Classical fields approximation for bosons at nonzero temperatures

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) R1–R37 doi:10.1088/0953-4075/40/2/R01

TOPICAL REVIEW

Classical fields approximation for bosons at nonzerotemperatures

Mirosław Brewczyk1, Mariusz Gajda2 and Kazimierz Rzazewski3

1 Instytut Fizyki Teoretycznej, Uniwersytet w Białymstoku, ul. Lipowa 41, 15-424 Białystok,Poland2 Instytut Fizyki PAN, Al. Lotnikow 32/46, 02-668 Warsaw, Poland3 Centrum Fizyki Teoretycznej PAN, Al. Lotnikow 32/46, 02-668 Warsaw, Poland

Received 7 November 2005, in final form 21 November 2006Published 10 January 2007Online at stacks.iop.org/JPhysB/40/R1

AbstractExperiments with Bose–Einstein condensates of dilute atomic gases requiretemperatures as low as hundreds of nanokelvins but obviously cannot beperformed at zero absolute temperature. So the approximate theory of sucha gas at nonzero temperatures is needed. In this topical review we describea classical field approximation which satisfies this need. As modes of light,also modes of atomic field may be treated as classical waves, provided theycontain sufficiently many quanta. We present a detailed description of theclassical field approximation stressing the significant role of the observationprocess as the necessary interface between our calculations and measurements.We also discuss in detail the determination of temperature in our approach andstress its limitations. We also review several applications of the classical fieldapproximation to dynamical processes involving atomic condensates.

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1. Introduction

Almost 80 years ago, Einstein [1], extending the ideas of Bose from photons to massiveparticles, predicted that at low temperatures the majority of identical, noninteracting particlesof integer spin would assemble in the lowest state of the trapping potential forming a coherent,macroscopically populated matter wave. What was most striking, the macroscopic populationof the ground state was expected at temperatures for which the available energy allows nearlyall particles to be distributed among the excited states of the potential.

Although the superfluidity of liquid helium was associated with the phenomenon of Bose–Einstein condensation, it was clear that the properties of liquid helium are very different thanthat of the ideal Bose gas due to its high density and resulting strong interaction.

It took 70 years of development of experimental atomic physics to realize nearly perfectlyEinstein’s original idea. Following progress in slowing and trapping of neutral atoms, the

0953-4075/07/020001+37$30.00 © 2007 IOP Publishing Ltd Printed in the UK R1

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condensation of dilute atomic gases was achieved in 1995 [2]. It triggered an avalanche ofboth experimental and theoretical papers.

To match the experiments with its nearly perfect isolation of a finite atomic sampleboth in terms of exchange of energy and exchange of particles even the statistical physics ofthe ideal Bose gas had to be revisited. As was already noticed by Schrodinger [3], withinthe commonly used grand canonical ensemble the condensate exhibits large and unphysicalfluctuations even at zero absolute temperature. In fact the experimental situation is probablybest approximated by the microcanonical statistical ensemble. So there was need for thesolution of thermodynamics in this fundamental ensemble [4].

While the dilute gas BEC experiments come close to the ideal gas case, the residualinteraction is very important. It provides a mechanism of thermalization necessary forsuccessful evaporative cooling and, in general, for the establishment of the stationary state ofthe cloud.

The most successful tool of the theory of weakly interacting Bose gas is the nonlinearSchrodinger equation, known in this case as the Gross–Pitaevskii equation [5]. It describes theorder parameter of the condensed system in the mean field approximation. Its lowest energystate is interpreted as the wavefunction of all atoms at zero temperature. Linearization aroundthe ground state gives normal modes of the elementary collective excitations—quasiparticlesof the system [6]. Thus for very low temperatures one usually assumes the macroscopicpopulation of the ground state and the standard Bose thermal state of the quasiparticle modes.It is clear that such an approximation suffers on two fronts: first, it violates the conservationof number of particles. Second, it is valid only at very low temperatures, where the thermaldepletion of the condensate is negligible. The first weakness is eradicated by the so-callednumber conserving Bogoliubov approximation [7]. It is harder to do away with the restrictionof very low temperature.

Ambitious attempts to solve exactly the quantum thermodynamics [8] are numericallyso complex that their applications are typically limited to something like 100 atoms in one-dimensional (1D) geometry. But, in 1D, there is no phase transition and real experiments onBEC are performed with 105–106 atoms.

Hence, approximate methods are necessary. The best developed are the two gas models[9]. In such models one assumes from the very beginning that the atomic system at nonzerotemperature consists of two distinct parts: quantum mechanical condensate and classicalthermal cloud. Of course, the two components do interact. Although effective, two gasmodels have an obvious weakness: in real life the splitting of the atomic cloud into thecondensate and the thermal cloud is a result of Bose statistics and of interactions. So a unifiedtreatment of all atoms that results in the formation of two components is desirable.

In this topical review we present such an approximate, unified, nonperturbative treatment,known as the classical fields approximation. It has its roots in the theory of electromagneticwaves. Ever since the discovery of wave solutions of Maxwell equations, physicists haveused classical wave theory to describe phenomena involving light. Only a limited numberof such phenomena require the quantum theory of light. The main reason for the validity ofthe classical wave picture is the large number of photons in relevant modes in most realisticsituations.

We note that a large number of atoms is just a hallmark of BEC experiments. So theclassical field is the most natural concept for the BEC matter waves. In section 2 we presentthe classical fields approximation for the weakly interacting massive bosons.

Electromagnetic waves are often only partially coherent. But even in this case the electricand magnetic fields have well defined values at each spatial point and at every instance of time.The partial coherence merely means that this value changes rapidly from one point to the next

Topical Review R3

one. Hence, the measurements, with their limited spatial and temporal resolution, require acoarse graining, a procedure that smoothes out the rapid changes of the incoherent fields. Sucha coarse graining, in a natural way, eliminates part of the information contained in the detailedspatiotemporal behaviour of the field. In the realm of quantum theory it makes a mixedstate out of the pure one. The analogous need of coarse graining occurs for classical matterwaves at finite temperatures. The procedure of coarse graining is also described in section 2.

As we remember, the need to quantize the light waves first arose in connection with thethermal states. Planck noticed that photons are needed to tame the ultraviolet divergence of thethermodynamics of classical electromagnetic waves. The analogous divergence plagues alsothe classical fields approximation to matter waves. The short wavelength cut-off is necessary.A natural question is what is its rational/optimal value.

The classical fields approximation, as presented in this review, describes thethermodynamic equilibrium of the isolated atomic system as the steady state evolution ofits single copy. Hence, via an ergodic hypothesis it is related to the microcanonical statisticalensemble. In this ensemble the total energy and the total number of particles are the controlparameters. The temperature is just a derived, fluctuating variable. Its value should be obtainedvia differentiation of the entropy with respect to energy. While some authors, using the ergodichypothesis, follow this method [10], we find it useful to proceed differently. In section 3 weshow how to introduce the temperature into our approximation by first using the equipartitionof energy between quasiparticles of the system to compute the ratio of temperature to the totalnumber of atoms and then identifying the absolute temperature of the weakly interacting gaswith that of the ideal gas of the same number of atoms. This way we simultaneously fix all therelevant control parameters of the classical field method, including the short wavelength cut-off.

The classical field approximation is valid for all temperatures up to the critical one. At leastin the case of atoms confined in a box with periodic boundary conditions one can effectivelyanalyse the quasiparticle excitations. In this non-perturbative approach they not only displaytemperature dependent shifts with respect to the conventional Bogoliubov formula, but arealso broadened. Their width has an obvious interpretation as the inverse of the lifetime. Allthis is the subject of section 3.

In section 4 we describe several direct applications of the classical field approximationto finite temperature dynamical phenomena involving a condensate. They include (1)photoassociation of molecules, (2) decay of the vortex (3) splitting of doubly charged vortices,(3) fragmentation of 1D condensates, (4) superfluidity and (5) spinor condensates.

Finally, in section 5 we present conclusions summarizing the main advantages of theclassical fields method as well as its limitations. We also discuss a possible generalization ofthe approach.

In this topical review we mostly present our version of the classical field approximation.Its origins may be traced to papers [11], where mainly the formation of the condensatewas considered. The classical field approximation as described here, motivated by the opticalanalogy, and corresponding to the microcanonical ensemble, has been simultaneously proposedin [12–14]. The canonical ensemble version of classical fields, stressing also more formaljustification via the truncated Wigner function, was introduced in [15]. The classical fieldsapproximation was also rediscovered in [16] in an approach based on the weak turbulencetheory.

2. The approximation

Theoretical description of a stationary Bose–Einstein condensate of a weakly interacting gas isa quite complicated and subtle problem even at zero temperature. Harder still is its description

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at finite temperatures. As stressed in the introduction, we intend to describe an approximateapproach in which the splitting of the system into two distinct fractions—the condensate andthe thermal cloud—is not introduced a priori but is a result of Bose statistics, interactions and,as we shall see, the measurement process. The best suited language for this kind of study isthe language of field theory. The N-particle Bose system is assigned a bosonic field operator�(r, t) that destroys a particle at position r at time t and obeys standard equal time bosoniccommutation relations with its conjugated partner �+(r, t) creating a particle at r at time t:

[�(r, t), �+(r′, t)] = δ3(r − r′). (1)

The N-particle Hamiltonian can be expressed in terms of the field operator

H =∫

d3r �+(r)H0(r)�(r) +1

2

∫ ∫d3r d3r ′�+(r)�+(r′)U(r − r′)�(r′)�(r), (2)

where H0(r) is a single particle Hamiltonian while U(r − r′) is a two-body interactionpotential. In realistic situations of Bose condensates of dilute atomic gases in magneticand optical dipole traps the two-body interaction is approximated by a contact potential

U(r − r′) = gδ3(r − r′), (3)

where g characterizes the atom–atom interaction in the low-energy, s-wave approximation.Expressed in terms of the s-wave scattering length a, it takes the form

g = 4πh2a

m. (4)

A Heisenberg equation originating from the Hamiltonian (2) acquires the form

ih∂

∂t� = H0� + g�+��. (5)

A full operator solution of (5) is impossible. There are different approaches to the timeevolution of an interacting condensate. They lead to numerical algorithms, which are hard toimplement [8]. None of them is in fact capable of handling a realistic case of a large numberof particles at a relatively high temperature. But this is only the first obstacle which oneencounters while studying the interacting Bose system at finite temperatures. It is not easy toextract relevant physical information such as, for example, a mean occupation and fluctuationsof a condensate—even having obtained the field operator. To talk about the population of thecondensate we need to know what is the condensate. According to the celebrated definition[17], the condensate can be assigned a condensate wavefunction which is an eigenvectorcorresponding to a dominant eigenvalue of a one-particle density matrix. This matrix can beexpressed with the help of field operators,

〈�+(r, t)�(r′, t)〉 = Nρ(1)(r, r′; t), (6)

and the average is taken in the initial state of a many-body system. In the general case,diagonalization of a density matrix is needed in order to determine other modes occupied bymany particles. Can we do anything to determine the highly occupied, thus relevant modeseven before we know the one particle density matrix? Of help would be a symmetry of theproblem. This is the case of a gas trapped in a box with periodic boundary conditions. Thesymmetry of the confining potential uniquely enforces a form of eigenstates of the thermalequilibrium one-particle density matrix regardless of interaction. Such a density matrix mustbe periodic with the periodicity of the box and must be translationally invariant, and thusshould only depend on the difference of positions:

ρ(1)(r, r′; t) = ρ(1)(r − r′; t) =∑

p

λpf∗p (r)fp(r′). (7)

Topical Review R5

The above condition indicates that eigenfunctions are simply plane waves,

fp(r) = 1

L3/2exp(−ip · r/h), (8)

and the momentum p takes quantized values p = (2πh/L)n with ni = 0,±1,±2, . . . (i =x, y, z), and L being a length of the cubic box. A stationary state of a Bose–Einstein condensatecorresponds to the p = 0 state of the box, unless a superfluid flow requires otherwise (seesection 4). Expansion of the field operator in the basis of the box eigenfunctions,

�(r, t) =∑

p

fp(r)ap(t), (9)

introduces operators ap(a+

p

)that annihilate (create) a particle in the state fp(r) at time t,

and np = a+p ap counts particles occupying this state. In this case of a box a one-particle

Hamiltonian is

H0 = p2

2m, (10)

and the dynamical equation for the operators ap(t) acquire the following form:

dap(t)

dt= −i

p2

2mhap − i

2g

V h

∑q1,q2

a+q1

aq2 ap+q1−q2 . (11)

The two-body interaction energy g/V and time t are expressed in units of ε = h2

2mL2 andτ0 = h/ε, respectively. V = L3 is the volume of the box. In the following, if not explicitlystated, we choose ε, τ0 and L to be the natural units of the problem. Obviously someapproximations are necessary if one wants to solve the nonlinear operator equations (11). Theclassical field approximation consists of replacing all creation and annihilation operators ofhighly occupied modes by c-numbers ap → √

Nαp. Such replacement is legitimate if thepopulation of a given mode is greater than its quantum fluctuations, i.e. if np = Nα∗

pαp > 1.At zero temperature it is only the p = 0 mode for which this condition is fulfilled. So atzero temperature we are left with just the ground state wavefunction of the correspondingGross–Pitaevskii equation. All nearly empty modes are neglected in the classical fieldsapproximation. At higher temperatures, however, many modes are macroscopically populatedand then thermal fluctuations dominate over the quantum ones. We stress that the thermalfluctuations are retained in the classical field approximation.

The classical fields approximation leads to a finite set of coupled nonlinear equations,which have to be solved numerically:

dαp(t)

dt= −i

p2

2mhαp − i

2gN

V h

∑q1,q2

α+q1

αq2αp+q1−q2 . (12)

Dynamical equations (12) can be viewed as equations for coupled macroscopically occupied‘mean fields’ αp. Our approximation is an extension of the Bogoliubov approach [6]. Thisapproach is valid at low temperatures because it assumes that only the p = 0 mode ismacroscopically occupied and therefore only the ‘condensed’ mode, described by a complexfield, is taken into consideration. Evidently, for a finite system at higher temperatures othermodes can be treated in the same fashion. Let us observe that our approximate dynamics,although nonperturbative, conserves both the total energy and the number of particles. Itcorresponds, therefore, to a genuine microcanonical description.

Our first goal is to describe the equilibrium state of an interacting system. We testednumerically that all ‘typical’ initial conditions with given values of all conserved quantities—the energy, the momentum and the particle number (see section 4 for additional conditions

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Figure 1. Typical time evolution of the relative occupation of the p = 0 mode (left panel) togetherwith its histogram (right panel). Note very small fluctuations of the relative occupation.

leading to a superfluid flow)—lead to an equivalently steady state. The one-dimensionalversion of this dynamics is completely integrable [18]. However, in a higher number ofdimensions or in the case of more realistic trapping potentials, the evolution may be chaotic. Westart the evolution choosing the initial ‘Bose–Einstein-like’ distribution of populations |αp|2.In determining this distribution, we neglect all interactions; thus the initial condition evidentlydoes not correspond to the equilibrium distribution of the interacting system. Moreover, thereis still a freedom in selecting phases of the coupled fields. In our approach, each mode isassigned an initial, randomly chosen, phase. Any subsequent dynamics depends on initialphases and, in a sense, describes a single realization of the quantum system. Microcanonicalexpectation values would require the average over the initial phases. Instead of doing so, wetrace the system for a sufficiently long time and compute time averages. In fact, time averagesrather than ensemble ones are measured experimentally. In addition, the two are equal if asystem is ergodic.

In our three-dimensional calculations we assumed that N rubidium (87Rb) atoms areheld in a cubic box with size equal to the Thomas–Fermi radius of the same systemtrapped in a harmonic trap of frequency 2π × 80 Hz. The two-body interaction strengthis g/(L3ε) = (2π)2 2.52 × 10−4. We performed our calculations with 729 modes. We tracedthe evolution of the system on a timescale equal approximately to 1 s. The calculations showthat after a few milliseconds the system reaches a state of dynamical equilibrium regardless ofthe choice of initial phases. The occupation of the p = 0 mode stabilizes at some value, andon a larger timescale it only fluctuates around that value (figure 1).

In figure 2 we present a typical time evolution of the p �= 0 mode along with its histogram.Two important features could be observed. First of all, we stress a striking difference betweenthe p = 0 mode (a condensate) and higher momentum modes. The amplitude of the condensatemode acquires a steady-state value and its relative fluctuations are small. In contrast, relativefluctuations of higher modes reach 100%.

The histogram of steady-state values of the occupation of high momentum thermal modesdiffers from the corresponding histogram of the p = 0 mode (left panel of figure 1). Wesee that the latter is well approximated by the exponential function, a clear characteristic ofthermal statistical property. In figure 3 we present the histogram of populations of the p = 0mode for much higher energy. Clearly its thermal character indicates that the system is aboveits critical temperature in this case.

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Figure 2. Typical time evolution of the relative population of the p = (1, 0, 0) mode (left panel)along with its histogram (right panel). Note that the probability distribution is well approximatedby the exponential function.

Figure 3. Histogram of populations of the p = 0 mode for very high energy. Its thermal characterclearly indicates that the system is above its critical temperature

The method presented here is very powerful. A careful reader, however, might beconfused. By replacing creation and annihilation operators by c-numbers a+

p , ap → α∗p, αp,

we simultaneously substituted the field operator by a ‘classical wavefunction’

�(r) →√

N�(r) =∑

p

αpfp(r), (13)

with a finite number of Fourier components, and the set of equations (12) is in fact nothingelse but the celebrated Gross–Pitaevskii equation written in the basis of plane waves. Thisequation traditionally describes a pure condensate at zero temperature. The correspondingone-particle density matrix

ρ(1)(r, r′, t) = 1

N�∗(r, t)�(r′, t), (14)

projects onto a pure state �(r, t) which is macroscopically occupied by N atoms. What aboutour arguments that it is |α0|2 which gives a relative population of a condensate? What lookslike a paradox is quickly resolved if we invoke a measurement process.

R8 Topical Review

The wavefunction �(r, t) varies rapidly both in space and time. The complex amplitudesof different modes oscillate very quickly, even when the evolution reaches a stage of dynamicalequilibrium. Of course, the detection process is never instantaneous nor does it have perfectspatial resolution. Optical methods are used to monitor an atomic Bose–Einstein condensate—a CCD camera monitors light passing through a condensate. An exposition time, texp, variesfrom a few microseconds up to several hundreds of microseconds and the size of the pixel isnever smaller than some 5 microns, thus observed is a time and space average of a diagonalpart of a one-particle reduced density matrix:

ρ = 〈ρ(1)(r, r′; t)〉,V = 1

V N

∫

dtexp

∫V

d3R �∗(r + R, t + texp)�(r′ + R, t + texp), (15)

where and V are the temporal and spatial resolutions of the detector.This coarse graining ‘spoils’ the purity of a dynamically evolving state. Rapidly oscillating

phases of different mean fields average to zero the terms in the expansion of (15) containingall products of two different modes and only diagonal elements survive:

ρ =∑

p

np

Nf ∗

p (r)fp(r′). (16)

Moreover, if the system reaches a state of dynamical equilibrium averaged occupations ofdifferent modes do not depend on time (they depend only on the total energy and the numberof particles) and one-particle density matrix acquires a stationary form with time independentmean occupation fractions.

It is obvious that the same classical fields approximation can be used in a more realisticcase of a condensate trapped in a harmonic potential. In such a case the correspondingdynamical equation for the classical field �(r) acquires the form

ih∂�(r)

∂t=

(− h2

2m +

1

2mω2r2 + g|�(r)|2

)�(r), (17)

where ω is the frequency of the harmonic trap. We need to specify the initial condition ascomplex amplitudes of a finite number of mode functions and let the nonlinear dynamics ofthe Gross–Pitaevskii equation do the thermalization. By averaging the product of a time-dependent solution of the Gross–Pitaevskii equation �∗(r, t)�(r′, t) over a finite observationtime and volume, we obtain the one-particle density matrix. In the figure 4(a) the one-particledensity of the interacting Bose gas trapped in a spherically symmetric harmonic trap is shownat some instance of time. Note the irregular behaviour, which (figure 4(b)) goes away uponcoarse graining. In figure 4(b) one can easily see a broad thermal cloud and a sharp peak ofthe Bose condensed atoms on top of it. It strongly resembles the results of real experiments.

In this section we have described the classical fields approximation. As defined here, itleaves serious questions unanswered. It does not tell us what is the optimal choice of the finitemodes that may be treated classically. We are going to address this issue in the next section.

3. Excitation spectrum

The results and arguments presented so far give quite strong indications that the classical fieldapproximation is well suited to describe the Bose-condensed system at finite temperatures.As mentioned at the end of the previous section, the method, to be useful, must give thepossibility of making quantitative predictions about the system. For example, one has to beable to assign relevant physical parameters characterizing the system such as temperature andparticle number [19–21].

Topical Review R9

(a) (b)

Figure 4. Picture of a 2D intersection of the single particle instantaneous density of the Bose gasin a spherically symmetric harmonic trap at energy higher than that corresponding to the groundstate: (a) instantaneous distribution; (b) after the coarse graining procedure. Both components:the condensate and the thermal cloud are clearly visible [19].

To this end, we should carefully analyse all underlying assumptions to properly select thevalues of those parameters which are the important ingredients of the method.

Therefore, as the first application of the classical method we use the celebrated example ofthe homogeneous system in a box with periodic boundary conditions. This system is a textbookexample and a lot is known about its physical features; therefore this is the perfect choiceto start with. Analysis of homogeneous system will help us to find an operational procedurefor the determination of the essential parameters of the method as well as to examine itsquantitative predictions.

One of the most important findings discussed in section 2 is that almost any initialcondition of a given number of atoms and given energy, after the transient time, leads to thesame steady state. In this way, we are able to produce a numerical version of the fluctuating,interacting Bose gas with macroscopic occupation of small momenta modes. Having producedsuch a state, we can study the thermodynamics of a uniform weakly interacting Bose gas inequilibrium.

The c-numbers amplitudes satisfy the equation

∂αp

∂t= −i

1

2p2αp − igN

∑l,m

α∗pαmαp+l−m, (18)

where the coupling constant g equals 4πas/L (as being the scattering length). Let us observethat due to the chosen normalization,

∑p α∗

pαp = 1, the number of particles enters thedynamical equation explicitly through the product of gN . The quantity np = αpα

∗p is

obviously the relative occupation of the mode with momentum p. Conservation of thetotal number of particles N = N

∑p np is equivalent to the conservation of the norm of

the classical field. The second constant of motion of the nonlinear equations (18), i.e.the total energy of the system expressed by the c-number amplitudes, acquires the formE = N

∑p(p

2/2)np + (gN/2)∑

p,l,m α∗pα

∗l αmαp+l−m.

As we have already mentioned in section 2, instead of solving equations (18) it is moreconvenient to transform them to the position representation by introducing the classical field:

�(r, t) =pmax∑

p

fp(r)αp(t). (19)

In the position representation the dynamical equations take a form of finite-grid version of the

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time-dependent Gross–Pitaevskii equation (in [20, 24]; this equation is called the projectedGross–Pitaevskii equation or finite temperature Gross–Pitaevskii equation) for a system ofparticles in a box with periodic boundary conditions. The equation is written as

i∂�

∂t= −1

2∇2� + gN |�|2�. (20)

This equation has to be solved on a rectangular grid and the grid spacing x is related to thevalue of the cut-off momentum pmax = π/x. The cut-off momentum has to be properlyadjusted depending on the value of the total energy. It is a very important parameter ofthe method. This fact was first recognized in [12], where a very formal procedure basedon introduction of the operator projecting onto classical modes was introduced. We shallelaborate extensively on the role of the cut-off of high frequency modes in the following partof this section.

Note that the nonlinear term of the GP equation can generate momentum components upto three times larger than those which are present in the original classical field. One mightexpect, therefore, that the nonlinear term must be calculated on a grid three times larger thanthe one used for calculating the wavefunction. Fortunately, as has been shown in [20], thefinal result is not sensitive to the grid size used for obtaining the nonlinear term, and both theoriginal wavefunction and the nonlinear term can be calculated on the same grid.

As described in section (2), the initial wavefunction is generated from the ground-state solution by its random disturbance followed by the normalization. The strength ofthe disturbance determines the total energy per particle. Both the energy and the particlenumber with properly adjusted cut-off momentum determine the unique equilibrium state.

We propagate the high energy solution of the Gross–Pitaevskii equation and, at each timestep, do the decomposition of the wavefunction into plane waves as they are the eigenstates ofthe single particle density matrix. In such a way, we gain the time dependence of amplitudesof natural modes, αp(t). By analysing these amplitudes in the frequency domain (Fouriertransform of αp(t) with respect to time) we get the spectrum of single particle eigenmodes[21].

The spectrum of the p = (0, 0, 0) mode, i.e. the condensate, is composed of a singlenarrow peak centred at frequency µ. The frequency µ plays the role of a chemical potential.In our units it is equal to single-particle energy in condensate phase. In figure 7 we show thechemical potential as a function of a condensate occupation (which is a monotonic functionof temperature). We present two curves, both corresponding to the same value of the effectivecoupling gN = 731.5, but for two different grid sizes. The number of grid points is equal tothe number of macroscopically occupied modes. If we increase the number of macroscopicallyoccupied modes while keeping the population of the highest mode constant, we inevitablyincrease the total number of particles in the system, and simultaneously decrease the value ofg (because gN is fixed) [22]. Therefore, the curve corresponding to the larger grid (circles)describes a larger system with weaker interactions, as compared to the curve marked byboxes.

In figure 8 we present the chemical potential as a function of the interaction strength g atfixed condensate fraction n0 = 0.55 and fixed effective coupling gN = 731.5. According tothe two gas model [23], the chemical potential should be constant in such a case and its value,for chosen parameters, equals µ = 1060. This value is indicated in figure 8 by a horizontaldashed line. Our results show that chemical potential depends on interaction strength not onlythrough the product gN . We expect that two gas model result can be reached in the limit ofg → 0; however, calculation in this regime is a demanding task. In the inset we show thechemical potential versus condensate population (temperature) for fixed values of both particle

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Figure 5. Fourier transform (in arbitrary units) of the p = (0, 0, 1) and p = (3, 3, 3) modes forgN = 731.5 and n0 = 0.79 (low temperature). Note the existence of two groups of frequencies[21].

Figure 6. Fourier transform (in arbitrary units) of the p = (0, 0, 1) and p = (3, 3, 3) modes forgN = 731.5 and n0 = 0.07 (high temperature) [21].

number N and interaction strength g. The difference between the two gas model predictionand our result grows with temperature.

The spectrum of higher modes is plotted in figures 5 and 6 in the case of low and highenergies, respectively, for modes (0, 0, 1) and (3, 3, 3). Typically two groups of frequenciespositioned symmetrically with respect to the condensate frequency µ are visible. The width ofeach group gets broader when the energy of the system increases. Assigning to each group the

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Figure 7. Chemical potential as a function of the condensate population for the effective couplingstrength gN = 731.5. Circles and boxes show numerical results obtained on the grids 36×36×36and 16 × 16 × 16, respectively. The solid line comes from the formula derived using the two gasmodel [21].

Figure 8. Chemical potential as a function of the interaction strength g alone for gN = 731.5and the condensate occupation n0 = 0.55 (solid line is added to guide the eye). The horizontaldashed line indicates the value of the chemical potential obtained from the two gas model formula:µ = gN(2 − n0) ≈ 1060. The inset shows just like figure 7, the chemical potential as a functionof the condensate population but for a constant number of atoms N = 235 000 [21].

corresponding central (or mean) frequency and neglecting, in the first approximation, the finitewidth of each group, we see that every plane wave amplitude oscillates with two frequenciesonly:

αp(t) ≈ e−iµt (Ap e−iεpt + B∗p eiεpt ), (21)

where Ap and B∗p are the amplitudes of positive and negative frequency component,

respectively. The separation between the two central frequencies increases with the modeenergy. Moreover, increasing the energy of the mode leads also to the suppression of thelower frequencies group, i.e. B∗

p ≈ 0. By taking appropriate linear combination of amplitudesαp and α∗

−p one can find normal modes of the system, i.e., modes oscillating with only onefrequency:

βp(t) = βp(0) e−i(µ+εp). (22)

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Amplitudes βp are related to particle amplitudes αp by the following transformation,

βp eiµt = upαp + v∗pα

∗−p, (23)

and up and vp have to satisfy the following ‘normalization’ condition: |up|2 − |vp|2 = 1.This condition has roots in the quantum Bose field theory where the amplitudes βp becomebosonic operators. It is not difficult to discover that the described transformation to thenormal modes βp is the transformation to Bogoliubov quasiparticle amplitudes. They can beregarded as elementary excitations of the system. More precisely, equation (22) is approximate.The Bogoliubov mode oscillates here with many frequencies concentrated around the meanfrequency µ + εp. The centre of the higher frequencies group defines the energy of thequasiparticles while the width of this group is related to its lifetime. This is the result of aresidual interaction between them. The Bogoliubov transformation for the finite temperaturesystem requires some care. Because of a finite width of both frequency groups observedin the spectrum the transformation has to be done separately for each individual frequencycomponent. Let us note that for each positive component µ + ε (where ε corresponds tothe frequency within the peak) there exists a companion at the frequency µ − ε. The phasedifference of these components is equal to the corresponding one for the mode −p. Because ofthis relation, the Bogoliubov transformation is simplified. The resulting frequency spectrumof the quasiparticles peaks at µ + εp. Moreover, relative occupation of the quasiparticle modecan by obtained by integrating over the quasiparticle spectrum:

np =∫

dω|βp(ω)|2. (24)

In fact only in the phonon range must the occupation of the quasiparticles be obtained with thehelp of the Bogoliubov transformation. In the particle region the negative frequency group isstrongly suppressed and βp(ω) ≈ αp(ω).

The excitation energy obtained numerically is well described by the gapless Bogoliubov-like formula:

εp =√(

p2

2+ gNn0

)2

− (gNn0)2, (25)

where n0 is the condensate fraction. For small momenta p <√

gNn0, the excitationspectrum at finite temperature has phonon-like character εp = pc, and the velocity of soundc = √

gNn0 decreases with diminishing population of the condensate mode. The phononspectrum continuously approaches the standard particle like spectrum in the limit of largemomenta, εp = p2/2. This simple generalization of the Bogoliubov formula is known asthe Popov approximation [23]. The excellent agreement between the Bogoliubov–Popovexpression (25) and the numerical spectrum manifests itself in figure 9, where we plot thequasiparticle energies for various condensate populations and interaction strengths. Numericalpoints (marked by boxes, circles and triangles) are calculated as the mean frequency values(with respect to µ) corresponding to higher frequency groups, whereas the solid lines comefrom (25).

Having energies of elementary excitations (25) and their populations (24) we can checkhow much energy is gathered in every normal mode. If a classical system is in thermalequilibrium then the energy in each mode should be the same. The equipartition of energyis a manifestation of the thermal equilibrium of a classical system. In a quantum systemonly low energy modes share the same amount of energy. Occupation of high energy modesdecreases exponentially with modes energy. So does the energy accumulated in these modes.A border between the two regions corresponds to modes of energies of the order of kBT ,

R14 Topical Review

Figure 9. Dispersion relation for various interaction strengths and temperatures. Numerical resultsare marked by boxes (gN = 731.5, n0 = 0.95), circles (gN = 73.15, n0 = 0.13) and triangles(gN = 73.15, n0 = 0.93). Solid lines are obtained from the expression (25). The inset showsexcellent agreement between the numerical data and formula (25) for all modes in the case of highenergies [21].

Figure 10. Quasiparticle energies as a function of the inverse of the population for the systemof N = 235 000 (gN = 731.5) atoms for various energies. The upper set of data (circles) wasobtained on the grid 36 × 36 × 36 and corresponds to n0 = 0.07, the middle one (triangles) is thecase of n0 = 0.79 and 24 × 24 × 24 grid, whereas the lower set (marked by boxes) results fromthe 16 × 16 × 16 grid and gives n0 = 0.95 [21].

where T is the temperature and kB is the Boltzmann constant. In our numerical simulationsthese high momentum modes are absent due to the momentum cut-off. In the studied case theequipartition relation reads [19, 20]

npεp = T

N, (26)

where the temperature T is expressed in units of ε/kB.This is the case shown in figure 10 where we plot the energy of quasiparticles as a function

of inverted relative population. Due to the equipartition relation this dependence should belinear and the slope of the line is proportional to the temperature of the system, T/N .

The equipartition of energy is the key feature which proves that the classical field is in thestate of thermal equilibrium. Moreover, the relation (26) is the basis for determination of the

Topical Review R15

temperature. As a matter of fact, the determination of the temperature of the system in theclassical field method is much more subtle issue that it seems at the first glance. Note thatnumerical solution gives a value of the left-hand side of equation (26), i.e. the value of

τ = T

N. (27)

To obtain the temperature one should know a value of particle number. Note that the numberof particles enters the dynamical equation via the product gN only and the actual value of Nused in the propagation is not uniquely specified. This property is the well-known symmetryof the standard Gross–Pitaevskii equation at zero temperature. All systems with the samevalue of gN have the same ground state. This is no longer true at higher temperatures. Thereason is that, in the latter case, there are two additional parameters, the energy and the numberof classical modes (directly related to the cut-off momentum). All these parameters cannotbe chosen independently if the assumption of macroscopic occupation has to be fulfilled. Toshow the dependence of the temperature on the number of classical modes, let us consider aset of equilibrium solutions of the dynamical equations characterized by the same energy andthe same value of gN and by a different number of macroscopically occupied modes. Becauseof the equipartition, which characterizes all these solutions, the same total energy is sharedby different numbers of modes. Evidently, the larger the number of modes, the smaller theenergy per mode, i.e. the lower the temperature of the system. Counter intuitively, the largerthe number of modes used in the numerical simulations, the lower the equilibrium temperatureand more particles tend to accumulate in the lowest p = 0 mode [21].

The classical field method can be applied to amplitudes of highly occupied modes only.In the numerical implementation only modes of momentum not exceeding the value of thecut-off momentum pmax are taken into account. The cut-off momentum has to be properlyadjusted depending on the value of the total energy. In the first approximation its value can beobtained via the relation specifying occupation of the least populated pmax mode:

Nnpmax = 1. (28)

All results presented in this paper, except if explicitly stated, are based on the above condition.This condition ensures that the number of atoms occupying a given momentum state is notsmaller than one and can be considered as macroscopic. This choice of the cut-off momentumis not unique and is the result of a compromise: the validity of the classical approximationis limited to modes with occupation large compared to 1, but on the other hand the methodpretends to describe the majority of particles in the system. It seems to be natural that the lessoccupied mode (the mode of the highest momentum) should contain on average about oneparticle. Moreover, equation (28) allows us to determine the absolute (not scaled) value of thetemperature of the system. Indeed, condition (28) applied to the equipartition relation gives

εpmax = T . (29)

Knowing the numerical value of relative occupation of the cut-off mode npmax one can obtainthe actual value of the total number of particles N = 1/npmax in the system. Evidently, thenumber of particles and temperature are not control parameters of the classical fields method.In the numerical implementation one can control values of total energy and cut-off momentum.The particle number and temperature are determined a posteriori.

Having determined the temperature, in figure 11 we plot the condensate occupation forvarious temperatures for the system built of N = 235 000 atoms. The dashed line in figure 11shows the population of an ideal gas of the same density. Although we cannot preciselydetermine the value of the critical temperature of finite size interacting gas, figure 11 clearlyshows that the shift of the critical temperature is positive.

R16 Topical Review

Figure 11. Condensate occupation as a function of temperature for a uniform weakly interactingBose gas; the scattering length equals as = 25 nm (gN = 731.5, N = 235 000). The dashed lineshows the fraction of atoms in the condensate for the ideal gas of the same density (the criticaltemperature is about T0 = 7.1 nK. The solid line is the fit: 1 − (T /9.85)1.80 [21].

The above described method of assigning the temperature of the system cannot be usedfor precise determination of the magnitude of the shift of the critical temperature. Thereason is that in our approach the temperature is proportional to the assumed value of thepopulation of the cut-off momentum mode which we set arbitrarily to one. Smaller valuesof the population of the cut-off momentum mode give smaller magnitudes of the criticaltemperature shift (compare figure 15). Evidently, without some additional assumptions theclassical fields method can give only order of magnitude estimation of the temperature of thesystem. The issue of limitations of the method as well as possible solutions to the problemof a more precise determination of the temperature of the system will be discussed in moredetail in the following part of this section. Here we add that Davies and Morgan [24] usingthe classical fields method were able to determine the shift of the critical temperature quiteaccurately by taking into account higher energy modes and assuming that they are subject toideal gas distribution.

Equipartition of energy is the extremely important feature of the classical fields method. Itproves thermal equilibrium and allows the temperature of the system to be determined. Thereare also other ways of establishing the temperature [25]. They might be even more effectivebut without equipartition they would be meaningless.

Finally, we would like to discuss the damping of the quasiparticle modes. There existsan analytical result for the decay rates of elementary excitations in a uniform Bose gas. Athigh temperatures (kBT gN/V ) Landau processes dominate and the decay rate for thephonon-like part of the spectrum follows the formula [26]

γp = 3π3/2

2

kBT

gN/V

(N

Va3

s

)1/2εp

h. (30)

Since phonon energies depend linearly on the momentum, so do the decay rates. The ratioγp/p is a function of temperature given by ∼√

n0T , where n0 is the condensate fraction.In figure 12 we compare numerical results with the analytical expression (30). In order todetermine the damping rates we average the spectrum of phonon-like modes over the anglesin the momentum space and fit to the Lorentzian curve. The FWHM of the Lorentzian fitis just the damping rate. We see that both approaches agree quite well except for two first

Topical Review R17

Figure 12. Comparison of the ratio γp/p calculated numerically (circles) with that coming fromthe analytical formula (30) (and given by boxes) for various temperatures and gN = 731.5, [21].

Figure 13. The decay rates as a function of momentum for gN = 731.5 and higher temperature(n0 = 0.48). Both phonon-like and particle-like parts of spectrum are considered. The solid linefollows the analytical formula (30) [21].

points. These points correspond to very low temperatures (approximately 2000 and 4000 inunits of h2/(mL2)) and certainly the condition of the validity of the analytical formula (30)is not fulfilled in this case since gN/V is as high as 731.5. However, in our method wecan go beyond the phonon-like excitations and find the decay rates for particle-like part ofthe spectrum. In figure 13 we plot the damping rates for all momenta in the case of highertemperature. The solid line follows the analytical expression (30). The numerical results andformula (30) agree only for small momenta as expected (see figure 12). The damping rates ofparticle-like excitations show nonlinear dependence on the momentum.

Our example of a homogeneous system proves the usefulness of the classical fieldsmethod. Knowing the details of implementation of the method, we are ready now to discussfoundation and limitations.

The classical fields method is based on the heuristic substitution of Bose operators byc-number amplitudes. This substitution can be easily justified at very low temperatures sincepractically all particles occupy the condensate mode. On the other hand, it might seemquestionable at temperatures close to the critical temperature. Nevertheless the classical fields

R18 Topical Review

Figure 14. The energy density spectrum for gN = 900 and T = 1000 obtained from theBose–Einstein distribution with the quasiparticle spectrum given by equation (25)—full line. Forcomparison, the same quantity resulting from equipartition of energy for two different populationsof the highest momentum mode is shown: npmax = 1—dashed line, npmax = 0.6—dotted-dashedline. Note that the latter is very close to the position of maximum of the energy density distributionwhich corresponds to npmax = 0.63.

method works quite well up to the condensation point—see figure 11. It is well known that theBose–Einstein condensation is a quantum phenomenon and the critical temperature dependson the Planck constant. The Planck constant is usually introduced into theory through thecommutation relation for the field operators. Those however are violated in the classicalfields approach. Nevertheless this approach evidently preserves some bosonic features of thesystem. Therefore it is justified to ask how come that some quantum features are present inthe classical field method.

In order to answer this question we will recall the way in which Planck introducedhis famous constant h into theory of electromagnetic radiation at thermal equilibrium. Theclassical theory gives the energy density of black-body radiation in the form of the Rayleigh–Jeans formula. This expression predicts quadratic growth of energy density with the frequencyof radiation and agrees with the quantum black-body energy density in the region of smallfrequencies only. Experimental data clearly show that after reaching a maximum the energydensity decreases exponentially in the region of high frequencies. The position of thismaximum grows linearly with temperature. In order to account for exponential decreaseand to determine the position of the maximum, Planck had to introduce the universal action.The position of maximum of energy density, up to a numerical factor of the order of one,equals hωmax ≈ kBT .

Let us use a similar reasoning in the case studied. Our approach is based on the dynamicsof a classical field. Interactions bring the field to the thermal equilibrium characterized by theequipartition of energy. This equipartition results in quadratic growth of energy density dueto the volume of momentum space. This is the inevitable result of the classical theory. Theexponential decrease of energy density at high frequencies cannot be obtained in the classicaldescription. To mimic this exponential decay in the classical field method, the occupation ofhigh energy modes is set to zero due to the cut-off.

In figure 14 we show the energy density spectrum dU/dω obtained from the Bose–Einstein distribution (in the thermodynamic limit) assuming quasiparticle spectrum in theform of εp =

√(p2/2 + gNn0)2 − (gNn0)2 (equation (25)). The dashed and dashed-dotted

lines correspond to the energy density spectra obtained by assuming equipartition of energy

Topical Review R19

for two different values of the cut-off momentum: npmax = 1—dashed line; npmax = 0.6—dotted-dashed line.

By analogy with the black-body distribution, the position of the cut-off should be close tothe maximum of the quantum distribution, i.e. 0.63εpmax ≈ kBT . Note that this is the conditionassumed on the basis of macroscopic occupation arguments (equation (29)). The differencesand analogies between the classical fields energy density spectrum and the correspondingquantum one shown in figure 14 visualize the quality of the classical fields method. In ouropinion, the agreement is quite satisfactory.

The above discussion shows the important role of the cut-off of high energy modes.It is necessary if the classical field method pretends to mimic the quantum distribution[27]. Moreover, its value (i.e. the value of the number of classical modes) must grow withtemperature. The discussion, on the other hand, shows the limitation of the method. It clearlyshows that the position of the cut-off momentum is determined by the fundamental principlesup to a numerical factor of the order of unity. Therefore equilibrium temperature as well asparticle number are also given with the same accuracy.

This accuracy of the classical field method might seem unsatisfactory. Therefore, ournext step is to impose some additional criterion which will allow us to reach better precision.This additional criterion cannot be unique. The only constraint is that cut-off energy shouldbe of the order of the system temperature.

In our search for the realistic criterion defining the cut-off we will utilize someexperimental facts. Namely, experiments with weakly interacting trapped atomic gases showthat interactions change only very weakly the dependence of the condensate fraction on thetemperature. The critical temperature of an interacting Bose gas is very close to that of theideal Bose gas, also there is a significant disagreement between various authors over its precisevalue [28]. For typical systems with atomic density about 1013 cm−3 and the scattering lengthabout a = 6 nm the correction to critical temperature is about 5% only. Therefore it seems tobe very natural to equate the temperature of the interacting system with the given condensatefraction n0 to the temperature of the ideal gas with the same condensate fraction. The latter isgiven by

T ideal = 2πh2

mkB

2/3 (1 − n0ζ(3/2)

N

L3

)2/3

, (31)

where ζ is the Riemann function. Comparison of the ideal gas formula with equation (27)allows us to determine the number of particles for which equipartition gives the sametemperature as the ideal gas formula:

N = (1 − n0)2

ζ 2(

32

)π3τ 3

. (32)

This way we can determine all physical parameters of the system. There is now no guaranteehowever that all modes taken into account are macroscopically occupied. Let us recall that thecontrol parameters of the classical fields method are the energy and the cut-off momentum.

We are going to show that the above conditions for the total particle number andtemperature can be directly related to the occupation of the cut-off momentum mode, Nnpmax .To this end, we study a set of equilibrium solutions of equation (20); however, we proceedin a slightly different way than sketched above. We set the number of particles to the valueof N = 235 000 and interaction strength to g = (2π)218.515. We vary total energy andthe number of classical modes. The system equilibrates to the temperature given by theequipartition relation (26). As the value of N is set, we are able to determine absolutepopulations of all classical modes. In figure 15 we present the relative occupation of the

R20 Topical Review

Figure 15. Condensate fraction versus temperature for various numerical grids. The interactionstrength g = (2π)218.515 and the number of atoms N is set to 235 000. Numbers describing thepoints in the figure are the population cut-off Nnmax (left numbers) and the grid size ngrid (numbersin brackets). The population cut-off is known only approximately due to large fluctuations in highlyexcited modes. The optimal value of the cut-off, for which these numerical results agree with thecorresponding ideal Bose gas is about 0.6–0.7.

condensate mode n0 as a function of temperature determined from the equipartition relation(26). Squares correspond to numerical solutions. Numbers in brackets specify the absolutepopulation of the cut-off mode and number of classical modes used in one spatial dimension(total number of classical modes used is equal to the third power of the number of modes ina single principal direction). The numerical points coincide with the corresponding ideal gasformula if the absolute occupation of the highest momentum classical mode is about n0 = 0.6–0.7 in the whole range of relevant temperatures. Such choice of the cut-off population ensuresthat almost all particles of the system are accumulated in the classical modes.

The existence of the universal population of the cut-off mode is in agreement with ourgeneral remarks about analogies with the Planck law. Moreover, the value of the populationof the highest momenta mode agrees perfectly with condition determining the position of themaximum of the energy density spectrum, i.e. 0.63εpmax = T . The last result gives a simplerecipe to the problem of optimal choice of the cut-off momentum and allows for assigningthe equilibrium temperature of the system. The recipe is as follows: (i) for a given valueof gN , choose some value of the total energy and some number of classical modes (cut-offmomentum); (ii) when the system reaches the thermal equilibrium determine τ = T/N fromthe equipartition relation; (iii) having numerical value of relative occupation of the cut-offmode npmax determine the number of particles using relation Nnpmax = 0.6 and then the valueof g; (iv) find the temperature of the system T = Nτ = 0.6εpmax . This temperature is veryclose to the temperature of the ideal gas with the same condensate fraction and total numberof particles.

It might happen that the obtained solution does not correspond to the desired value of theparticle number and interaction strength. That means that a new solution corresponding todifferent energy and different grid size has to be found. After gaining some experience it isnot very difficult to properly adjust the control parameters.

Evidently, the above prescription precludes the determination of the shift of the criticaltemperature due to interactions. There are, however, other ways of implementation of theclassical field method. The other solution to the problem has been proposed by Davis

Topical Review R21

and Morgan [24]. In order to determine the temperature of the system they map the hightemperature version of the Gross–Pitaevskii equation (18) onto the Hamilton equation ofmotion for classical position and momentum variables. The temperature of this classicalsystem can be expressed in terms of a phase space expectation value of a suitable function ofthe canonical position and momentum coordinates [25]. The method allows for very accuratedetermination of the ‘scaled’ temperature τ = T/N and agrees quite well with the methodbased on the equipartition of energy. In order to obtain the shift of the critical temperature,the authors of [24] took a relatively small cut-off momentum, pmax = 15(2π/L), whichensures large occupation of classical modes. Such choice of the cut-off momentum leaveshowever quite a large part of the system outside the classical modes. In order to accountfor these particles one can assume that above cut-off, particles are subject to the ideal gasdistribution. The occupation of the high energy modes decreases exponentially and it is notdifficult to observe that the procedure of [24] is another way of forcing the classical fieldsto mimic Bose–Einstein distribution at high excitation energies. This approach shows thatthe critical temperature for condensation in a homogeneous Bose gas on a lattice with a UVcut-off increases with interaction strength.

There also exists a very interesting version of the classical fields method based on thecanonical ensemble [15]. As in this ensemble the temperature, not the energy, is a controlparameter, the method is free of the difficulties in relating the energy of the system to thetemperature that are extensively discussed here. Instead, one has to propagate numerically anensemble of properly chosen initial conditions. These initial conditions correspond to differentenergies which should be selected according to the Boltzmann factor e−E/kBT . Each initialclassical field is a sum of two parts: the zero temperature solution of the Gross–Pitaevskiiequation for a given fraction of condensed atoms, and some excited particle component. Thisexcited component is generated according to the truncated Wigner distribution [15]. Eachsingle initial condition is therefore a single realization of the system and nonlinear dynamicsdrives the field to the equilibrium state. The expectation values of observables have to beaveraged over the ensemble. Similarly to the microcanonical version of the classical fieldmethod, the canonical approach requires precise selection of the number of classical modes.Note, however, that in experiments with ultra-cold trapped atoms, the atomic systems arealmost perfectly isolated from the environment because a heat reservoir which is as cold as thecondensate does not exist. This suggests that the microcanonical description is more adequatefor such systems.

So far we have discussed the choice of the optimal grid in the case of atoms confined ina box. In more realistic case of the harmonic trap, there is an additional condition found in[14]. For such a trapping potential, the maximal kinetic and potential energy should be equal.

4. Dynamical applications

4.1. Photoassociation of molecules

As the first application of the classical fields approximation we analyse the multimodedynamics of a system of coupled atomic and molecular Bose gases. In 2000 a major steptowards the achievement of a molecular condensate was made in the group of Heinzen [30],who produced first ultracold molecules by photoassociating atoms in an atomic condensate intoa selected internal molecular state. Although three years later the first molecular condensatewas obtained in a degenerate mixture of fermionic atoms with the help of Feshbach resonances[31], the photoassociation technique can still be regarded as a potential route to molecularcondensates with arbitrary molecules (as opposed to the ones built of fermionic atoms) in

R22 Topical Review

Figure 16. Population of the atomic (solid line) and molecular (dotted line) condensates calculatedusing a 2-mode model without atomic collisions. The dashed line is the quantum correction in thefirst excited atomic mode. The parameters are � = 18.432 and N = 105 [32].

arbitrary internal states (as opposed to very weakly bound molecular states condensed in[31]).

The classical fields approximation helps to understand the dynamics of the mixedatomic–molecular systems. In particular, it demonstrates that the 2-mode approach (onlythe condensates are involved) is inadequate as a description of experiments on stimulatedproduction of cold molecules in atomic Bose–Einstein condensates. It shows that for typicalparameters instead of predicted oscillations between the two condensates, excited atomicand molecular states are significantly populated leading to a complete depletion of the initialcondensate on a very short timescale.

The second-quantized Hamiltonian for the hybrid atomic–molecular system confined ina box with periodic boundary conditions may be written as follows [32]:

H =∫

d3r

[�+ p2

2m� + �+ p2

4m�

]+

√V h�

∫d3r[�+�2 + ��+2]

+V hg

2

∫d3r�+�+��, (33)

where � and � are the atomic and molecular field operators, respectively, and � parametrizesa coupling between the two fields. The 2-mode model corresponding to the Hamiltonian(33) can be solved analytically in the absence of atomic collisions and in this case it showsthe conversion from atomic to molecular condensate (see figure 16). However, it turns outthat quantum corrections to such a 2-mode model are divergent for many low-lying states (anexample is given in figure 16). The cure to this problem is just a multimode model derivedfrom (33) based on the classical fields approximation.

A striking discovery of a multimode model is that after a short time both the atomicand molecular condensates are completely depleted (see figure 17) and all particles occupyexcited modes in roughly equal proportions. Typically, only one oscillation in the condensatepopulation survives. Therefore, if one wants to convert an atomic condensate to its molecularcounterpart using the photoassociation, it is necessary to precisely tailor the duration of thecoupling pulse so that the maximum of the molecular condensate mode is picked. Thepopulations of condensate modes as well as all excited modes (of each kind) do not depend onthe initial condition (initial phases) if the number of modes is big enough. This effect resultsfrom self-averaging caused by very many random phases present in the system. Effectivelosses from the condensates are completely due to a Hamiltonian evolution and not becauseof any phenomenologically introduced loss processes.

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Figure 17. Populations of atomic (thick solid line) and molecular (thick dashed line) condensatesand the sums of populations of all excited atomic (thin solid line) and molecular (thin dashed line)modes in the model with 2622 states [32].

Figure 18. Populations of atomic (thick solid line) and molecular (thick dashed line) condensatesand the sums of populations of all excited atomic (thin solid line) and molecular (thin dashed line)modes in a very small box—recovery of the 2-mode dynamics [32].

In order to recover the characteristic oscillatory dynamics of the 2-mode limit within amultimode model, one needs to detune the excited modes. This can be achieved by decreasingthe size of the box (i.e., enlarging a spacing between the excited modes) while keeping the totaldensity fixed (in order not to alter the scattering and coupling parameters). Experimentally,this could be realized by putting the system into an optical lattice. Numerical results areshown in figure 18, where the volume of the box as well as the number of particles has beendecreased by a factor 153 in comparison with the case of figure 17.

4.2. Dissipative dynamics of a vortex

The weakly interacting Bose gas can exhibit the quantized vortices [33–35] revealing in thisway its superfluid properties. The quantization of the circulation, originally predicted byOnsager and Feynman in the context of rotating superfluid 4He [36], is a manifestation of theexistence of a macroscopic wavefunction. Direct 2π -winding of the phase of the condensatewavefunction was experimentally demonstrated by using the interferometric technique [37].The peculiar behaviour of quantized vortices shows, in a way, that they cannot just disappear.They can, however, leave the condensate or annihilate with other vortices having oppositecirculations. In section 4.3 we consider another possibility available for multiply chargedvortices, a way of decaying into vortices with smaller charges.

R24 Topical Review

Figure 19. Sample trajectory of a vortex core. The intensity of lines and dots fades away withtime. Oscillatory units for the length are used [22].

Most experiments with vortices have been performed at essentially zero temperaturecondensates. For example, Madison et al [34] generated vortices in a condensate stirred by alaser beam while the thermal component was below their detection limit. They observed theloss of vortices on the timescale of 500–1000 ms suggesting the possible mechanism to bethe instability induced by the trap anisotropy or the coupling with residual thermal atoms.On the other hand, Anderson et al [38] have seen no trend in moving the vortex towards thecondensate surface within 1 s. The formation and decay of vortex lattices at finite temperatureshave been investigated in [39], showing striking differences between the processes with respectto the temperature and the need of physics beyond the ‘standard’ Gross–Pitaevskii equation.

Therefore, in this section we apply the classical fields approximation to investigate theinstability of a singly quantized vortex induced by the thermal component [22]. As an example,we consider the condensate of 105 87Rb atoms confined in a spherically symmetric trap offrequency 2π × 100 Hz. To generate a vortex, a phase pattern corresponding to a singlequantum of circulation is imprinted on a finite-temperature equilibrium state of an interactingBose gas. Such phase imprinting could be realized experimentally by shining far off-resonantlight on the condensate through a glass plate whose absorption coefficient varies continuouslywith an azimuthal angle. As a result, the atoms acquire a phase depending on their location[40] and the vortex is developed while the density and the phase try to adapt to each other.We place the core of the vortex off the trap centre. The vortex created in this way startsmoving around the trap centre, slowly spiralling out to the surface of the condensate. Thetrajectory of the vortex core is traced by time-averaging the gas density over a short time.A sample trajectory of the vortex is shown in figure 19. In this case, the phase imprintinghas been performed on the equilibrium state with about 48% atoms in the condensate. Theinitial displacement of the vortex core from the trap centre was 2.66 osc units, whereas theThomas–Fermi radius of a condensate containing 48 000 87Rb atoms in our trap is 5.2 oscunits. The vortex disappears reaching roughly this distance from the trap centre. Figure 20

Topical Review R25

Figure 20. Time dependence of the distance of the vortex core from the centre of the trap. Thetime unit is a trap period T [22].

Figure 21. Dependence of the vortex lifetime τ (in units of trap period) on its initial displacementfrom the trap centre xmin (in oscillatory units). The solid line is a fit to equation (34) [22].

shows how the distance of the vortex core from the trap centre changes in time. It is clear thatthe vortex drifts towards the condensate surface with roughly constant velocity although itsmotion is slightly perturbed by the interactions with thermal atoms.

So far the initial displacement of the vortex core was fixed. In the following, we investigatethe dependence of the vortex lifetime on its initial position in a trap. Fedichev and Shlyapnikov[41] have shown that it is the interaction of the thermal cloud with the vortex that provides amechanism of energy dissipation. As a result, an off-centre vortex spirals out to the condensatesurface where it decays through the creation of turbulent excitations. For typical experimentalparameters, they estimate the vortex lifetime to be in the range 0.1–10 s. According to them,

τ ∼ ln(R/xmin), (34)

where R is the spatial size of the condensate and xmin is the initial displacement of the vortexcore from the trap centre. In figure 21 we plot the vortex lifetime as a function of its initialposition as well as a fit to the function (34). One can see that it does agree very well with thepredicted logarithmic functional dependence. However, vortex lifetimes resulting from ourdata are longer by a factor of roughly 5 than the simple estimates of Fedichev and Shlyapnikov.

R26 Topical Review

Figure 22. Decay time of a doubly quantized vortex as a function of the energy per particle.The successive curves correspond to different values of the nonlinearity parameter defined as anz,where a is the scattering length and nz is an axial density at the centre of the trap. From top tobottom, anz equals 12.2, 10.0, 7.2, 5.9, 3.9 and 2.3, respectively [45].

The on-line version of this review contains a movie showing an off-centre vortex decayingdue to thermal noise (available at stacks.iop.org/JPhysB/40/R1).

4.3. Decay of multiply charged vortex

Now, we focus our attention on the problem of instabilities of doubly quantized vortices in acondensate. In experiments [42, 43] doubly charged vortices have been imprinted in a Bose–Einstein condensate of 23Na atoms by using a novel approach theoretically described in [44]. Inthis new method, as opposed to dynamical phase-imprinting techniques, vortices are generatedby adiabatically reversing the magnetic bias field along the trap axis. Resulting vortices displaywinding numbers 2 or 4 depending on the hyperfine state the sodium condensate was preparedin (|1,−1〉 and |2, +2〉, respectively) [42]. The authors of [43] observed the decay of doublyquantized vortex into singly quantized vortices in tens of milliseconds. It turns out also thatthe lifetime of a doubly quantized vortex is a monotonic function of the interaction strength.A natural question arises: what kind of instability leads to the observed decay times?

To answer this question, we have performed numerical simulations using the three-dimensional Gross–Pitaevskii equation as described in section 3, i.e., with a classical thermalnoise [45]. We find that the process of splitting of a doubly quantized vortex is very sensitiveto the presence of thermal atoms (see figure 22). Increasing the number of such atoms by even10–15% reduces the lifetime of the vortex below 100 ms and makes it comparable with theexperimental values [43]. This is demonstrated in figure 23 where we compare the decay timeof a doubly quantized vortex for different levels of thermal noise. The pictures are obtainednumerically according to the tomographic imaging technique as described in [43]. The firsttwo rows show how important are the thermal fluctuations. Introducing the thermal noise ona very low level (the energy of the system is increased by less than 1%) already decreases thetime needed to split the vortex by half. Increasing the level of the thermal noise causes furtherlowering of the decay time, as necessary to get an agreement with the experiment.

Finally, in figure 24 we present the data which, we believe, reproduce the main result ofthe experimental work [43]. Since the presence of less than 15% of thermal atoms cannotbe detected by using currently available experimental techniques, we plot three sets of data,each of them corresponding to different levels of thermal noise. It is clear from figure 24

Topical Review R27

Figure 23. Tomographic images of the condensate that illustrate the dependence of the decay timeon the energy of the system. The interaction strength equals anz = 3.9 and the energy per particleis 234, 235 and 331hωz from top to bottom, respectively. The lowest panel shows the separationof two singly quantized vortices in the case where E/N = 235hωz (squares) and E/N = 331hωz

(triangles) [45].

that increasing thermal fluctuations but being still below (or at the edge of) the detection limitwe get quantitative agreement with experimental results. Figure 24 also shows that the decaytime gets larger when the interaction strength increases. We have investigated other sources ofinstability. For example, the trap anisotropy caused by the gravitational sag makes the lifetimeof a doubly quantized vortex finite but still bigger than the experimental values. Therefore, weclaim that the thermal noise is responsible for the decay of doubly quantized vortices observedin the experiment of [43].

4.4. Quasicondensates

In this section we study the phase coherence of a condensate at finite temperatures. The phasecoherence of three-dimensional condensates at temperatures below the critical temperaturewas studied experimentally using the interference technique [46]. It turned out, however, thatlow-dimensional systems behave in a qualitatively different way. In particular, condensates inone-dimensional traps as well as in elongated three-dimensional traps exhibit, as predicted byPetrov et al [47], strong spatial phase fluctuations (caused by the low-energy excitations)while the density fluctuations are suppressed. Condensates with fluctuating phase (i.e.,quasicondensates) have different coherence properties, for example, their coherence length issmaller than the condensate length. Only decreasing the temperature low enough turns thequasicondensate into the true condensate. Phase-fluctuating condensates have been observedin a series of experiments [48].

First, we investigate the phase fluctuations of a condensate in a pure one-dimensionalharmonic trap [49]. According to the analytical treatment of [47], in the long-wavelength

R28 Topical Review

Figure 24. Decay time of doubly quantized vortex as a function of the density. Three sets of pointscorrespond to different values of the energy brought in the system while introducing the thermalnoise. From top to bottom the relative increase of the energy equals 40% (squares), 85% (triangles)and 135% (stars). The condensate depletion amounts to 6%, 13% and 20%, respectively [45].

limit (low-energy excitations limit) the normalized one-particle density matrix near the centreof the trap decays as

g(1)(z, z′) = 〈ψ †(z)ψ(z′)〉√|ψ(z)|2|ψ(z′)|2

∼ e−|z−z′ |/2Lφ (35)

where the radius of the phase coherence length Lφ is

Lφ = LT 1D

φ

T(36)

and the characteristic temperature T 1Dφ = Tdhωz/µ (where Td = Nhωz/kB is the degeneracy

temperature). When the temperature of a condensate is above T 1Dφ , the phase fluctuations

destroy the long-range phase coherence of the condensate and the phase coherence length getssmaller than the size of the condensate. One obtains in this case a quasicondensate. BelowT 1D

φ , the phase fluctuations are strongly reduced. The decrease of the temperature well belowT 1D

φ turns the quasicondensate continuously to the true condensate. In figure 25 we showthe time-averaged normalized one-particle density matrix for z′ = −z calculated within theclassical fields approximation. The parameters are taken in such a way that the temperature ofthe system is larger than the characteristic temperature T 1D

φ . Fitting the exponential functionexp(−z/2Lφ) to the numerical data we determine the coherence length Lφ ≈ 1 osc unitswhich is much less than the condensate length L ≈ 10 osc units.

Our calculations in a one-dimensional trap show that a number of excited states are highlyoccupied well below the characteristic temperature [50]

T 1Dc = hωz

kB

N

log(2N). (37)

In this temperature region the condensate phase fluctuates and the one-particle density matrixhas several eigenvalues of a comparable size. In figure 26 we confront our numerical datawith the analytical prediction (36). The coherence length increases as the temperature of thecondensate decreases becoming eventually (i.e., at the characteristic temperature T 1D

φ ) equalto the size of the condensate; but, even then, our calculations show, the phase still fluctuates.

Topical Review R29

(a)

(b)

Figure 25. Time-averaged density of the condensate in a one-dimensional trap (a) and the time-averaged normalized one-particle density matrix for z′ = −z (b). The correlation functionis decreasing as the distance between z and z′ is increasing. The energy per atom equalsEtot/N = 34.3hωz (it corresponds to about T = 50hωz/kB) and the length unit is

√h/mωz

[49].

Figure 26. Phase coherence length as a function of the energy per atom in a pure one-dimensionalharmonic trap. The dots represent the numerical data and the horizontal dashed line is the value ofthe length of the condensate in the Thomas–Fermi regime (L ≈ 10 osc units) [49].

R30 Topical Review

Figure 27. Phase coherence length as a function of the trap anisotropy. Initial temperature equalsT = 16hωρ/kB. The vertical dotted line (ωρ/ωz around 8.3) represents the crossover region froma three-dimensional to quasi-one-dimensional system [49].

We have also investigated the phase fluctuations of the condensate in three-dimensionalelongated traps. In this case, however, the coherence length depends, as predicted in [47], notonly on the temperature of the system but also on the geometry of the trap. When the aspectratio of the trap is fixed and large enough, we find again that the coherence length decreaseswhen the temperature gets higher and can be smaller than the condensate length even wellbelow the critical temperature. Next, by adiabatically opening the trap we studied the crossoverbetween the three-dimensional traps and the quasi-one-dimensional traps. In figure 27 we plotthe phase coherence length as a function of trap anisotropy while the temperature of the gasis kept constant. It shows that the coherence length first decreases with increasing the aspectratio up to 8.3 (the region of weak anisotropy) but then for larger anisotropy it continuouslyincreases. Such a behaviour is a signature of the crossover between the three-dimensional andone-dimensional geometry.

4.5. Superfluidity—determination of the critical velocity

In experiments with gaseous Bose–Einstein condensates the existence of quantized vorticesand the properties of scissor modes [33, 34, 51–53] are usually considered as the mainmanifestations of superfluidity. On the other hand, little is known about superfluidity in atranslational motion of the atomic condensate [54–60]. The dissipationless movement of anexternal impurity has been studied both theoretically and experimentally [61, 62]. Still, theproperties of a frictionless non-rotating flow of the condensate with respect to the thermalcloud have been mostly unexplored. Until now, there have been no experiments that wouldillustrate such a flow. However, recent advances in obtaining a condensate in ring-shaped traps[63] could provide the possibility of addressing this issue experimentally.

Therefore, in this section we investigate the dynamics of a superflow between a condensateand a thermal cloud in a weakly interacting Bose gas using the classical fields approximation.We focus on a simplified geometry of a three-dimensional box potential with periodic boundaryconditions which is the best setup to isolate superfluidity from other phenomena. Thisgeometry may be also considered as a simplified model for the novel horizontal ring-shaped

Topical Review R31

(a)

(b)

Figure 28. Fraction of the condensate versus time in the subcritical regime with the initial relativevelocity v = 0.16c in the case where only condensate is pushed (a) and condensate with phononmodes are pushed together (b) [64].

traps. We prepare initial states which will allow us to study the relative motion of the condensateand the thermal cloud. These states correspond to the non-equilibrium situation that arisesfrom applying a momentum kick to the condensate. In order to take into account possibleexperimental difficulties in targeting selectively the condensate mode, we also consider asituation of pushing the condensate together with some thermal atoms.

The Landau model of superfluidity predicts the critical velocity for the frictionless flowto be equal to the min(εp/p). In the case of a weakly interacting Bose gas this gives theLandau critical velocity equal to the sound velocity c (c = √

gNn0), which is just the slopeof the excitation spectrum in the phonon part. In experiments—both in superfluid helium andBose–Einstein condensate in dilute gases the critical velocity tends to be somewhat smaller(usually of the order of 0.1–0.2c) and it depends on the type of experiment (i.e. whetherit is a flow through an external potential or a flow of external impurities). The commonexplanation for this phenomenon invokes vortex nucleation [57] but the phenomenon is stilllargely unexplored, especially in a dynamical situation.

In our numerical experiment [64] we observe that the critical velocity equals vcr = 0.21c.For initial relative velocities below the critical velocity, the flow is frictionless and the systemreaches a superfluid thermal equilibrium. In a re-thermalization process, however, we observea depletion of the condensate (see figure 28(a)) that leads to a reduction of the relative velocitybetween the condensate and the thermal cloud, so that the highest observed stationary superflowhas velocity v

eqcr = 0.12c. This process should not be interpreted as a friction since depleted

atoms have the condensate velocity instead of a velocity of the thermal cloud. Applying themomentum kick to both the condensate and to the phonon modes reduces the re-thermalizationtime and increases slightly the equilibrium condensate fraction (figure 28(b)) but it does noteffect the resulting relative velocity. Such a weak dependence of equilibrium dynamics onhow the system is excited makes it easier to prepare an experiment with a stable flow of the twofractions, as the condensate does not need to be targeted very selectively with the momentumkick.

The stationary superflow exhibits the shifted energy spectrum of quasiparticle excitations(figure 29). As expected, it can be well described by the tilted Popov formula

εp =√

((p − p0)2

2+ gNn0)2 − (gNn0)2 + (p − p0)p0, (38)

where p0 represents the momentum of the condensate mode. Relative energies of quasiparticlesmoving faster than the condensate are shifted upwards, and those of quasiparticles moving

R32 Topical Review

Figure 29. Energy spectrum of quasiparticle excitations in a direction along the momentumkick with the condensate in p0 = (2, 0, 0) mode versus the relative momenta with respect to thecondensate. Dots depict a fit with a tilted Bogoliubov formula and stars represent numerical resultsfor the equilibrium condensate fraction n0 = 0.53 and veq = 0.1c, [64].

Figure 30. Population of the condensate versus time for the supercritical flow for the initialrelative velocity of the condensate and the thermal v = 0.60c. The mode initially occupied bythe condensate becomes depleted due to a friction with the thermal cloud (I) and the condensatereappears in a slower moving mode |p0| = 1 (II) [64].

slower than the condensate are shifted downwards with respect to the stationary state, dueto the Doppler-like shift. This shift, however, depends on the reference frame, due to theGalilean transformation of the momentum p0. This is because in the case of a weaklyinteracting condensate the energy spectrum depends on atomic populations and also becausein the system there exists a mode which is distinguished (the condensate). This is contraryto the energy spectrum of a classical gas, which does not change its shape upon Galileantransformation. Moreover, we find that the superflow fulfils the equipartition, i.e., there existsthe reference frame in which npεp = T/N .

For the relative velocities above the critical velocity the condensate experiences a dragand slows down. It becomes depleted from the initially occupied mode to finally reappearin a different mode (figure 30). The drag manifests itself in changing the velocity ofcondensed atoms, which does not occur in a subcritical regime. This is an important differencebetween this process and the re-thermalization, where the condensate experienced temporaryfluctuations of its size resulting in a slight depletion only. In our simulations the time duringwhich the system reaches thermal equilibrium is of the order of 5 ms and is longer than in a

Topical Review R33

subcritical regime. It turns out that the equilibrium relative velocity between the condensateand the thermal cloud is close to zero, which is in contrast to an intuitive expectation that thecondensate should slow down only to velocities slightly below the critical one.

4.6. Spinor condensates at a nonzero temperature

Finally, we investigate a temperature properties of a spinor condensate of 87Rb atoms in anF = 1 hyperfine state. In the second quantization notation, the Hamiltonian of the system(which is a generalized form of (2)) is given by [65]

H =∫

d3r[ψ

†i (r)H0ψ i(r) +

c0

2ψ

†j (r)ψ

†i (r)ψ i(r)ψj (r) +

c2

2ψ

†k(r)ψ

†i (r)Fij Fklψj (r)ψ l(r)

],

(39)

where the repeated indices (each of them going through the values +1, 0, and −1) are to besummed over. The field operator ψ i(r) annihilates an atom in the hyperfine state |F = 1, i〉at point r. The first term in (39) is the single-particle Hamiltonian that consists of the kineticenergy part and the trapping potential assumed to be independent of the hyperfine state. Theterms with coefficients c0 and c2 describe the spin-independent and spin-dependent parts of thecontact interactions, respectively—c0 and c2 can be expressed with the help of the scatteringlengths a0 and a2 which determine the collision of atoms in a channel of total spin 0 and 2.One has c0 = 4πh2(a0 + 2a2)/3m and c2 = 4πh2(a2 − a0)/3m [65], where m is the mass ofan atom and a0 = 5.387 nm and a2 = 5.313 nm [66]. F are the spin-1 matrices.

According to the classical fields approximation, we replace the field operators ψ i(r) bythe classical wavefunctions ψi(r). The equation of motion for these wavefunctions is writtenas

ih∂

∂t

ψ1

ψ0

ψ−1

= H

ψ1

ψ0

ψ−1

. (40)

The diagonal part of H is given by H11 = H0 + (c0 + c2)|ψ1|2 + (c0 + c2)|ψ0|2 +(c0 − c2)|ψ−1|2,H00 = H0 + (c0 + c2)|ψ1|2 + c0|ψ0|2 + (c0 + c2)|ψ−1|2,H−1−1 = H0 +(c0 − c2)|ψ1|2 + (c0 + c2)|ψ0|2 + (c0 + c2)|ψ−1|2. The off-diagonal terms that describe thecollisions not preserving the projection of spin of each atom (although the total spin projectionis conserved) equal H10 = c2ψ

∗−1ψ0,H0−1 = c2ψ

∗0 ψ1. Moreover, H1−1 = 0.

To study the properties of a spinor condensate at finite temperatures we prepare initially theatomic sample in the mF = 0 component [67]. The Bose gas is confined in a pancake-shapedtrap with the aspect ratio ωz/ωr = 100 and the radial frequency ωr = 2π × 100 Hz. Such ageometry allows us to simplify calculations. In fact, we solve numerically equation (40) in atwo-dimensional space [68] modifying appropriately the interaction strengths. Obviously, dueto the spin changing collisions other Zeeman states (mF = ±1) will be populated. Therefore,it is interesting to investigate the interplay between the processes responsible for the transferof atoms from one component to the other and those leading to the equilibrium of a spinorcondensate. In figure 31 we plot the kinetic energy of each spinor component as a functionof time in the case where the thermal noise introduced to the mF = 0 component leads toapproximately 20% condensate depletion in this component. It turns out that after some timethe kinetic energy becomes equally distributed among spinor components (see figure 31). Thiscondition defines the equilibrium of a spinor condensate and the time needed for the system toenter this regime depends on the initial level of the thermal noise (the higher thermal energy thelonger relaxation time). However, approaching the equilibrium is rather a step-like process,not a uniform one. This is because the atomic cloud becomes fragmented and the overlap

R34 Topical Review

Figure 31. Typical behaviour of a kinetic energy of spinor components. Step-like change ofthe kinetic energy leads eventually to the equipartition of thermal energy among the components.Curves correspond to mF = 0 (red colour) and mF = ±1 (black colour) Zeeman states.

Figure 32. Fraction of total thermal (left panel) and total condensed (right panel) atoms in eachspinor component. Red, green and black (almost completely covered by green) colours denote themF = 0,mF = +1 and mF = −1 Zeeman states, respectively.

of different spin components changes significantly in time. Of course, the steps in figure 31correspond to the maximal overlap.

Spin changing collisions between atoms in the mF = 0 state produce equally atomsin states mF = +1 and mF = −1. Initially, both condensed and thermal atoms go to themF = ±1 components and back (see figure 32). This behaviour changes after the equilibriumis established, i.e., after 0.6 s. From now on, the fraction of thermal atoms in each componentbecomes 1/3 (figure 32, left panel). Since the total number of thermal atoms is also constant(in fact, it fluctuates around some value) it means that the number of thermal atoms in eachcomponent does not change. At the same time a non-trivial spin dynamics for condensed atomsis observed accompanied by the change of the number of condensed atoms in each component(figure 32, right panel). In other words, the spinor condensate decouples from the thermalcloud. In [68] we shall also investigate the coherence properties of the spin components.

5. Conclusions

We have described the classical field approximation for weakly interacting bosons at nonzerotemperatures. We stressed the complex relation between the detailed dynamics of the classicalsystem and the observation. The extraction of measured properties of the system requiresa coarse-graining procedure imitating the finite spatio-temporal resolution of the realisticexperiment.

Topical Review R35

We argued for a particular choice of the necessary cut-off parameter of the method based onthe correspondence with the statistical physics of the ideal Bose gas. This way we also proposea particular way of solving the dilemma of defining the temperature within the classical fieldapproximation. This dilemma is natural for the microcanonical approach via time evolution ofa single copy of the system considered. Note, however, that a single realization of the evolvingquantum system is studied in BEC experiments. Hence, the competing approach, in whicha canonical ensemble of initial conditions is used may smear-out observed features, such asspontaneous fragmentations in the evolving spinor condensates.

In the previous chapter we have also outlined a number of applications of the methodto dynamical processes involving condensates. The method, although approximate, isconceptually very simple, computationally only moderately complex (full 3D dynamics istractable) and yet effective.

Of course classical fields approximation has its limitations. For instance, it is not capableof accounting for higher order spatial correlations of the quantum fields. In the interplaybetween molecular and atomic fields accompanying photoassociation of molecules, two atomsemerging from a decaying molecule are highly correlated—they must be close. This correlationwould require extending the classical equations to include higher cumulants. One can easilywrite down suitable equations but they are much more difficult to solve numerically [69].

Another challenging and as yet unsolved problem would be to force the classical fields toimitate Bose rather than classical distribution of energies of normal modes. A step similar tothe one taken by Planck when he introduced photons and possibly still avoiding full quantumfield theory would be most desirable. It would eliminate the ultraviolet problem and eliminatethe low wavelength cut-off.

Acknowledgments

The authors are very grateful to few ‘generations’ of graduate students and postdocs whocarried out most of the numerical work and who worked with enthusiasm and commitmentat different stages of formulation and application of the classical fields method. These areK Goral, H Schmidt, F Floegel, P Borowski, D Kadio, L Zawitkowski, and K Gawryluk.We owe special thanks to J Mostowski for his valuable discussions on the foundation ofthe classical fields method and to E Witkowska for contributing figure 14. The work wassupported by the Polish Ministry of Scientific Research and Information Technology undergrant no PBZ-MIN-008/P03/2003. We also acknowledge the support of the ESF QUDEDISnetwork. Most of the results have been obtained using computers at the InterdisciplinaryCentre for Mathematical and Computational Modeling of Warsaw University.

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