Jan 10, 2007 - temperatures. MirosÅaw Brewczyk1, Mariusz Gajda2 and Kazimierz RzaËzewski3. 1 Instytut Fizyki Teoretycznej, Uniwersytet w BiaÅymstoku, ul.
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 nonzero temperatures Mirosław Brewczyk1, Mariusz Gajda2 and Kazimierz Rza¸z˙ ewski3 1 Instytut Fizyki Teoretycznej, Uniwersytet w Białymstoku, ul. Lipowa 41, 15-424 Białystok, Poland 2 Instytut Fizyki PAN, Al. Lotnik´ ow 32/46, 02-668 Warsaw, Poland 3 Centrum Fizyki Teoretycznej PAN, Al. Lotnik´ ow 32/46, 02-668 Warsaw, Poland
Received 7 November 2005, in final form 21 November 2006 Published 10 January 2007 Online at stacks.iop.org/JPhysB/40/R1 Abstract Experiments with Bose–Einstein condensates of dilute atomic gases require temperatures as low as hundreds of nanokelvins but obviously cannot be performed at zero absolute temperature. So the approximate theory of such a gas at nonzero temperatures is needed. In this topical review we describe a classical field approximation which satisfies this need. As modes of light, also modes of atomic field may be treated as classical waves, provided they contain sufficiently many quanta. We present a detailed description of the classical field approximation stressing the significant role of the observation process as the necessary interface between our calculations and measurements. We also discuss in detail the determination of temperature in our approach and stress its limitations. We also review several applications of the classical field approximation to dynamical processes involving atomic condensates. M This article features online multimedia enhancements (Some figures in this article are in colour only in the electronic version)
1. Introduction Almost 80 years ago, Einstein [1], extending the ideas of Bose from photons to massive particles, predicted that at low temperatures the majority of identical, noninteracting particles of 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 population of the ground state was expected at temperatures for which the available energy allows nearly all 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 than that 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 perfectly Einstein’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
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condensation of dilute atomic gases was achieved in 1995 [2]. It triggered an avalanche of both experimental and theoretical papers. To match the experiments with its nearly perfect isolation of a finite atomic sample both in terms of exchange of energy and exchange of particles even the statistical physics of the ideal Bose gas had to be revisited. As was already noticed by Schr¨odinger [3], within the commonly used grand canonical ensemble the condensate exhibits large and unphysical fluctuations even at zero absolute temperature. In fact the experimental situation is probably best approximated by the microcanonical statistical ensemble. So there was need for the solution of thermodynamics in this fundamental ensemble [4]. While the dilute gas BEC experiments come close to the ideal gas case, the residual interaction is very important. It provides a mechanism of thermalization necessary for successful evaporative cooling and, in general, for the establishment of the stationary state of the cloud. The most successful tool of the theory of weakly interacting Bose gas is the nonlinear Schr¨odinger equation, known in this case as the Gross–Pitaevskii equation [5]. It describes the order parameter of the condensed system in the mean field approximation. Its lowest energy state is interpreted as the wavefunction of all atoms at zero temperature. Linearization around the ground state gives normal modes of the elementary collective excitations—quasiparticles of the system [6]. Thus for very low temperatures one usually assumes the macroscopic population 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 conservation of number of particles. Second, it is valid only at very low temperatures, where the thermal depletion of the condensate is negligible. The first weakness is eradicated by the so-called number conserving Bogoliubov approximation [7]. It is harder to do away with the restriction of very low temperature. Ambitious attempts to solve exactly the quantum thermodynamics [8] are numerically so complex that their applications are typically limited to something like 100 atoms in onedimensional (1D) geometry. But, in 1D, there is no phase transition and real experiments on BEC 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 nonzero temperature consists of two distinct parts: quantum mechanical condensate and classical thermal cloud. Of course, the two components do interact. Although effective, two gas models have an obvious weakness: in real life the splitting of the atomic cloud into the condensate and the thermal cloud is a result of Bose statistics and of interactions. So a unified treatment 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 electromagnetic waves. Ever since the discovery of wave solutions of Maxwell equations, physicists have used classical wave theory to describe phenomena involving light. Only a limited number of such phenomena require the quantum theory of light. The main reason for the validity of the classical wave picture is the large number of photons in relevant modes in most realistic situations. We note that a large number of atoms is just a hallmark of BEC experiments. So the classical field is the most natural concept for the BEC matter waves. In section 2 we present the classical fields approximation for the weakly interacting massive bosons. Electromagnetic waves are often only partially coherent. But even in this case the electric and 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
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one. Hence, the measurements, with their limited spatial and temporal resolution, require a coarse graining, a procedure that smoothes out the rapid changes of the incoherent fields. Such a coarse graining, in a natural way, eliminates part of the information contained in the detailed spatiotemporal behaviour of the field. In the realm of quantum theory it makes a mixed state out of the pure one. The analogous need of coarse graining occurs for classical matter waves 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 the thermal states. Planck noticed that photons are needed to tame the ultraviolet divergence of the thermodynamics of classical electromagnetic waves. The analogous divergence plagues also the 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 the thermodynamic equilibrium of the isolated atomic system as the steady state evolution of its single copy. Hence, via an ergodic hypothesis it is related to the microcanonical statistical ensemble. In this ensemble the total energy and the total number of particles are the control parameters. The temperature is just a derived, fluctuating variable. Its value should be obtained via differentiation of the entropy with respect to energy. While some authors, using the ergodic hypothesis, follow this method [10], we find it useful to proceed differently. In section 3 we show how to introduce the temperature into our approximation by first using the equipartition of energy between quasiparticles of the system to compute the ratio of temperature to the total number of atoms and then identifying the absolute temperature of the weakly interacting gas with that of the ideal gas of the same number of atoms. This way we simultaneously fix all the relevant 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 least in the case of atoms confined in a box with periodic boundary conditions one can effectively analyse the quasiparticle excitations. In this non-perturbative approach they not only display temperature dependent shifts with respect to the conventional Bogoliubov formula, but are also broadened. Their width has an obvious interpretation as the inverse of the lifetime. All this is the subject of section 3. In section 4 we describe several direct applications of the classical field approximation to 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 the classical fields method as well as its limitations. We also discuss a possible generalization of the 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 condensate was considered. The classical field approximation as described here, motivated by the optical analogy, and corresponding to the microcanonical ensemble, has been simultaneously proposed in [12–14]. The canonical ensemble version of classical fields, stressing also more formal justification via the truncated Wigner function, was introduced in [15]. The classical fields approximation was also rediscovered in [16] in an approach based on the weak turbulence theory. 2. The approximation Theoretical description of a stationary Bose–Einstein condensate of a weakly interacting gas is a 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 approximate approach in which the splitting of the system into two distinct fractions—the condensate and the 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 is the language of field theory. The N-particle Bose system is assigned a bosonic field operator ˆ (r, t) that destroys a particle at position r at time t and obeys standard equal time bosonic ˆ + (r, t) creating a particle at r at time t: commutation relations with its conjugated partner ˆ ˆ + (r , t)] = δ 3 (r − r ). (1) [(r, t), The N-particle Hamiltonian can be expressed in terms of the field operator 1 3 ˆ+ ˆ ˆ + (r )U (r − r )(r ˆ )(r), ˆ ˆ + (r) H = d r (r)H0 (r)(r) + (2) d3 r d3 r 2 where H0 (r) is a single particle Hamiltonian while U (r − r ) is a two-body interaction potential. In realistic situations of Bose condensates of dilute atomic gases in magnetic and optical dipole traps the two-body interaction is approximated by a contact potential U (r − r ) = gδ 3 (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 4π h ¯ 2a . (4) m A Heisenberg equation originating from the Hamiltonian (2) acquires the form ∂ ˆ ˆ + g ˆ + ˆ . ˆ i¯h (5) = H0 ∂t A full operator solution of (5) is impossible. There are different approaches to the time evolution of an interacting condensate. They lead to numerical algorithms, which are hard to implement [8]. None of them is in fact capable of handling a realistic case of a large number of particles at a relatively high temperature. But this is only the first obstacle which one encounters while studying the interacting Bose system at finite temperatures. It is not easy to extract relevant physical information such as, for example, a mean occupation and fluctuations of a condensate—even having obtained the field operator. To talk about the population of the condensate 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 eigenvector corresponding to a dominant eigenvalue of a one-particle density matrix. This matrix can be expressed with the help of field operators, ˆ + (r, t)(r , t) = Nρ (1) (r, r ; t), (6) g=
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 by many particles. Can we do anything to determine the highly occupied, thus relevant modes even before we know the one particle density matrix? Of help would be a symmetry of the problem. This is the case of a gas trapped in a box with periodic boundary conditions. The symmetry of the confining potential uniquely enforces a form of eigenstates of the thermal equilibrium one-particle density matrix regardless of interaction. Such a density matrix must be periodic with the periodicity of the box and must be translationally invariant, and thus should only depend on the difference of positions: λp fp∗ (r)fp (r ). (7) ρ (1) (r, r ; t) = ρ (1) (r − r ; t) = p
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The above condition indicates that eigenfunctions are simply plane waves, 1 fp (r) = 3/2 exp(−ip · r/¯h), (8) L and the momentum p takes quantized values p = (2π h ¯ /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 condensate corresponds to the p = 0 state of the box, unless a superfluid flow requires otherwise (see section 4). Expansion of the field operator in the basis of the box eigenfunctions, ˆ fp (r)aˆ p (t), (9) (r, t) = p
introduces operators aˆ p aˆ p+ that annihilate (create) a particle in the state fp (r) at time t, and nˆ p = aˆ p+ aˆ p counts particles occupying this state. In this case of a box a one-particle Hamiltonian is p2 , (10) H0 = 2m and the dynamical equation for the operators aˆ p (t) acquire the following form: daˆ p (t) p2 2g + = −i aˆ p − i aˆ aˆ q aˆ p+q1 −q2 . (11) dt 2m¯h Vh ¯ q ,q q1 2 1
2
2
h ¯ The two-body interaction energy g/V and time t are expressed in units of ε = 2mL 2 and 3 ¯ /ε, respectively. V = L is the volume of the box. In the following, if not explicitly τ0 = h stated, we choose ε, τ0 and L to be the natural units of the problem. Obviously some approximations are necessary if one wants to solve the nonlinear operator equations (11). The classical field approximation consists of replacing √ all creation and annihilation operators of highly occupied modes by c-numbers aˆ p → N αp . Such replacement is legitimate if the population of a given mode is greater than its quantum fluctuations, i.e. if np = N αp∗ αp > 1. At zero temperature it is only the p = 0 mode for which this condition is fulfilled. So at zero temperature we are left with just the ground state wavefunction of the corresponding Gross–Pitaevskii equation. All nearly empty modes are neglected in the classical fields approximation. At higher temperatures, however, many modes are macroscopically populated and then thermal fluctuations dominate over the quantum ones. We stress that the thermal fluctuations 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) p2 2gN + = −i αp − i α αq αp+q1 −q2 . (12) dt 2m¯h Vh ¯ q ,q q1 2 1
2
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]. This approach is valid at low temperatures because it assumes that only the p = 0 mode is macroscopically occupied and therefore only the ‘condensed’ mode, described by a complex field, is taken into consideration. Evidently, for a finite system at higher temperatures other modes 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. It corresponds, therefore, to a genuine microcanonical description. Our first goal is to describe the equilibrium state of an interacting system. We tested numerically 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) together with 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-dimensional version of this dynamics is completely integrable [18]. However, in a higher number of dimensions or in the case of more realistic trapping potentials, the evolution may be chaotic. We start 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 evidently does not correspond to the equilibrium distribution of the interacting system. Moreover, there is still a freedom in selecting phases of the coupled fields. In our approach, each mode is assigned an initial, randomly chosen, phase. Any subsequent dynamics depends on initial phases and, in a sense, describes a single realization of the quantum system. Microcanonical expectation values would require the average over the initial phases. Instead of doing so, we trace the system for a sufficiently long time and compute time averages. In fact, time averages rather than ensemble ones are measured experimentally. In addition, the two are equal if a system is ergodic. In our three-dimensional calculations we assumed that N rubidium (87 Rb) atoms are held in a cubic box with size equal to the Thomas–Fermi radius of the same system trapped in a harmonic trap of frequency 2π × 80 Hz. The two-body interaction strength is g/(L3 ε) = (2π )2 2.52 × 10−4 . We performed our calculations with 729 modes. We traced the evolution of the system on a timescale equal approximately to 1 s. The calculations show that after a few milliseconds the system reaches a state of dynamical equilibrium regardless of the choice of initial phases. The occupation of the p = 0 mode stabilizes at some value, and on 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 between the p = 0 mode (a condensate) and higher momentum modes. The amplitude of the condensate mode acquires a steady-state value and its relative fluctuations are small. In contrast, relative fluctuations of higher modes reach 100%. The histogram of steady-state values of the occupation of high momentum thermal modes differs from the corresponding histogram of the p = 0 mode (left panel of figure 1). We see that the latter is well approximated by the exponential function, a clear characteristic of thermal statistical property. In figure 3 we present the histogram of populations of the p = 0 mode for much higher energy. Clearly its thermal character indicates that the system is above its 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 approximated by the exponential function.
Figure 3. Histogram of populations of the p = 0 mode for very high energy. Its thermal character clearly indicates that the system is above its critical temperature
The method presented here is very powerful. A careful reader, however, might be confused. By replacing creation and annihilation operators by c-numbers aˆ p+ , aˆ p → αp∗ , αp , we simultaneously substituted the field operator by a ‘classical wavefunction’ √ ˆ αp fp (r), (13) (r) → N (r) = p
with a finite number of Fourier components, and the set of equations (12) is in fact nothing else but the celebrated Gross–Pitaevskii equation written in the basis of plane waves. This equation traditionally describes a pure condensate at zero temperature. The corresponding one-particle density matrix 1 ∗ (r, t)(r , t), (14) N projects onto a pure state (r, t) which is macroscopically occupied by N atoms. What about our arguments that it is |α0 |2 which gives a relative population of a condensate? What looks like a paradox is quickly resolved if we invoke a measurement process. ρ (1) (r, r , t) =
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The wavefunction (r, t) varies rapidly both in space and time. The complex amplitudes of different modes oscillate very quickly, even when the evolution reaches a stage of dynamical equilibrium. Of course, the detection process is never instantaneous nor does it have perfect spatial 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 , varies from a few microseconds up to several hundreds of microseconds and the size of the pixel is never smaller than some 5 microns, thus observed is a time and space average of a diagonal part of a one-particle reduced density matrix: 1 (1) ρ = ρ (r, r ; t) ,V = dtexp d3 R ∗ (r + R, t + texp )(r + R, t + texp ), (15)
V N
V 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) containing all products of two different modes and only diagonal elements survive: np fp∗ (r)fp (r ). ρ= (16) N p Moreover, if the system reaches a state of dynamical equilibrium averaged occupations of different modes do not depend on time (they depend only on the total energy and the number of particles) and one-particle density matrix acquires a stationary form with time independent mean occupation fractions. It is obvious that the same classical fields approximation can be used in a more realistic case of a condensate trapped in a harmonic potential. In such a case the corresponding dynamical equation for the classical field (r) acquires the form h ¯2 1 ∂(r) (17) = −
+ mω2 r2 + g|(r)|2 (r), i¯h ∂t 2m 2 where ω is the frequency of the harmonic trap. We need to specify the initial condition as complex amplitudes of a finite number of mode functions and let the nonlinear dynamics of the Gross–Pitaevskii equation do the thermalization. By averaging the product of a timedependent solution of the Gross–Pitaevskii equation ∗ (r, t)(r , t) over a finite observation time and volume, we obtain the one-particle density matrix. In the figure 4(a) the one-particle density of the interacting Bose gas trapped in a spherically symmetric harmonic trap is shown at some instance of time. Note the irregular behaviour, which (figure 4(b)) goes away upon coarse graining. In figure 4(b) one can easily see a broad thermal cloud and a sharp peak of the 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, it leaves serious questions unanswered. It does not tell us what is the optimal choice of the finite modes 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 field approximation 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 the possibility of making quantitative predictions about the system. For example, one has to be able to assign relevant physical parameters characterizing the system such as temperature and particle number [19–21].
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(a)
(b)
Figure 4. Picture of a 2D intersection of the single particle instantaneous density of the Bose gas in a spherically symmetric harmonic trap at energy higher than that corresponding to the ground state: (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 the values 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 of the homogeneous system in a box with periodic boundary conditions. This system is a textbook example and a lot is known about its physical features; therefore this is the perfect choice to start with. Analysis of homogeneous system will help us to find an operational procedure for the determination of the essential parameters of the method as well as to examine its quantitative predictions. One of the most important findings discussed in section 2 is that almost any initial condition of a given number of atoms and given energy, after the transient time, leads to the same 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 produced such a state, we can study the thermodynamics of a uniform weakly interacting Bose gas in equilibrium. The c-numbers amplitudes satisfy the equation ∂αp 1 = −i p2 αp − igN αp∗ αm αp+l−m , (18) ∂t 2 l,m where the coupling constant g equals 4π as /L (as being the scattering length). Let us observe ∗ that due to the chosen normalization, p αp αp = 1, the number of particles enters the dynamical equation explicitly through the product of gN . The quantity np = αp αp∗ is obviously the relative occupation of the mode with momentum p. Conservation of the total 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 totalenergy of the system expressed by the c-number amplitudes, acquires the form E = N p (p2 /2)np + (gN/2) p,l,m αp∗ αl∗ αm αp+l−m . As we have already mentioned in section 2, instead of solving equations (18) it is more convenient to transform them to the position representation by introducing the classical field: (r, t) =
pmax
fp (r)αp (t).
(19)
p
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 projected Gross–Pitaevskii equation or finite temperature Gross–Pitaevskii equation) for a system of particles in a box with periodic boundary conditions. The equation is written as i
1 ∂ = − ∇ 2 + gN||2 . ∂t 2
(20)
This equation has to be solved on a rectangular grid and the grid spacing x is related to the value of the cut-off momentum pmax = π/ x. The cut-off momentum has to be properly adjusted depending on the value of the total energy. It is a very important parameter of the method. This fact was first recognized in [12], where a very formal procedure based on introduction of the operator projecting onto classical modes was introduced. We shall elaborate extensively on the role of the cut-off of high frequency modes in the following part of this section. Note that the nonlinear term of the GP equation can generate momentum components up to three times larger than those which are present in the original classical field. One might expect, therefore, that the nonlinear term must be calculated on a grid three times larger than the one used for calculating the wavefunction. Fortunately, as has been shown in [20], the final result is not sensitive to the grid size used for obtaining the nonlinear term, and both the original wavefunction and the nonlinear term can be calculated on the same grid. As described in section (2), the initial wavefunction is generated from the groundstate solution by its random disturbance followed by the normalization. The strength of the disturbance determines the total energy per particle. Both the energy and the particle number 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 time step, do the decomposition of the wavefunction into plane waves as they are the eigenstates of the single particle density matrix. In such a way, we gain the time dependence of amplitudes of natural modes, αp (t). By analysing these amplitudes in the frequency domain (Fourier transform 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 single narrow 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 the chemical potential as a function of a condensate occupation (which is a monotonic function of temperature). We present two curves, both corresponding to the same value of the effective coupling gN = 731.5, but for two different grid sizes. The number of grid points is equal to the number of macroscopically occupied modes. If we increase the number of macroscopically occupied modes while keeping the population of the highest mode constant, we inevitably increase the total number of particles in the system, and simultaneously decrease the value of g (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 by boxes. In figure 8 we present the chemical potential as a function of the interaction strength g at fixed condensate fraction n0 = 0.55 and fixed effective coupling gN = 731.5. According to the 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 horizontal dashed line. Our results show that chemical potential depends on interaction strength not only through the product gN. We expect that two gas model result can be reached in the limit of g → 0; however, calculation in this regime is a demanding task. In the inset we show the chemical 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 for gN = 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 for gN = 731.5 and n0 = 0.07 (high temperature) [21].
number N and interaction strength g. The difference between the two gas model prediction and our result grows with temperature. The spectrum of higher modes is plotted in figures 5 and 6 in the case of low and high energies, respectively, for modes (0, 0, 1) and (3, 3, 3). Typically two groups of frequencies positioned symmetrically with respect to the condensate frequency µ are visible. The width of each 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 coupling strength gN = 731.5. Circles and boxes show numerical results obtained on the grids 36 × 36 × 36 and 16 × 16 × 16, respectively. The solid line comes from the formula derived using the two gas model [21].
Figure 8. Chemical potential as a function of the interaction strength g alone for gN = 731.5 and the condensate occupation n0 = 0.55 (solid line is added to guide the eye). The horizontal dashed 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 function of 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 finite width of each group, we see that every plane wave amplitude oscillates with two frequencies only: αp (t) ≈ e−iµt (Ap e−i p t + Bp∗ ei p t ),
(21)
where Ap and Bp∗ are the amplitudes of positive and negative frequency component, respectively. The separation between the two central frequencies increases with the mode energy. Moreover, increasing the energy of the mode leads also to the suppression of the lower frequencies group, i.e. Bp∗ ≈ 0. By taking appropriate linear combination of amplitudes ∗ one can find normal modes of the system, i.e., modes oscillating with only one αp and α−p frequency: βp (t) = βp (0) e−i(µ+ p ) .
(22)
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Amplitudes βp are related to particle amplitudes αp by the following transformation, ∗ βp eiµt = up αp + vp∗ α−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 become bosonic operators. It is not difficult to discover that the described transformation to the normal modes βp is the transformation to Bogoliubov quasiparticle amplitudes. They can be regarded as elementary excitations of the system. More precisely, equation (22) is approximate. The Bogoliubov mode oscillates here with many frequencies concentrated around the mean frequency µ + p . The centre of the higher frequencies group defines the energy of the quasiparticles while the width of this group is related to its lifetime. This is the result of a residual interaction between them. The Bogoliubov transformation for the finite temperature system requires some care. Because of a finite width of both frequency groups observed in the spectrum the transformation has to be done separately for each individual frequency component. Let us note that for each positive component µ + (where corresponds to the frequency within the peak) there exists a companion at the frequency µ − . The phase difference of these components is equal to the corresponding one for the mode −p. Because of this relation, the Bogoliubov transformation is simplified. The resulting frequency spectrum of the quasiparticles peaks at µ + p . Moreover, relative occupation of the quasiparticle mode can 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 the help of the Bogoliubov transformation. In the particle region the negative frequency group is strongly suppressed and βp (ω) ≈ αp (ω). The excitation energy obtained numerically is well described by the gapless Bogoliubovlike formula: 2 2 p + gN n0 − (gN n0 )2 , (25) p = 2 √ where n0 is the condensate fraction. For small momenta p < gN n0 , the excitation spectrum √ at finite temperature has phonon-like character p = pc, and the velocity of sound c = gN n0 decreases with diminishing population of the condensate mode. The phonon spectrum continuously approaches the standard particle like spectrum in the limit of large momenta, p = p2 /2. This simple generalization of the Bogoliubov formula is known as the Popov approximation [23]. The excellent agreement between the Bogoliubov–Popov expression (25) and the numerical spectrum manifests itself in figure 9, where we plot the quasiparticle energies for various condensate populations and interaction strengths. Numerical points (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 come from (25). Having energies of elementary excitations (25) and their populations (24) we can check how much energy is gathered in every normal mode. If a classical system is in thermal equilibrium then the energy in each mode should be the same. The equipartition of energy is a manifestation of the thermal equilibrium of a classical system. In a quantum system only low energy modes share the same amount of energy. Occupation of high energy modes decreases 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 kB T ,
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Figure 9. Dispersion relation for various interaction strengths and temperatures. Numerical results are 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 shows excellent agreement between the numerical data and formula (25) for all modes in the case of high energies [21].
Figure 10. Quasiparticle energies as a function of the inverse of the population for the system of N = 235 000 (gN = 731.5) atoms for various energies. The upper set of data (circles) was obtained on the grid 36 × 36 × 36 and corresponds to n0 = 0.07, the middle one (triangles) is the case of n0 = 0.79 and 24 × 24 × 24 grid, whereas the lower set (marked by boxes) results from the 16 × 16 × 16 grid and gives n0 = 0.95 [21].
where T is the temperature and kB is the Boltzmann constant. In our numerical simulations these high momentum modes are absent due to the momentum cut-off. In the studied case the equipartition relation reads [19, 20] T (26) np p = , N 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 be linear 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 the state of thermal equilibrium. Moreover, the relation (26) is the basis for determination of the
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temperature. As a matter of fact, the determination of the temperature of the system in the classical field method is much more subtle issue that it seems at the first glance. Note that numerical solution gives a value of the left-hand side of equation (26), i.e. the value of T (27) τ= . N To obtain the temperature one should know a value of particle number. Note that the number of particles enters the dynamical equation via the product gN only and the actual value of N used in the propagation is not uniquely specified. This property is the well-known symmetry of the standard Gross–Pitaevskii equation at zero temperature. All systems with the same value of gN have the same ground state. This is no longer true at higher temperatures. The reason is that, in the latter case, there are two additional parameters, the energy and the number of classical modes (directly related to the cut-off momentum). All these parameters cannot be chosen independently if the assumption of macroscopic occupation has to be fulfilled. To show the dependence of the temperature on the number of classical modes, let us consider a set of equilibrium solutions of the dynamical equations characterized by the same energy and the same value of gN and by a different number of macroscopically occupied modes. Because of the equipartition, which characterizes all these solutions, the same total energy is shared by different numbers of modes. Evidently, the larger the number of modes, the smaller the energy per mode, i.e. the lower the temperature of the system. Counter intuitively, the larger the number of modes used in the numerical simulations, the lower the equilibrium temperature and 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 the cut-off momentum pmax are taken into account. The cut-off momentum has to be properly adjusted depending on the value of the total energy. In the first approximation its value can be obtained via the relation specifying occupation of the least populated pmax mode: N npmax = 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 not smaller than one and can be considered as macroscopic. This choice of the cut-off momentum is not unique and is the result of a compromise: the validity of the classical approximation is limited to modes with occupation large compared to 1, but on the other hand the method pretends to describe the majority of particles in the system. It seems to be natural that the less occupied mode (the mode of the highest momentum) should contain on average about one particle. Moreover, equation (28) allows us to determine the absolute (not scaled) value of the temperature of the system. Indeed, condition (28) applied to the equipartition relation gives pmax = T .
(29)
Knowing the numerical value of relative occupation of the cut-off mode npmax one can obtain the actual value of the total number of particles N = 1/npmax in the system. Evidently, the number 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 for various temperatures for the system built of N = 235 000 atoms. The dashed line in figure 11 shows the population of an ideal gas of the same density. Although we cannot precisely determine the value of the critical temperature of finite size interacting gas, figure 11 clearly shows that the shift of the critical temperature is positive.
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Figure 11. Condensate occupation as a function of temperature for a uniform weakly interacting Bose gas; the scattering length equals as = 25 nm (gN = 731.5, N = 235 000). The dashed line shows the fraction of atoms in the condensate for the ideal gas of the same density (the critical temperature 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 used for precise determination of the magnitude of the shift of the critical temperature. The reason is that in our approach the temperature is proportional to the assumed value of the population of the cut-off momentum mode which we set arbitrarily to one. Smaller values of the population of the cut-off momentum mode give smaller magnitudes of the critical temperature shift (compare figure 15). Evidently, without some additional assumptions the classical fields method can give only order of magnitude estimation of the temperature of the system. The issue of limitations of the method as well as possible solutions to the problem of a more precise determination of the temperature of the system will be discussed in more detail in the following part of this section. Here we add that Davies and Morgan [24] using the classical fields method were able to determine the shift of the critical temperature quite accurately by taking into account higher energy modes and assuming that they are subject to ideal gas distribution. Equipartition of energy is the extremely important feature of the classical fields method. It proves thermal equilibrium and allows the temperature of the system to be determined. There are also other ways of establishing the temperature [25]. They might be even more effective but without equipartition they would be meaningless. Finally, we would like to discuss the damping of the quasiparticle modes. There exists an analytical result for the decay rates of elementary excitations in a uniform Bose gas. At high temperatures (kB T gN/V ) Landau processes dominate and the decay rate for the phonon-like part of the spectrum follows the formula [26] N 3 1/2 p 3π 3/2 kB T as . (30) γp = 2 gN/V V h ¯ Since phonon energies depend linearly on the momentum, so do the decay rates. The ratio √ γp /p is a function of temperature given by ∼ n0 T , where n0 is the condensate fraction. In figure 12 we compare numerical results with the analytical expression (30). In order to determine the damping rates we average the spectrum of phonon-like modes over the angles in the momentum space and fit to the Lorentzian curve. The FWHM of the Lorentzian fit is just the damping rate. We see that both approaches agree quite well except for two first
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Figure 12. Comparison of the ratio γp /p calculated numerically (circles) with that coming from the 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 line follows the analytical formula (30) [21].
points. These points correspond to very low temperatures (approximately 2000 and 4000 in units of h ¯ 2 /(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 we can go beyond the phonon-like excitations and find the decay rates for particle-like part of the spectrum. In figure 13 we plot the damping rates for all momenta in the case of higher temperature. The solid line follows the analytical expression (30). The numerical results and formula (30) agree only for small momenta as expected (see figure 12). The damping rates of particle-like excitations show nonlinear dependence on the momentum. Our example of a homogeneous system proves the usefulness of the classical fields method. Knowing the details of implementation of the method, we are ready now to discuss foundation and limitations. The classical fields method is based on the heuristic substitution of Bose operators by c-number amplitudes. This substitution can be easily justified at very low temperatures since practically all particles occupy the condensate mode. On the other hand, it might seem questionable at temperatures close to the critical temperature. Nevertheless the classical fields
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Figure 14. The energy density spectrum for gN = 900 and T = 1000 obtained from the Bose–Einstein distribution with the quasiparticle spectrum given by equation (25)—full line. For comparison, the same quantity resulting from equipartition of energy for two different populations of the highest momentum mode is shown: npmax = 1—dashed line, npmax = 0.6—dotted-dashed line. Note that the latter is very close to the position of maximum of the energy density distribution which corresponds to npmax = 0.63.
method works quite well up to the condensation point—see figure 11. It is well known that the Bose–Einstein condensation is a quantum phenomenon and the critical temperature depends on the Planck constant. The Planck constant is usually introduced into theory through the commutation relation for the field operators. Those however are violated in the classical fields approach. Nevertheless this approach evidently preserves some bosonic features of the system. Therefore it is justified to ask how come that some quantum features are present in the classical field method. In order to answer this question we will recall the way in which Planck introduced his famous constant h ¯ into theory of electromagnetic radiation at thermal equilibrium. The classical 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 frequency of radiation and agrees with the quantum black-body energy density in the region of small frequencies only. Experimental data clearly show that after reaching a maximum the energy density decreases exponentially in the region of high frequencies. The position of this maximum grows linearly with temperature. In order to account for exponential decrease and 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 h ¯ ωmax ≈ kB T . Let us use a similar reasoning in the case studied. Our approach is based on the dynamics of a classical field. Interactions bring the field to the thermal equilibrium characterized by the equipartition of energy. This equipartition results in quadratic growth of energy density due to the volume of momentum space. This is the inevitable result of the classical theory. The exponential decrease of energy density at high frequencies cannot be obtained in the classical description. To mimic this exponential decay in the classical field method, the occupation of high 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 the form of p = (p2 /2 + gN n0 )2 − (gN n0 )2 (equation (25)). The dashed and dashed-dotted lines correspond to the energy density spectra obtained by assuming equipartition of energy
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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 to the maximum of the quantum distribution, i.e. 0.63 pmax ≈ kB T . Note that this is the condition assumed on the basis of macroscopic occupation arguments (equation (29)). The differences and analogies between the classical fields energy density spectrum and the corresponding quantum one shown in figure 14 visualize the quality of the classical fields method. In our opinion, 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 with temperature. The discussion, on the other hand, shows the limitation of the method. It clearly shows that the position of the cut-off momentum is determined by the fundamental principles up to a numerical factor of the order of unity. Therefore equilibrium temperature as well as particle number are also given with the same accuracy. This accuracy of the classical field method might seem unsatisfactory. Therefore, our next 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 should be of the order of the system temperature. In our search for the realistic criterion defining the cut-off we will utilize some experimental facts. Namely, experiments with weakly interacting trapped atomic gases show that interactions change only very weakly the dependence of the condensate fraction on the temperature. The critical temperature of an interacting Bose gas is very close to that of the ideal Bose gas, also there is a significant disagreement between various authors over its precise value [28]. For typical systems with atomic density about 1013 cm−3 and the scattering length about a = 6 nm the correction to critical temperature is about 5% only. Therefore it seems to be very natural to equate the temperature of the interacting system with the given condensate fraction n0 to the temperature of the ideal gas with the same condensate fraction. The latter is given by 2/3 2π h ¯2 N 2/3 1 − n0 ζ (3/2) 3 , (31) T ideal = mkB L 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 same temperature as the ideal gas formula: N=
(1 − n0 )2 . ζ 2 32 π 3 τ 3
(32)
This way we can determine all physical parameters of the system. There is now no guarantee however that all modes taken into account are macroscopically occupied. Let us recall that the control 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 and temperature can be directly related to the occupation of the cut-off momentum mode, N npmax . To this end, we study a set of equilibrium solutions of equation (20); however, we proceed in a slightly different way than sketched above. We set the number of particles to the value of N = 235 000 and interaction strength to g = (2π )2 18.515. We vary total energy and the number of classical modes. The system equilibrates to the temperature given by the equipartition relation (26). As the value of N is set, we are able to determine absolute populations of all classical modes. In figure 15 we present the relative occupation of the
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Figure 15. Condensate fraction versus temperature for various numerical grids. The interaction strength g = (2π )2 18.515 and the number of atoms N is set to 235 000. Numbers describing the points in the figure are the population cut-off N nmax (left numbers) and the grid size ngrid (numbers in brackets). The population cut-off is known only approximately due to large fluctuations in highly excited modes. The optimal value of the cut-off, for which these numerical results agree with the corresponding 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 absolute population 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 in a single principal direction). The numerical points coincide with the corresponding ideal gas formula 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 ensures that 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 our general remarks about analogies with the Planck law. Moreover, the value of the population of the highest momenta mode agrees perfectly with condition determining the position of the maximum of the energy density spectrum, i.e. 0.63 pmax = T . The last result gives a simple recipe to the problem of optimal choice of the cut-off momentum and allows for assigning the equilibrium temperature of the system. The recipe is as follows: (i) for a given value of gN, choose some value of the total energy and some number of classical modes (cut-off momentum); (ii) when the system reaches the thermal equilibrium determine τ = T /N from the equipartition relation; (iii) having numerical value of relative occupation of the cut-off mode npmax determine the number of particles using relation N npmax = 0.6 and then the value of g; (iv) find the temperature of the system T = N τ = 0.6 pmax . This temperature is very close to the temperature of the ideal gas with the same condensate fraction and total number of particles. It might happen that the obtained solution does not correspond to the desired value of the particle number and interaction strength. That means that a new solution corresponding to different energy and different grid size has to be found. After gaining some experience it is not very difficult to properly adjust the control parameters. Evidently, the above prescription precludes the determination of the shift of the critical temperature due to interactions. There are, however, other ways of implementation of the classical field method. The other solution to the problem has been proposed by Davis
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and Morgan [24]. In order to determine the temperature of the system they map the high temperature version of the Gross–Pitaevskii equation (18) onto the Hamilton equation of motion for classical position and momentum variables. The temperature of this classical system can be expressed in terms of a phase space expectation value of a suitable function of the canonical position and momentum coordinates [25]. The method allows for very accurate determination of the ‘scaled’ temperature τ = T /N and agrees quite well with the method based 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), which ensures large occupation of classical modes. Such choice of the cut-off momentum leaves however quite a large part of the system outside the classical modes. In order to account for these particles one can assume that above cut-off, particles are subject to the ideal gas distribution. The occupation of the high energy modes decreases exponentially and it is not difficult to observe that the procedure of [24] is another way of forcing the classical fields to mimic Bose–Einstein distribution at high excitation energies. This approach shows that the critical temperature for condensation in a homogeneous Bose gas on a lattice with a UV cut-off increases with interaction strength. There also exists a very interesting version of the classical fields method based on the canonical ensemble [15]. As in this ensemble the temperature, not the energy, is a control parameter, the method is free of the difficulties in relating the energy of the system to the temperature that are extensively discussed here. Instead, one has to propagate numerically an ensemble of properly chosen initial conditions. These initial conditions correspond to different energies which should be selected according to the Boltzmann factor e−E/kB T . Each initial classical field is a sum of two parts: the zero temperature solution of the Gross–Pitaevskii equation for a given fraction of condensed atoms, and some excited particle component. This excited component is generated according to the truncated Wigner distribution [15]. Each single initial condition is therefore a single realization of the system and nonlinear dynamics drives the field to the equilibrium state. The expectation values of observables have to be averaged over the ensemble. Similarly to the microcanonical version of the classical field method, 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 are almost perfectly isolated from the environment because a heat reservoir which is as cold as the condensate does not exist. This suggests that the microcanonical description is more adequate for such systems. So far we have discussed the choice of the optimal grid in the case of atoms confined in a 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 multimode dynamics of a system of coupled atomic and molecular Bose gases. In 2000 a major step towards 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 into a selected internal molecular state. Although three years later the first molecular condensate was 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 molecular condensates with arbitrary molecules (as opposed to the ones built of fermionic atoms) in
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Figure 16. Population of the atomic (solid line) and molecular (dotted line) condensates calculated using a 2-mode model without atomic collisions. The dashed line is the quantum correction in the first 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 mixed atomic–molecular systems. In particular, it demonstrates that the 2-mode approach (only the condensates are involved) is inadequate as a description of experiments on stimulated production of cold molecules in atomic Bose–Einstein condensates. It shows that for typical parameters instead of predicted oscillations between the two condensates, excited atomic and molecular states are significantly populated leading to a complete depletion of the initial condensate on a very short timescale. The second-quantized Hamiltonian for the hybrid atomic–molecular system confined in a box with periodic boundary conditions may be written as follows [32]:
√ p2 p2 + + + Vh ¯ d3 r[+ 2 + +2 ] H = d3 r + 2m 4m Vh ¯g (33) + d3 r + + , 2 where and are the atomic and molecular field operators, respectively, and parametrizes a 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 shows the conversion from atomic to molecular condensate (see figure 16). However, it turns out that quantum corrections to such a 2-mode model are divergent for many low-lying states (an example is given in figure 16). The cure to this problem is just a multimode model derived from (33) based on the classical fields approximation. A striking discovery of a multimode model is that after a short time both the atomic and molecular condensates are completely depleted (see figure 17) and all particles occupy excited modes in roughly equal proportions. Typically, only one oscillation in the condensate population survives. Therefore, if one wants to convert an atomic condensate to its molecular counterpart using the photoassociation, it is necessary to precisely tailor the duration of the coupling pulse so that the maximum of the molecular condensate mode is picked. The populations of condensate modes as well as all excited modes (of each kind) do not depend on the initial condition (initial phases) if the number of modes is big enough. This effect results from self-averaging caused by very many random phases present in the system. Effective losses from the condensates are completely due to a Hamiltonian evolution and not because of any phenomenologically introduced loss processes.
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Figure 17. Populations of atomic (thick solid line) and molecular (thick dashed line) condensates and 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) condensates and 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 a multimode model, one needs to detune the excited modes. This can be achieved by decreasing the size of the box (i.e., enlarging a spacing between the excited modes) while keeping the total density 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 are shown in figure 18, where the volume of the box as well as the number of particles has been decreased 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 this way its superfluid properties. The quantization of the circulation, originally predicted by Onsager and Feynman in the context of rotating superfluid 4 He [36], is a manifestation of the existence of a macroscopic wavefunction. Direct 2π -winding of the phase of the condensate wavefunction 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 opposite circulations. In section 4.3 we consider another possibility available for multiply charged vortices, a way of decaying into vortices with smaller charges.
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Figure 19. Sample trajectory of a vortex core. The intensity of lines and dots fades away with time. Oscillatory units for the length are used [22].
Most experiments with vortices have been performed at essentially zero temperature condensates. For example, Madison et al [34] generated vortices in a condensate stirred by a laser beam while the thermal component was below their detection limit. They observed the loss of vortices on the timescale of 500–1000 ms suggesting the possible mechanism to be the 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 the condensate surface within 1 s. The formation and decay of vortex lattices at finite temperatures have been investigated in [39], showing striking differences between the processes with respect to 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 the instability of a singly quantized vortex induced by the thermal component [22]. As an example, we consider the condensate of 105 87 Rb atoms confined in a spherically symmetric trap of frequency 2π × 100 Hz. To generate a vortex, a phase pattern corresponding to a single quantum of circulation is imprinted on a finite-temperature equilibrium state of an interacting Bose gas. Such phase imprinting could be realized experimentally by shining far off-resonant light on the condensate through a glass plate whose absorption coefficient varies continuously with 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 starts moving around the trap centre, slowly spiralling out to the surface of the condensate. The trajectory 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 imprinting has been performed on the equilibrium state with about 48% atoms in the condensate. The initial displacement of the vortex core from the trap centre was 2.66 osc units, whereas the Thomas–Fermi radius of a condensate containing 48 000 87 Rb atoms in our trap is 5.2 osc units. The vortex disappears reaching roughly this distance from the trap centre. Figure 20
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Figure 20. Time dependence of the distance of the vortex core from the centre of the trap. The time unit is a trap period T [22].
Figure 21. Dependence of the vortex lifetime τ (in units of trap period) on its initial displacement from 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 that the vortex drifts towards the condensate surface with roughly constant velocity although its motion is slightly perturbed by the interactions with thermal atoms. So far the initial displacement of the vortex core was fixed. In the following, we investigate the 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 a mechanism of energy dissipation. As a result, an off-centre vortex spirals out to the condensate surface where it decays through the creation of turbulent excitations. For typical experimental parameters, 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 vortex core from the trap centre. In figure 21 we plot the vortex lifetime as a function of its initial position as well as a fit to the function (34). One can see that it does agree very well with the predicted logarithmic functional dependence. However, vortex lifetimes resulting from our data are longer by a factor of roughly 5 than the simple estimates of Fedichev and Shlyapnikov.
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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 to bottom, 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 decaying due 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 a condensate. In experiments [42, 43] doubly charged vortices have been imprinted in a Bose– Einstein condensate of 23 Na atoms by using a novel approach theoretically described in [44]. In this new method, as opposed to dynamical phase-imprinting techniques, vortices are generated by adiabatically reversing the magnetic bias field along the trap axis. Resulting vortices display winding numbers 2 or 4 depending on the hyperfine state the sodium condensate was prepared in (|1, −1 and |2, +2, respectively) [42]. The authors of [43] observed the decay of doubly quantized vortex into singly quantized vortices in tens of milliseconds. It turns out also that the 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 threedimensional Gross–Pitaevskii equation as described in section 3, i.e., with a classical thermal noise [45]. We find that the process of splitting of a doubly quantized vortex is very sensitive to the presence of thermal atoms (see figure 22). Increasing the number of such atoms by even 10–15% reduces the lifetime of the vortex below 100 ms and makes it comparable with the experimental values [43]. This is demonstrated in figure 23 where we compare the decay time of a doubly quantized vortex for different levels of thermal noise. The pictures are obtained numerically according to the tomographic imaging technique as described in [43]. The first two rows show how important are the thermal fluctuations. Introducing the thermal noise on a very low level (the energy of the system is increased by less than 1%) already decreases the time needed to split the vortex by half. Increasing the level of the thermal noise causes further lowering 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 of the experimental work [43]. Since the presence of less than 15% of thermal atoms cannot be 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
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Figure 23. Tomographic images of the condensate that illustrate the dependence of the decay time on the energy of the system. The interaction strength equals anz = 3.9 and the energy per particle is 234, 235 and 331¯hωz from top to bottom, respectively. The lowest panel shows the separation of two singly quantized vortices in the case where E/N = 235¯hωz (squares) and E/N = 331¯hωz (triangles) [45].
that increasing thermal fluctuations but being still below (or at the edge of) the detection limit we get quantitative agreement with experimental results. Figure 24 also shows that the decay time gets larger when the interaction strength increases. We have investigated other sources of instability. For example, the trap anisotropy caused by the gravitational sag makes the lifetime of a doubly quantized vortex finite but still bigger than the experimental values. Therefore, we claim that the thermal noise is responsible for the decay of doubly quantized vortices observed in the experiment of [43]. 4.4. Quasicondensates In this section we study the phase coherence of a condensate at finite temperatures. The phase coherence of three-dimensional condensates at temperatures below the critical temperature was studied experimentally using the interference technique [46]. It turned out, however, that low-dimensional systems behave in a qualitatively different way. In particular, condensates in one-dimensional traps as well as in elongated three-dimensional traps exhibit, as predicted by Petrov 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 is smaller than the condensate length. Only decreasing the temperature low enough turns the quasicondensate into the true condensate. Phase-fluctuating condensates have been observed in a series of experiments [48]. First, we investigate the phase fluctuations of a condensate in a pure one-dimensional harmonic trap [49]. According to the analytical treatment of [47], in the long-wavelength
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Figure 24. Decay time of doubly quantized vortex as a function of the density. Three sets of points correspond to different values of the energy brought in the system while introducing the thermal noise. 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 centre of the trap decays as (z ) † (z)ψ ψ g (1) (z, z ) = ∼ e−|z−z |/2Lφ 2 2 (z)| |ψ (z )| |ψ
(35)
where the radius of the phase coherence length Lφ is Lφ = L
Tφ1D T
(36)
and the characteristic temperature Tφ1D = Tdh ¯ ωz /µ (where Td = N h ¯ ωz /kB is the degeneracy temperature). When the temperature of a condensate is above Tφ1D , the phase fluctuations destroy the long-range phase coherence of the condensate and the phase coherence length gets smaller than the size of the condensate. One obtains in this case a quasicondensate. Below Tφ1D , the phase fluctuations are strongly reduced. The decrease of the temperature well below Tφ1D turns the quasicondensate continuously to the true condensate. In figure 25 we show the time-averaged normalized one-particle density matrix for z = −z calculated within the classical fields approximation. The parameters are taken in such a way that the temperature of the system is larger than the characteristic temperature Tφ1D . Fitting the exponential function exp(−z/2Lφ ) to the numerical data we determine the coherence length Lφ ≈ 1 osc units which 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 highly occupied well below the characteristic temperature [50] Tc1D =
N h ¯ ωz . kB log(2N )
(37)
In this temperature region the condensate phase fluctuates and the one-particle density matrix has several eigenvalues of a comparable size. In figure 26 we confront our numerical data with the analytical prediction (36). The coherence length increases as the temperature of the condensate decreases becoming eventually (i.e., at the characteristic temperature Tφ1D ) equal to the size of the condensate; but, even then, our calculations show, the phase still fluctuates.
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(a)
(b)
Figure 25. Time-averaged density of the condensate in a one-dimensional trap (a) and the timeaveraged normalized one-particle density matrix for z = −z (b). The correlation function is decreasing as the distance between z and z is increasing. The energy per atom √ equals Etot /N = 34.3¯hωz (it corresponds to about T = 50¯hωz /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-dimensional harmonic trap. The dots represent the numerical data and the horizontal dashed line is the value of the length of the condensate in the Thomas–Fermi regime (L ≈ 10 osc units) [49].
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Figure 27. Phase coherence length as a function of the trap anisotropy. Initial temperature equals T = 16¯hωρ /kB . The vertical dotted line (ωρ /ωz around 8.3) represents the crossover region from a three-dimensional to quasi-one-dimensional system [49].
We have also investigated the phase fluctuations of the condensate in three-dimensional elongated traps. In this case, however, the coherence length depends, as predicted in [47], not only on the temperature of the system but also on the geometry of the trap. When the aspect ratio of the trap is fixed and large enough, we find again that the coherence length decreases when the temperature gets higher and can be smaller than the condensate length even well below the critical temperature. Next, by adiabatically opening the trap we studied the crossover between the three-dimensional traps and the quasi-one-dimensional traps. In figure 27 we plot the phase coherence length as a function of trap anisotropy while the temperature of the gas is kept constant. It shows that the coherence length first decreases with increasing the aspect ratio up to 8.3 (the region of weak anisotropy) but then for larger anisotropy it continuously increases. Such a behaviour is a signature of the crossover between the three-dimensional and one-dimensional geometry. 4.5. Superfluidity—determination of the critical velocity In experiments with gaseous Bose–Einstein condensates the existence of quantized vortices and the properties of scissor modes [33, 34, 51–53] are usually considered as the main manifestations of superfluidity. On the other hand, little is known about superfluidity in a translational motion of the atomic condensate [54–60]. The dissipationless movement of an external impurity has been studied both theoretically and experimentally [61, 62]. Still, the properties of a frictionless non-rotating flow of the condensate with respect to the thermal cloud have been mostly unexplored. Until now, there have been no experiments that would illustrate 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 condensate and 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 boundary conditions which is the best setup to isolate superfluidity from other phenomena. This geometry may be also considered as a simplified model for the novel horizontal ring-shaped
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Figure 28. Fraction of the condensate versus time in the subcritical regime with the initial relative velocity v = 0.16c in the case where only condensate is pushed (a) and condensate with phonon modes are pushed together (b) [64].
traps. We prepare initial states which will allow us to study the relative motion of the condensate and the thermal cloud. These states correspond to the non-equilibrium situation that arises from applying a momentum kick to the condensate. In order to take into account possible experimental difficulties in targeting selectively the condensate mode, we also consider a situation of pushing the condensate together with some thermal atoms. The Landau model of superfluidity predicts the critical velocity for the frictionless flow Bose gas this gives the to be equal to the min( p /p). In the case of a weakly interacting √ Landau critical velocity equal to the sound velocity c (c = gN n0 ), which is just the slope of the excitation spectrum in the phonon part. In experiments—both in superfluid helium and Bose–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. whether it is a flow through an external potential or a flow of external impurities). The common explanation for this phenomenon invokes vortex nucleation [57] but the phenomenon is still largely 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 system reaches a superfluid thermal equilibrium. In a re-thermalization process, however, we observe a depletion of the condensate (see figure 28(a)) that leads to a reduction of the relative velocity between the condensate and the thermal cloud, so that the highest observed stationary superflow eq has velocity vcr = 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 the momentum kick to both the condensate and to the phonon modes reduces the re-thermalization time and increases slightly the equilibrium condensate fraction (figure 28(b)) but it does not effect the resulting relative velocity. Such a weak dependence of equilibrium dynamics on how the system is excited makes it easier to prepare an experiment with a stable flow of the two fractions, as the condensate does not need to be targeted very selectively with the momentum kick. 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 (p − p0 )2 + gN n0 )2 − (gN n0 )2 + (p − p0 )p0 , (38) p = ( 2 where p0 represents the momentum of the condensate mode. Relative energies of quasiparticles moving faster than the condensate are shifted upwards, and those of quasiparticles moving
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Figure 29. Energy spectrum of quasiparticle excitations in a direction along the momentum kick with the condensate in p0 = (2, 0, 0) mode versus the relative momenta with respect to the condensate. Dots depict a fit with a tilted Bogoliubov formula and stars represent numerical results for the equilibrium condensate fraction n0 = 0.53 and v eq = 0.1c, [64].
Figure 30. Population of the condensate versus time for the supercritical flow for the initial relative velocity of the condensate and the thermal v = 0.60c. The mode initially occupied by the condensate becomes depleted due to a friction with the thermal cloud (I) and the condensate reappears in a slower moving mode | p0 | = 1 (II) [64].
slower than the condensate are shifted downwards with respect to the stationary state, due to the Doppler-like shift. This shift, however, depends on the reference frame, due to the Galilean transformation of the momentum p0 . This is because in the case of a weakly interacting condensate the energy spectrum depends on atomic populations and also because in the system there exists a mode which is distinguished (the condensate). This is contrary to the energy spectrum of a classical gas, which does not change its shape upon Galilean transformation. Moreover, we find that the superflow fulfils the equipartition, i.e., there exists the reference frame in which np p = T /N. For the relative velocities above the critical velocity the condensate experiences a drag and slows down. It becomes depleted from the initially occupied mode to finally reappear in a different mode (figure 30). The drag manifests itself in changing the velocity of condensed atoms, which does not occur in a subcritical regime. This is an important difference between this process and the re-thermalization, where the condensate experienced temporary fluctuations of its size resulting in a slight depletion only. In our simulations the time during which the system reaches thermal equilibrium is of the order of 5 ms and is longer than in a
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subcritical regime. It turns out that the equilibrium relative velocity between the condensate and the thermal cloud is close to zero, which is in contrast to an intuitive expectation that the condensate 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 87 Rb atoms in an F = 1 hyperfine state. In the second quantization notation, the Hamiltonian of the system (which is a generalized form of (2)) is given by [65] c0 c2 H = d3 r ψˆ †i (r)H0 ψˆ i (r) + ψˆ †j (r)ψˆ †i (r)ψˆ i (r)ψˆ j (r) + ψˆ †k (r)ψˆ †i (r)Fij Fkl ψˆ j (r)ψˆ l (r) , 2 2 (39) where the repeated indices (each of them going through the values +1, 0, and −1) are to be summed over. The field operator ψˆ 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 kinetic energy part and the trapping potential assumed to be independent of the hyperfine state. The terms with coefficients c0 and c2 describe the spin-independent and spin-dependent parts of the contact interactions, respectively—c0 and c2 can be expressed with the help of the scattering lengths a0 and a2 which determine the collision of atoms in a channel of total spin 0 and 2. ¯ 2 (a0 + 2a2 )/3m and c2 = 4π h ¯ 2 (a2 − a0 )/3m [65], where m is the mass of One has c0 = 4π h an 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 ψˆ i (r) by the classical wavefunctions ψi (r). The equation of motion for these wavefunctions is written as ψ1 ψ ∂ 1 ψ0 = H ψ0 . (40) i¯h ∂t ψ−1 ψ−1 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 the collisions not preserving the projection of spin of each atom (although the total spin projection ∗ ψ0 , H0−1 = c2 ψ0∗ ψ1 . Moreover, H1−1 = 0. is conserved) equal H10 = c2 ψ−1 To study the properties of a spinor condensate at finite temperatures we prepare initially the atomic sample in the mF = 0 component [67]. The Bose gas is confined in a pancake-shaped trap with the aspect ratio ωz /ωr = 100 and the radial frequency ωr = 2π × 100 Hz. Such a geometry allows us to simplify calculations. In fact, we solve numerically equation (40) in a two-dimensional space [68] modifying appropriately the interaction strengths. Obviously, due to 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 transfer of atoms from one component to the other and those leading to the equilibrium of a spinor condensate. In figure 31 we plot the kinetic energy of each spinor component as a function of time in the case where the thermal noise introduced to the mF = 0 component leads to approximately 20% condensate depletion in this component. It turns out that after some time the kinetic energy becomes equally distributed among spinor components (see figure 31). This condition defines the equilibrium of a spinor condensate and the time needed for the system to enter this regime depends on the initial level of the thermal noise (the higher thermal energy the longer 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
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Figure 31. Typical behaviour of a kinetic energy of spinor components. Step-like change of the 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 each spinor component. Red, green and black (almost completely covered by green) colours denote the mF = 0, mF = +1 and mF = −1 Zeeman states, respectively.
of different spin components changes significantly in time. Of course, the steps in figure 31 correspond to the maximal overlap. Spin changing collisions between atoms in the mF = 0 state produce equally atoms in states mF = +1 and mF = −1. Initially, both condensed and thermal atoms go to the mF = ±1 components and back (see figure 32). This behaviour changes after the equilibrium is established, i.e., after 0.6 s. From now on, the fraction of thermal atoms in each component becomes 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 each component does not change. At the same time a non-trivial spin dynamics for condensed atoms is 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 thermal cloud. 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 nonzero temperatures. We stressed the complex relation between the detailed dynamics of the classical system and the observation. The extraction of measured properties of the system requires a coarse-graining procedure imitating the finite spatio-temporal resolution of the realistic experiment.
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We argued for a particular choice of the necessary cut-off parameter of the method based on the correspondence with the statistical physics of the ideal Bose gas. This way we also propose a particular way of solving the dilemma of defining the temperature within the classical field approximation. This dilemma is natural for the microcanonical approach via time evolution of a single copy of the system considered. Note, however, that a single realization of the evolving quantum system is studied in BEC experiments. Hence, the competing approach, in which a canonical ensemble of initial conditions is used may smear-out observed features, such as spontaneous fragmentations in the evolving spinor condensates. In the previous chapter we have also outlined a number of applications of the method to dynamical processes involving condensates. The method, although approximate, is conceptually very simple, computationally only moderately complex (full 3D dynamics is tractable) and yet effective. Of course classical fields approximation has its limitations. For instance, it is not capable of accounting for higher order spatial correlations of the quantum fields. In the interplay between molecular and atomic fields accompanying photoassociation of molecules, two atoms emerging from a decaying molecule are highly correlated—they must be close. This correlation would require extending the classical equations to include higher cumulants. One can easily write 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 to imitate Bose rather than classical distribution of energies of normal modes. A step similar to the one taken by Planck when he introduced photons and possibly still avoiding full quantum field theory would be most desirable. It would eliminate the ultraviolet problem and eliminate the low wavelength cut-off. Acknowledgments The authors are very grateful to few ‘generations’ of graduate students and postdocs who carried out most of the numerical work and who worked with enthusiasm and commitment at different stages of formulation and application of the classical fields method. These are K G´oral, 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 of the classical fields method and to E Witkowska for contributing figure 14. The work was supported by the Polish Ministry of Scientific Research and Information Technology under grant no PBZ-MIN-008/P03/2003. We also acknowledge the support of the ESF QUDEDIS network. Most of the results have been obtained using computers at the Interdisciplinary Centre for Mathematical and Computational Modeling of Warsaw University. References [1] Bose S N 1924 Z. Phys. 26 178 Einstein A 1924 Sitzber. Kgl. Preuss. Akad. Wiss. 22 261 [2] Anderson M H, Ensher J H, Mattews M R, Wieman C E and Cornell E A 1995 Science 269 198 Davies K B, Mewes M-O, Andrews M R, van Druten N J, Durfee D S, Kurn D M and Ketterle W 1995 Phys. Rev. Lett. 75 3969 [3] Schr¨odinger E 1989 Statistical Thermodynamics (New York: Dover) Fujiwara I, ter Haar D and Wergeland H 1970 J. Stat. Phys. 2 329 [4] Navez P, Bitouk D, Gajda M, Idziaszek Z and Rza¸z˙ ewski K 1997 Phys. Rev. Lett. 79 1789 [5] Pitaevskii L P 1961 Zh. Eksp. Teor. Fiz. 40 646 Gross E P 1961 Nuovo Cimento 20 454 [6] Bogoliubov N N 1947 J. Phys. Moscow 11 231
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