Scenario of Strongly Non-Equilibrated Bose-Einstein Condensation

PHYSICAL REVIEW A 66, 013603 ͑2002͒

Scenario of strongly nonequilibrated Bose-Einstein condensation

Natalia G. Berloff1,* and Boris V. Svistunov2,† 1Department of Mathematics, University of California, Los Angeles, California 90095-1555 2Russian Research Center ``Kurchatov Institute,'' 123182 Moscow, Russia ͑Received 10 July 2001; revised manuscript received 3 December 2001; published 15 July 2002͒ Large scale numerical simulations of the Gross-Pitaevskii equation are used to elucidate the self-evolution of a Bose gas from a strongly nonequilibrium initial state. The stages of the process confirm and refine the theoretical scenario of Bose-Einstein condensation developed by Svistunov and co-workers ͓J. Mosc. Phys. Soc. 1, 373 ͑1991͒; Sov. Phys. JETP 75, 387 ͑1992͒; 78, 187 ͑1994͔͒: the system evolves from the regime of weak turbulence to superfluid turbulence via states of strong turbulence in the long-wavelength region of energy space.

DOI: 10.1103/PhysRevA.66.013603 PACS number͑s͒: 03.75.Fi, 02.60.Cb, 05.45.Ϫa, 47.20.Ky

I. INTRODUCTION short-range order, which is a state of superfluid turbulence A. Statement of the problem with quasicondensate local correlation properties. The notions of weak turbulence and superfluid turbulence The experimental realization of Bose-Einstein conden- are crucial to our understanding of ordering kinetics. In the sates ͑BEC͒ in dilute alkali-metal and hydrogen gases ͓1͔ regime of weak turbulence ͑for an introduction to weak tur- and more recently in a gas of metastable helium ͓2͔ has bulence theory for the GPE, see Ref. ͓14͔͒, the ‘‘single- stimulated great interest in the dynamics of BEC. In the case particle’’ modes of the field are almost independent due to of a pure condensate, both the equilibrium and dynamical weak nonlinearity of the system. The smallness of the corre- properties of the system can be described by the Gross- lations between harmonics in the regime of weak turbulence Pitaevskii equation ͑GPE͓͒3͔͑in nonlinear physics this implies the absence of any order. On the other hand, the equation is known as the defocusing nonlinear Schro¨dinger regime of superfluid turbulence ͑for an introduction, see Ref. equation͒. The GPE has been remarkably successful in pre- ͓15͔͒ is the regime of strong coherence where the local cor- dicting the condensate shape in an external potential, the relation properties correspond to the superfluid state, but dynamics of the expanding condensate cloud, and the motion long-range order is absent because of the presence of a cha- of quantized vortices; it is also a popular qualitative model of otic vortex tangle and nonequilibrium long-wave phonons superfluid helium. ͓8͔. In the case of a weakly interacting gas, local superfluid An important and often overlooked feature of the GPE is order is synonymous with the existence of quasicondensate that it gives an accurate microscopic description of the for- correlation properties ͓7͔. In a macroscopically large system, mation of a BEC from a strongly degenerate gas of weakly interacting bosons ͓4,5͔. By large scale numerical simula- the crossover from weak turbulence to superfluid turbulence tions of the GPE it is possible, in principle, to reveal all the is a key ordering process. Indeed, in the regime of weak stages of this evolution from weak turbulence to superfluid turbulence there is no order at all, while in the regime of turbulence with a tangle of quantized vortices, as was argued superfluid turbulence ͑local͒ superfluid order has already by Svistunov, Kagan, and Shlyapnikov ͓6–8͔͑for a brief been formed. Meanwhile, rigorous theoretical as well as nu- review, see Ref. ͓9͔͒. This task has up to now remained un- merical or experimental studies of this stage of evolution fulfilled, although some important steps in this direction have been lacking. The general conclusions concerning this were made in Refs. ͓10,11͔. We would also like to mention stage ͓7,8͔ were made on the basis of a qualitative analysis the description of the equilibrium fluctuations of the conden- that naturally contained ad hoc elements. The difficulty with sate and highly occupied noncondensate modes using the an accurate analysis of the transition from weak turbulence time-dependent GPE ͓12,13͔. to superfluid turbulence comes from the fact that the evolu- The goal of this paper is to obtain a conclusive description tion between these two qualitatively different states takes of the process of strongly nonequilibrium BEC formation in place in the regime of strong turbulence, which is hardly a macroscopically large uniform weakly interacting Bose gas amenable to analytical treatment. Large computational re- using the GPE. We are especially interested in tracing the sources are necessary for a numerical analysis of this stage development of the so-called coherent regime ͓4,6–8͔ at a since the problem involves significantly different length certain stage of evolution. According to the theoretical pre- scales and, therefore, requires high spatial resolution. dictions ͓7,8͔, this regime sets in after the breakdown of the In the present paper we demonstrate that this problem can regime of weak turbulence in a low-energy region of wave be unambiguously solved with a powerful enough computer. number space. It corresponds to the formation of superfluid Our numerics clearly reveal the dramatic process of transformation from weak turbulence to superfluid turbulence and thus fills in a serious gap in the rigorous theoretic description *Electronic address: [email protected] of strongly nonequilibrated BEC formation kinetics in a †Electronic address: [email protected] macroscopic system.

The paper is organized as follows. In Sec. I B we discuss dering process takes place. Even if the occupation numbers the relevance of the time-dependent GPE to the description are of order unity in the initial state, so that the classical of BEC formation kinetics and its relation to the other for- matter ®eld description is not yet applicable, the evolution, malisms. In Sec. I C we give some important details of the which can be described at this stage by the standard Boltz- evolution scenario that we are going to observe. In Sec. II we mann quantum kinetic equation, inevitably results in the ap- describe our numerical procedure. In Sec. III we present the pearance of large occupation numbers in the low-energy re- results of our simulations. In Sec. IV we conclude by outlin- gion of the particle distribution ͑see, e.g., Ref. ͓17͔͒. The ing the observed evolution scenario and making a comment blowup scenario ͓6͔ indicates that only the low-energy part on the case of a con®ned gas. of the ®eld is initially involved in the process. Therefore, one can switch from the kinetic equation to the matter ®eld de- B. Time-dependent Gross-Pitaevskii equation and BEC scription for the long-wavelength component of the ®eld at a formation kinetics certain moment of the evolution when the occupation num- In this section we will discuss the question of applicabil- bers become appropriately large. As the time scale of the ity of the GPE to BEC formation kinetics, and its connec- formation of the local quasicondensate correlations is much tions to otherÐfully quantumÐtreatments. This discussion smaller than any other characteristic time scale of evolution is especially relevant in the wake of a recent controversy on ͓7͔, the cutoff of the high-frequency modes, associated with the applicability of the classical-®eld description to a non- the matter ®eld description, is not important. By the time the condensed bosonic ®eld. interactions ͑particle exchange͒ between the high- and low- A general analysis of the kinetics of a weakly interacting frequency modes became signi®cant, the local super¯uid or- bosonic ®eld was performed in Ref. ͓16͔. In terms of the der had already been developed. The interaction wavelengths coherent-state formalism, it was demonstrated that if the oc- are of the order of the typical thermal de Broglie wavelength cupation numbers are large and somewhat uncertain ͑with and therefore these interactions are essentially local with re- the absolute value of the uncertainty being much larger than spect to the quasicondensate and can be described in terms of unity and with the relative value of the uncertainty being the kinetic equation ͓6,17͔. arbitrarily small͒, then the system evolves as an ensemble of The thesis of the applicability of the matter ®eld descrip- classical ®elds with corresponding classical-®eld action. ͑For tion at large occupation numbers was justi®ed by the analysis an elementary demonstration of this fact for a weakly inter- of Ref. ͓16͔. Later, Stoof questioned the validity of this thesis acting Bose gas and especially for a discussion of the struc- by introducing the concept of ``quantum nucleation'' of the ture of the initial state, see Ref. ͓5͔.͒ This has a direct anal- condensate as a result of an essentially quantum instability ogy with the electromagnetic ®eld: ͑i͒ the density matrix of a ͓18͔; the path-integral version of the Keldysh formalism was completely disordered weakly interacting Bose gas with used to substantiate this concept. For a criticism of the con- large and somewhat uncertain occupation numbers is almost cept of ``quantum nucleation'' see Refs. ͓5,19͔. diagonal in the coherent-state representation, so that the ini- It is important to emphasize, however, that the path- tial state can be viewed as a mixture or statistical ensemble integral approach developed in ͓18͔ appears to be the most of coherent states; ͑ii͒ to leading order each coherent state fundamental, powerful, and universal way of deriving the evolves along its classical trajectory, which in our case is basic equations for the dynamics of a weakly interacting given by the GPE Bose gas. In particular, we believe that the demonstration of the applicability of the time-dependent GPE to the descrip- -␺ ប2 tion of highly occupied single-particle modes of a nonconץ iប ϭϪ ٌ2␺ϩU͉␺͉2␺, ͑1͒ t 2m densed gas within this formalism would be the most naturalץ since the effective action for the bosonic ®eld is simply the classical-®eld action of the GPE. Basically, one simply has to where ␺ is the complex-valued classical ®eld that speci®es make sure that for the modes with large and somewhat un- the index of the coherent state, m is the mass of the boson, certain occupation numbers the main contribution to the path Uϭ4␲ប2a/m is the strength of the ␦-function interaction integral comes from the close vicinity of the classical trajec- ͑pseudo͒potential, and a is the scattering length. ͑Note that in tories with the quantum corrections being relevant only at a strongly interacting system it is impossible to divide single- large enough times of evolution. particle modes into highly occupied and essentially empty An interesting all-quantum description of the BEC kinet- ones, so the requirement of weak interactions is essential ics was implemented in Ref. ͓11͔. This technique is based on here. In a strongly interacting system there are always quan- associating the quantum-®eld density matrix in the coherent- tum modes with occupation numbers of order unity that are state representation with a correlator of a pair of classical coupled to the rest of the system.͒ Therefore, the behavior of ®elds, whose evolution is governed by a system of two the quantum ®eld is equivalent to that of an ensemble of coupled nonlinear equations with stochastic terms. Using this classical matter ®elds. method the authors performed a numerical simulation of It is important to emphasize that in the context of strongly BEC formation in a trapped gas of a moderate size. We be- nonequilibrium BEC formation kinetics the condition of lieve ͑in particular, in view of the general results of Refs. large occupation numbers is self-consistent: the evolution ͓5,16͔͒ that this approach might be further developed ana- leads to an explosive increase of occupation numbers in the lytically to demonstrate explicit overlapping with the other low-energy region of wave number space ͓6͔ where the or- treatments and with the time-dependent GPE. Indeed, the

013603-2 SCENARIO OF STRONGLY NONEQUILIBRATED BOSE-... PHYSICAL REVIEW A 66, 013603 ͑2002͒ form of the system of two coupled equations of Ref. ͓11͔ is where ⑀ϭប2k2/2m. The dimensional constants A and B re- reminiscent of that of the GPE. This suggests that under the late to each other by (␣Ϫ1)m3U2A2ϭ␭␲3ប7B2(␣Ϫ1), condition of large occupation numbers the system can be where the parameters ␣ and ␭ were determined by numerical decoupled, leading to the GPE for the diagonal part of the analysis as ␣Ϸ1.24 ͓17͔ and ␭Ϸ1.2 ͓6͔. The form of the density matrix with relative smallness of the nondiagonal function f was also determined numerically in Ref. ͓6͔. The terms. If the standard Boltzmann equation is applicable, so solution ͑3͒±͑5͒ has only one free parameter, say A, which that the system can be viewed as an ensemble of weakly depends on the conditions of the nonuniversal dynamics pre- coupled elementary modes ͓20͔, it is natural to expect that ceding the appearance of self-similarity. This dynamics is the equations of Ref. ͓11͔ should lead to the kinetic equation. sensitive to the details of the initial condition or/and cooling A natural way for deriving this kinetic equation from the mechanism as well as to the spontaneous-scattering terms in dynamical equations of Ref. ͓11͔ is to utilize the standard the kinetic equation, which cannot be neglected until the oc- formalism of the weak turbulence theory. In the case of the cupation numbers are large enough. When the self-similar GPE, the weak turbulence approximation leads to the quantum-®eld Boltzmann kinetic equation without spontane- regime sets in at a certain step of evolution all the particular ous scattering processes ͑see, e.g., Ref. ͓14͔; note also that it details of the previous evolution are absorbed in the single is the simplest way to make sure that the GPE is immediately parameter A. applicable once the occupation numbers are large͒. It is natu- The self-similar solution ͑3͒±͑5͒ describes a wave in en- ral to expect that in the full-quantum treatment of Ref. ͓11͔ ergy space propagating from high to lower energies. The the weak turbulence procedure over the dynamical equations energy ⑀0(t) de®nes the ``head'' of the wave. The wave propagates in a blow-up fashion: ⑀ (t)→0 and n (t)→ϱ as would result in the complete quantum-®eld Boltzmann ki- 0 ⑀0 netic equation with the spontaneous processes retained. Un- t→t . In reality the validity of the kinetic equations associ- * fortunately, we are not aware of such investigations of the ated with the random-phase approximation breaks down equations of Ref. ͓11͔, which might be very instructive for shortly before the blowup time t . This moment marks the * the general understanding of the dynamics of a weakly inter- beginning of a qualitatively different stage in the evolution acting Bose gas. of the coherent regime: strong turbulence evolves into a quasicondensate state. In the coherent regime the phases of the C. Initial state and evolution scenario complex amplitudes ak of the ®eld ␺ become strongly correlated and the periods of their oscillations are then compa- In what follows we consider the evolution of Eq. ͑1͒ start- rable with the evolution times of the occupation numbers. ing with a strongly nonequilibrium initial condition The formation of the quasicondensate is manifested by the appearance of a well-de®ned tangle of quantized vortices ϭ ϭ • ␺͑r,t 0͒ ͚ ak exp͑ik r͒, ͑2͒ and, therefore, by the beginning of the ®nal stage of the k evolution: super¯uid turbulence. In this regime the vortex tangle starts to relax over macroscopically large times. where the phases of the complex amplitudes ak are distributed randomly. Such an initial condition follows from the microscopic quantum-mechanical analysis of the state of a II. NUMERICAL PROCEDURE weakly interacting Bose gas in the kinetic regime ͓5͔. Theo- retical investigations of the relaxation of such an initial state A. Finite-difference scheme toward the equilibrium con®guration were performed by We performed a large scale numerical integration of a Svistunov and co-workers ͓6±8͔. The analysis revealed a dimensionless form of the GPE: number of stages in the evolution. Initially the system is in ␺ץ the weak turbulence regime and thus can be described by the Ϫ2i ϭٌ2␺ϩ͉␺͉2␺, ͑6͒ tץ Boltzmann kinetic equation. The kinetic equation is obtained as the random-phase approximation of Eq. ͑1͒ for occupation * numbers nk de®ned by ͗akakЈ͘Ϸnk␦kkЈ. Alternatively, the weak turbulence kinetic equation follows from the general starting with a strongly nonequilibrium initial condition. Our quantum Boltzmann kinetic equation if one neglects sponta- calculations were done in a periodic box N3, with Nϭ256, neous scattering as compared to stimulated scattering ͑be- using a fourth-order ͑with respect to the spatial variables͒ cause of the large occupation numbers͒. Svistunov ͓6͔ and ®nite-difference scheme. The scheme corresponds to the later Semikoz and Tkachev ͓17͔ considered the self-similar Hamiltonian system in the discrete variables ␺ijk : solution of the Boltzmann kinetic equation:

Ϫ n t͒ϭA⑀ ␣ t͒f ⑀/⑀ ͒, tрt , ͑3͒ Hץ ␺ijkץ 0 ͑ ͑ 0 ͑⑀ * i ϭ , ͑7͒ tץ ␺*ijkץ ⑀ ͑t͒ϭB͑t Ϫt͒1/2(␣Ϫ1), ͑4͒ 0 *

f͑x͒→xϪ␣ at x→ϱ, f ͑0͒ϭ1, ͑5͒ where

013603-3 NATALIA G. BERLOFF AND BORIS V. SVISTUNOV PHYSICAL REVIEW A 66, 013603 ͑2002͒

1 When choosing the parameters of the initial condition ͑2͒ ϭ Ϫ 1 Ϫ ϩ Ϫ H ͚ ␺*ijk͓ 12 ͑␺iϩ2,j,k ␺iϪ2,j,k ␺i, jϩ2,k ␺i, jϪ2,k speci®ed by the complex Fourier amplitudes ͑11͒, we have to 2 ijk take ⑀0 small enough to be free from the systematic error of ϩ Ϫ ϩ 4 Ϫ ϩ large ®nite differences. On the other hand, taking ⑀ too ␺i, j,kϩ2 ␺i, j,kϪ2͒ 3 ͑␺iϩ1,j,k ␺iϪ1,j,k ␺i, jϩ1,k 0 small reduces the physical size of the system. Let us de®ne 1 4 Ϫ␺ Ϫ ϩ␺ ϩ Ϫ␺ Ϫ ͔͒Ϫ ␺ ͑8͒ ϭ i, j 1,k i, j,k 1 i, j,k 1 2 ͉ ijk͉ one period of the amplitude oscillation as t p 2␲/⑀0 and the ϭ number of periods before the blowup as P t /t p . When ͑in the numerics we set the space step in each direction of the * choosing the value of n0 in combination with ⑀0, we would grid as dxϭdyϭdzϭ1͒. like to avoid having P too small when the time scale of the Equation ͑7͒ conserves the energy H and the total particle kinetic regime becomes too short, or having P too large 2 number ͚ijk͉␺ijk͉ exactly. In time stepping, the leapfrog when the ®nite-size effects ͑the discreteness of the k) domi- scheme was implemented: nate the calculation. Given the maximal available grid size ϭ ϭ N 256, we found that it is optimal to take n0 15 and ⑀0 nϩ1Ϫ nϪ1 n ϭ ,H 1/18, so that t pϷ113ץ ␺ijk ␺ijk i ϭ , ͑9͒ ͪ *␺ץ ͩ 2dt ijk ϭ 3 Ϫ 2 2 t 4␭␲ /͑␣ 1͒⑀0n0Ϸ893, ͑12͒ with dtϭ0.03. To prevent the even-odd instability of the * leapfrog iterations, we introduce the backward Euler step and PϷ8. nϩ1Ϫ n nϩ1 Hץ ␺ijk ␺ijk i ϭ ͑10͒ ͪ *␺ץ ͩ dt ijk III. DATA PROCESSING AND RESULTS every 104 time steps. The leapfrog scheme is nondissipative, The instantaneous values of the occupation numbers ϭ 2 so the only loss of energy and of the total particle number nk(t) ͉ak(t)͉ are extremely ``noisy'' functions of time. To occurs during the backward Euler step and, since we take be able to draw some quantitative comparisons and conclu- this step very rarely, these losses are insigni®cant. sions we need either to perform some averaging or to deal The code was tested against known solutions of the GPE: with some coarse-grained self-averaging characteristics of vortex rings and rarefaction pulses ͓21͔. The simulations the particle distribution. Taking the second option, we intro- were performed on a Sun Enterprise 450 server and took duce shells in momentum space. By the ith shell (i ϭ about three months to complete for the main set of calcula- 1,2,3,...) we understand the set of momenta satisfying Ϫ Ͻ tions discussed below. the condition i 1рlog2(k/2␲) i. The idea behind this de®nition is that each shell represents some typical momentum ͑wavelength͒ scale and thus allows us to introduce a B. Initial condition coarse-grained characteristic of the occupation numbers cor- To eliminate the computationally expensive ͑and the least responding to a given scale; namely, for each shell i we physically interesting͒ transient regime, we started directly introduce the mean occupation number ␩i(t) ϭ (shell i) from the self-similar solution Eq. ͑2͒ with Eqs. ͑3͒±͑5͒,so ͚ nk(t)/Mi , where M i is the number of harmonics that in the ith shell. The harmonic kϭ0, which plays a special role ͑at the very end of the evolution͒, is not assigned to any ϭ ak ͱ␰kn0 f ͑⑀/⑀0͒ exp͓i␾k͔, ͑11͒ shell. Another instructive coarse-grained characteristic of the where ␰k and ␾k are random numbers. ͑Note that in the particle distribution is the integral distribution function Fk ϭ simulations the momentum k is the momentum of the lattice ͚kЈрknkЈ which shows how many particles have momenta Fourier transform.͒ The phase ␾k is uniformly distributed on not exceeding k. We use the function Fk to keep track of the ͓0,2␲͔ in accordance with the basic statement of the theory formation of the quasicondensate and to determine the wave of weak turbulence and with the explicit microscopic analy- number span of the above-the-condensate particles. This in- sis of corresponding quantum ®eld states ͓5͔. The choice of formation is used, in particular, for ®ltering out the high- ␰k is rather arbitrary: we only ®x its mean value to be equal frequency harmonics in order to interpret the results of our to unity, introducing, therefore, the parameter n0. The weak numerical calculations in the super¯uid turbulence regime. turbulence evolution is invariant to the details of the statistics With the above-introduced quantities we now turn to the of ͉ak͉. By the time the system enters the regime of strong analysis of the results of our numerical simulations. The self- turbulence, the proper statistics is established automatically similar character of the evolution is clearly observed in Fig. since each harmonic participates in a large number of scat- 1. The insets in Fig. 1 give a comparison of the theoretical tering events. We tried different distributions for ␰k and saw prediction of the evolution of the occupation number func- no systematic difference in the evolution picture. The main tion n⑀(t) de®ned by Eq. ͑3͒ and the evolution of the ®rst set of our simulations was done with the distribution function and the second shells. The agreement with the theoretical ϭ Ϫ w(␰k) exp( ␰k) ͑heuristically suggested by equilibrium predictions ͓6͔ is quite good for tϽ600. After that the nu- Gibbs statistics of harmonics in the noninteracting model͒. merical solution deviates from the self-similar theoretical so-

013603-4 SCENARIO OF STRONGLY NONEQUILIBRATED BOSE-... PHYSICAL REVIEW A 66, 013603 ͑2002͒

ϭ FIG. 1. The time evolution of ␩i(t 100j) in the weak turbulence regime for jϭ0,...,6.Intheinsets we show the theoretical FIG. 3. Evolution of topological defects in the phase of the self-similar solution ͑3͒±͑5͒͑solid line͒ and the solution obtained long-wavelength part Ä␺ of the ®eld ␺ in the computational box through the numerical integration of Eq. ͑6͒͑dashed line͒ for the 2563. The defects are visualized by isosurfaces ͉Ä␺͉2ϭ0.05 ͉Ä␺͉2 . shells iϭ1 ͑a͒ and iϭ2 ͑b͒. ͗ ͘ High-frequency spatial waves are suppressed by the factor max͕1 Ϫk2/k2,0͖, where the cutoff wave number is chosen according to the lution, which is the manifestation of the onset of the strong c phenomenological formula k ϭ9Ϫt/1000. turbulence stage of evolution. c As follows from the dimensional analysis ͑see, e.g., Ref. comes sharper and sharper as the evolution continues. Note ͓9͔͒, the characteristic time t0 and the characteristic wave that by de®nition of the function Fk the height of the shoul- vector k0 at the beginning of the strong turbulence regime der is equal to the number of quasicondensate particles. are given by the relations From Fig. 2 we estimate k0 as the characteristic wave num- ϩ Ϫ ber at which F (tϭ600) changes its slope, so that k ϳ15, t Ϫt ϳC ͓ប2␣ 5/m3U2A2͔1/(2␣ 1), ͑13͒ k 0 * 0 0 which implies that k ϳC ͓AU͑m/ប͒␣ϩ1͔1/(2␣Ϫ1), ͑14͒ 0 1 C1ϳC0ϳ40. ͑15͒ where C0 and C1 are some dimensionless constants. Our Within the coherent regime the momentum distribution of numerical results ͑Fig. 1͒ indicate that the characteristic time the harmonics yields a rather incomplete picture of the evo- of the begining of the strong turbulence regime is t0ϳ600 lution and it becomes reasonable to follow the ordering pro- which together with the theoretical blowup time ͑12͒ gives cess in coordinate space. It is important to trace the topologi- t Ϫt ϳ300 and implies that C ϳ40. After the formation of * 0 0 cal defects in the phase of the long-wavelength part of the the quasicondensate (tϾ1000), the distribution of particles complex matter ®eld ␺ since the transformation of these de- acquires a bimodal shape, which is seen in Fig. 2. The salient fects into a tangle of well-separated vortex lines is the most characteristic of the distribution is the shoulder, which be- essential feature of super¯uid short-range ordering ͓8͔.To this end we ®rst ®lter out the high-frequency harmonics by → Ϫ 2 2 performing the transformation ak akmax͕1 k /kc,0͖, where kc is a cutoff wave number. When the function Fk has a pronounced quasicondensate shoulder, the natural choice is to take kc somewhat larger than the momentum of the shoulder in order to remove the above-the-condensate part of the ®eld ␺. In the regime of weak turbulence, when there is no quasicondensate, the procedure of ®ltering is ambiguous: the distribution is not bimodal, so there is no special low momentum kc ; also, the structure of the defects in the ®ltered ®eld essentially depends on the cutoff parameter and thus has no physical meaning. The results of visualizing the topological defects are presented in Fig. 3. The formation of a tangle of well-separated vortices and the decay of super¯uid turbulence are clearly FIG. 2. Evolution of the integral distribution of particles Fk ϭ seen. This is the key point of our simulation. To the best of ͚kЈрknkЈ . Notice the appearance of a ``shoulder'' of Fk indica- ϭ tive of quasicondensate formation. The evolution of nk 0 is pre- our knowledge, this is the ®rst unambiguous demonstration sented in the inset. Note the strong ¯uctuations typical for the evo- of the formation of the state of super¯uid turbulence in the lution of a single harmonic. The ¯uctuations are also seen in the course of self-evolution of a weakly interacting Bose gas. graph of the ®rst shell ͓see inset ͑a͒ of Fig. 1͔. This result forms a solid basis for the analysis of the further

013603-5 NATALIA G. BERLOFF AND BORIS V. SVISTUNOV PHYSICAL REVIEW A 66, 013603 ͑2002͒

FIG. 5. Comparison of two isosurfaces obtained by different ®ltering techniques. The solution at tϭ4000 is obtained by numerical integration of Eq. ͑6͒ in the periodic box with Nϭ128. The isosurface ͉Ä␺͉2ϭ0.05͉͗Ä␺͉2͘ is plotted in ͑a͒ using high-frequency ϭ Ã 2ϭ Ã 2 ®ltering with kc 5. The isosurface ͉␺͉ 0.2͉͗␺͉ ͘ is plotted in ͑b͒, where ␺Ã is de®ned by Eq. ͑16͒.

Ã ϭ Ϫ Ϫ 2 ␺ijk͑t͒ ͵ ␺ijk͑␶͒exp͓ ͑␶ t͒ /100͔d␶. ͑16͒

FIG. 4. Evolution of topological defects in the phase of the The width of the Gaussian kernel in Eq. ͑16͒ is chosen in long-wavelength part Ä␺ of the ®eld ␺ in the computational box such a way that the ͑disordered͒ high-frequency part of the 1283. The defects are visualized by isosurfaces ͉Ä␺͉2ϭ0.05͉͗Ä␺͉2͘. ®eld ␺ is averaged out, revealing the strongly correlated low- High-frequency spatial waves are suppressed by the factor max͕1 frequency part ␺Ã . Figure 5 compares the density isosurfaces Ϫ 2 2 k /kc,0͖, where the cutoff wave number is chosen according to the obtained by two different methods: by high-frequency sup- ϭ Ϫ phenomenological formula kc 9 t/1000. pression ͓Fig. 5͑a͔͒ and by time averaging ͓Fig. 5͑b͔͒. In the stages of long-range ordering in terms of the well-developed latter case we reveal the actual shape of the vortex core and theory of super¯uid turbulence that was performed in Ref. resolve the rarefaction pulses. ͓8͔͑see also Ref. ͓22͔͒. The characteristic time of the evolution of the vortex IV. CONCLUSION tangle depends on the typical interline spacing R as 2 We have performed large scale numerical simulations of R /ln(R/a0), where a0 is the vortex core size ͑see, e.g., ͓8͔͒. During the ®nal stage of evolution, when R is of the order of the process of strongly nonequilibrated Bose-Einstein con- the linear size of the computational box, the slowing down of densation in a uniform weakly interacting Bose gas. In the the relaxation process makes numerical simulation of the ®- limit of weak interactions under the condition of strong nal stage of the vortex tangle decay enormously expensive in enough deviation from equilibrium, the key stage of ordering a large computational box. For example, according to the dynamicsÐsuper¯uid turbulence formationÐis universal above-mentioned estimate of the relaxation time, to achieve and corresponds to the process of self-ordering of a classical the complete disappearance of the vortex tangle in our N matter ®eld whose dynamics is governed by the time- ϭ256 system we would need several years. To observe this dependent Gross-Pitaevskii equation ͑defocusing nonlinear ®nal stage of the vortex tangle decay, we repeated the calcu- SchroÈdinger equation͒. The universality implies indepen- lations for a smaller computational box with Nϭ128 ͑reduc- dence of the evolution of the details of initial processes such ing in this way the computational time by a factor of ϳ32 as, for example, the cooling mechanism and rate as well as of ϭ23ϫ22); see Fig. 4. Parameters of the initial condition are quantum effects such as spontaneous scattering. All the in- ϭ ϭ ⑀ 1/2 and n0 2␲, so that the number of periods before the formation about the evolution preceding the universal stage blowup is PϷ5. A single vortex ring remains at tϭ4000 as is absorbed in the single parameter A that de®nes the scaling a result of the turbulence decay; see Fig. 5͑a͒. of the characteristic time and wave number in accordance The above-mentioned ®ltering method allows us to visu- with Eqs. ͑13͒±͑15͒. alize the position of the core of a quantized vortex line, but The most important features of the BEC formation sce- not the actual size of the core, since we force the solution to nario observed in our simulation are as follows. The low- be represented by a relatively small number of harmonics. To energy part of the quantum ®eld, characterized by large oc- get a better representation of the actual size of the core as cupation numbers and described by a classical complex well as to resolve another objects of interestÐrarefaction matter ®eld ␺ obeying Eq. ͑1͒, initially evolves in a weak pulses ͓21͔, which are likely to appear in the course of trans- turbulent self-similar fashion according to Eqs. ͑3͒±͑5͒. The formation of strong turbulence into super¯uid occupation numbers at small energies become progressively turbulenceÐwe implement a different type of ®ltering based larger. At the characteristic time moment t0, given by Eq. on time averaging. We introduce a Gaussian-weighted time ͑13͒, close to the formal blowup time t of the solution * average of the ®eld ␺: ͑3͒±͑5͒, the self-similarity of the energy distribution breaks

013603-6 SCENARIO OF STRONGLY NONEQUILIBRATED BOSE-... PHYSICAL REVIEW A 66, 013603 ͑2002͒ down. The distribution gradually becomes bimodal; the low- ine condensate ͓20,24͔. Clearly, our numerical approach can energy quasicondensate part of the ®eld sets to a state of be extended to the case of a trapped Bose gas by simply super¯uid turbulence characterized by a tangle of vortex including a term with an external potential in Eq. ͑1͒. Such a lines. The further evolution of the quasicondensate is inde- simulation could provide a deeper interpretation of the ®rst pendent of the rest of the system ͑apart from a permanent experiments on the kinetics of BEC formation ͓25,26͔, an- ¯ux of the particles into the quasicondensate͒ and basically is swering, in particular, the question of whether the process the process of relaxation of super¯uid turbulence. All vortex involves the formation of a vortex tangle; and if not, under lines relax in a macroscopically large time. what conditions one may expect formation of super¯uid tur- In the present paper we considered the case of macro- bulence ͑quasicondensate͒ in a realistic experimental situa- scopically large uniform system. As far as the case of a tion. trapped gas is concerned, the situation becomes sensitive to the competition between ®nite size and nonlinear effects. If ACKNOWLEDGMENTS nonlinear effects dominate, the basic physics of the ordering process is predicted to be analogous to that revealed by our N.G.B. was supported by the NSF under Grant No. DMS- simulation ͓23͔. If ®nite-size effects dominate ͑which means 0104288. B.V.S. acknowledges support from the Russian that the initial size of the condensate is smaller than the Foundation for Basic Research under Grant No. 01-02-16508 corresponding healing length, so that, for example, vortices and from the Netherlands Organization for Scienti®c Re- cannot arise in principle ͓23͔͒, the ordering kinetics is sub- search ͑NWO͒. The authors are very grateful to Professor stantially simpli®ed, being reduced to the growth of a genu- Paul Roberts for fruitful discussions.

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