Reversible and irreversible evolution of a condensed bosonic gas

Transcription

1 PHYSICAL REVIEW A, VOLUME 63, Reversibe and irreversibe evoution of a condensed bosonic gas R. Waser, J. Cooper, and M. Hoand JILA, Nationa Institute of Standards and Technoogy, and University of Coorado, Bouder, Coorado Received 17 Apri 2000; pubished 12 December 2000 We have formuated a kinetic theory for a condensed atomic gas in a trap, i.e., a generaized Gross-Pitaevskii equation, as we as a quantum-botzmann equation for the norma and anomaous fuctuations R. Waser et a., Phys. Rev. A 59, In this paper, the theory is appied to the case of an isotropic configuration and we present numerica and anaytica resuts for the reversibe rea-time propagation, as we as irreversibe evoution towards equiibrium. DOI: /PhysRevA PACS numbers: Fi, Jp, Db, Ln I. INTRODUCTION More than 70 years ago, Bose and Einstein proposed a provocative hypothesis that at utraow temperatures a nove state of matter shoud exist. They predicted this state coud be attained by cooing an ordinary gas towards absoute zero. At a we-defined point in this process, a spontaneous transition shoud occur and change the state of matter from an unordered ensembe of individua partices into one coective entity. This singe object, now devoid of its manypartice character, ought to evove as a coective matter wave. With the discovery of superfuidity in iquid heium in 1938 and its subsequent expanation in terms of Bose- Einstein condensation BEC, the hypothesis had been firmy estabished. In turn, this phenomenon has had a major impact on the deveopment of modern quantum physics. Today, BEC is fundamenta to our understanding of many owtemperature phenomena and it is the cornerstone of many quantitative expanations. However, up to 1995, condensation of a weaky interacting, atomic Bose-Einstein gas had never been achieved, as such. With the ground-breaking accompishment of condensing atomic 87 Rb by Corne and Wieman et a. 1, of sodium by Kettere et a. 2, and ithium by Huet et a. 3, a chapter of quantum statistica physics has been opened. It is now possibe to study in a tabe-top experiment quantum statistica effects of materia objects on a human scae up to 5 mm the very phenomena that govern the otherwise microscopic physics of nucear matter, macroscopic quantum iquids, or astronomica objects, such as neutron stars. Today, more bosonic akai-meta eements have crossed the transition temperature, in particuar atomic hydrogen 4 as we as 85 Rb 5, and many more vasty improved experiments have been carried out. For exampe, it is now possibe to examine muticomponent condensates 6,7, to create vortices 8,9, and to prepare topoogica modes 10. For a ist of current experiments see Ref. 11, or the review artice in Ref. 12. However, the technoogica breakthrough of combining aser cooing with evaporative cooing is not imited to bosonic species ony. Most recenty, the fermionic isotope of potassium 40 K has aso been cooed successfuy beow the Fermi temperature 13. Instigated by these spectacuar experiments, strongy renewed interest has deveoped in their quantitative description. Whie cod quantum gases had been studied extensivey in the s, they were mainy considered as precursor theories for strongy interacting systems, such as iquid heium. Thus, most of the avaiabe resuts were focused on spatiay uniform systems in therma equiibrium. Exceent accounts of these standard resuts can be found, for exampe, in the textbooks and monographs However, the spatia nonuniformity, the therma isoation resuting from the confinement in a utrahigh vacuum trap, as we as the arge disparity of coision and reaxation time scaes, are indispensabe ingredients for a quantitative description of today s experiments. To account for these differences that distinguish the present experimenta situation from the homogeneous Bose- Einstein gas 20 23, a growing number of equiibrium and nonequiibrium kinetic theories have been recenty presented However, the effort to go beyond the mean-fied description of the Gross-Pitaevskii equation 30 is considerabe. Thus, the research for a unified description of the equiibrium and nonequiibrium situation is sti very active. In this paper, we expore numericay and anayticay some of the impications of the reversibe and irreversibe evoution of a condensed gas immersed in the noncondensate coud. The points discussed are organized as foows. Section II revisits the main resuts of our kinetic theory 31, i.e., the two-partice Hamitonian and the energy and number conserving coisiona kinetic equations for the condensate, as we as the norma and anomaous fuctuations. In Sec. III, we speciaize these kinetic equations for a competey isotropic situation. Based on these prerequisites, we discuss in Sec. IV the resuts of propagating the coisioness mean-fied and the Hartree-Fock-Bogoiubov HFB equations in rea time. Finay, in Sec. V, we study the evoution of an ergodic distribution towards equiibrium in the presence of coisions. II. KINETIC MASTER EQUATIONS A. Master variabes The kinetic master equation of the weaky interacting diute atomic gas describes the couped evoution of the condensed fraction immersed in the quantum fuctuations. In this context, we associate the condensate with a c number fied x (t) that represents the expectation vaue of the quantum /2000/631/ /$ The American Physica Society

2 R. WALSER, J. COOPER, AND M. HOLLAND PHYSICAL REVIEW A fied â x (t). The fied operator â x removes a partice from point x and satisfies the scaar, equa-time commutation reation, â x,â y xy, of a boson. The position representation x used above is not necessariy the most suitabe basis to formuate a kinetic theory. It proves to be more usefu to postpone the choice of a particuar representation and to formuate the theory in terms of a genera singe-partice basis i 1 that spans the same singe-partice Hibert space: 1 â x i1 â i1 xi 1. 2 In the case of an unstructured scaar atomic condensate, three externa quantum abes (i 1 ) are sufficient to describe its motiona state in space, competey. 1 In this manner, we can expand any fied as â i1 i1 i Here we have simpified the notation by dropping the name of the dummy variabe, i.e., i 1 1, and by assuming impicit summation over repeated indices, as usua. In an anaogous fashion, we can describe the norma density of the atomic gas f â â f (c) f by a Hermitian tensor operator of rank 1,1: f f 12 12, f c 2 * Moreover, we wi aways decompose any quantum average into a mean-fied contribution and the remaining fuctuations. Simiary, we define the anomaous averages mââ m (c) m as symmetric tensors of rank 2,0, 4 m m 12 12, m c , 5 and their symmetric conjugates as nm 12 * 12. B. Dynamica evoution The kinetic evoution of a weaky interacting gas is primariy governed by the motion of the individua partices in the externa trapping potentia and by binary coisions. Simutaneous coisions of more than two partices are unikey events in a diute gas. Consequenty, we wi disregard such processes and use the foowing number-conserving Hamitonian operator: ĤĤ 0 Ĥ 1 H 012 â 1 â â 1 â 2 â 3 â 4. 1 This is readiy generaized to accommodate mutipe interna eectronic configurations if i 1 encompasses more quantum abes accordingy, i.e., i 1 n 1, 1,m 1 ;F 1,M 1,... 6 Here, Ĥ (0) denotes a singe-partice Hamitonian operator with matrix eements H (0)12 1p 2 /(2m)V ext (x)2. For the externa trapping potentia, we assume a threedimensiona isotropic harmonic osciator, V ext (x)m 2 (x 2 y 2 z 2 )/2. In most of the present experiments with arge, stabe condensates, the two-body interaction potentias V bin (x 1 x 2 ) are repusive and of short range. From such potentias, we can obtain two-partice matrix eements as S1 2V binx 1 x 2 3 4, Ony the symmetric part of the two-partice matrix eement 1234 is physicay reevant. Therefore, we have expicity S symmetrized it. In the ow kinetic energy range that we are interested in, s-wave scattering is the dominant twopartice scattering event Thus, by discarding a detais of the two-partice potentia, we can describe the interaction strength with a singe parameter V 0 reated to the scattering ength a s by V a s /m. This imit corresponds to a singuar interaction potentia, i.e., V bin (x 1,x 2 ) V 0 (x 1 x 2 ). In the case of this deta potentia, one finds for the two-body matrix eements: V 0 2 d 3 x1x2xx3x4, 9 which need not be symmetrized, as they are symmetric aready. However, considering the caveats that are reated to the singuar functiona form of the two-partice potentia, 35, we wi ony rey on the existence and symmetry of the two-partice matrix eements as defined in Eq. 7. C. Mean-fied equations Based on these assumptions, we have derived a set of kinetic equations that describe the dynamica evoution of the condensate fraction immersed in a coud of noncondensate partices. By discarding a of the interactions except for the condensate s sef-interaction, they reduce to the famiiar Gross-Pitaevskii GP equation for the mean fied. However, due to the presence of anomaous fuctuations m, this noninear, but otherwise unitary GP equation acquires a contribution proportiona to the time-reversed or compex conjugated fied *. To represent these equations compacty, it is usefu to arrange them in a 22 matrix form. Moreover, we transform this fied equation to a frame corotating with a positive frequency defined by (t)exp(it) (t). However, in order not to overoad the notation, we wi suppress the overine in the foowing generaized GP equation: d dt i. 10 The two-component state vector (,*) T, introduced above keeps track of the forward and time-reversed compo

3 REVERSIBLE AND IRREVERSIBLE EVOLUTION OF A... PHYSICAL REVIEW A nents of the mean fied. It is symmetric under time reversa, i.e., 1 *. The Paui matrix 1 achieves the exchange of upper and ower components and is defined in Appendix A. Two distinct processes govern the rea-time evoution of the mean fied. First, there is the generaized GP propagator that is defined as N A. 11 A * N * The two contributions that define this sympectic propagator are a norma Hermitian Hamitonian operator N H 0 1U f c2u f, as we as a symmetric anomaous couping strength A V m It is easy to identify N with the we-known unitary GP propagator that accounts for the free evoution of the meanfied (H (0) ), its sef-interaction U f (c), as we as the energy shift U f caused by the presence of the noncondensate coud. However, due to the existence of the anomaous fuctuations there is aso a couping through A to the timereversed fied. For convenience, we have introduced two auxiiary operators U f and V m. Expicity, they are defined in terms of the two-body matrix eements, such as U f f , and a first-order anomaous couping strength V m m Second, there are a of the coisiona second-order damping rates and energy shifts that are given by N A A *, 16 N * and the time-reversed contribution 1 * 1. It can be shown that they are equivaent to the extended Beiaev rates 36. The forward and backward transition rates N, A, A, and N, describe the bosonicay enhanced scattering of noncondensate partices into and out of the condensate. In turn, these transition rates are formed from various binary scattering processes, and are given by and N f f 1 f 2 f m ñ, N 1 f 1 f f 2 1 f m ñ, A m m ñ 2 f m 1 f, A m m ñ 2 1 f m f Within the Born-Markov approximation of kinetic theory, we define these eementa coision processes as fff f 3 1 f 42 f 42 13, fmf f 3 1 m 43 f 42 12, 21 fmn f 3 1 m 43 n 22 14, mmn m3 4 m 43 n During a binary coision event, two partices can conserve their energy ony approximatey. After a, the individua scattering event happens within a medium and the asymptotics cannot be reached within the finite duration of the coision. Thus, within the imits of the Born-Markov approximation, any second-order coision operator accrues a dispersive as we as a dissipative part from the compex vaued matrix eement: i It is essentiay nonzero ony if the energy difference (t) 2 (t) 3 (t) 4 (t) between the preand post-coision energies is smaer than an energy uncertainty : 1 im 0 i 1 ip. 23 On genera physica grounds, it can be argued that this uncertainty is bracketed by the binary coision rate, on one side, and the energy uncertainty arising from the finite duration of an individua coision event on the other side. As we have shown, one has aso the iberty to choose a more accurate intermediate propagator such that the singe-partice energies (t) and the eigenstates incorporate mean-fied shifts. D. Norma and anomaous fuctuations The norma and anomaous fuctuations f (t) and m (t) of a quantum fied are not independent quantities, but actuay they are the components a generaized singe-time density operator G (t): f m G ñ 1 f * The non-negativity of this covariance operator impies that the magnitude of the anomaous fuctuations is imited by the norma depetion through a Cauchy-Schwartz inequaity see Appendix B. In the genera context of Green function s,15,16 this singe-time density operator G (t) can aso be viewed as a particuar imit of a time-ordered T, two-time Green function G(,t), i.e., G (t) im t TG(,t). Consequenty, it is aso necessary to

4 R. WALSER, J. COOPER, AND M. HOLLAND PHYSICAL REVIEW A consider the opposite imit and to define a time-reversed, singe-time density operator through G (t) im t TG(,t). Expicity, this operator is given by 1 f m G 1 G * 1 G ñ f * With the hep of these definitions, we can now present the resuts of the kinetic theory as a generaized Botzmann equation for the singe-time density operator G (t) as d dt G ig G G H.c. 26 In anaogy to the previous discussion of the mean-fied dynamics, we again find that the evoution of the density operator is rued by two types of propagators. First, there is the Hartree-Fock-Bogoiubov HFB sef-energy operator that can be obtained aso by variationa methods 16. In detai, this sympectic sef-energy is given by N A, 27 A * N * where we have introduced Hermitian Hamitonian operators and symmetric anomaous couping potentias as N H 0 2U f c2u f, A V m c m It is important to note the different weighing factors of the mean-fied potentia in Eqs. 12 and 28, as we as the appearance of the anomaous condensate density m (c) in Eq. 29. This HFB operator is the usua starting point of any finite-temperature cacuations. Depending on additiona considerations, i.e., gapess vs conserving approximations see Refs. 22 and 37 40, the anomaous coupings V m are usuay discarded from Eqs. 13 and 29. However, since we do go beyond a first-order cacuation, we need to retain a contributions for consistency. Second, the Botzmann equation, Eq. 26, introduces forward and backward coision operators and. They are responsibe for partice transfer out of and into the condensate on one hand, and ead to therma equiibration within the noncondensate coud, on the other hand. These forward and backward coision operator are defined by N A A *, 30 N * and 1 * 1, where N f f c f 1 f f f c 1 f f f f c 2 f f c m ñ f mc ñ f m n c, 31 and N 1 f f c 1 f f 1 f f c f 1 f 1 f f c 2 1 f f c m ñ 1 f m c ñ 1 f m n c, 32 A m m c m ñ m m c ñ m m n c 2 f f c m 1 f f mc 1 f f m f c, 33 A m m c m ñ m m c ñ m m n c 2 1 f f c m f 1 f m c f 1 f m f c. 34 It is interesting to note that a of the coision processes that contribute to the Botzmann equation, Eq. 26, are of the same basic structure as the coision operators in the GP equation, Eq. 10. In particuar, one can generate a of the processes and by functiona differentiation from and. This very fact is actuay the key principe to the functiona-anaytic Green functions method described in Ref. 15 and, for exampe, eads to the gapess Beiaev approximation 22,41. E. Conservation aws 1. Number The tota partice number Nˆ is a conserved quantity if the atoms evove under the generic two-partice Hamitonian operator Ĥ given by Eq. 6, i.e., Ĥ,Nˆ 0. This conservation aw impies that the system is invariant under a goba phase change â â exp(i). By using this continuous symmetry, i.e., exp(i), f f, and m m exp(2i), it is easy to see that kinetic Eqs. 10 and 26 are aso expicity number conserving at a times: Nˆ ttr f c ttr f tconst. 35 Nevertheess, it is important to note that there are aways coherent and incoherent processes present that do transfer partices between the condensate and the noncondensate couds, continuousy. 2. Energy In the absence of any time-dependent externa driving fieds, such as optica asers or magnetic rf fieds, the overa energy Ĥ must be conserved as we. To find the expectation vaue of the tota system energy EĤTrĤ(t), one can use the same power-series expansion of the coarsegrained many-partice density matrix (t) that eads to the kinetic equations. Thus, within the imits of the Born- Markov approximation and the systematic appication of Wick s theorem, we have obtained first- and second-order (0) (1) contributions for the energy ETrĤ( (t) (t) ) O 3. Expicity, this energy functiona EE (c) E f Em, is given as

5 REVERSIBLE AND IRREVERSIBLE EVOLUTION OF A... PHYSICAL REVIEW A E c Tr H [1U f c2u f i( N N )] f c TrT m 1 2 Tm mm, [V m i( A A )n c, E f Tr H U f c2u f i N N f, Em Tr 1 2 V m c m i A ñ For exampe, the same first-order resuts can be found in Ref. 16, derived by a variationa procedure. III. A COMPLETELY ISOTROPIC SYSTEM In the previous section, we have reviewed the main resuts of the kinetic theory that describes the couped evoution of the condensate immersed in the noncondensate. The forma derivation did not rey on a particuar trapping geometry, nor a specia form for the binary interaction potentia. In order to gain deeper understanding of the intricate interactions, we wi now speciaize the theory to the most simpe, though reaistic, three-dimensiona mode: a competey isotropic configuration, a sphericay symmetric harmonic trapping potentia, and a binary short-range s-wave scattering potentia. A. Irreducibe tensor fieds Compete isotropy is easiy achieved for the mean fied by decomposing it in terms of a few zero anguar momentum partia waves. For this purpose, we wi use a set of basis states 1 that can be characterized by radia and anguar momentum quantum numbers n,,m, i.e., x1 n R 1 1 (r)y 1 m1 (,). However, in order to isoate the isotropic components of the noncondensate fuctuations, f and m, we need to generaize the concept of partia waves and introduce an irreducibe set of tensor fieds. Furthermore, by ony seecting the scaar component (0), we can enforce the desired radia symmetry. Thus, according to Refs. 42 and 43, we introduce irreducibe representations of tensor fieds of rank: 2,0, 1,1, 0,2 as T m 1 2 m1 1 2 m 2C 1 2 m1 n m,m 2 m 1 1 m 1 n 2 2 m 2, 2 39 S m 1 2 m1 1 2C 1 2 m1 n m,m 2 m 1 1 m 1 n 2 2 m 2. 2 The quantum abes carry additiona overines or underines to indicate whether a function depends ony on two of the three quantum abes, for exampe, 1 (n 1, 1 ) or 2 ( 2,m 2 ). With these definitions, it is easy to verify the foowing orthogonaity reationships: TrS m 1 2 Sm mm. 41 In the case of a scaar fied (0), one can simpify Eq. 39 by the foowing reation for the Cebsch-Gordan coefficients C m1 m m1,m 2 (1) 1 m 1/ Provided the basis states transform under a coordinate rotation, here denoted by R, according to the finite dimensiona representation of the rotation group D (R) 43, U R nm m nmd m mr; 42 it foows that the set of tensors T m m and S m m are irreducibe as we: U R U R 1 T m 1 2 m U R U R S m 1 2 m T m S m 1 2 Dm mr, 1 2 Dm mr. B. Isotropic two-partice matrix eement The most commony used mode for a short-range binary interaction potentia is the s-wave hard-core deta potentia V bin (x 1,x 2 )V 0 (x 1 x 2 ). This mode potentia is most suited to describe the ow energetic coision dynamics of two rea partices. However, it has to be used with caution in connection with infinite summation over virtua, high-energy states. It is cear that the energy independent scattering approximation fais above a certain energy range when the spatia scae of variation of the high-energetic wave functions begin to sampe the detaied form of the interaction potentia. Thus, the true vaue of the interaction matrix eement ought to decrease much faster with energy than the vaue obtained from the simpe hard-core deta-potentia approximation. It is we known that indiscriminate use of the energy independent approximation eads to a nonphysica utravioet divergence 28,35,44. Considering these imitations, we wi use the s-wave scattering matrix eement that is obtained from the energyindependent approximation, i.e., 1234 V 0 d 3 x1x2xx3x4, 2 45 ony for energies beow a certain eve and truncate it appropriatey otherwise. n By using the basis states x1r 1 1 (r)y 1 m1 (,), we can decompose the matrix eement 1234 into a reduced radia part and a purey geometric factor Y , where Y ,

6 R. WALSER, J. COOPER, AND M. HOLLAND PHYSICAL REVIEW A V 0 r n dr R 1 n 1 rr 2 n 2 rr 3 n 3 rr 4 4 r 47 The first-order norma mean-fied potentia U f anomaous couping strength V m are then and the and U f g f 3 2 T , 53 Y d 2 Y 1 * m1y 2 * m2y 3 m3 Y 4 m4 V m g m 3 4 S C 3 4 m3 m 4 m 1 m 2 C C C m1 m 2 m 1 m 2 4 1/2 421 i1 2 i 1. Finay, the norma and anomaous coisiona contributions simpify to fff g f 3 1 f 4 2 f 4 2 T , 55 In the context of evauating the coision integras, it wi be necessary to consider products of two matrix eements summed over a energeticay accessibe subeves. Within the isotropic mode, a magnetic subeves are energeticay degenerate. Moreover, spherica symmetry demands an equa popuation distribution and rues out the existence of coherences within magnetic submanifods. Thus, it wi be required to know the magnitude of Y averaged over a the magnetic quantum numbers. For ater reference, we wi now introduce such convenienty scaed factors as g /2 /2, g g 1 2 g Y m 1 m 2 m 3 m g1 2 g 0 49 C C These couping strengths g in Eq. 50 measure the amount and principe connectivity between precoision and post-coision anguar momenta submanifods ( 3, 4 ) ( 1, 2 ). In particuar, it estabishes a parity seection rue such that the coefficients are nonvanishing ony if the sum of the anguar momenta is even. In addition, transitions are aowed ony if the anguar momentum is within a range of max( 1 2, 3 4 )min( 1 2, 3 4 ). C. Scaar component of states and energies With the hep of the auxiiary resuts estabished in the previous section, we are now abe perform the desired mutipoe decomposition of the kinetic equations. The postuate of compete isotropy then impies that we can focus on the scaar component of the fied (0) excusivey. This is,0 m,0 1 1, m m 1 2 S , f f 1 2 T , ññ1 2 S Anaogousy, we can decompose a norma operators, such as the bare singe-partice Hamitonian operator, as H 0 0 H 0 T fmn g f 3 1 m 4 3 n 2 2 T , fmf g f 3 1 m 4 3 f 4 2 S , mmn g m 3 4 m 4 3 n 2 2 S , IV. REVERSIBLE EVOLUTION In this section, we wi examine severa imiting situations of the reversibe evoution in order to eucidate the compex behavior of the condensed gas. Since canonica transforms and Hartree-Fock-Bogoiubov HFB operators are crucia for an understanding of the reversibe evoution, we wi review the main resuts 16. Subsequenty, we are going to examine the stationary equiibrium, as we as the reversibe rea-time evoution of the condensed gas. A. Structure of the generaized density matrix The definition of a generaized density matrix G, i.e., either G or G, was given in Eqs. 24 and 25. Its specific structure impies various important physica properties. First of a, we have to assume that there is a basis that diagonaizes this (2n2n)-dimensiona fuctuation matrix. Exacty n of its 2n eigenvaues correspond to the positive occupation numbers of finding a partice or, more generay, a quasipartice in a certain mode. For a given, but otherwise arbitrary, G matrix, one can construct this basis by studying the transformation aw of the density matrix under a canonica transformation T see Appendix A, GTGT. 59 It is important to note that this is not the transformation aw of a genera matrix under coordinate change. This woud require that T T 1. However, by ony using the properties of the sympectic transformations, one can show that a canonica eigenvaue probem is defined by

7 REVERSIBLE AND IRREVERSIBLE EVOLUTION OF A... PHYSICAL REVIEW A GT T 3 G. 60 The soution of this eigenvaue probem yieds the eigenvector matrix T and the corresponding diagona eigenvaue matrix 3 G. Here, we have introduced standard Paui spin matrices 1 and 3, which exchange upper and ower component of a 2n dimensiona vector, or fip the sign of the ower segment, respectivey. A normaizabe states can be rescaed such that T 3 T 3. Now, we are abe to reconstruct the positive G matrix GVPV, 61 from its eigenvectors V 3 T and the diagona, positive occupation number matrix P 3 G 3. Second, an important feature of an admissibe fuctuation matrix is its consistency with the commutation reation, i.e., â 1 â 2 â 2 â 1 12 and â 1 â 2 â 2 â 1. This can be expressed compacty as 1 G* 1 G By invoking the properties of a unitary sympectic transformation, one can show that the eements of the diagona occupation number matrix P are not 2n independent variabes. Actuay haf of them are determined by the other haf, P (n1,...,2n) 1P (1,...,n),or 1 P 1 P In other words, by separating the occupation numbers P and the eigenvector matrix V into a first and second haf, i.e., (P,1P )P and (V,V )V, one can then decompose a genera fuctuation matrix as GV P V V 1P V. B. Structure of the Hartree-Fock-Bogoiubov operator 64 The sympectic HFB operator arises not ony naturay in kinetic theories or variationa cacuations, but in many other contexts invoving stabiity anaysis. In the case of bosonic fieds, the sef-energy operator is of the generic form: N A. 65 A * N * In here, N stands for a Hermitian operator N N and A denotes an anomaous couping term that has to be symmetric A T A. The reative size of the operators N and A determines the character of the energy spectrum. It can either be rea vaued with pairs of positive and negative eigenenergies, or one finds a douby degenerate zero eigenvaue, if the energy difference between the smaest positive and highest negative vanishes gapess spectrum. In the genera case, there is a mixed spectrum consisting of pairs of rea sign reversed as we as pairs of compex conjugated eigenvaues. The eigenvectors W are normaizabe with respect to the indefinite norm W 2 W 3 W, except for those that beong to zero or compex eigenvaues. It is important to note that this energy basis W is in genera distinct from the instantaneous basis V that diagonaizes the fuctuation matrix G in Eq. 61. They do coincide ony in equiibrium. The mathematica properties of the eigenstates W can be derived easiy from the intrinsic symmetries of the HFB operator: 1 * 1, Thus, if W and E are the soutions of the right eigenvaue probem, WWE, 68 it foows directy from Eq. 66 that W 1 W*, is aso a right eigenvector but corresponds to the eigenvaue Ē E*. Starting from the second symmetry in Eq. 67 and the right eigenvaue probem of Eq. 68, it is easy to construct the eft eigenvectors W W 3 that correspond to the eigenvaues ẼE*: W E*W. 69 Finay, from a combination of the resuts for the right and eft eigenvectors, it foows that the eigenvectors are orthogona with respect to the metric 3 : 0E*EW E 3 W E, 70 if E*E. On the other hand, this reation impies aso that eigenvectors that beong to compex eigenvaues must have zero norm. The situation of a douby degenerate zero-energy eigenvaue E0 needs specia attention. One can view this case as a imit when two nondegenerate states approach each other. However, as the energy gap decreases, the two eigenstates become more and more coinear. Thus, in the imit of a vanishing energy separation, the dimension of the spanned vector space coapses from 2 to 1 and becomes defective. In the present context however, we did not encounter this situation see Ref. 16 for detais. C. Stationary soution of the Hartree-Fock-Bogoiubov equations In spite of the compex noninear interactions taking pace within the atomic gas, the kinetic evoution is competey reversibe if we disregard a coisionay induced redistributions of quasipartices. Thus, for the moment, we wi eiminate the coision operators and from the kinetic Eqs. 10 and 26 and we wi study the coisioness stationary equiibrium, as we as the rea-time evoution in this section. With these assumptions, we are eft with the foowing set of stationary equations for the mean fied and the fuctuations G : 0, 0G G

8 R. WALSER, J. COOPER, AND M. HOLLAND PHYSICAL REVIEW A FIG. 1. Sef-consistent potentia energy densities of the condensate Hamiton operator N soid, the noncondensate Hamiton operator N dashed dot, and the bare harmonic-osciator potentia dashes versus radius. Energy and ength are scaed in the natura units for a harmonic osciator. The sef-energies of the condensate and the noncondensate are nonineary couped and impicity incude the rotation frequency of the mean-fied. However, the equiibrium soution to Eqs. 71 and 72 is not fuy determined as it stands. From the resuts of the previous section, we know that any fuctuation matrix G that is diagona with respect to the positive and negative energy eigenvectors W(W,W )of, wi be a stationary and compete soution of Eq. 72 G W P W W 1P W. 73 By choosing a canonica Bose-Einstein distribution P(EE 0 )1/exp(E)1 for the quasipartices above the nondegenerate ground state E 0 0 and a vanishing groundstate occupation number P(E 0 )0, we obtain a variationay minima energy soution for the tota system at some inverse temperature 16. G EE0 PEW E W E 1PEW E W E. 74 In order to understand the sef-consistent equiibrium soution of Eqs. 71 and 72, it is usefu to examine first the potentias that govern the evoution of the condensate, as we as the noncondensate. In Fig. 1, we depict the potentia energy densities of the norma Hamitonian operators N and N versus radius that arise for the zero anguar momentum manifod 0, i.e., V ext 1U f (c) and V ext 2U f (c), respectivey. They are compared to the bare isotropic harmonic-osciator potentia V ext r 2 /2, for reference. Here and in a of the subsequent resuts, we wi use the experimenta data of a typica 87 Rb condensate 45. A physica parameters are scaed in the natura units for a harmonic osciator, i.e., the anguar frequency 2200 Hz, the atomic mass m amu, the ground-state size a H FIG. 2. Position density of the mean-fied f (c) (r) soid, the Thomas-Fermi approximation f (c) TF (r) dashed-dot, as we as the norma fuctuations f (r) soid and the anomaous fuctuations m (r) dashed-dot. Density and ength are scaed in the natura units for a harmonic osciator. /(m 87 ) 1/2 763 nm, the s-wave scattering ength a S 5.82 nm a H, a very ow temperature of k B T 0.2, and a condensate number chosen as N (c) The isotropic partice densities are normaized to N 0 dr r 2 f (r). From the effective couping parameter a S /a H, one obtains an estimate of the mean-fied energy shift as TF (15N (c) ) 2/5 / This gives an exceent approximation of the sef-consistent chemica potentia of 8.52, as can be seen in Fig. 1. The position densities of the condensate f (c) (r), as we as the oca densities of the norma and anomaous fuctuations, i.e., f (r) and m (r), are represented in Fig. 2. First of a, it has to be noted that the soutions are rea vaued. This important fact foows from the detaied structure of the stationary Eqs. 71 and 72, which are invariant under a goba phase change. Second, if we focus on the condensate density, one can see that it cosey foows the Thomas-Fermi approximation f (c) TF (r)(r 2 TF r 2 )/(2), for radii ess than r TF This imit is vaid in the strong-couping regime N (c) 1, where the kinetic energy is a negigibe contribution compared to the externa trapping potentia and the sef-energies. The sef-consistent soutions for the norma and anomaous densities are depicted in the ower haf of Fig. 2. Whie a norma densities are necessariy positive, the anomaous fuctuations carry a negative sign. The anomaous fuctuations are the response of the noncondensate medium to a phase-coherent mean fied. In anaogy to the poarization of an atom that is subjected to an eectric fied, it tries to compensate for the externa perturbation. In the evauation of the norma and anomaous fuctuations, we have truncated the finite-temperature sums beyond the radia and anguar momentum quantum numbers n r 14 and 6. This eads to fuy converged vaues of the norma fuctuations. However, it has to be noted that the vaues of the anomaous fuctuations are sti subject to change further decrease. As ong as one keeps adding vacuum contribu

9 REVERSIBLE AND IRREVERSIBLE EVOLUTION OF A... PHYSICAL REVIEW A FIG. 3. Radia eigenfunctions R n r (r) of the particeike basis states 1 associated with the noncondensate Hamitonian operator H (0) 2U f (c) versus radius. Depicted are a few representative states for the radia and anguar quantum numbers n r 1, 2, 10 and 0,2. tions at an unatered strength of the s-wave matrix-eement n r r r r 1 1 n 22 n 33 n 44 see Eq. 46, this woud ead to the we known utravioet divergence 28,35,44. Thus, a judicious truncation, or aternativey an energy renormaization is needed to remove the nonphysica divergence that arises soey from the energy-independent approximation of the scattering ampitudes. In Fig. 3, we show a few seected radia eigenfunctions n x1 R r (r)y m (,) of the noncondensate Hamitonian operator (H (0) 2U f (c)) The owest energy state n R r1 0 (r) is ocaized at the rim of the condensate r TF It has a smaer spatia extent than the condensate and consequenty a higher energy. A s-wave functions (0) have a finite vaue at the origin in contrast to the 0 states that must be vanishing at r0. An eigenfunction that is characterized by quantum numbers (n r,) has n r 1 nodes. Eigenfunctions corresponding to higher anguar momenta are shifted outwards due to the increased anguar momentum barrier (1)/r 2. The corresponding eigenenergies are depicted in Fig. 5. In the context of spatiay homogeneous condensed matter systems, these eigenfunctions are associated with particeike excitations. The other reevant set of eigenstates arises from the condensate Hamitonian operator, i.e., the stationary GP equation (H (0) 1U f (c))1 (c) 1 1 (c). The owest sefconsistent energy eigenstate defines the condensate wave function. A seection of these eigenstates are shown in Fig. 4. As these states correspond to the ow energetic excitation modes of the condensate they are referred to as phononike. The eigenenergies of the isotropic (0) modes are shown in Fig. 5. We have compied the four important positive energy spectra that arise in the probem in Fig. 5. In essence, these are the spectra of the condensate and the noncondensate Hamitonians, N and N, as we as their generaization in FIG. 4. Radia eigenfunctions R n r (r) of the phononike basis states 1 (c) of the mean-fied Hamitonian operator H (0) 1U f (c) versus radius. Depicted are a few representative states for the radia and anguar quantum numbers n r 1, 2, 10, and 0,2. terms of the HFB sef-energy operators and. It can be seen that the s-wave energies of N and are virtuay identica. In contrast to this, one finds that the excitation frequencies of the fuctuations are characteristicay shifted downwards from the energies of the noncondensate N.Itis aso important to note that the sef-energy of the fuctuations N incudes the energy shifts of the noncondensate itsef. These numerica resuts compare we within the imits of vaidity with the perturbative and semicassica approximations of Refs The spectrum of eigenvaues of exhibits a characteristic energy gap above the condensate energy eve. It is we known that this gap energy vanishes asymptoticay for a FIG. 5. Energy spectra E n r versus radia and anguar quantum numbers 1n r 8 and 06 for the phononike states of N 0, ony, the particeike states of N, as we as the positive part of the energy spectrum of 0, ony and HFB sef- energy. Dimensioness energies are measured in natura harmonicosciator units

10 R. WALSER, J. COOPER, AND M. HOLLAND PHYSICAL REVIEW A homogeneous system in the thermodynamic imit. By deiberatey excuding the anomaous couping V m strength from a first-order theory Popov approximation or by incuding the second-order Beiaev correction of Eq. 26 in the sefenergies, one can obtain a gapess approximation 22. D. Time-dependent soution of the Hartree-Fock-Bogoiubov equations After studying some aspects of the stationary soutions of the generaized HFB equations Eqs. 71 and 72, such as the oca densities, the eigenstates, or the energy spectra, we wi now investigate the reversibe rea time evoution of the couped condensate and noncondensate system, i.e., d dt i, d dt G ig ig In contrast to the compete kinetic Eqs. 10 and 26, which account for coisionay induced energy shifts and irreversibe popuation transfer, Eqs. 75 and 76 contain ony the first-order reversibe processes. As we wi show in the foowing, this restriction impies constant occupation numbers P. In Sec. IV A, we have shown that any admissibe fuctuation matrix G has to be of the form G V PV V 1PV, 77 where P represents the positive occupation numbers of the eigenstates V. This property is not ony to be satisfied in equiibrium where the eigenstates coincide with the HFB states see Eq. 74, but in a instances. By formay integrating the reversibe kinetic equations, we can show that this structure of the fuctuation matrix is preserved at a times. Thus, Eqs. 75 and 76 define a consistent initia vaue probem. This simpe but important fact can be demonstrated easiy by defining the forma soution in terms of a time-ordered exponentia: Tt,t 0 T exp i t0 t dtt. 78 It is obvious that the propagator matrix T(t,t 0 ) can be constructed in two steps by T(t,t 0 )T(t,t 1 )T(t 1,t 0 ) since the sef-energy is oca in time semigroup property. Moreover, it foows from the structure of the generator that T(t,t 0 )isa proper sympectic transformation 3 T(t,t 0 ) 3 T(t,t 0 ) at a times. Consequenty, a occupation numbers of the genera soution to the noninear, initia vaue probem are constants of motion G ttt,t 0 G t 0 Tt,t 0 V tpt 0 V tv t 1Pt 0 V t, 79 FIG. 6. Rea-time evoution of the tota partice number N N c Ñ soid, the number of partices in the condensate N c dash-dot, the noncondensate partices number Ñ soid and the trace over the anomaous fuctuations Trm dashed-dot. At t 1, the equiibrium soution is suddeny distorted by setting m (t 1)0. Dimensioness time is measured in natura harmonicosciator periods. with time-evoved basis states V(t)T(t,t 0 )V. Since T(t,t 0 ) represents a genuine sympectic transform, the eigenstates V (t) 3 V(t)V 3 V 3 remain orthogona. In the foowing figures, we iustrate these fundamenta facts that effectivey define reversibiity for any noninear system in the generic form of Eqs. 75 and 76. In particuar, we wi use the sef-consistent, finite-temperature soution for (t0)() and G (t0)g () as an initia vaue for the time propagation at t0. It is obvious that this choice does not induce any change during the subsequent rea-time evoution. However, at t1, we suddeny distort this equiibrium soution by setting the anomaous component of G (t1 ) to zero, i.e., m (t1 )0 and then propagate forward up to t4. It is important to note that the new G (t1 ) is sti a vaid fuctuation matrix. In Fig. 6, we have depicted the number of partices that occupy the condensate N (c) Tr f (c) (t), the noncondensate ÑTr f (t), and the tota partice number NN (c) Ñ versus time. Time is measured in units of the harmonicosciator period T2/. In contrast to these numbers that are genuine singe-partice properties, the anomaous fuctuations are a physica measure of the degree of two-partice correations or squeezing.49 For exampe, the tota partice number fuctuations (Nˆ Nˆ ) 2 or, more specificay, the normay ordered density fuctuations : fˆ(x,x),fˆ(y,y): woud refect the degree of squeezing of the quadrature components aong certain directions x,y. Whie we have not evauated such observabes here, we have incuded the averaged strength of the anomaous fuctuations Trm to represent their size. The most important feature in Fig. 6 is the exact conservation of the tota partice number during a phases of the evoution. The instantaneous change in m (t1) does not affect it directy. But it can be seen that the reative partition

11 REVERSIBLE AND IRREVERSIBLE EVOLUTION OF A... PHYSICAL REVIEW A FIG. 7. Rea-time evoution of the tota system energy EE (c) E f Em soid, the energy of the partices in the condensate E (c) dashed-dot, the norma noncondensate energy E f and the anomaous energy Em. Att1, the equiibrium soution is suddeny distorted by setting m (t1)0. Dimensioness energy is measured in natura harmonic osciator units. ing of the partices between condensate and norma noncondensate is massivey distorted by this sudden infux of energy and partices start to osciate coherenty between the couped subsystems. In Fig. 7, we show the individua first-order contributions to the tota system energy EE (c) E f Em as described in Eqs Again the most notabe feature is the exact conservation of tota energy during the time evoution. Due to the sudden change in the anomaous fuctuations at t1, the overa energy increases instantaneousy by more than 400. In Fig. 8, we show the compete spectrum of occupation numbers P(1n r 14,06;t) that occur in the instantaneous fuctuation matrix G (t), as defined by Eq. 79. From the ogarithmic pot that covers 13 decades, it can be seen that the occupation numbers are indeed numericay exact constants of motion. Due to the instantaneous change at t1, many of the high-energy occupation numbers that are quasidegenerate before the change, spit up into a mutitude of nondegenerate eves afterwards. Minute changes in the occupation numbers P10 10 at the end of the integration period identify the precision oss of the numerica cacuation. The numerica resuts discussed in this section were obtained by discretizing the generaized GP Eq. 75 and the reversibe part of the Botzmann Eq. 76 with a standard finite eement method 50 based on b-spines 51,52. The use of these finite-support, piecewise poynomia basis functions resuts in matrix representations of the kinetic and potentia energy operators that are banded. Very efficient inear agebra agorithms can be empoyed in this case LAPACK Ref. 53. In particuar, we used a set of 400 b-spines on an equidistant radia grid from 0r256r TF. Eventuay, we represented the condensate wave function Fig. 2, the particeike basis states 1 Fig. 3, as we as the phononike basis states 1 (c) Fig. 4 in this b-spine basis. Finay, a subset of quantum states was chosen either 1 or 1 (c) with (n r 1,0):2(n r 1)182 to evauate the finite-temperature sums or to propagate in rea time. We have verified numericay that no particuar advantage can be obtained from either choice, as ong as a of the reevant energies scaes are resoved. The conservation of the tota partice number N Fig. 6, the tota energy E Fig. 7 as we as the instantaneous occupation numbers P Fig. 8 support this argument. V. THE IRREVERSIBLE EVOLUTION A. An ergodic equiibrium soution of the master equation In the foowing discussion of the irreversibe evoution of the kinetic equations, we wi again try to eucidate the main physics by additiona simpifications. In particuar, we wi assume ergodicity for the norma fuctuations f f 1, and the anomaous fuctuations m 0. This physica approximation is appropriate for most kinetic temperatures, except for a region cose to T0. Within this imit, we are abe to estabish an important resut for the stationary behavior of the condensed atomic gas, i.e., a canonica Bose-Einstein distribution for the noncondensate partices coexisting with an energeticay ower-ying, coherent condensate mode. By imposing the restriction of vanishing anomaous fuctuations upon the kinetic equations, Eqs. 10 and 26, we are eft with the foowing equations of motion: d dt i N N N, 80 FIG. 8. Rea-time evoution of the instantaneous occupation numbers P(1n r 14,06) that characterize the fuctuation matrix G (t). At t1, G (t) is suddeny distorted by setting m (t1)0. d dt f N 1 f N f H.c. 81 It is worth mentioning that in the rea-time evoution the rotating frame frequency is sti an adjustabe parameter and not necessariy synonymous with the chemica potentia

12 R. WALSER, J. COOPER, AND M. HOLLAND PHYSICAL REVIEW A First, et us concentrate on the equation for the mean-fied ampitude. From Eq. 80, we can see that the mean-fied evoution consists of two distinct parts: a Hermitian, numberconserving contribution N, and a part that accounts for condensate number changing coisions out of and into the noncondensate, i.e., N N. In stationarity, both processes have to vanish identicay 0H 0 1U f c2u f, 0 f f 1 f 1 f 1 f f Given the constraint on the condensate partice number, it is, in principe, straightforward to sove the Hermitian eigenvaue probem of Eq. 82 and obtain an eigenvaue. The ater equation Eq. 83 poses a much more chaenging constraint on the couped system. In order to maintain a stationary state, the mean-fied has to be orthogona to a number changing processes. This means that the norma fuctuations have to adjust sef-consistenty with respect to the condensate wave function. It is interesting to note that the soutions of this system are infinitey degenerate with respect to a goba phase rotation. In other words, the condensate s phase is not pinned down by any restoring force and is free to drift, consequenty. However, the ater, vector-vaued condition of Eq. 83 is satisfied identicay if f 1 1 f f 4 f 1 1 f 1 f 2 f 4 1 f 2 1 f 4, 84 vanishes for a components 1. Provided that energy conservation is satisfied exacty, i.e., 1 2 4, it is straightforward to verify that a canonica Bose-Einstein distribution 1 f e 1, 85 with an inverse temperature, is the equiibrium soution. In addition to this functiona form of the distribution that is dictated by detaied baance, it is required that a of the excitation energies are above the condensate energy, i.e., 1. Thus the eigenenergy spectrum exhibits a finite gap. From Fig. 5 it can be seen that both the positive energy spectrum of the HFB operator as we as the eigenenergies of N, are suitabe candidates within this approximation. Second, the generaized Botzmann equation, Eq. 81, is stationary if 0 f f 1 f 2 f c f 1 f f f f c1 f 1 f 1 f f 2 f c 1 f f 1 f 1 f f cf. 86 Within the ergodic approximation, the Hermitian part is satisfied identicay. On the other hand, there are two distinct types of coisiona reaxation processes in Eq. 86. There are the number-conserving in and out rates of the conventiona quantum-botzmann equation, i.e., f f (1 f )(1 f ) (1 f )(1 f ) f f. Obviousy, both rates match identicay under the detaied baance conditions of Eq f 1 1 f 2 1 f 4 1 f 2 f 1 1 f 1 f 2 f 4 1 f 2 1 f 4 f 2 1 f Both of the two remaining distinct processes in Eq. 86, i.e., 2 f (c) f (1 f )(1 f )2 f (c) (1 f ) f f, as we as the process f f f (c)(1 f ) (1 f )(1 f ) f (c)f, invove a condensate partice in the precoision or post-coision channes. Thus, rea partices wi be transfered between the condensate and the noncondensate, unti the rates are baanced. Anaogous to the arguments that ead to Eq. 84, it can be shown that a canonica Bose-Einstein distribution is attained in equiibrium. VI. CONCLUSIONS AND OUTLOOK In this paper, we have studied aspects of the reversibe and irreversibe evoution of a condensed atomic gas immersed in the noncondensate. By speciaizing the kinetic equations 31 for a simpe isotropic mode, we were abe to anayze the equiibrium soution, as we as the dynamic nonequiibrium behavior numericay. In particuar, we obtained the excitation spectra of a finite-temperature equiibrium. Moreover, we demonstrated the reversibiity of the timedependent HFB equations far from equiibrium and in the coisioness regime. This is tantamount to noting that the instantaneous occupation numbers of the HFB modes are constants of motion. Finay, we studied the coisiona regime for an ergodic distribution of quasipartices and showed that detaied baance is obtained in the fu quantum kinetic theory with a sef-consistent canonica Bose-Einstein distribution. Based on this isotropic mode, we can aso obtain the coision rates that ead to a sef-consistent equiibrium. However, such an anaysis is sti work in progress and resuts wi be presented in future pubications. ACKNOWLEDGMENTS R.W. acknowedges gratefuy financia support by the U.S. Department of Energy and the Austrian Academy of Sciences through an APART grant. This work aso benefited greaty by the BEC seminars of Professor C. Wieman, Professor E. Corne, and Professor D. Jin, as we as discussions with J. Wachter

13 REVERSIBLE AND IRREVERSIBLE EVOLUTION OF A... PHYSICAL REVIEW A APPENDIX A: CANONICAL TRANSFORMATIONS A canonica transformation is an inhomogeneous inear combination of creation and destruction operators that preserves the commutation reation 16. In particuar, if â and â denotes a pair of Hermitian conjugated bosonic operators, such that â 1,â 2 1,2, A1 then any affine inear transformation defines a new set of operators b and b by b b T â â d. A2 In an n-dimensiona vector space, T represents a (2n2n) dimensiona matrix and d isa(2n)-dimensiona vector. Such a transformation is canonica if the new pair of operators aso satisfies the commutation reation: b 1,b 2 1,2. A3 More specificay, the transformation is unitary canonica if the new operators are Hermitian conjugate pairs, i.e., b b. By inserting Eq. A2 into Eq. A3, one finds that the transformation matrices are a representation of the sympectic group Sp(2n): In addition, it can be shown that T* 1 T 1 and T 1 3 T 3. Here, we have introduced the (2n)-dimensiona Paui matrices 1 and 3 as , A5 APPENDIX B: CAUCHY-SCHWARTZ INEQUALITY For a positive semidefinite density operator and a genera operator Lˆ it foows that the expectation vaue Lˆ Lˆ TrLˆ Lˆ 0 B1 is never negative. Consequenty, the covariance matrix G of Eq. 24 must be positive semidefinite u G u0, as we. This can be easiy seen, by considering a inear combination of two arbitrary operators  and Bˆ, i.e., LÂBˆ.By minimizing the positive expression Eq. B1, one obtains the Cauchy-Schwartz inequaity as  Bˆ Bˆ Bˆ  ÂBˆ. B2 In particuar, for the specia choice of Ââ 1 1 and Bˆ â 2 2 *, this impies that the magnitude of the anomaous fuctuations is imited by T 3 T 3. A4 1 f 11 f 22 m B3 1 M. H. Anderson et a., Science 269, K. B. Davis et a., Phys. Rev. Lett. 75, C. C. Bradey, C. A. Sackett, and R. G. Huet, Phys. Rev. Lett. 78, D. G. Fried et a., Phys. Rev. Lett. 81, S. L. Cornish et a., Phys. Rev. Lett. 85, B. D. Esry and C. H. Greene, Nature London 392, J. Wiiams et a., Phys. Rev. A 59, R J. Wiiams and M. Hoand, Nature London 401, M. Matthews et a., Phys. Rev. Lett. 83, J. Wiiams et a., Phys. Rev. A 61, W. Kettere, D. Durfee, and D. Stamper-Kurn, in Proceedings of the Internationa Schoo of Physics Enrico Fermi, Course CXL, Soc. Itaiana di Fisica, Boogna, Itay, edited by M. Inguscio, S. Stringari, and C. Wieman IOS, Amsterdam, B. DeMarco and D. Jin, Science 285, A. I. Akhiezer and S. V. Peetminskii, Methods of Statistica Physics Pergamon, Oxford, Engand, L. Kadanoff and G. Baym, Quantum Statistica Mechanics, Frontiers in Physics Benjamin, New York, J. P. Baizot and G. Ripka, Quantum Theory of Finite Systems MIT Press, Cambridge, Massachusetts, P. Nozières and D. Pines, The Theory of Quantum Liquids: Superfuid Bose Liquids Addison-Wesey, New York, 1990, Vo. II. 18 A. Griffin, D. W. Snoke, and S. Stringari, Bose-Einstein Condensation Cambridge University Press, Cambridge, Engand, D. Zubarev, V. Morozov, and G. Röpke, Statistica Mechanics of Nonequiibrium Processes Akademie Verag, Berin, 1997, Vo. 1, Basic Concepts, Kinetic Theory. 20 S. Beiaev, Zh. Eksp. Teor. Fiz. 34, Sov. Phys. JETP 34, J. Kane and L. Kadanoff, J. Math. Phys. 6, P. Hohenberg and P. Martin, Ann. Phys. N.Y. 34, T. R. Kirkpatrick and J. R. Dorfman, J. Low Temp. Phys. 58, N. Proukakis and K. Burnett, Phys. Rev. A 57, , and references therein. 25 C. W. Gardiner and P. Zoer, Phys. Rev. A 61, , and references therein. 26 P. O. Fedichev and G. V. Shyapnikov, Phys. Rev. A 58, S. A. Morgan, e-print cond-mat/ H. T. C. Stoof, J. Low Temp. Phys. 114, E. Zaremba, T. Nikuni, and A. Griffin, J. Low Temp. Phys. 116, F. Dafovo, S. Giorgini, L. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71,