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arXiv:cond-mat/0209637 · Journal of Optics B Quantum and Semiclassical Optics

Probing the classical field approximation thermodynamics and decaying vortices

Harry Schmidt, Krzysztof G ral, Filip Floegel, Mariusz Gajda, Kazimierz Rza̧żewski

Published Apr 1, 2003• 40 citations
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Abstract

We review our version of the classical field approximation to the dynamics of a finite temperature Bose gas. In the case of a periodic box potential, we investigate the role of the high momentum cut-off, essential in the method. In particular, we show that the cut-off going to infinity limit decribes the particle number going to infinity with the scattering length going to zero. In this weak interaction limit, the relative population of the condensate tends to unity. We also show that the cross-over energy, at which the probability distribution of the condensate occupation changes its character, grows with a growing scattering length. In the more physical case of the condensate in the harmonic trap we investigate the dissipative dynamics of a vortex. We compare the decay time and the velocities of the vortex with the available analytic estimates.

DOI
10.1088/1464-4266/5/2/364

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We review our version of the classical field approximation to the dynamics of a finite temperature Bose gas. In the case of a periodic box potential, we investigate the role of the high momentum cut-off, essential in the method. In particular, we show that the cut-off going to infinity limit decribes the particle number going to infinity with the scattering length going to zero. In this weak interaction limit, the relative population of the condensate tends to unity. We also show that the cross-over energy, at which the probability distribution of the condensate occupation changes its character, grows with a growing scattering length. In the more physical case of the condensate in the harmonic trap we investigate the dissipative dynamics of a vortex. We compare the decay time and the velocities of the vortex with the available analytic estimates.
A classical field approximation to the finite temperature microcanonical thermodynamics of weakly interacting Bose gases is applied to the idealized case of atoms confined in a box with periodic boundary conditions. The whole isolated system is described by a set of classical amplitudes. We show that if the system is in equilibrium the classical amplitudes follow the equipartition theorem which allows us to assign a unique temperature to each given energy state. We also analyse the spectral properties of classical amplitudes and obtain nonperturbative results for the chemical potential as well as temperature-dependent Bogoliubov frequencies and their damping rates. Where available, we make comparison with the analytical estimates of these quantities.
For a weakly interacting Bose gas rotating in a harmonic trap we relate the yrast states of small systems (that can be treated exactly) to the thermodynamic limit (derived within the mean-field approximation). For a few dozens of atoms, the yrast line shows distinct quasiperiodic oscillations with increasing angular momentum that originate from the internal structure of the exact many-body states. These finite-size effects disappear in the thermodynamic limit, where the Gross-Pitaevskii approximation provides the exact energy to leading order in the number of particles NNN. However, the exact yrast states reveal significant structure not captured by the mean-field approximation: Even in the limit of large NNN, the corresponding mean-field solution accounts for only a fraction of the total weight of the exact quantum state.
We consider the evolution and dissipation of vortex rings in a condensate at nonzero temperatures in the context of the classical field approximation, based on the defocusing nonlinear Schrödinger equation. The temperature in such a system is fully determined by the total number density and the number density of the condensate. The collisions with noncondensed particles reduce the radius of a vortex ring until it completely disappears. We obtain a universal decay law for a vortex line length and relate it to mutual friction coefficients in the fundamental equation of vortex motion in superfluids.
We discuss dissipative dynamics of a vortex state in a trapped Bose-condensed gas at finite temperature and draw a scenario of decay of this state in a static trap. The interaction of the vortex with the thermal cloud transfers energy from the vortex to the cloud and induces the motion of the vortex core to the border of the condensate. Once the vortex reaches the border, it immediately decays through the creation of excitations. We calculate the characteristic lifetime of a vortex state and address the question of how the dissipative dynamics of vortices can be studied experimentally.
Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.
We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω{\omega}ω , the temperature T, and the particle number N are related by N∼(T/ω)3→∞{N \sim (T / \omega)^{3} \to\infty}N∼(T/ω)3→∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.