Quantum Kinetic Theory I: A Quantum Kinetic Master Equation for Condensation of

a weakly interacting Bose gas without a trapping potential

C.W. Gardiner1 and P. Zoller21 Physics Department, Victoria University, Wellington, New Zealand

2 Institut f�ur Theoretische Physik, Universit�at Innsbruck, 6020 Innsbruck, Austria

A Quantum Kinetic Master Equation (QKME) for bosonic atoms is formulated. It is a quantum

stochastic equation for the kinetics of a dilute quantum Bose gas, and describes the behavior and

formation of Bose condensation. The key assumption in deriving the QKME is a Markov approx-

imation for the atomic collision terms. In the present paper the basic structure of the theory is

developed, and approximations are stated and justi�ed to delineate the region of validity of the

theory. Limiting cases of the QKME include the Quantum Boltzmann master equation and the

Uehling-Uhlenbeck equation, as well as an equation analogous to the Gross-Pitaevskii equation.

I. INTRODUCTION

Recent observations of Bose-Einstein condensation (BEC)in a dilute gas of magnetically trapped alkali atoms [1{6],and in excitonic systems [7] has stimulated new theoret-ical e�orts to describe the dynamics and signatures ofweakly interacting Bose gases. In contrast to super uidHelium, which for many years has been the only exper-imental example of BEC, these weakly interacting Bosegases are much more amenable to a theoretical analysis.Recent theoretical work has focused on a description ofBEC in a trapping potential and dynamics of formationof a Bose condensate [8{39].In this paper we want to develop techniques to de-

scribe the behavior and formation of the Bose condensatein which we will combine the simplicity of a quantumstochastic methods used in quantum optics [40] with arealistic treatment of the interatomic interactions whichare known to play a major role in the dynamics of the con-densing system [41]. Such a description must necessarilyinvolve aspects of a kinetic theory as well as quantummechanics.The �rst realistic formulation of kinetic theory was the

Boltzmann Equation (BE) whose usefulness and successto this day is universally accepted. The so-called Quan-tum Boltzmann or Uehling-Uhlenbeck equation (QBE)[42] introduces corrections for quantum statistics into theBoltzmann collision term, but since it deals only withone-particle distribution functions we cannot expect thisequation to give a realistic treatment of the quantum me-chanical aspects which must occur in a Bose condensate.A di�erent description is provided by the time depen-

dent Gross-Pitaevskii equation (GP equation) [43] whichcan be viewed as an equation for the condensate wavefunction (order parameter for the Bose condensate). Thisequation is clearly a simpli�ed description in that it in-cludes no quantum uctuations, or thermal or irreversiblee�ects, but it may well be valid in the situation of a largenumber of condensate particles. Both of these equations

1

contain essential aspects of the problem we are study-ing. However, in practice the process of creating a Bosecondensate in a trap by means of evaporative coolingstarts in a regime covered by a kinetic equation and �n-ishes in a regime where the GP is thought to be valid.What is needed is uni�ed description which covers thewhole range | a true quantum kinetic theory. In thispaper we will develop a quantum kinetic master equation

(QKME) which is not a mere modi�cation of the BE butextends its philosophy into quantum mechanics and de-rives a quantum stochastic equation of the kind similar tothose derived in quantum optics.The QKME which we derive is thus a consistent ex-

tension of the philosophy of the BE to a regime in whichquantum mechanical e�ects are not a minor correctionto classical results, but play a major role in the full de-scription of the system.In applying the QKME to the problem of Bose conden-

sation, although quantum mechanics plays major role,the signi�cant quantum aspects are restricted to a fewmodes, the remaining modes being able to be describein a manner similar to the classical Boltzmann equa-tion. This is a situation which is very familiar in quan-tum optics, where we very often �nd that it is neces-sary to describe a few optical modes fully quantum me-chanically, while treating the rest of the systems as aheat bath. However, the methods necessary for quan-tum kinetic theory contain considerably more technicaldi�culty, largely because of the interaction between theatoms. The nearest optical analogy is multimode prop-agation in a medium with a strong nonlinear refractiveindex [44].

II. THE NEED FOR A QUANTUM KINETIC

THEORY

Our goal is to derive a quantum kinetic master equa-tion for a gas of N weakly interacting Bose atoms. Themaster equation is formulated as an evolution equationfor the N -atom density matrix, giving a quantum me-chanical generalization of the Quantum Boltzmann equa-tion. This provides a fully quantum mechanical descrip-tion of the kinetics of a Bose gas, including the regimeof Bose condensation. In particular, such an equation iscapable of describing the formation of the Bose conden-sate. The quantum mechanical processes which must beincluded in this treatment are the atomic motion (trans-port) and coherent and incoherent interactions (colli-sions) between the atoms. In this section we give a qual-itative overview over the main results to be derived inlater sections.

A. Essential elements of the theory

There are three principal elements, as follows.

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1. Coherent and incoherent dynamics

The derivation of a quantum master equation impliesthat the system dynamics can divided into into coherent

and incoherent physical processes, where the incoherentdynamics is modeled as a quantum stochastic Markovianprocess. This builds on ideas, which have been developedand applied successfully in quantum optics as a methodof describing coherent and incoherent processes simulta-neously (e.g. in laser theory).In a weakly interacting Bose gas, we will treat the col-

lisions between the atoms resulting in a large momentum

transfer as random and weak incoherent processes, re-sponsible for the \noise" in the system. On the otherhand, forward or near forward scattering is, if the wave-length is su�ciently long, largely a coherent process, giv-ing rise to dispersive e�ects.

2. Dynamics of the condensate

The (coherent) dynamics of the Bose condensate istreated explicitly and separate from the description ofthe non-condensed modes which play the role of noiseand feeding terms for the Bose condensate. Again, thisis guided by laser theory where the coherent laser mode isseparated from the weakly populated incoherent modeswhich are typically treated as a heat bath.

3. The \cell" description

The theory is formulated in terms of a quantum me-chanical phase space description for with coarse grainedposition and momentum variables, based on division ofphase space into \cells" of volume h3. This seems tobe the most natural formulation, since collisions are bestdescribed in terms of momentum, whereas transport isbest understood in terms of position. We will introducethis formulation by means of a wavelet description, whichgives an exact exact description of the dynamics; approx-imations appear as a result of the procedure for describ-ing the kinetics, and the conditions for the validity of ourapproximations then put some conditions on the precisenature of the cells chosen.

B. The scope of this paper

In developing the physical picture, there are three mainstages through which we must proceed in order to achievea full quantum kinetic theory applicable to the presentexperiments on Bose-Einstein condensation.

i) It is necessary �rst to develop the theory for the weaklyinteracting Bose gas with no trapping potential, whosestationary solution is well approximated by an ideal Bosegas.

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ii) The next stage is to take account of the modi�cations(these may be very signi�cant) to the dynamics, whichare induced by the presence of a large proportion of atomsbeing in the condensate. This can be developed in termsof Bogoliubov quasiparticles [45,46], or more generally.

iii) Finally, we introduce the trapping potential. Thereare two extreme situations to consider. If the trap isvery broad, then the situation is qualitatively not verydi�erent from the case of no trap, though the appropri-ate modi�cations can be quite intricate. In the case ofa very tight trap, we have a completely di�erent treat-ment, based on the kinetics of transitions between di�er-ent trap levels. There will also be an intermediate regime,in which it is relevant to treat only the lower trap levelsexplicitly, while upper levels are thermalized.

This paper deals with i) only; ii) and iii) will be treated inour second paper QKII. Thus this paper develops all thebasic ideas and methodology, while QKII will develop there�nements necessary to deal with strong condensationwith and without a trapping potential.

1. The physical picture

This phase space description is one in which particlesare represented as wave packets interacting with eachother. From the point of view of wavepackets there arethe following processes: Transport: that is the motionthrough space of a wavepacket. However we also under-stand that from a wavepacket point of view, transport is aprocess by which the wavepacket moves with unchangedshape|the wavepacket spreading inherent in the quan-tum mechanics of particle motion is not itself viewed asa part of transport. Wavepacket spreading occurs in ad-dition to what one would normally call transport, andmust contribute to the production of coherence whichcan arise in a Bose condensed system. It is purely quan-tum mechanical, but arises of course from exactly thesame quantum mechanical source as transport, as de-�ned here. Motion in a trap is readily included in thesearguments to the extent the trapping potential can beconsidered as slowly varying. Collisions: here we meanlocalized events in which the momentum of wavepacketswill change instantaneously, with a consequent changein the wavefunction. The localization cannot of coursebe exact, because wavepackets themselves are extendedobjects.

2. The size of the cells

Quantum mechanics itself does not give any preferredsize of the momentum space or coordinate space cellsseparately. Physically, we can characterize the choiceof cells by a wavenumber � (or a cell size lc = �=�(compare Fig. 1), and this will naturally provide a \cut"

4

in momentum space; changes of momentum greater than�h� are seen as collisions, while smaller changes are seenas changes which take place in coordinate space. Thevalue of � is constrained by two considerations.

(a) We want to have a description in terms of \en-ergy levels" within a box inside each cell. Thismeans that the wavefunction of a particularparticle must maintain its coherence over thelength lc of the cell. The coherence lengthwhich arises from collisions should normallybe of the same order of magnitude as the meanfree path �mfp = v� , where v is the speed ofthe particle,and � is the mean time betweencollisions. Thus this requirement yields thecondition lc � �mfp.

(b) We will be wanting to develop a Markoviandescription of collisions, which depends on thedetermination of a characteristic decoherencetime, which is essentially the bandwidth of fre-quencies of particles in the system, that is,thermal correlation time �h=2�kT . This shouldbe much less than the the characteristic timefor evolution of the the phase-space density,and this yields the condition that the meanfree path �mfp � �T = h=

p2mkT , which is

the wavelength of a typical particle, known asthe thermal wavelength.

(c) As well as the condition that the bandwidth offrequencies must be su�ciently large, we mustalso have a su�cient density of states to ensurethat the correlation function inherent does in-deed become very small for times greater thanthe correlation time. Essentially this requiresthat the maximum energy di�erence betweencells should be much less than kT , leading tothe requirement lc � lT � h=

p2mkT , that is

the cells much be very much larger than thethermal wavelength.

(d) The method is based on a requirement thatcorrelations which involve momentum di�er-ences greater than � are negligible. Thismeans that the \collisions", that is, processeswhich cause a momentum change larger thanj�j, are seen as inducing incoherence. Thederivation of the QKME which we use doesthis by means of a projection formalism, andthe condition for the validity of the procedureis found to be

a�mfp�T � l3c (1)

where a is the scattering length for the inter-action between the particles.

(e) There is also a weak condensation condition

for the validity of the methods used in thispaper, and this can be written in terms of the

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cell size lc, the scattering length a, and theparticle density � as lc �

p�=8a�.

These criteria are easily met, for example in the ex-periment of [3] one �nds that � = 0:4619m, while thethermal wavelength is lT = 4:8� 10�7m.On the other hand in that experiment the trap itself

has a size of the order of magnitude as lT . Thus to de-scribe the behavior of the wavefunction within the trap,one would want lc to be �ner than this. This �rst paperdoes not deal with trapping potentials, but we will showhow to overcome this di�culty in the QKII. The essenceof the solution is that it is not necessary to have the samesize cells for all ranges of momentum|they can be �nerin space at the lower momentum ranges.Figs. 2 and 3 illustrate the phase space description

and the various physical processes. The �gures show thephase space cells corresponding to di�erent momentumbands K with width � and position cells r. A stateof the system is given by specifying a set of occupationnumbers fn(K; r)g of the cells (Fig. 2). Transport andwave packet spreading corresponds to coherent quantumprocesses connecting cells in a given momentum band(horizontal arrows in Fig. 3a). Note that these coherentprocesses do not change a given momentum distributionN(K) = (N(K1); N(K2); : : :) with N(K) =

Pr n(K; r)

the number of atoms in a momentum band. In a sim-ilar way, coherent processes corresponding to a forwardscattering, or motion in a slowly varying trapping po-tential give rise to smooth shifts between the cells ofneighboring K (Fig. 3a and b). Incoherent collision pro-cesses between two atoms, on the other hand, are asso-ciated with a \quantum jump like" momentum changeK1;K2 ! K3;K4 between widely separated in momen-tum bands (however, essentially con�ned to within a sin-gle spatial cell r) (see Fig. 3c). These transitions conservemomentum,

K1 +K2 = K3 +K4 +O(�); (2)

up to an uncertainty � associated with the width of theband, and energy

E1 +E2 = E3 +E4 (Ei = �h2K2=2m): (3)

Collisions cause transitions between di�erent momentumdistributions

N(K) = (: : : ; n1; n2; n3; n4; : : :)

! N0(K) = (: : : ; n1 � 1; n2 � 1; n3 + 1; n4 + 1; : : :) (4)

which we write in the form

N(K)! N0(K) = N(K) + e: (5)

C. The quantum kinetic master equation

The derivation of the master equation in Sec. IVC be-low assumes that the full quantum coherence is kept for

6

di�erent spatial con�gurations (occupation numbers ofthe phase space cells) within one K-band, and the as-sumption that atomic coherences between di�erent mo-mentum bands are eliminated in the Born and Markovapproximation. The mathematical procedure of deriv-ing a master equation in Sec. III is based on projectingthe N -atom density operator � on states associated witha given momentum distributions N(K). For a given cellsize there is a regime where this diagonally projected den-sity matrix vN obeys a closed equation

_vN(t) = � i

�h

"XK

Zd3x

��hK

2m

�� jK(x) +

XK

Zd3x

yK(x)

���h2r2

2m

� K(x); vN(t)

#(6a)

� i

2�h

24 XK1 6=K2

(U(K1;K2;K1;K2) + U(K1;K2;K2;K1)) +XK1

U(K1;K1;K1;K1); vN(t)

35 (6b)

+�

�h

Xe

�(�E(e))f2U(e)vN�e(t)Uy(e)� Uy(e)U(e)vN(t)� vN(t)Uy(e)U(e)g: (6c)

The �rst line in (6a) describes atomic motion betweenthe di�erent phase space cells within a given K{band.The �rst term is a streaming term with current operatorjK(x) (for a formal de�nition see (42 below), and the sec-ond term corresponds to wavepacket spreading (the �eldoperator for a K{band will be de�ned in Eq. (eq1.16)).The second line is (coherent) forward scattering, and thelast describes the redistribution by collisions accordingto a U(e)-operator, which is expressible in terms of theinteraction potential, and is explicitly de�ned in (47). Adetailed derivation of the equation will follow in Sec. IVbelow.

1. Wavefunction stochastic di�erential equation (SDE)

interpretation

It is interesting to interpret (6a) from the point of viewof evolution of a stochastic N -atom wavefunction wherethe wide angle collisions are described as quantum jumps[47]. Note that (6a) can be written in the form

_vN(t) = � i

�hHe�vN(t) +

i

�hvN(t)H

ye� (7)

+2�

�h

Xe

�(�E(e)) U(e)vN�e(t)Uy(e) : (8)

Let us de�ne an atomic state vector j N(t)i with N(K)a given con�guration of occupation numbers. In the timeevolution of this state vector a collision is associated withan instantaneous jump

j N0+e(t+ dt)i / U(e)j N(t)i (9)

Here the U -matrix plays the role of the jump operator,connecting con�gurationsN(K)! N(K+e). The prob-ability for a collision is in the time interval (t; t + dt] is,given a normalized statevector j N(K)(t)i at time t, by

7

P(t;t+dt] =2�

�h

Xe

�(�E(e))jjU(e)j N(t)ijj2 dt (10)

�Xe

P(t;t+dt](e) (11)

The time evolution between the collisions is governed bythe nonhermitian Hamiltonian He� ,

j N(K)(t)i = e�iHeff t=�hj N(K)(0)i : (12)

Physically, (12) describes the streaming and forwardscattering between the jumps, and the nonhermitian isassociated with loss due to collisions. Note that He�

preserves the distribution N(K) (see Figs. 2a and b). Astochastic average over these quantum trajectories givethe density matrix, vN(t) = hhj N(t)ih N(t)j=jj N(t)jj2ii.

D. Limiting cases

The Quantum Kinetic Master Equation (6a) includesas a limiting cases both the Quantum Boltzmann masterequation and the Uehling-Uhlenbeck equation, as well asan equation analogous to the Gross-Pitaevskii equation.These results will be derived in Sec. V).

1. Quantum Boltzmann master equation

If we assume a single spatial cell, i.e. the size of thesystem equals the dimension of the system (compareFigs. 1 and 2), the transport terms trivially disappearand Eq. (6a) reduces (Sec. V) to an equation which weshall call the Quantum Boltzmann master equation:

_wn = � �

�h2

X1234

�(�E(1234))jT1234j2

�nn1n2(n3 + 1)(n4 + 1)[wn � wn+e]

+(n1 + 1)(n2 + 1)n3n4[wn � wn�e]o: (13)

This is a rate equation connecting the occupation prob-abilities of a given con�guration n � N, wn, where thefactors nk+1 re ect quantum statistics for the collisionaltransition rates. It obviously has a limited validity, butit is attractive because of its simplicity and ease of sim-ulation. [48].

2. Uehling-Uhlenbeck equation

For spatially inhomogeneous systems we derive from(6a) the Uehling-Uhlenbeck equation (Sec. (VB)). Let usde�ne a single particle distribution function fK(x) whereK labels a momentum band and x is a (continuous) spa-tial coordinate (for a precise de�nition see Eq. (112)). We

8

interpret fK(x) as a joint momentum position distribu-tion function, similar to the Wigner function. Assuminga factorization of the N{atom distributions (an assump-tion not valid in the BEC regime) one obtains from (6a)the kinetic equation

@

@tfK(x)) � �hK � rx

mfK(x) +

2juj2h2

Z Z Zd3K2d

3K3d3K4

��(K+K2 �K3 �K4)�(! + !2 � !3 � !4)

�ffK(x)fK2(x)[fK3

(x) + 1][fK4(x) + 1]

�[fK(x) + 1][fK2(x) + 1]fK3

(x)fK4(x)g

(14)

which now includes a streaming term. The collisionalterm in Eq. (14) as been approximated as s-wave scat-tering.

3. Condensate master equation

Finally, to illustrate the kinetics of Bose condensationwe will derive in Sec. (VC) a simple approximation ofthe QKME based on the assumption that the densityoperator of the total system can be factorized into a con-densate density operator �0(t) for the K = 0 Bose con-densate band, and an operator for the non{condensedmodes K 6= 0 which is assumed to be in thermal equi-librium. We obtain the equation, which we shall call thecondensate master equation:

_�0(t) = � i

�h

�Zd3x

y0(x)

���h2r2

2m

� 0(x) +

1

2u

Zd3x

y0(x)

y0(x) 0(x) 0(x) ; �0

�(15)

+

24(ug(0)X

K6=0

�nK)

Zd3x

y0(x) 0(x) ; �0

35

+

Zd3x

Zd3x0G(�)(x� x0; T; �)f2 0(x)�0 y0(x0)� �0

y0(x

0) 0(x) � y0(x

0) 0(x)�0g

+

Zd3x

Zd3x0G(+)(x� x0; T; �)f2 y0(x)�0 0(x0)� �0 0(x

0) y0(x) � 0(x0) y0(x)�0g;

(16)

Here 0(x) refers to the atomic destruction operator forthe K = 0 band with the Bose condensate. The �rst linein Eq. (15) gives the dynamic of the Bose condensateincluding the nonlinear interaction term proportional to

y0

y0 0 0, and an interaction term of the atom in the

Bose condensate with the above condensate particleswith thermal occupation numbers g(0)

P�nK6=0 (where

g(0) is a normalization factor to be de�ned in Eq. (39)).For zero temperature, and when the �eld operators 0(x)are replaced by c{numbers, Eq. (15) is, of course, equiva-lent to the Gross-Pitaevskii equation. For �nite temper-ature, solutions of the corresponding Hartree Fock equa-tions have been discussed in references [49,20,21]. Thesecond and third line in Eq. (15) give dissipative loss andfeeding terms for the condensate due collisions with the

9

non-condensed atoms. The relevant collisions are of theform K1;K2 $ K3;K = 0, where atoms are transferredto the Bose mode under conservation of momentum andenergy. In Eq. (15) these transition rates are denoted toG(�)(x�x0; T; �) (to be de�ned in Sec. 138); they involvethe occupation numbers of the K 6= 0 modes accordingto the given temperature and chemical potential

4. Validity of the limiting cases

These three cases are all very simpli�ed, and more re-alistic applications of the QKME will require much morecare. But each one them illustrates a di�erent aspect ofthe problem of quantum kinetics, which gives them greatvalue as an aid to intuition.

III. DESCRIPTION OF THE SYSTEM

We consider a set of Spin-0 Bose particles, describedby the Hamiltonian

H = H0 +HI +HT ; (17)

in which

H0 =

Zd3x y(x)

�� �h2

2mr2

� (x) (18)

and

HI =1

2

Zd3x

Zd3x0 y(x) y(x0)u(x� x0) (x0) (x): (19)

Thus, the operators (x) have the commutators

[ (x); y(x0)] = �(x� x0); (20)

[ (x); (x0)] = [ y(x); y(x0)] = 0: (21)

The potential function u(x � x0) is a c-number and isas usual not the true interatomic potential, but rather ashort range potential|approximately a delta function|which reproduces the correct scattering length. This en-ables the Born approximation to be applied, and thussimpli�es the mathematics considerably. [50]The term HT arises from a trapping potential, and is

written as

HT =

Zd3xVT (x)

y(x) (x): (22)

In this paper we will restrict our considerations to thecase of no trapping potential, so VT will be set equal tozero. The modi�cations necessary if there is a nonzerotrapping potential will be dealt with in a future publica-tion.Thus, this is the standard second quantized theory of

an interacting Bose gas of particles with mass m.

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A. Possible phase space descriptions

1. The Wigner function

The original work of Bose and Einstein [51] was con-ceived before the invention of quantum mechanics byHeisenberg and Schr�odinger, and was based on a simplephase space description, in which both momentum andcoordinate space were discretized to give phase space cellsof volume h3. The state of the system was then speci-�ed by the number of quanta in each phase space cell.The treatments in modern elementary textbooks use es-sentially the same arguments, but use a single large boxin coordinate space, so that the discretizing comes aboutfrom the discrete energy levels in this box.We will want to develop a formulation which is as close

to the Boltzmann equation as is permitted by the quan-tum mechanical nature of the problem. This requiresa phase-space description of the �eld (x). Although itmight seem that the Wigner function provides the appro-priate method, it is in fact impossible to write a phase-space Wigner function for a multiparticle system whichmanifestly exhibits the symmetry of the Bose wavefunc-tion. A simple demonstration of this problem can bemade for a two particle system whose Wigner function isobtained from the Fourier transforms with respect to y1,y2, of

Trf (x1 + y1) (x2 + y2) y(x1 � y1) y(x2 � y2)�0g� ~W (x1;y1;x2

;y2); (23)

where �0 is the density operator of the vacuum. Thus,the Wigner function is (where N is an appropriate nor-malization factor)

W (x1;k1;x2;k2) =

NZe2i(k1�y1+k2�y2) ~W (x

1;y1;x2;y2)d

3y1d3y2: (24)

TheWigner function has the symmetry 1$ 2 ; but this isnot the full Bose symmetry, which requires full symmetryunder the independent exchanges:

x1 + y1=2$ x2 + y2=2; (25)

x1 � y1=2$ x2 � y2=2 :The Wigner function symmetry 1 $ 2 is given by thesimultaneous exchanges (25), not the independent ex-changes. The problem obviously persists for all numbersof particles. Where the Wigner function has been used inkinetic theory [52] the Bose symmetry takes the form of avery complicated integral operator relation, whose use isvery impractical. Although an exact evolution equationwould preserve Bose symmetry, the absence of manifest

Bose symmetry makes it extremely di�cult to developapproximation methods which preserve Bose symmetry.In fact it is only in a very dilute gas limit, in which essen-tially only the one-particle Wigner function occurs thatthe Wigner function has been used, and for this of courseBose symmetry is not an issue.

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2. Wavelet expansion of �eld operators

The most natural way to give a phase space descrip-tion of a Bose gas is to follow the method of Bose andEinstein|that is, to divide space into cells of volume�V , and for each cell to introduce a set of basis wave-functions which vanish outside the cell, but are quan-tized within the cell. Unfortunately, even from the pointof view of the non interacting Hamiltonian (18), suchwavefunctions have in�nite energy, arising from the sharptransition from inside to outside the cell. This yields awavefunction which is not twice di�erentiable, as requiredby (18), so that the spread in momentum p is so largethat the mean of p2 is diverges.To avoid this problem, the transition from inside the

cell to outside it must be made smoother, and this is aproblem which is like that which arises in the study ofwavelets [53]. We introduce a set of wavelet functions (inone dimension)

vK(x; r) =1p4��

Z K+�

K��

eik(x�r) dk (26)

� eiK(x�r)

p��

sin�(x� r)

x� r: (27)

If

r = n�=�; n = 0;�1;�2; ::: (28)

these functions have the property of orthogonality:Z 1

�1

dx v�K(x; r)vK0 (x; r) = �KK0�rr0 (29)

and they are also complete. From the de�nition (26),it can be seen that vK(x; r) has momenta in the range(�h(K ��); �h(K +�)), and from of (27), it can be seenthat wavefunction vK(x; r) is localized to a certain extentat the point x = r � n�=�.These wavefunctions correspond to a complete set of

phase cells; we see that if the uncertainties are de�ned asthe intervals between the discrete values of the momentaand position, then

�x = �=� and �p = 2��h) �x �p = h (30)

However, the uncertainty as de�ned by a variance in x isclearly in�nite, re ecting the fact that the wavelet func-tions (26),(27) are not well localized. The quantities �x,�p represent rather the spacing between the phase cells|not the uncertainties in x and p. In terms of the variablex, these wavelet functions represent the smoothest pos-sible functions we could choose, because by constructionthe momentum spread is bounded, so that the waveletfunctions are in fact in�nitely di�erentiable.The �eld operators can now be expanded as (in an

obvious three-dimensional generalization)

12

(x) =XK

Xr

v�K(x; r)a(r;K) (31)

and the commutation relation for a(r;K) is

[a(r;K); ay(r0;K0)] = �rr0�KK0 : (32)

The states of the Bose gas are now speci�ed by the eigen-values of the number operators N(r;K), which is a trulyquantum mechanical version of the original idea of Bose.Notice that there is no need in this description for �

to be unique|one can choose a di�erent value of � foreach K-band if one wishes, since momentum bands areorthogonal independently of their size.

3. Momentum resolved �eld operators

For many parts of our discussion, it is convenient toresolve �eld operators into only the di�erent ranges ofmomentum; thus we write

(x) =XK

e�iK�x K(x) (33)

where

K(x) = eiK�xXr

v�K(x; r)a(r;K) (34)

= eiK�xZDK(x� x0) (x0) d3x0 (35)

where

DK(x � x0) = e�iK:(x�x0)g(x� x0) (36)

where

g(x) =1

�3

�sin�x

x

��sin�y

y

��sin�z

z

�: (37)

We also have the commutation relation

[ K(x); yK0(x0)] = �KK0g(x� x0): (38)

The resolution into momentum bands thus yields a rathernonlocal description. Notice also that we can write

g(x� x0) = 1

(2�)3

Z �

��

eik�(x�x0) d3k

=Xr

e�iK�(x�x0)vK(x; r)v

�K(x

0; r); (39)

which is an expression of the completeness of the waveletfunctions within a K-band, and is useful in some of thecomputations used in deriving the Uehling-Uhlenbeckequation in sect.VB

13

4. Free Hamiltonian in terms of momentum resolved �eld

operators

Because the functions used in the expansion of e�iK�x K(x)

are orthogonal to those used for e�iK0�x K0(x), it is ob-

vious thatZd3x

yK(x) K0 (x)e�i(K

0�K)�x = 0 if K 6= K0: (40)

Using this fact, it is straightforward to show that

H0 =XK

�h2K2

2m

Zd3x

yK(x) K(x) (41a)

+XK

Zd3x �hK � jK(x) (41b)

+XK

Zd3x

yK(x)

���h2r2

2m

� K(x) (41c)

� Ha +Hb +Hc

Here we have used a \probability current" for the mo-mentum K de�ned in a way analogous to that normallyemployed

jK(x) � � i�h

2m

n yK(r)r K(r) �r yK(r) K(r)

o: (42)

The resolution into three parts has a simple interpreta-tion. The resolution into a full wavelet description showsthat we can write

Ha =XK;r

�h2K2

2mayK(r)aK(r) (43)

corresponding to an energy �h2K2=2m for each quantumat each phase space location K; r. This is the Hamil-tonian corresponding to the original ideas of Bose andEinstein.The term Hb corresponds to transport, as shown by

computing the commutator�Hb; K(x)

�= i�hvK � r K(x) (44)

where vK = �hK=m. This is obviously transport corre-sponding to the velocity appropriate to the central mo-mentum of the momentum band K.Finally, the last term Hc corresponds to wavepacket

spreading through

�Hc; K(x)

�= � �h2

2mr2 K(x) (45)

Thus the wavelet description naturally gives the separa-tion into the three parts fundamental to a phase spacedescription of the processes.

14

5. Interaction Hamiltonian in terms of the momentum

resolved �eld operators

It is trivial that the interaction part, HI , can be writ-ten

HI =1

2

XK1;K2;K3;K4

U(K1;K2;K3;K4); (46)

in which

U(K1;K2;K3;K4) =Zd3x

Zd3x0 e(iK1�x+iK2�x

0�iK3�x�iK4�x0)

� yK1(x) yK2

(x0)u(x� x0) K3(x0) K4

(x): (47)

Furthermore, it is easy to see that

U(K1;K2;K3;K4) = 0 (48)

unless

K1 +K2 =K3 +K4 +O(�) (49)

i.e., the quasimomentumK1 is conserved to within some-thing not much bigger than �.

IV. TREATMENT OF COLLISIONS

A. Eigenstates of Ha and its corresponding

Liouvillian

Suppose we use the wavelet basis; then we can label theeigenstates of Ha (as given in (41a)) as jni where n isa vector of elements ni, the number of quanta belongingto the phase space wavelet i, which has quantum number(Ki; ri). The eigenstates of the corresponding Liouvillianare given by the outer product jnihmj, and the eigenvalueis only zero if the energy of jni is the same as that of jmi.

B. Projectors

We will de�ne a projector onto all eigenstates with thesame number of quanta in each K, irrespective of r; thuswe de�ne

pNjni = jni ifXr

n(K; r) = N(K) and K 6= 0;

= 0 otherwise. (50)

This kind of projector identi�es all con�gurations withthe same distribution inK, but ignoresK = 0 and leavesundisturbed the distribution over r.Suppose now that � is the density operator for the

system of particles; we de�ne a projector on � by

15

PN� = pN � pN � vN: (51)

Thus the projected � is an operator in position space,but acts only with a space that has the con�guration Nof particles over the momentum space cells|except anycon�guration ofK = 0 particles is permitted (we call thisK diagonal). It follows also that, for any �

PNHb� = HbPN� (52a)

PNHc� = HcPN� (52b)

and that for Ha there is the stronger result�Ha;PN�

�= 0 (53)

which is true by construction.The thrust of this paper is now to develop a closed

equation of motion for the projected parts of �(t), namelyvN(t), and to show that we can use these parts to repre-sent the physics in which we are interested. Thus we willbe making the approximation

�(t) �XN

vN(t); (54)

which will require the assumption that we can neglectthe remainder: thus

w(t) � 1�

XN

PN!�(t)

� 0: (55)

A density operator of the form (54) will be called K-diagonal. For such K-diagonal density operators, the al-gebra in deriving the equations of motion is considerablysimpli�ed by results like

PMf yK(x) K0 (x0)vNg = 0

unless K = K0 and M =N. (56)

Notice however that in this equation, we do not requirethat x = x0; thus the designation K-diagonal is appro-priate. Another kind of identity is

PMf yK(x)vN K0(x0)g = 0

unless K =K0 and M = N+ b. (57)

where b is a vector in the same space as M and N, andwhose only non-zero component is b(K) = 1, so thatM and N are identical apart from the change M(K) =N(K) + 1.There are many other relations like these, but all are

essentially of the same kind, that the K-diagonal prop-erty is preserved only by matching a creation �eld oper-ator with a destruction �eld operator with the same K,though not necessarily the same x, either on the same orthe other side of the density operator. If the matching isdone on the same side, the con�guration N is preserved,if the matching is done on opposite sides the con�gura-tion changes.

16

C. Derivation of the quantum kinetic master

equation

The equation of motion for the density operator is

_� = � i

�h[Ha +Hb +Hc; �]� i

�h[HI ; �] (58)

= La�+ Lb�+ Lc�+ L2�: (59)

We de�ne the Laplace transform

~�(s) =

Z 1

0

e�st�(t)dt (60)

and correspondingly

~vN(s) = PN~�(s) (61)

~w(s) = Q~�(s) �"1�

XN

PN#~�(s); (62)

so that

s~vN(s)� vN(0)

= (Lb + Lc)~vN(s) + PNL2(X

M

~vM(s) + ~w(s)

)(63)

and

s ~w(s)� w(0)

= fLa + Lb + Lcg ~w(s) +QL2(X

M

~vM(s) + ~w(s)

): (64)

We assume w(0) = 0; that is the state that is already K-diagonal. This assumption cannot be made without somejusti�cation, which we shall postpone to Sect. IVD3.Obviously we can choose any initial condition we wish;the justi�cation required is that when we solve the equa-tions of motion it will be possible to show that w(t)rapidly becomes negligible, and negligible at all times,so that any time can be chosen as an initial time.From this we can then readily derive

s~vN(s)� vN(0) = (Lb + Lc)~vN(s) + PNL2XM

~vM(s)

+PNL2[s�La �Lb �Lc �QL2]�1QL2XM

~vM(s): (65)

This equation is exact; the physical content comes intothe assumptions concerning what parts can be consideredto be negligible. We invert the Laplace transform in (65),and consider the parts separately.

1. Streaming and Quantum Terms

These are the names we use for Lb and Lc as de�nedin (59), (41b, 41c). They keep the same form with noapproximations.

17

2. The Forward Scattering Terms

The term P(N)L2PM

~vM(s) can be simpli�ed by not-

ing that vM(s) is K-diagonal, and that P(N) projectsonto a K-diagonal density operator. Since L2 is a com-mutator; i.e.,

L2XM

vM(s) = � i

�h

XM

[HI ; vM(s)] (66)

this can only yield a K-diagonal term from those partsof HI which do not change the K-distribution; that is,the terms in HI given by HF , which we de�ne as

HF � 1

2

XK1 6=K2

U(K1;K2;K1;K2)

+1

2

XK1 6=K2

U(K1;K2;K2;K1)

+1

2

XK1

U(K1;K1;K1;K1): (67)

This thus gives only transitions between the same K dis-tribution, and this can be regarded as forward scattering.These forward scattering terms give rise to the so-calledmean �eld e�ects, that is the average e�ect on the motionof one particle of the interaction with all other particles.

3. The Collision Terms

We now introduce a Born approximation, which amountsto the neglect of Lb+Lc+QL2 in the [ ]�1 term in (65),leaving a contribution to _vN(t) given by

XM

PNL2Z t

0

d� expfLa�gQL2vM(t� �): (68)

Notice however that QL2 contains terms which depend

on 0, y0, and when there is signi�cant condensation

these terms can become very large. Thus this approx-imation is valid only in the case of weak condensation.This means that we approximate the free evolution oper-ator by discretizing the eigenvalues to those correspond-ing to the centers of the momentum bands. This leads toan approximate measure of the validity of the neglect ofthe terms QL2 in comparison to the term retained, La,which is given by requiring that the modi�cation by thepresence of any condensate to the excitation spectrumshould be negligible for energy greater than �h2�2=2m.Using the Bogoliubov theory, this leads to the conclusionthat we must have the condition on the cell size

lc =�

�� ��hp

2m�u

=

r�

8a�(69)

18

where � is the density of particles. We now want tomake a Markov approximation, which involves two steps.Firstly we neglect the � dependence in the last term, andsecondly we let the upper limit in the integral becomein�nite. For this approximation to be valid there must bea smooth distribution over the available energy states, sothat when we sum over allM, the resultant range of � inwhich the integrand is non-zero is very much smaller thenthe characteristic time over which vM(t) evolves. This isa requirement on the kind of vM(t) being considered.Now de�ne the vector in the space of M;N; called

e(1234) which can be written

e(1234)K1= e(1234)K2

= �e(1234)K3= �e(1234)K4

= 1:

(70)

and all other components are zero, and let

�h!K =�h2K2

2m(71)

We also introduce the notation

�!e = !4 + !3 � !1 � !2; (72)

Before we examine the range of validity of themMarkov conditions, we can see that using them (68) nowbecomes

� 1

4�h2PN

X1234

10203040

��U(10203040);

Z 1

0

d� exp(i�!e�)Q�U(4321);

XM

vM(t)���

:

(73)

Since the state vM(t) is K-diagonal, and we project ontoK-diagonal states, we must have

1 = 10; 2 = 20 or 1 = 20; 2 = 10

and

3 = 30; 4 = 40 or 3 = 40; 4 = 30

but since the operator U(1234) is symmetric in (12) and(34) this simply gives 4 times the result obtained fromsetting (1234) = (10203040).Notice that no \forward scattering terms" arise in (73).

These have already been explicitly separated from thescattering terms and included in the term HF de�ned in(67), and are eliminated from (73) by the Q projectionoperator.We can now carry out the time integral so that

(73)becomes

� lim�!0

Xe

�UeU

yevN + vNU

yeUe � UevN�eU

ye � UyevN+eUe

�h2(�+ i�!e)

�:

(74)

19

Here we have used U(1234)y = U(4321). We now use

1

z + i�! P

1

z� i��(z) (75)

and rearrange, so that (74) becomes

� i

�h2

Xe

P1

�!e[UeU

ye ; vN]

+�

�h2

Xe

�(�!e)f2UevN�e(t)Uye � UyeUevN � vNUyeUeg: (76)

As is usual, we �nd a level shift term, which is a com-mutator, and a purely dissipative term. Adding together(76), (66), (59), we get the full Quantum Kinetic MasterEquation (QKME)

_vN(t) = � i

�h

"XK

Zd3x

��hK

2m

�� jK(x); vN(t)

#(77a)

� i

�h

"XK

Zd3x yK(x)

���h2r2

2m

� K(x); vN(t)

#(77b)

� i

2�h

24 XK1 6=K2

�U(K1;K2;K1;K2) + U(K1;K2;K2;K1)

�+XK1

U(K1;K1;K1;K1); vN(t)

35 (77c)

� i

�h2

Xe

P1

�!(e)

�U(e)Uy(e); vN(t)

�(77d)

+�

�h2

Xe

�(�!(e))f2U(e)vN�e(t)Uy(e)� U y(e)U(e)vN(t)� vN(t)Uy(e)U(e)g: (77e)

D. Approximations

1. Weak condensation condition

Approximations only occur in the derivation of thecollision and level shift terms. The major approxima-tion is the neglect of Lb + Lc + QL2 compared to La.We choose � so that eigenvalues of La (apart from zeroeigenvalues) are substantially larger than typical mea-sures of the size of QL2 on the particular states involved.Since the projectors PN do not a�ect the K = 0 com-position of the states, the operator QL2 includes the

term (u=2)Rd3x y0

y0 0 0, which can become very large

when there is signi�cant condensation, hence this approx-imation will only be valid in the case of weak condensa-tion, the condition for which is given by (69); that is

lc =�

�� ��hp

2m�u

=

r�

8a�(78)

We will deal with the issues of strong condensation in asubsequent paper.

20

2. The Markov Approximation

The Markov approximation is really a kind of pertur-

bative result. The change from the integralR t0d� !R1

0d� requires the assumption that the terms that turn

up have a broad spectrum of frequencies which are almostcontinuous. The frequencies which do turn up are of theform !e, and the range of these is of the order kT=�h, whilea typical separation between the frequencies correspond-ing to adjacent transitions with a typical magnitude ofmomentum

p2mkT is of order of magnitude �

p2kT=m,

which is the inverse of the time required for a particle totraverse a cell at a typical thermal velocity.The condition that we can regard the frequency spec-

trum as almost continuous is that the separation betweenthe frequencies is very much less than their range, that is,the density of states is su�ciently high. Quantitatively,this leads to

lc � �T � h=p2mkT: (79)

If the density operators vN are such that the contri-butions from sums over the di�erent e are all smoothlyvarying functions of the contributing frequencies, thenthe time dependence given by summing up over all tran-sitions in (73) will drop of over a characteristic time givenby the inverse of the bandwidth, i.e, �h=kT , and this timemust be very much less than the typical time scale ofevolution of the the distribution, which is of the order ofmagnitude of the time between collisions. This leads tothe requirement on the mean free path �mfp

�mfp � �T : (80)

Finally, the QKME itself gives a level broadening,which is implicitly small compared to the distance be-tween levels. The condition for this must be that thereis little likelihood of a collision as a particle traverses asingle cell, which means of course that

�mfp � lc (81)

.All these conditions can be consistently satis�ed.Coming back now to the quantum kinetic master equa-

tion we see that the summationsPe are over the discrete

momentum ranges K1;K2;K3;K4, and the energy con-servation delta function is sharp. This sharpness is reallyan artifact which arises from the extension of the upperlimit of the time integral to in�nity. This integral canbe cut o� at any time very much larger than the charac-teristic decay time of the kernel, �h=kT . This means thatthe delta function can be taken as being broadened cor-respondingly, and it is easy to see that this broadeningcan encompass a signi�cant number of momentum bands.Nevertheless, this broadened delta function can be writ-ten as an exact delta function if the remainder of theintegrand is smooth, and the sums written as integralsusing the continuum approximation

PK

! Rd3K.

21

However, we have already assumed this smoothness inestablishing the validity of the Markov approximation sothis procedure is consistent with the Markov approxima-tion.

3. The neglect of w(0)

We will assume w(0) = 0, and then make an estimateof the size of w(t) and �nd under what conditions that itis negligible for all future times. From (64) it follows, oninverting Laplace transforms, that

w(t) =

Z t

0

exp f(La + Lb + Lc +QL2)(t� t0)g

�XM

vM(t0)dt0: (82)

We can now approximate this by keeping La, as in thederivation of the QKME, and replacing the e�ect of theall the other terms, including the interactions, by a phe-nomenological decay term, so that we write approxi-mately

w(t) �Z t

0

ef(La� )(t�t0)gXM

vM(t0)dt0 (83)

= � i

�h

Xe

XM

e(i�!e� )(t�t0)[U(e); vM(t0)]dt0: (84)

As in the QKME derivation, we make the replacementvM(t0)! vM(t), and now we can do the integration overt0 to get approximately

w(t) � � i

�h

Xe

XM

1

� i�!e[U(e); vM(t)]: (85)

For any givenM and e we get a speci�c non K-diagonalterm. The largest of these will occur when �!e = 0. Toestimate their size, we go to the de�nition (47) of U(e).Substituting the expansion on wavelet functions into thisequation, and assuming the approximation u(x) = u�(x),we �nd that the largest terms that occur are of the formZ

d3x=; vK1(x; r)eiK1 �xvK2

(x; r))eiK2 �x

�v�K3

(x; r))e�iK3�xv�K4

(x; r))e�iK4 �x

�ay(K1; r)ay(K2; r)a(K3; r)a(K4; r) (86)

in which all wavelet functions refer to the same cell. Us-ing the de�nitions of the wavelet functions, the coe�cientof the operator part in this equation can be estimated tobe of order of magnitude 1=l3c The e�ect of the creationand destruction operators is of order of magnitude 1, sowe can now estimate the size of the term in w(t) com-pared to those in vM(t) to be of order of magnitude givenby

22

w(t) � a�h

l3c

�Xe;M;r

[ay(K1; r)ay(K2; r)a(K3; r)a(K4; r); vM(t)]: (87)

The size of the initial coe�cient can be made clearest bytaking to be given by vT =�mfp, in terms of which thecoe�cient can be simpli�ed, leading to the condition forthe validity of the neglect of non-K-diagonal terms:

a�mfp�T

l3c� 1: (88)

This condition can be satis�ed simultaneously with thetwo other conditions (79) and (80). For example, forsodium at T = 2�K, the scattering length is a = 4:9nm,leading to

�mfp = 0:42m (89)

�T = 4:8� 10�7m (90)

a = 4:9� 10�9m (91)

(a�mfp�T )1=3 � 10�5m (92)

Thus if lc is chosen in the range somewhat greater than10�5m this kind of treatment will be valid.The condition (69) is best viewed as a condition on the

condensate density �; insertion of the sodium data intoit gives the requirement that �� 1018m�3, which can besatis�ed, but will not always be satis�ed in current ex-periments. In fact this density corresponds to about oneatom per cube of volume �3T using the above data, whichindicates that for an accurate treatment of condensation,the weak condensation assumption will be invalid in thiscase.It should be borne in mind that all of the above has

been in the absence of a trapping potential|the questionwill be revisited in our forthcoming paper dealing withstrong condensation.

4. Born Approximation

The derivation of the collision terms relies on the Bornapproximation, and the collision rates appearing in theQKME (6a) are proportional to the Born scattering crosssection. This implies that the potential should be weak,i.e. a small perturbation on the energy scale associatedwith the momentum coarse graining. In derivations of thequantum Boltzmann equation it is shown [50] that in abinary collision approximation, which assumes a low den-sity but not necessarily a weak potential (�a3 � 1), theBorn amplitude should be replaced by the two particle T -matrix element (proportional to the scattering amplitudea in low energy s-wave scattering). Essentially the samesituation should pertain in this case and this means thatwe replace the exact interatomic potential with one forwhich the Born approximation is valid, and whose scat-tering length is the same as that of the exact potential.

23

However, a proof of this assertion in our formulation isat present lacking.

5. Level shifts

What are normally called \collisional level shifts" ap-pear in our formalism in two guises. There are the for-ward scattering terms which generate HF , as de�ned in(67), and there are the higher order terms which fromthe derivation of the collisional terms in (77d). In theweak condensation situation which we want to treat inthis paper, we expect the density to be so low that thesecan be ignored, but in our forthcoming work it will notbe possible to do so without more careful justi�cation.

E. Stationary Solution

Let us assume the level shift term is small|negligiblein fact. Then it is clear, since none of the other terms canactually change the eigenvalue of Ha, (the total coarsegrained kinetic energy) or the total number of particles,that any function of

Ha =Xr;K

�h2K2

2mayK(r)aK(r) (93)

and

N =Xr;K

ayK(r)aK(r) (94)

will provide a stationary solution|clearly microcanoni-cal, canonical, and grand canonical versions will exist.Thus the result is the statistical mechanical result|

but coarse grained in momentum space|the size � ofthe momentum coarse graining to be such that HI isnegligible compared with it on the states of interest.

1. Correlation Functions in the Stationary Solution

Since the stationary solution can be written as a func-tion ofHa andN , the correlation functions of the waveletcreation and destruction operators can be written as

hayK(r) aK0(r0)i = �NK �K;K0 �r;r0 : (95)

Assuming the grand canonical form

�s(T; �) = expf�(Ha � �N)=kTg; (96)

which is of quantum Gaussian form, we can easily seethat

�NK(T; �) =

�exp

��h!K � �

kT

�� 1

��1: (97)

24

The �eld operator correlation function is then

h yK(x) K0 (x0)i = �KK0g(x� x0) �NK(T; �) (98)

and similarly

h K(x) yK0(x0)i = �K;K0g(x� x0)[1 +NK(T; �)]: (99)

V. SIMPLEST APPLICATIONS OF THE

QUANTUM KINETIC MASTER EQUATION

The situation in which something like the \molecu-lar chaos" assumption of Boltzmann is valid is always ofgreat interest, and it guides the intuition of most physi-cists in kinetic theory. We will consider in this sectiontwo simpli�ed versions of the QKME which are valid inthis situation, and one equation which permits the exis-tence of coherent e�ects, which cannot arise in a molec-ular chaos regime.The Uehling-Uhlenbeck equation, sometimes called the

quantum Boltzmann equation, is a simple modi�cation ofthe Boltzmann equation in which the phase space e�ectsarising from quantum statistics are included. This equa-tion will be derived by assuming the distribution is notvery di�erent from equilibrium, and by factorizing higherorder correlation functions.The situation in which we eliminate all dependence on

position is obviously an extreme simpli�cation, and cor-responds to the system being composed of only one spa-tial cell. However, in this situation the QKME reducesto a stochastic master equation of great simplicity andintuitive appeal. This equation does not appear ever tohave been treated in the literature, so we have given itthe name \quantum Boltzmann master equation".Finally, a very simpli�ed quantum mechanical master

equation for the K = 0 band can be derived by assum-ing all other bands are thermalized. This equation willbe called the \condensate Master equation", and we areable to give an appealing description of the initiation ofthe condensation process, and of the growth of the con-densate as long as the weak condensation condition isvalid.

A. The quantum Boltzmann master equation

A very simple stochastic master equation can be de-rived by assuming that l = �=� is equal to the size of thewhole system, so that r takes on only one value. In thiscase we �nd that the K-bands are all one-dimensional, sothat all the commutator terms, (77a{77d) are zero. Sincein this case there is really no di�erence between n andN, we can write

vn(t) = jnihnjwn(t) (100)

25

so that wn(t) is the occupation probability of the state n.We then derive what we will call the quantum Boltzmann

master equation

_wn = � �

�h2

X1234

�(!4 + !3 � !2 � !1)jU1234j2

�nn1n2(n3 + 1)(n4 + 1)[wn � wn+e]

+(n1 + 1)(n2 + 1)n3n4[wn � wn�e]o

(101)

For clarity of argument we will also write (100) in theform

_w(n) =X1234

nt+1234(n� e)w(n� e)� t+1234(n)w(n)

o

+X1234

nt�1234(n+ e)w(n+ e)� t�1234(n)w(n)

o(102)

where t�1234(n) = 0 unless !1 + !2 = !3 + !4, and if thisis satis�ed

t+1234(n) = 1234(n1 + 1)(n2 + 1)n3n4

t�1234(n) = 1234n1n2(n3 + 1)(n4 + 1) (103)

and

1234 =�jU1234j2

�h2: (104)

1. The Boltzmann master equation

If we drop the terms 1+ni in the quantum Boltzmannmaster equation we get an equation called the BoltzmannMaster Equation, which has been previously consideredby Van Kampen [54] and Gardiner [55] based on stochas-tic arguments. The equation we have given includes nostreaming terms, since we have only one spatial cell, andthus our treatment assumes that the energy levels are dis-cretely spaced. This means that our normalization vol-ume is so small that the broadening of the levels whicharises from collisions is signi�cantly less than the spacingbetween levels.

2. Stationary solutions

Apart from the ambiguities caused by conserved quan-tities, the solutions of equations like (101) are unique.There is a consistent detailed balance stationary solu-tion, in which

t+1234(n� e)ws(n� e) = t�1234(n)ws(n) (105)

and from this we see, using the explicit form (103) of t�,that

26

ws(n� e)ws(n)

=t�1234(n)

t+1234(n� e)= 1 (106)

so that for all n able to be connected by a collision 1+2$3 + 4,

ws(n) = constant: (107)

Thus ws(n) is a function only of conserved quantities,total kinetic energy, momentum and the total numberof particles. We can thus choose the grand canonicalsolution given by

ws(n) / exp

��E � �N � u �P

kT

�(108)

where E is the total kinetic energy, N the total numberof particles, and P the total momentum of the particles,while u is the mean velocity of the system. The station-ary distribution is in this case factorizable

ws(n) =Yi

�e�

�h!i����hKi�u

kT

�ni(109)

3. Factorized equation for the mean occupation numbers

We can straightforwardly derive the equation for hnai =Pn naw(n)

h _nai = 4X234

a234

n� hnan2(n3 + 1)(n4 + 1)i

+h(na + 1)(n2 + 1)n3n4io

(110)

If we assume that the averages inside this equation canbe factorized, which should be valid if we are not too farfrom equilibrium, and for compactness set hnii ! ni, weget (without the streaming terms) the Uehling-Uhlenbeckequation [42];

_na = 4X234

a234

n� nan2(n3 + 1)(n4 + 1)

+(na + 1)(n2 + 1)n3n4

o(111)

The quantum Boltzmann master equation is a very sim-ple equation whose simulation is entirely feasible, andresults of such simulations will be presented elsewhere.However, as shown by our derivation, it is expected thatit will not give an accurate description of situations inwhich there is signi�cant condensation. Nevertheless itcould well give valuable insights into the region just belowthe threshold of condensation, which will be investigatedin QKIII.

27

B. The Uehling-Uhlenbeck equation

We will now show that we can derive from the quan-tum kinetic master equation (77a{77e) the more conven-tional kinetic equation known as the Uehling-Uhlenbeckequation [42]. This equation is essentially the same as(111), but with streaming terms added as well, which inour derivation will arise naturally out of the streamingterms of the QKME. For this we consider the one particledistribution function to be de�ned as

fK(x) =� ��

�3Trf� yK(x) K(x)g (112)

=� ��

�3XN

TrfvN yK(x) K(x)g: (113)

Notice that the de�nitions (31,34) imply that

(x) =XK

e�iK�x K(x) (114)

so that the usual probability density function is, becausethe vN are K-diagonal,

��

�

�3

Trf� y(x) (x)g =XK

fK(x) (115)

so that fK(x) can be viewed as a phase space density perphase space volume h3.

1. Streaming terms

We consider the various terms in the quantum kineticmaster equation. The term arising from (77a) is obtainedby considering�

K(x);

Zd3y jK(y)

�(116)

which can be written� K(x);

Zd3y

yK(y)

�� i�hm

�r K(y)

�(117)

=

Zd3y g(x� y)

�� i�hm

�ry K(y) (118)

and now integrating by parts,

(118) = � i�hmrx�Z

d3y g(x� y) K(y)�

(119)

but the integral is simply the projection of K(y) on tothe K subspace; thus

(118) = � i

�hmr K(x): (120)

28

Using this kind of technique, it is straightforward toshow that the contributions to _fK(x) from (77a, 77e)become

_fK(x)ja = �hK

m� rxf(K;x) (121)

_fK(x)jb =� ��

�3r � jK(x) (122)

where

jK(x) =i�h

2m

D yK(x)r K(x) � [r yK(x)] K(x)

E: (123)

The term (121) is the usual Boltzmann streaming term.The term (122) is a purely quantum mechanical termarising from the spreading of the wavepacket.In order to get some idea of the signi�cance of this and

some terms to come, we make a local equilibrium ansatz,in which we write

h yK(x) K0 (x0)i = g(x� x0) fK�x+ x0

2

��K;K0 : (124)

This is a natural generalization of (98) to a slightly nonequilibrium situation, and since g(0) = (�=�)3, it is con-sistent with (112). Using this form we �nd quite straight-forwardly that jK(x) = 0: The extent to which this termdoes not vanish must be related to the extent to whichthe mean momentum in the band (K ��; K +�) isdi�erent from �hK, and this di�erence is of order of mag-nitude �, which will only be appreciable for very smallK, and in particular for K = 0.

2. Forward scattering terms

If we work out the mean value of the commutator, we�nd for the term arising from (77c)

� i

�h

n2

Zd3x0

XK0

h yK(x0) yK0(x0) K0(x0) K(x)ig(x � x0):

�2Zd3x0h yK(x) yK0

(x0) K0(x0) K(x0)ig(x� x0)

o:

(125)

If g(x� x0) was a perfect delta function, this term wouldvanish, thus the degree to which it does not vanish de-pends on the smoothness of the averages as functions ofx;x0.Using the local equilibrium ansatz (124), and a Gaus-

sian factorization assumption on the four point correla-tions, it is easy to show that this term vanishes, and istherefore presumably very small when the state is closeto equilibrium. Thus we can say _fK(x)jc � 0.

29

3. Collision terms

In evaluating the contribution from (77e) we will get anumber of terms, of which a typical one is proportionalto

�X

K2;K3;K4

Zd3y d3y0 g(x� y0)h yK(x) yK2

(y0) K3(y0) K4

(y0) yK4(y) yK3

(y) K2(y) yK(x)i

��(!K + !K2� !K3

� !K4)ei(K+K2�K3�K4)�(y�y

0): (126)

If we factorize as in local equilibrium, and use (124), weget terms like

XK2;K3;K4

Zd3y d3y0 g(x� y0)g(x� y)[g(y0 � y)]3fK

�x+ y

2

�fK2

�y0 + y

2

�

�[fK3

�y0 + y

2

�+ 1][fK4

�y0 + y

2

�+ 1]�(!K + !K2

� !K3� !K4

)ei(K+K2�K3�K4)�(y�y0): (127)

Assuming the functions are relatively smooth, so thatwe can set (because of the peaked nature of g(x� y0),g(x� y) and g(y0 � y)

x+ y

2� y0+y

2� x (128)

we getXK2;K3;K4

fK(x)fK2(x)[fK3

(x) + 1][fK4(x) + 1]�(!K + !K2

� !K3� !K4

)

�Zd3y

Zd3y0g(y0)g(y)[g(y0 � y)]3ei(K+K2�K3�K4)�(y�y

0): (129)

The last factor is in fact able to be evaluated exactly,and the result is a momentum conservation functionM�(K+K2 �K3 �K4), where

M� (Q) =

��

�

�3 3Yi=1

f 23�Qi;0 +

16�Qi;� + 1

6�Qi;��g: (130)

Thus \momentum" K is not exactly conserved, since thequantity K in K(x) represents only the midpoint of therange of possible momenta; it is possible for three mo-menta, each from such a range, to add up to a valueeither within the range, or in either of the ranges aboveor below. The exact proportions of each are shown bythe calculation to be 2

3 : 16 : 1

6 . However, assuming thatfK(x) are smooth functions of K on the scale of �, wecan ignore the slight spreading, and make the transferfrom a summation to an integration: This yields

XK2;K3;K4

��

�

�3

M�(K+K2 �K3 �K4)! 1

(2�)3

Zd3K2

Zd3K3

Zd3K4

Z�(K+K2 �K3 �K4)

(131)

so that this form eventually becomes

30

� u

2�h2

Zd3K2

Zd3K3

Zd3K4 �(K+K2 �K3 �K4)�(! + !2 � !3 � !4)fK(x)fK2

(x)[fK3(x) + 1][fK4

(x) + 1]

(132)

Here we have made the approximation

u(x� x0) = u�(x� x0) (133)

Taking all terms into account we eventually get a term

_f(K;x)je = 2juj2h2

Zd3K2

Zd3K3

Zd3K4�(K+K2 �K3 �K4)�(! + !2 � !3 � !4)

�ffK(x)fK2(x)[fK3

(x) + 1][fK4(x) + 1]� [fK(x) + 1][fK2

(x) + 1]fK3(x)fK4

(x)g (134)

which is the collision term of the equation known as theUehling-Uhlenbeck equation.

4. Summary

In the situation that we have an approximate localequilibrium, and that multi�eld correlation functions canbe factorized as if the density operator is Gaussian, we de-rive the Uehling-Uhlenbeck equation as an approximatekinetic equation.

@f

@t(K;x) � �hK � rx

mf(K;x) +

2juj2h2

Zd3K2

Zd3K3

Zd3K4�(K+K2 �K3 �K4)�(! + !2 � !3 � !4)

�ffK(x)fK2(x)[fK3

(x) + 1][fK4(x) + 1]� [fK(x) + 1][fK2

(x) + 1]fK3(x)fK4

(x)g(135)

In situations of interest for Bose Condensation it wouldno longer be valid to make the factorizations necessaryfor the collision term as above, nor factorization in thelocal equilibrium assumption (124).

C. Equations for the condensate density operator

Let us now assume that the scattering of non conden-sate modes is strong, and that we are only interested inthe behavior of the condensate. We can then assume thatthe total density operator can be factored into a thermalnon-condensate part, and a condensate density operator�0(t) as a �rst approximation at least. Thus

�(t) =XN

vN(t)! �B(�; T ) �0(t) (136)

This situation is not that employed in current experi-ments, where the condensate grows by taking atoms outof the bath of warmer atoms, but is given for illustrativepurposes only|it describes the situation in which thecondensate mode is put in contact with a bath composedof all the atoms in the other modes, which are held at agiven temperature and chemical potential. More realistictreatments of condensate growth are left to a later paper.

31

We trace over the non-condensate modes and the as-sumption that the non-condensate is thermalized willmean that all terms not involving condensate operatorsin the QKME will be vanish. The only non-vanishing dis-

sipative terms will involve matched y0 and 0 operators,so that we arrive at the condensate master equation:

_�0(t) = � i

�h

�Zd3x

y0(x)

���h2r2

2m

� 0(x) +

1

2U(0000); �0(t)

�� i

�h(ug(0)

XK6=0

�nK)

�Zd3x

y0(x) 0(x); �0

�

+

Zd3x

Zd3x0G(�)(x� x0; T; �)f2 0(x)�0 y0(x0)� �0

y0(x

0) 0(x) � y0(x0) 0(x)�0g

+

Zd3x

Zd3x0G(+)(x� x0; T; �)f2 y0(x)�0 0(x0)� �0 0(x

0) y0(x) � 0(x0) y0(x)�0g; (137)

in which the quantities G(�) are given by

G(+)(x� x0; T; �) = TrB

(X123

�

�h2Zy(1; 2; 3;x)Z(1; 2; 3;x0)�B(T; �)

)

G(�)(x� x0; T; �) = TrB

(X123

�

�h2Z(1; 2; 3;x)Zy(1; 2; 3;x0)�B(T; �)

):

(138)

with the de�nition

Z(K1;K2;K3;x) = ue�i(K1+K2�K3)�x K1(x) K2

(x) yK3(x) (139)

Using thermal averages we �nd

G(+)(x� x0; T; �) = �u2

�h2fX123

[g(x� x0)]3ei(K1+K2�K3)�(x�x0)�nK1

�nK2(�nK3

+ 1)�(!1 + !2 � !3)g

G(�)(x� x0; T; �) = �u2

�h2fX123

[g(x� x0)]3ei(K1+K2�K3)�(x�x0)(�nK1

+ 1)(�nK2+ 1)�nK3

�(!1 + !2 � !3)g (140)

Suppose we now set, as in (97)

�nK =�e(�h!K��)=kT � 1

��1; (K 6=0) (141)

then it follows that

�nK1�nK2

(1 + �nK3) = e�=kT (�nK1

+ 1)(�nK2+ 1)�nK3

; (142)

so that

G(+)(x� x0; T; �) = e�=kTG(�)(x� x0; T; �): (143)

1. Quantum stochastic di�erential equation form

We can write a Quantum stochastic di�erential equa-tion (QSDE) equivalent to the master equation (137) byusing the methods in Chap.5 of [40]. This equation takesthe form

32

i�hd 0(y; t) =

(� �h2

2mr2 0(y; t) �

0@ug(0)X

K6=0

�nK

1A 0(y; t) + u

Zd3x g(x� y) y0(x; t) 0(x; t) 0(x; t)

+

Zd3x

Zd3x0g(x0 � y)fG(+)(x� x0; T; �)�G(�)(x� x0; T; �)g 0(x; t)

)

+

Zd3x g(x� y) dW (x; t) (144)

in which dW (x; t) is a quantum white noise incrementwhich satis�es the conditions

dW (x; t)dW y(x0; t) = 2G(�)(x� x0; T; �) (145a)

dW y(x; t)dW (x0; t) = 2G(+)(x � x0; T; �) (145b)

dW (x; t)dW (x0; t) = 0 (145c)

dW y(x; t)dW y(x0; t) = 0 (145d)

2. Gain and loss

In order to get some idea of the predictions of the mas-ter equation (137) we can write equations for the averages

h 0(y)i � �(y); (146)

h y0(y) 0(y)i � ��(y): (147)

These become

@�(y)

@t= � i

�h

8<:� �h2

2mr2�(y) �

0@ug(0)X

K6=0

�nK

1A�(y) + u

Zd3x g(x� y)h y0(x) 0(x) 0(x)i

9=;

+

Zd3x

Zd3x0g(x0 � y)fG(+)(x� x0; T; �)�G(�)(x� x0; T; �)g�(x)

@��(y)

@t= �r � j(y) +

Zd3xfG(+)(y � x; T; �)�G(�)(y � x; T; �)g

�h y0(x) 0(y) + y0(y) 0(x)i + 2

Zd3x G(+)(x; T; �)g(x): (148)

Here j(y) is the probability current. The equation (148)for �(y) is a rather familiar non-linear Schr�odinger equa-tion form, but contains the second line as well, which isan explicit gain-loss term. We can use the relation (143)to write this in the formZd3x

Zd3x0g(x0�y)(e�=kT � 1)G(+)(x� x0; T; �)�(y) (149)

and clearly when � = 0, this vanishes.If we consider spatially homogeneous solutions, then

we can write �(x)! �, and we �nd

@�

@t= � i

�h

�ug(0)

Xk6=0

�nK

��� i

�hu

Zd3x g(x� y)h y0(x) 0(x) 0(x)i

+�(e�=kT � 1)u2

2�h2�5

Zd3K1

Zd3K2�nK1

�nK2(1 + �nK1+K2

)�(!K1+ !K2

� !K1+K2):

(150)

33

3. Stationary solutions.

If � < 0, as must be the case above the condensationtemperature, the last part clearly represents a loss term.There will be no growth of a non zero � value. If weassume a Gaussian solution, we can write

h y0(y) 0(y) 0(y)i ! 2�h y0(y) 0(y)i (151)

and it is clear that the only stationary solution will be� = 0.Similarly, we can get an approximate closed equation

from (148) by assuming

h y0(x) 0(y)i =g(x� y)g(0)

��; (152)

which assumes the equilibrium shape (99) of the correla-tion function. This leads to

@��

@t= 2

Zd3xG+(y � x; T; �)g(x� y)�(e�=kT � 1)��=g(0) + 1

: (153)

We then �nd the stationary solution

�� =g(0)

1� e�=kT=

l�3

1� e�=kT; (154)

which gives the total mean number per volume as l3 as1=(1�e�=kT ), the usual statistical mechanical result when� < 0. When � = 0 there is no longer a stationary stateof Eq..(148); �� will grow inde�nitely. However there isstill no gain, so no coherent phase appears. Of course inpractice � is not exactly zero, so inde�nite growth willnot occur.Thus we see that the ideal Bose condensate result arises

purely out of noise; that no coherent phase appears atall. This result is only valid in the weak condensationlimit; we will see in a subsequent paper that higher orderterms can lead to an e�ective gain, and that a non-zerostationary value of � can arise.

4. Gain and a coherent phase arising from a modi�ed

distribution of energies in the non-condensate

It is of interest to see whether we could produce a non-zero value of � by manipulating the nature of the energydistribution of the non-condensate. If we do assume thatthe non-condensing modes are in a non-equilibrium sta-tionary state, so that instead of �B(T; �) we have an ar-bitrary density operator �ne, the gain in (148) becomes

34

u2

2�h2�5

Zd3K1d

3K2�(!K1+ !K2

� !K1+K2)

��nK1

�nK2[1 + �nK1+K2

]� [1 + �nK1][1 + �nK2

]�nK1+K2

�:

(155)

We can get gain if (but not only if) every team in the fg is positive. Assuming that �n(K) is only a function of !K,this will happen when

�n(!1)

1 + �n(!1)

�n(!2)

1 + �n(!2)

1 + �n(!1 + !2)

�n(!1 + !2)> 1 (156)

or, de�ning

F (!) = log

��n(!)

1 + �n(!)

�(157)

F (!1) + F (!2) > F (!1 + !2): (158)

This is the condition that F (!) is a convex downwards function as in the solid line in the diagram. For example,

F (!) =��h! � �!2

kT(159)

gives

F (!1) + F (!2)� F (!1 + !2) =2�!1!2kT

> 0 (160)

and the net gain is

u2

2�h2�5

Zd3K1d

3K2�(!K1+ !K2

� !K1+K2)

� exp

�2�!K1

!K2

kT

��nK1

�nK2[1 + �nK1+K2

]: (161)

VI. CONCLUSIONS

Quantum kinetic theory is both a genuine kinetic theory and a genuine quantum theory. The kinetic aspect arisesfrom the decorrelation between di�erent momentum bands, and this is essentially our version of the molecular chaoscondition. However, because eachK-band has a range of momenta within it, from which coherences are not eliminated,there is still the possibility of long wavelength quantum coherence. In practice such coherence is not expected to beof signi�cance except in the Bose condensate, but the formulation we have developed enables a correct formulation ofthe method by which incoherence can be transferred into the condensate.We have formulated quantum kinetic theory in terms of the Quantum Kinetic Master Equation for bosonic atoms.

It is a quantum stochastic equation for the kinetics of a dilute quantum degenerate bose gas, and it is valid in theregime of Bose condensation, as well as in regimes where no condensation takes place.The QKME is a genuine N -atom equation, which is intermediate between the full description in terms of the

complete density matrix and the familiar kinetic equations for single particle distribution functions (like the quantumBoltzmann equation). It contains as limiting cases both the Uehling-Uhlenbeck (or quantum Boltzmann equation)and the Gross-Pitaevskii equation.The key assumption in deriving the QKME is a Markov approximation for the atomic collision terms. This is valid

to the extent we are interested in a situation where in addition to the Bose condensate we have a thermal fraction ofatoms which e�ectively acts like a heat bath with a (short) thermal correlation time �h=kT , which is taken as zero inthe Markov approximation.The present paper has developed the basic structure of the theory, has stated and justi�ed the approximations

which are needed, and has thus delineated the region of validity of the theory. Extensions to include a trappingpotential and to account for a large fraction of condensed atoms giving rise to Bogoliubov type excitation spectrumare essentially technical, because the basic formulation carries through with little conceptual change, although the

35

equations which are appropriate in these cases are signi�cantly changed. These aspects, which are essential for theapplication of the QKME method to realistic experiments, will be dealt with in QKII [56].Acknowledgment: We thank K. Burnett and J.I. Cirac for discussions. C.W.G. thanks the Marsden Fund for

�nancial support under contract number GDN-501. P.Z. was supported in part by the �Osterreichische Fonds zurF�orderung der wissenschaftlichen Forschung.

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