Microcanonical temperature for a classical field: Application to Bose-Einstein condensation

Morgan, S. A.

doi:10.1103/physreva.68.053615

Cited by 68 publications

(108 citation statements)

References 38 publications

(84 reference statements)

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“…However, the so-called classical field method introduces a methodology that effectively redeploys the GrossPitaevskii equation to account for the effects of quantum and thermal fluctuations [12]. It has previously been shown, for example, that populating randomised incoherent excitations that subsequently undergo conservative dynamics leads to the thermalisation of the system [68][69][70].…”

Section: Connection With Classical Field Methodologymentioning

confidence: 99%

Emergence of Order from Turbulence in an Isolated Planar Superfluid

2014

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We study the relaxation dynamics of an isolated zero temperature quasi-two-dimensional superfluid Bose-Einstein condensate (BEC) that is imprinted with a spatially random distribution of quantum vortices. Following a period of vortex annihilation, we find that the remaining vortices self-organise into two macroscopic coherent 'Onsager vortex' clusters that are stable indefinitely. We demonstrate that this occurs due to a novel physical mechanism -the evaporative heating of the vortices -that results in a negative temperature phase transition in the vortex degrees of freedom. At the end of our simulations the system is trapped in a non-thermal state. Our computational results provide a pathway to observing Onsager vortex states in a superfluid Bose gas.The question of how thermodynamics arises from unitary quantum evolution [1] has been debated since the early days of quantum mechanics. The closest laboratory realisation of an isolated quantum system of many particles is perhaps the ultracold quantum gas, and recent progress in the control and manipulation of these systems mean that the question is no longer simply an academic one [2]. A particular focus has been one-dimensional Bose gases as described by the Lieb-Liniger model [3], as the integrability of the model suggests that it may be prevented from attaining thermal equilibrium following a quench [4]. Indeed, two groundbreaking experiments on the dynamics of one-dimensional Bose gases [5,6] sparked a rush of further activity due to their seemingly contradictory results on whether the experiments returned to standard thermal equilibrium. These experiments have generated significant recent theoretical interest in the nonequilibrium dynamics and relaxation of idealised isolated quantum systems following a disturbance [2,[7][8][9].The quantum relaxation of higher dimensional isolated quantum systems is difficult to address computationally due to their exponential complexity. Recent experiments have quenched the interatomic interaction strength of 3D BECs in oblate and spherical harmonic traps and followed the subsequent dynamics [10,11], including apparent saturation of the momentum distribution [10]. However the relevance of quantum dynamics in their relaxation is not clear. In some situations the classical field approximation [12,13] can provide insight to quench dynamics. Indeed, recent theoretical work has considered the classical equilibration dynamics of 2D superfluids following a quench from the perspectives of turbulence and non-thermal fixed points [14][15][16][17][18][19].The thermalisation of isolated classical systems is generally understood in terms of ergodicity and chaotic dynamics [20]. Even though the equations of motion of a physical system are entirely reversible, if a system with a sufficiently large number of degrees of freedom begins in an 'atypical' state -such as a gas with all particles in one half of the container -it will quickly relax to a more 'typical' state consistent with thermal equilibrium as predicted by statistical mechanics...

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Section: Connection With Classical Field Methodologymentioning

confidence: 99%

Emergence of Order from Turbulence in an Isolated Planar Superfluid

2014

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“…6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30 However, it has brought a rather poor description of condensed Bose particles at finite temperatures. For example, we still do not have a complete agreement on the Bose-Einstein condensation temperature T c of the homogeneous weakly interacting Bose gas under constant density n, 31,32,33,34,35,36,37,38,39,40,41,42,43 not to mention its thermodynamic properties over 0 ≤ T ≤ T c ; see Refs. 42 and 43 for a review on the T c calculations.…”

Section: Introductionmentioning

confidence: 99%

Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Kita

2006

J. Phys. Soc. Jpn.

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This paper presents a conserving gapless mean-field theory for weakly interacting Bose gases. We first construct a mean-field Luttinger-Ward thermodynamic functional in terms of the condensate wave function Ψ and the Nambu Green's functionĜ for the quasiparticle field. Imposing its stationarity respect to Ψ andĜ yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: "conserving" and "gapless." The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length a and particle mass m to clarify its basic thermodynamic properties under two complementary conditions of constant density n and constant pressure p. The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near Tc inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature Tc shows quite a different interaction dependence between the n-fixed and p-fixed cases. In the former case Tc increases from the ideal gas value T0 as Tc/T0 = 1+2.33an 1/3 , whereas it decreases in the latter as Tc/T0 = 1−3.84a(mp/2π 2 ) 1/5 . Temperature dependences of basic thermodynamic quantities are clarified explicitly.

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“…For a cylindrically symmetric harmonic trap these are the total number of particles, the energy, and the component of the angular momentum along the symmetry axis. Once in equilibrium, we use the assumption of ergodicity to accurately determine the condensate fraction [26], and the temperature T and chemical potential b [22]. By varying the initial state energy we measure the dependence of condensate fraction on temperature.…”

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confidence: 99%

“…While in agreement, the theoretical results lie near the upper range of the experimental error bars. Previously one of us used the classical field projected Gross-Pitaevskii equation (PGPE) formalism [19][20][21] to give an estimate of the shift in T c of the homogeneous Bose gas [22], which was found to be in agreement with the Monte Carlo calculations [5,6]. The PGPE is a dynamical nonperturbative method, with the only approximation being that the highly occupied modes (hN k i 1) of the quantum Bose field are well approximated by a classical field evolved according to the GPE.…”

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confidence: 99%

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

2006

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We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature T c of a harmonically confined Bose gas as reported in Gerbier et al., Phys. Rev. Lett. 92, 030405 (2004). The PGPE method includes critical fluctuations and we find the results differ from various mean-field theories, and are in best agreement with experimental data. To unequivocally observe beyond mean-field effects, however, the experimental precision must either improve by an order of magnitude, or consider more strongly interacting systems. This is the first application of a classical field method to make quantitative comparison with experiment. The shift in critical temperature T c with interaction strength for the homogeneous Bose gas has been the subject of numerous studies spanning almost 50 years since the first calculations of Lee and Yang [1,2]. While there is a finite shift to the chemical potential in mean-field (MF) theory, the shift of the critical temperature is zero [3]. The leading order effect is due to long-wavelength critical fluctuations and is inherently nonperturbative. Using effective field theory it was determined that the shift is T c =T 0 c can 1=3 , where n is the particle number density, a is the s-wave scattering length, and c is a constant of order unity [4]. Until recently, results for the value of c disagreed by an order of magnitude and even sign, as summarized in Fig. 1

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Microcanonical temperature for a classical field: Application to Bose-Einstein condensation

Cited by 68 publications

References 38 publications

Emergence of Order from Turbulence in an Isolated Planar Superfluid

Emergence of Order from Turbulence in an Isolated Planar Superfluid

Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

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