which is given by the integral of the two-time correla-tion function of the force operator. The goal of thissection is to indicate how one can handle such a calcula-tion in the MCWF method and to give some examples ofthis procedure.2 7
A. Master-Equation Approach to Correlation FunctionsIn the master-equation approach, correlation functionssuch as the one in Eq. (31) are calculated by usingthe quantum regression theorem5 : one expands A on thebasis of Xij = i)(jI, where i) and i) are members of abasis set of the system Hilbert space, and one calculatesthe value of the corresponding correlation functions:
The averages in Eq. (39) are taken first, for a given (t)),over the different outcomes for the evolution between 0and T of x± (T)), lx(T)), and second, over the different out-comes for the evolution between 0 and t of ) itself.
To prove that this procedure gives the same results asthe ones obtained from the master equation and the use ofthe quantum regression theorem, we consider the quanti-ties Kij, defined as
and we check that the average Kij of these quantities overthe different outcomes of the Monte Carlo evolution indeedequals Cftj). For = 0 this is easily checked from theexpressions (35) and (36) for x±(°)) and Jx_(0)):
Because they are linear combinations of one-time aver-ages and therefore follow from Eq. (34), the evolution ofthe Kij(T) values is identical to the evolution of the Cij(t,T)values given in Eq. (33). Consequently KiYj(T) coincideswith Cij(t,T) for any T.
C. Examples of Correlation FunctionsWe now consider two examples of the calculation of corre-lation functions. The first one can be treated completelyanalytically. It consists in the calculation of the symmet-ric position correlation function of the damped harmonicoscillator of Eqs. (3) and (5):
C3(t,T) = (X(t + T)X(t) + X(t)X(t + T))
X = (b + b)/2. (43)
We note from the derivation of Eq. (41) that, for aHermitian operator B, the symmetric or antisymmetriccorrelation functions (A(t + T)B(t) ± B(t)A(t + T)) can bedetermined from mean values with only a pair of func-tions |X±(T)) or IX-(T)). We choose as initial state theground state of the oscillator 10(0)) = 0), which is thesteady state of Eq. (3). Therefore C8(t,T) does not dependon t, and the first step of the procedure outlined above,i.e., Monte Carlo evolution of 10) between 0 and t is trivial.At time t we construct the two new wave functions y± (0)),which in this simple case give
JX±(O) = V/710) ± (1/V)11).
z00z
U-
z0
uJwMacc000w
0z
(44)
We now have to evolve these wave functions and calculatethe average values c (r) of X. Fortunately, the stochasticprocess associated with the evolution of Ix±) has alreadybeen determined. From Eq. (26) we get
which agrees with the result from the master-equationtreatment.3
The second example deals with a laser-driven two-levelatom. We suppose here that the laser is strictly resonantso that the atom-laser coupling can be written in therotating-wave approximation:
Ho = (Iif/2)(S+ + S), (47)
where Ql is the Rabi frequency characterizing the atom-laser coupling. We want to calculate the dipole correla-tion function:
-0.5
C(t, T) = (S (t + T)S (t)). (48)
This calculation is done in steady state so that the Fouriertransform of Eq. (48) gives access to the fluorescencespectrum (for a resonant excitation in this case). We pro-ceed in the following way: we start at time t = 0 in theground state, and we let the MCWF evolve for a time suf-ficiently long to have several quantum jumps (spontaneousemissions). In a density-matrix description this guaran-tees that the steady state has been reached. In theMCWF approach it implies that there is no memory of ini-tial conditions. From /+(t)) obtained in this way we gen-erate n, times two pairs of states x+t(T)) and !x(T) asdefined above, and we calculate the average over those n,runs of the quantities (37) and (38). Then we repeat thiswhole procedure n2 times, each time getting a new 10(t))from the random evolution of k) between 0 and t. Theresults of this procedure are indicated in Fig. 1, where weshow the values of C(t, T) normalized to its value at T = 0for various choices of n, and n2, in comparison with the
0 2 3 4 5TIME IN UNITS OF `
Fig. 1. Solid curves: real part (upper curve) and imaginarypart of the dipole correlation function for a two-level atom(S+(t + )S(t))/(S+(t)S(t)), for various choices of the numbers niand n (see text). Dotted curves: we have indicated the exactresult obtained by using optical Bloch equations and the quantumregression theorem. The field parameters of the calculations area= lOr, = 0.
Molmer et al.
530 J. Opt. Soc. Am. B/Vol. 10, No. 3/March 1993
analytic predictions.2" The same number of Ix) functionsis used in all the calculations. As observed, the highervalues of n2 lead to the best results because they provide acloser description of the state at time t.
5. EXAMPLES
We now give several examples of quantum-optics problemsto which the MCWF formalism can be applied. This se-ries of examples is far from being exhaustive, and here wedo not intend to give explicit results but simply discuss themain lines of the method in each case.
A. Reservoir at Finite TemperatureWe have already discussed the case of a two-level system orof a harmonic oscillator with a spontaneous-emissionlikecoupling [Eq. (3)], i.e., a coupling to a bath of harmonicoscillators in its ground state (zero temperature). We caneasily extend this treatment to the case of a nonzero-temperature reservoir. In this case we know that the re-laxation operator reads as
-Trelax(PS) = -(F/2)[1 + n(cwo)]
X (oops + pso*ur - 2f pso-r)- (r/2)n(wo)( ocrps + Psoa c - 2opsor), (49)
where Aod± are still given by Eqs. (4) and (5) and wheren(cwo) stands for the mean number of excited quanta attemperature T at the resonance frequency coo of the two-level system or of the oscillator:
n(wo) = [exp() - 1-1 (50)
By comparison with Eq. (2) we see that in this case wehave to deal with two operators Cm:
C = {F[1 + n(wo)]}"2o-,
C2 = [n(w~o)]12of+.
(51)
(52)
The quantum jumps associated with those two operatorscorrespond to a decay by a spontaneous or a stimulatedemission [Eq. (51)] or to an excitation by the absorption ofa reservoir quantum of energy [Eq. (52)].
B. Relaxation of Type T2We consider here the case of a two-level system, but wenow suppose that the relaxation consists of a dephasingbetween ground- and excited-state amplitudes:
2Prelax(Ps) = (l/T2 )(PePsPg + PgPSPe) (53)
where Pg and Pe denote the projection operators on theground and the excited states. This relaxation operatordamps the nondiagonal terms of the density matrix with atime constant T2, but it does not change the populations ofthe ground or the excited state. In order to treat such arelaxation operator by the MCWF procedure, we rewriteEq. (53) as
This has the structure of Eq. (2) with a single Cm operator:
C = 1/\/(Pe - Pg)- (55)
The operator CIC, is then proportional to identity, andthis implies that the wave function remains unchangedwhen no quantum jump occurs, except for the free evolu-tion that is due to Hs. In a quantum jump the action ofthe operator C, on a wave function IN) = alg) + 61e) issimply to change the sign of a and to leave 6 unchanged.We note also that in this case the probability for a quan-tum jump p does not depend on 1I) since it is alwaysequal to 8t/2T2 .
Let us remark finally that it is quite difficult to asso-ciate a measurement process with the quantum jumps re-sulting from the action of C,. A description closer toreality is obtained if we simulate real dephasing collisionsof reservoir particles with the system, with each collisionhaving a random duration and a random strength, withmean values leading to relaxation equation (53). The factthat relaxation equation (53) can be brought into the formof Eq. (2), however, implies that such an additional pre-scription for simulating a wave-function evolution is notrequired at this point.
C. Spontaneous Emission With Zeeman DegeneracyWe now come back to the problem of spontaneous emissionof a two-level system, and we take into account the angu-lar momentum of the ground (Jg) and excited (Je) levels.In this case we choose a quantization axis z and, in orderto give the relaxation equation a simple form in theIJg, mg)z, IJe, me)z basis, we write it as
-Trelax(PS) = (F/2)(Peps + psPe)
+ r> (eq* S-)ps(eq s+)X (56)
where Eq is the standard basis associated with the z axis,
e = 2"-12(u, ± i),
e = UZIo
(57)
(58)
and where S+ and S- are raising and lowering operatorsproportional to the atomic dipole operator:
Eq SJg, mg)z = (1, Jg, q, mg; Je, me = mg + q)
X JeI me = mg + q)z,
Eq S IJe, me)z = 0,
Eq* S- = (q S+)t. (59)
In Eqs. (59) a Clebsch-Gordan coefficient enters in thecoupling of the ground- and the excited-state sublevels.From Eq. (56) we see that the MCWF procedure will in-volve three operators Cm:
(60)Cq = (F) 2 (Eq* S, q = 0 +1.
We note that the relation
Cqc = Fs+ S- = rPeq=-1
(61)
ensures that the relaxation operator (56) has the samestructure as Eq. (2). From the measurement point ofview presented in Section 3 this corresponds to a simula-tion in which not only the number of photons emitted dur-
Molmer et al.
Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. B 531
ing 8t is detected but also the angular momentum of thosephotons along a given axis z.
D. Momentum Diffusion in Brownian MotionWe now turn to a problem in which the atomic motionplays a role. We consider the motion of a free particle,with a Hamiltonian Hs given by
HS = P2/2M. (62)
We suppose that this particle interacts with a bath formedby small particles and that this interaction causes a diffu-sion of the momentum of the Brownian particle with arelaxation operator:
Yfreiax (Ps)
- -yPS + Yf d3qX(q)exp(iq R/l)ps exp(-iq R/).
(63)
In Eq. (63), R represents the Brownian particle positionoperator. The integral is taken over the momentum qtransferred in a collision, and X(q) is the normalized dis-tribution of those transfers of momentum. The quantityy is the collision rate.
In the master-equation approach this relaxation equa-tion leads to simple equations of evolution for the popula-tions II(p) of each momentum eigenstate Ip):
1t(p) = -yII(p) + Yf d3 qX(q)II(p - q). (64)
To apply the MCWF formalism, we check that Eq. (63) hasthe structure of Eq. (2) with an infinity of Cm operators:
(65)
This simple form for the Cm values ensures that for aninitial state equal to a momentum eigenstate Ipo), thewave function remains at any time a momentum eigen-state Ip). More precisely, at each step St there is a proba-bility 1 - y~t for remaining in this eigenstate and aprobability y8t for changing p to p + q, where q is deter-mined randomly according to the probability law X(q).We recover the Monte Carlo simulation that one wouldperform intuitively by applying to the Brownian particlerandom kicks with this probability distribution X(q).
Note that one can also start with a wave function that isa linear superposition of various momentum eigenstates:
kk(0)) = I ai(O)Ipio). (66)
In this case, I+(t)) will remain a superposition of momen-tum eigenstates Ipi), which are obtained from the Ipio) by atranslation that is the same for all states. On the otherhand, the free evolution of the coefficients ai(t) that is dueto Hs [Eq. (62)] is different for each state. On average,this leads to a damping of the nondiagonal density-matrixelements between the various momentum eigenstates 1p).Such a simulation will reveal the destruction of spatial co-herences by the collisions, and it will give access to thespatial diffusion coefficient.
E. Spontaneous Emission Including RecoilWe now come to the description of the center-of-mass mo-tion of an atom that is due to spontaneous emission.With the additional interaction with a laser wave this situ-ation is at the basis of laser cooling and is of great practicalimportance. The dissipation operator can be writtenas2930
1reiax(ps) = (F/2)(Peps + P?e)
+ (3/81r) d2l E exp(-ik R)(e* S )e-lk
X ps(e S+)exp(ik R), (67)
where the first line describes the decay of excited-statepopulations and coherences and of optical coherences byspontaneous emission and the second and third lines de-scribe the corresponding feeding of ground-state popula-tions and coherences. The integral runs over thedirection of the emitted photon, with a wave vector kpointing in the direction of the solid angle Q, and the sumincludes a basis set of two polarizations E orthogonal tothis wave vector; R is the atomic position operator.
We take
Cae =. (3/8r)1"2 exp(-ik R)(E* S), (68)
and we check that relax can be put in the form of Eq. (2),with an integral over and a summation over E, by usingthe following identity:
fd2 E Ct eCC = (3F/8qr)fd2HE ( S+)(e* S-)el~k elk
= (3F/8ir) f d2H(S+ S- - [(S+ k)
x (S- k)]/k2 )
= rs+- = Pe. (69)
The probability for getting a quantum jump in the timestep t is given, according to Eq. (69), by p = rFert,where He represents the total population of the excitedlevel. When a quantum jump occurs, i.e., when a photonis spontaneously emitted, we make a random choice todetermine its direction. This is done by using the normal-ized probability density for a given direction of emission H:
9(fl) = P(,El) + QP(H,e2), (70)
where (El, E2) is a polarization basis set orthogonal to thedirection H and where
Once the direction of the recoil is known, the polarizationE of the photon is chosen between the two possible resultsEl, e2 with the probabilities gP(Q. ei)/QP(H). Finally, C isapplied to the wave function I+(t)) in order to get the stateof the system after the quantum jump.
We note that this quite lengthy procedure can be greatlysimplified if one restricts it to a simplified spontaneous-emission diagram for which photons are emitted onlyalong a given set of coordinates u., u, u2. In this case we
Molmer et al.
Cq = [yX(q)] "' exp(iq - R/A).
532 J. Opt. Soc. Am. B/Vol. 10, No. 3/March 1993
can choose also ux, u,, u, as a basis set for the polarizationvectors: a photon propagating along u,, for instance, canhave polarization uY or u,. This leads to a replacement ofthe integral over Q in Eq. (67) by a sum of only six termsk = kui:
(E/2) E E exp(-ik R)0-uUyUz kie
X (e S-)ps(e * S+)exp(ik R). (72)
When one is not interested in the detailed effect of thespontaneous-emission pattern on the atomic dynamics,this approximation brings a significant simplification tothe MCWF procedure.
Another simplification occurs when only the atomic mo-tion in one dimension, for example, along the z axis, isconsidered, as is the case, for example, in one-dimensionallaser cooling calculations. We then take the trace ofEq. (67) over the x and y variables, which leaves us withthe relaxation equation2 9 3 '
4elax(PS) = (r/2)(Peps + psPe)1 q k+ r E dk'X,(k')exp(-ikZ)(q,* --
q--1 -k
X p(eq S+)exp(ik'Z). (73)
The vectors eq have been introduced in Eqs. (57) and (58),and Xq(k') is the normalized probability density for hav-ing a spontaneous photon with angular momentum hq andlinear momentum 1k' along the z axis:
X(k' ) = (3/8k) [1 + ( (74)
Xo(k') = (3/4k)[1 - (k)]- (75)
The Monte Carlo procedure corresponds in this caseto the detection of the momentum hk' of the emitted pho-ton along the z axis and of the angular momentum of thephoton along the same axis. The corresponding Cm opera-tors are
Ck", = [Xq(k')] lI2exp(-ikZ)(e* S). (76)
We use this form in Subsection 5.F, which is devoted toDoppler cooling.
F. Doppler CoolingWe now focus on the case of one-dimensional Dopplercooling of a two-level atom, for which we present some nu-merical results. This will give an illustration of the effec-tiveness of the MCWF method as compared with themaster-equation approach when the number of states isnot too low.
1. The ModelAn atom with a transition Jg B JC = Jg + 1 is placed in aa-+ polarized standing wave so that only the Zeeman sub-levels g, mg = Jg) and le, me = Je) play a role; for simplic-ity we denote these substates as g) and le) in what follows.Doppler cooling occurs for negative values of the detuning
8 = L - WA between the laser and the atomic frequen-cies; it originates from the fact that a moving atom iscloser to resonance with the counterpropagating compo-nent of the wave than with the copropagating one; theatom therefore feels a net radiation pressure force op-posed to its velocity.3 233 This picture works well at non-saturating laser intensities, where one can add the effectof the two waves independently. At higher intensitiesthis type of semi-classical analysis based on the calcula-tion of a damping force becomes more complicated,34 3 5 anda quantum treatment of the atomic external motion is agood alternative. Here we present the result of such ananalysis using both a master-equation and a MCWFapproach.
The Hamiltonian Hs using the rotating wave approxi-mation reads as25
Hs = (P2/2M) + MI cos(kZ)(S' + S-) - IiPe, (77)
where Z and P are the atomic position and momentumoperators and fQ is the Rabi frequency of each travelingwave forming the standing wave. We choose the initialwave function 10(0)) equal to Ig,p = 0). At a time t, 1I(t))can be written as
j40(t)) = 2 a,,(t)Ig,p = Po + 2nlk)
+ 3n(t)Ie,p = Po + (2n + 1)hk), (78)
where the momentum po depends on the random recoilsthat have occurred between 0 and t and remains constantbetween two quantum jumps. According to Subsection2.B, evolution of an and f3,n consists of sequences of twosteps. First, the wave function evolves linearly with thenon-Hermitian Hamiltonian H = Hs - irPe/2:
Then we randomly decide whether a quantum jump oc-curs. The probability Sp for a jump is proportional to thetotal excited-state population:
6p = FE 1/3n 12
5t. (81)
If no quantum jump occurs, we simply normalize the wavefunction. If a quantum jump occurs, the momentum 1k'along the z axis of the fluorescence photon is chosen ran-domly with the probability law X+ (k') [Eq. (74)], and thenew wave function is obtained by the action of
(82)Ck' = [X+(k')]" 2exp(-ik'Z)S-
on I4(t)). This leads to
an(t + t) = IJn(t)
p6,,(t + at) = 0,
(83)
(84)
Po Po + 11k - k', (85)
Molmer et al.
Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. B 533
I ~~50
V ~~~~~~~~~40-50 - 30
20
10
0 I0 40 80 120 160 20000 I. I . I.
0 400 800 1200 1600 2000
TIME IN UNITS OF D'
Fig. 2. Time evolution of (p2) in Doppler cooling. Time is mea-sured in units of the excited-state lifetime r , and momentum inunits of Sk. The detuning 6 and the Rabi frequency fl are givenby £1 = -8 = r/2. The atomic mass is such that F = 200hk 2 /M.The points represent the Monte Carlo results, obtained by averag-ing n = 500 MCWF evolutions. The error bars correspond to thestatistical error WP(2). We have also indicated by a curve the re-sults of the density-matrix approach. Both calculations involve200 quantum levels and take approximately the same computingtime on a scalar machine. In the inset we have detailed theshort-time regime corresponding to the diffraction of the atomicde Broglie wave by the laser standing wave.
where A is a normalization coefficient. We note that inthis way the recoil that is due to spontaneous emission istreated in an exact manner. In the master-equation ap-proach an exact treatment of the spontaneous recoil re-quires a discretization of atomic momenta on a grid with astep size smaller than hk. This increases the amount ofcalculation of the master equation with respect to theMCWF one, in addition to the N-versus-N2 argument men-tioned in Section 1.
In order to make a fair comparison between the two ap-proaches, we have chosen a coarse discretization for theatomic momentum, with a step size Sk, i.e., k' = -k,0, ork. An optimum representation of the diffusion rate thatis due to the directional distribution of spontaneouslyemitted photons is obtained by taking the probability lawfor the quantum jumps governed by C-k, CO, Ck to be1/5:3/5:1/5. The corresponding modified values of X+(k')are then also used in the master-equation calculations forthis problem.
2. Numerical ResultsWe have considered the case of sodium atoms (=200 hk 2 /M), for which the minimal Doppler cooling limit,obtained for 8 = -r/2 and << F, corresponds to Prms -
8.4 Sk. We have discretized the momentum between-50 hk and + 50 Sk, which corresponds to a basis with202 eigenstates total, with at any time 101 nonzero coeffi-cients a,, and fJ,, [see Eq. (78), wherepo is either an odd oran even multiple of Sk].
The results for the evolution of the sample mean (p2)defined as
(p2) Wt = _ I (¢,i'(t)Jp2|+' i)(t)), (86)n il
are given in Fig. 2, together with the results for (P2)(t)obtained by using the master-equation treatment. These
results correspond to the parameters fl = -6 = F/2. TheMCWF results have been obtained with the average ofn = 500 evolutions.
In Fig. 2 we have indicated the statistical error 5P2, onthe determination of (P2)c,). This quantity 6P&2, which isdefined in Section 7, gives an estimate of the quality ofthe result, and with n = 500 wave functions the signal-to-noise ratio in the range of 20 is quite satisfactory.
With a scalar machine we have found that the time re-quired for the calculation with 500 wave functions is equalto the time required for the master-equation evolution.With a vectorial compiler we have found that there is anadditional gain of a factor 15 in the benefit of the MCWFprocedure. Therefore, even for this relatively simpleone-dimensional problem with only 200 levels, theMCWF method is at least as efficient as the master-equation approach for determining cooling limits with agood precision.
In Fig. 2 we clearly see the existence of two regimes inthe evolution of (P2 )(t). For short times (t < 20017-1; seethe inset of Fig. 2) the number of spontaneous emissions issmall, and the physics involved is essentially the diffrac-tion of the plane atomic de Broglie wave by the gratingformed by the laser standing wave.3 6 For longer interac-tion times, dissipation comes into play3 738 and (P2)(t)tends to a steady-state value, of the order of (1111k)2 . Thisvalue for Prms is larger than the Doppler cooling limit(8.4 Sk) because of saturation effects.
Figure 3 shows the evolution of the momentum distribu-tion of a single MCWF The MCWF extends over approxi-mately 51k and explores as time goes on all the significantparts of the momentum space. In Fig. 4 we have giventhe evolution of the momentum distributions obtained bythe master-equation approach (Fig. 4a) and by the MCWFapproach after average (Fig. 4b). In Section 7 we willcompare the convergence of the MCWF method for globaloperators (i.e., p2) and for local operators (i.e., populationof a single state).
6. EQUIVALENT MONTE CARLOSIMULATIONS FOR A GIVEN MASTEREQUATION
In this section we discuss the existence of several differ-ent Monte Carlo approaches for a given relaxation opera-
t=200or-
t=500r- 1
,t-1O0lr
-_ t=50r-1
t=or--50 0 50
MOMENTUM (oik)
Fig. 3. Time evolution of the momentum distribution for a singleMonte Carlo wave function for the Doppler cooling situation de-scribed in Fig. 2. The MCWF extends over approximately 5kand explores as time goes on all significant parts of the momen-tum space.
Molmer et al.
534 J. Opt. Soc. Am. B/Vol. 10, No. 3/March 1993
a b
. t=2000r-
_ t=500r-1
- t=101-
t=0r- 1
t=or-1
-50 0 50 -50 0 50
MOMENTUM (k) MOMENTUM (k)Fig. 4. Time evolution of the atomic momentum distributions, obtained for the Doppler cooling situation described in Fig. 2. a, Result ofthe master-equation approach. b, Result of the average of 500 MCWF's.
tor 2frelax, leading in average to the same results but withpossibly different physical pictures.
Here we restrict ourselves to the class of Monte Carloapproaches that are related by a given invariance propertyof the relaxation operator -Trelax. Suppose that there ex-ists an operator T, acting in the Hilbert space of the sys-tem, such that
T[-Te.1a(ps)]T t = S-reax(TpsT t). (87)
This operator T can be, for instance, a rotation operator,and Eq. (87) is then fulfilled if the dissipation process isisotropic. Equation (87) can be transformed into
2relax(PS) = Ttrelax(TpSTt)T, (88)
which means that we can write the relaxation operatoralternatively as
Efreiax(pS) = 2 (D t Dmps + p sD Dm)2m
The first MCWF analysis can be done by using the opera-tors Cq defined in Eq. (60), obtained after writing the re-laxation equation with z as quantization axis. Supposethat the atom starts, for instance, in g, m, = -1). Theatom-laser coupling leads first to an increase of the popu-lation of the excited state le, m. = 0). A spontaneous pho-ton may then be emitted (Fig. 6a), which, depending on itsangular momentum q = ±1, puts the atom back intoIg, m = 1) (the transition le, m, = 0) - g, m, = 0) isforbidden because of the vanishing Clebsch-Gordan coef-ficient). However, it may also happen that no sponta-neous photon is detected after a long time, with thesuccessive steps of evolution owing to the non-HermitianHamiltonian H and renormalization of the resulting wavefunction causing a continuous rotation from g, m, = ±1)into IkNc) and therefore trapping the atomic populationinto this state (last part of the time sequence of Fig. 6a).
+ IDmpsD DIm
with
Dm = TtCm T.
(89)
e,J=1mZ=0 mz =1
a-(90)
We can therefore perform the Monte Carlo simulationeither with the set of operators Cm or with the set of oper-ators Dm. The physical pictures given by these two MonteCarlo simulations may be quite different from each other,although we know that their predictions concerning one-time averages or two-time correlation functions are thesame. The choice of a particular simulation should bemade by considering the convergence of the numerical cal-culation, as we show in Section 7, or for emphasizing aparticular physical aspect of the problem.
We now give an example of two equivalent Monte Carlosimulations for the same physical process. Consider a g,Jg = 1 - e, Je = 1 transition irradiated by two resonantlaser fields with the same intensity and polarized or+ ando, with respect to the z axis (Fig. 5a). It is known fromthe analysis by optical Bloch equations that the atomicpopulation is eventually trapped in a ground state that isnot coupled to the laser field. This is related to the darkresonance phenomenon. If we suppose that the two wavesare in phase, this dark state is
I'NC) = (Ig, m = -1) + g, m. = 1))/V. (91)
g,J=1
e,J=1 I
g,J=1 ,m Y=_1
mZ=0 mZ=1
mY=0
mY=0
b
mY=1
MY =1
Fig. 5. Configuration schemes leading to a dark resonance: ag, Jg = 1 - e, Je = 1 transition is irradiated by two waves respec-tively o-+ and o-- polarized along the z axis. a, If angular mo-mentum is quantized along the z axis, the dark resonanceappears as the formation of a nonabsorbing state, which is a lin-ear combination of g, mz = -1) and g, m. = 1). b, If angularmomentum is quantized along the axis parallel with the resultinglinear polarization of the light (y axis), the dark resonance corre-sponds to an optical pumping to the Ig, my = 0) state, which is notcoupled to the light because the Clebsch-Gordan coefficient con-necting g, Jg = 1, my = 0) and le, J, = 1, my = O) is zero.
a
-
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,_ \
Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. B 535
Note that the continuous rotation into the trapping Eis of the same type as the one seen in Section 3 forspontaneous emission of an oscillator in a linear supEsition of the ground and the excited state.
Now we can replace the set of operators Cq by a diffeset obtained in choosing a different quantization-For instance, suppose that we choose this new quanition axis y parallel to the resulting linear polarizaticthe laser light at the atom position. Because of the 7larization of the laser excitation along that axisFig. 5b) we now identify the trapping state as
kkNC) = g, my = 0),
and we perform the Monte Carlo simulation by usingset of operators Dq that are analogous to those giveEq. (60) but defined now with respect to the y axis.Fig. 6b we have plotted such a simulation involvingquantum jumps. We start again in the state Ig, m2 =that we expand on the Ig, my) basis. In the first pathe evolution a continuous rotation toward Ig, mytakes place. Then a first photon with qy = 0 is dete(this detection projects the wave function onto a super]tion of Ig, my = ±1) and the population of the trapstate g, my = 0) is 0. The atom then cycles betvIg, my = -1) and le, my = +1), and the population oftrapping state remains zero until one detects a seiphoton with a polarization q_ = ±1. This detecprojects the wave function into the trapping state.way the system enters into the dark resonance ismore readily understood as a kind of optical pumping
In Figs. 6c and 6d we check that the two simulatlead to the same average results. We observe smfluctuations in the results obtained with the quantizEalong the z axis; the reason for this difference in theity of the simulation is explained in Section 7.
7. GETTING GOOD STATISTICS WITH TEMONTE CARLO WAVE-FUNCTION METHC
for the calculation]
Vn >> AA(.(t/[(A)(01 . (96)
In the following we give an estimate of (AA)')(t), and wethen discuss the requirement on n given by the inequality(96) in terms of local or global operators.
B. Estimate of the Sample VarianceTo estimate (AA)')(t), we use the general inequality
(97)(OIAIO)2 -_ (1A 20).
(92) Expression (95) can be overestimated by
n in (AA)(2n)(t) _ z( E 9')I_) (tT .. / =
When n is large compared with 1, the two terms of theright-hand side of this expression have a finite limit. Thefirst term tends to (A2)(t), and the second tends to (A) 2 (t),
where the averages are now taken in the system densitymatrix at time t. The right-hand side of inequality (98)therefore tends to the variance (A)2 S(t) of the operatorA for a system in a state described by the density matrixps(t):
(AA)2p)(t) = Tr[ps(t)A2 ] - [Tr(psA)] 2 . (99)
We therefore have the following overestimate of (AA))(t)
(100)
Consequently, a sufficient condition on n deduced from in-equality (96) is
>> AA(pS(t)/[(A)(t)]. (101)
We note that inequality (100) becomes an identity if allthe (i)(t)) are eigenstates of the operator A. This is thecase for instance in the study of Brownian motion outlinedin Subsection 5.D, if we take 1(t)) = I) and if we choose A
(98)
A. Definition of a Signal-to-Noise RatioA primary goal of our procedure is to determine the aver-age value (A)(t) at time t of a given Hermitian system op-erator A, knowing the state of the system at time 0.Applying our MCWF method with a number n of simula-tions, we obtain the sample mean
1.0
0z 0.5
0.0
1.0
0tZ 05
(93)n
(A) (n)(t =-E (0(')(t0lAl(`()),ni=l
which will approximate (A)(t) with a statistical error 6A(W,related to the square root of the sample variance (A) by
When n is large compared with 1, the sample variancetends to a finite value that we denote AA(.). Conse-quently, the condition for having a good signal-to-noiseratio (A)/6A(,,) can be written as [if (A)(t) is expected to bezero, it may be replaced by the precision that one requires
0.00 2 4 6 8 0 2 4 6 8 10
TIME IN UNITS OF r-1 TIME IN UNITS OF l
Fig. 6. MCWF simulation of a dark resonance. The populationof the uncoupled state in a single MCWF evolution, correspondingto a measurement of the photon angular momentum along the zaxis, a, or along the y axis, b. The two types of evolution areclearly different. Average of 100 MCWF evolutions, with a mea-surement of the photon angular momentum along the z axis, c, oralong the y axis, d. Apart from fluctuations, the two simulationslead to the same result, as expected.
- T h I I I I K K K l I I
a _i
l . .
b
l I I I II I I I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I
Molmer et al.
(AA)2 (t) - (AA)2 )(t) .W (P
- I I I I I I I I I -
a -. . . . . I . . .
_ I I I I l, I I I
536 J. Opt. Soc. Am. B/Vol. 10, No. 3/March 1993
0 2 4 6 8 10
TIME IN UNITS OF r-1
Fig. 7. Time evolution of the variance AA(PS)(t) and of thesample variances AA()(t) obtained for the two simulations withthe y and z quantization axis for the dark resonance problem(n = 1000). The sample variance for the y axis choice is close toits upper bound AA(PS)(t), whereas the simulation with the choiceof the z axis leads to a noise that is more than two times smaller(also compare Figs. 6c and 6d).
equal to the projection operator on a given po): A=Ipo) (poa. For the case of Doppler cooling, taking A = P2 ,we have found in steady; state that AA(n,) = 0.8AA(pS,) forn 500; the difference between these two quantities isessentially determined by the typical width of the individ-ual MCWF's (cf. Fig. 3).
Let us also emphasize that, in contrast to AA(pS,.(t), thequantity A(x)(t) can vary for a given system with thechoice of the set of the Cm operators. We illustrate thispoint for the example of dark resonances discussed at theend of the previous section. We choose A = lkNc)(kNcI,whose average value gives the population of the trappingstate, and in Fig. 7 we plot the value of AA(pS,(t) and of theAA(,,(t) values obtained for the two simulations with theyand z quantization axes. The initial state is Ig, m, = -1)for the three curves. We have taken n = 1000 so thatAA(n)(t) AA()(t). In Fig. 7 one clearly sees that in-equality (100) is confirmed for this example. We also notethat the simulation with the y quantization axis has aAA(.) close to its upper bound; this is easily understoodsince in this case, as soon as one quantum jump occurs,the wave function is in an eigenstate of A, with the eigen-value 0 if the detected photon is 7r polarized or 1 if it is c-.polarized. The simulation with quantization along thez axis leads to AA(,,)(t) smaller by a factor of 2 and istherefore more appropriate if one looks for a rapidly con-verging procedure.
C. Local versus Global OperatorsIn order to discuss the requirements imposed by inequal-ity (101), we now split the various operators A into twokinds. First, there are local operators, such as the onesgiving the population of a particular state I j), A = li) (iand A2 = A. For those operators we expect that
1 1(A)(t) - ' (AM),(t) N ' (102)
where N denotes the total number of quantum levels in-volved in the simulation (we suppose here that N >> 1).
If we insert these values into inequality (101), we see thatthe number n of simulations that have to be performedmust be larger than the number of levels N:
local operators n >> N. (103)
Clearly one does not gain by using a Monte Carlo treat-ment in this case. The amount of calculation for deter-mining a single l+(t)) is reduced by a factor N with respectto the calculation of the density matrix ps(t), but one hasto repeat n MCWF runs to get good statistics, with n > N.
We note that inequality (100) is in fact an overestimateof AA(.) and that wave functions extending over severaleigenstates of the operator A of interest lead to smallervalues of AA(.) compared with AA(pS,. This may in par-ticular be the case for the local operators, and the typicalmomentum width of a Monte Carlo wave function in oursimulation of laser cooling (see Fig. 3) leads to a much bet-ter agreement between the momentum distributions inFig. 4a and 4b than expected from relations (101) and(102). To illustrate this, we calculated the fluctuations inthe population of the zero-momentum state, described bythe local operator A = p = O)(p = 01; we obtained attime t = 2000F1 the values (A)(n) = 4 x 10', (AA) n) =
7 x 10'. The ratio between these quantities is propor-tional to the typical width in momentum space of a wavefunction and weakens constraint (103), which was deducedfrom relations (101) and (102).
The MCWF treatment is more efficient if one deals withglobal operators, such as the population of a large group oflevels, or, for the description of laser cooling, the averagekinetic energy. For those operators we have
AA(,,t)W - (A)(t). (104)
For instance, if we consider the problem of a Brownianparticle thermalized with a reservoir at temperature Tand if we take A = P2 /2M, we get in steady state
(A) = (3/2)kBT, AA(pS,) = \/32kB T. (105)
In this case we see from inequality (101) that good statis-tics are obtained after n Monte Carlo runs as soon as
global operators n >> 1. (106)
If one requires, say, a 10% accuracy for the average ofglobal operators, inequality (106) dictates the choicen 100. Thus, when the number N of levels involved islarger than this number, the MCWF treatment may bemore efficient than the master-equation approach. Tocompare with laser cooling experiments, a 10% accuracywill usually be sufficient, and even for the simple case ofDoppler cooling we have noted that the stochastic evolu-tion of wave functions is competitive with the integrationof the density-matrix equation.
8. CONCLUSION
As we mentioned in the Introduction, there have recentlybeen a number of papers that discuss the applicability ofstochastic methods as an alternative to the usual treat-ment of master equations in quantum mechanics. Thesemay draw on different sources of inspiration, but they alllead to a description involving a stochastic wave-function
nL
Molmer et al.
Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. B 537
evolution, which may be either continuous" or involvequantum jumps.6' 7"110"1
In the present paper we have presented a stochastic evo-lution for the wave function of a system coupled to a reser-
voir generalizing that of Ref. 6. We have proved theequivalence of this Monte Carlo wave-function approachwith the master-equation treatment, and we have givenseveral examples where it can be applied. In addition, wehave considered in detail the efficiency with which expec-tation values of physical observables may be calculatedwith this method. In order to control the fluctuations in-herent in any simulation procedure, one must propagate alarge number of wave functions and, depending on thedimensionality of the system Hilbert space and on thetype of observable considered, the simulations may bemore or less efficient than the master-equation treatment.
As an example, we considered one-dimensional lasercooling of two state atoms, and these calculations bringpromises for our present attempts to treat more generalthree-dimensional cooling. Also, a method for calculatingtwo-time correlation functions was demonstrated; the ap-plicability of this method to the more complex problem ofthe spectrum from cooled atoms in optical molasses is nowbeing studied (see also Ref. 41).
Apart from a numerical procedure, the replacement ofthe density matrix by wave functions provides new in-sight; for example, it reveals mechanisms for the evolutionof the system that may manifest themselves less clearly inthe master-equation approach. We wish to emphasize,however, that by the MCWF procedure one does not getknowledge of how individual systems in an ensemble actu-ally evolve and how they contribute to the averages as ob-tained by the density matrix. At the level of individualwave functions the method only provides a model. As we
saw in Section 6, several different such models can apply,and none of these models can be claimed to yield a moretypical realization than the others. It is noteworthy,though, that if a particular detection scheme is invoked,according to our measurement interpretation of theMCWF procedure, the Monte Carlo wave function does
correspond to a true evolution of the system according tostandard quantum mechanics.
As seen from another perspective, this work confrontstwo very different definitions of the density matrix: (i) areduction, by means of a trace, of the state of the com-bined system + reservoir, which can no longer be de-scribed by a pure state in the small system, and (ii) astatistical description of an ensemble of systems populat-ing different states with a given probability law. Whichof the two underlying interpretations for the density ma-trix is used has no influence on the way in which meanvalues are obtained; this explains why the master equa-tion, which is certainly derived from the first definitionfor the density matrix, can be treated by our MCWFmethod, which relies on the second interpretation.
ACKNOWLEDGMENTS
We are grateful to C. Cohen-Tannoudji, A. Aspect,C. Salomon, W D. Phillips, P. D. Lett, and K. Berg-
S0rensen for many helpful discussions. The Laboratoirede Spectroscopie Hertzienne is a Unit6 de Recherche del'Ecole Normale Sup6rieure et de l'Universit6 Paris 6
and is associated with the Centre National de la Recher-che Scientifique.
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