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arXiv:cond-mat/0107515 · Journal of Physics B Atomic Molecular and Optical Physics

Dynamics of thermal Bose fields in the classical limit

Matthew J. Davis, R. J. Ballagh, Keith R. Burnett

Published Nov 13, 2001• 76 citations
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Abstract

We develop an approximate formalism suitable for performing simulations of the thermal dynamics of interacting Bose gases. The method is based on the observation that when the lowest-energy modes of the Bose field operator are highly occupied, they may be treated classically to a good approximation. We derive a finite-temperature Gross-Pitaevskii equation (FTGPE) for these modes which is coupled to an effective reservoir described by quantum kinetic theory. We discuss each of the terms that arise in this GPE, and their relevance to experimental systems. We then describe a simpler projected GPE that may be useful in simulating thermal Bose condensates. This classical method could be applied to other Bose fields.

DOI
10.1088/0953-4075/34/22/316

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We develop an approximate formalism suitable for performing simulations of the thermal dynamics of interacting Bose gases. The method is based on the observation that when the lowest-energy modes of the Bose field operator are highly occupied, they may be treated classically to a good approximation. We derive a finite-temperature Gross-Pitaevskii equation (FTGPE) for these modes which is coupled to an effective reservoir described by quantum kinetic theory. We discuss each of the terms that arise in this GPE, and their relevance to experimental systems. We then describe a simpler projected GPE that may be useful in simulating thermal Bose condensates. This classical method could be applied to other Bose fields.
We demonstrate that the time-dependent projected Gross-Pitaevskii equation (GPE) derived earlier [M. J. Davis, R. J. Ballagh, and K. Burnett, J. Phys. B 34, 4487 (2001)] can represent the highly occupied modes of a homogeneous, partially-condensed Bose gas. Contrary to the often held belief that the GPE is valid only at zero temperature, we find that this equation will evolve randomized initial wave functions to a state describing thermal equilibrium. In the case of small interaction strengths or low temperatures, our numerical results can be compared to the predictions of Bogoliubov theory and its perturbative extensions. This demonstrates the validity of the GPE in these limits and allows us to assign a temperature to the simulations unambiguously. However, the GPE method is nonperturbative, and we believe it can be used to describe the thermal properties of a Bose gas even when Bogoliubov theory fails. We suggest a different technique to measure the temperature of our simulations in these circumstances. Using this approach we determine the dependence of the condensate fraction and specific heat on temperature for several interaction strengths, and observe the appearance of vortex networks. Interesting behavior near the critical point is observed and discussed.
The Gross–Pitaevskii equation (GPE) describes the dynamics of superflows and Bose–Einstein Condensates (BEC) at very low temperature. When a truncation of Fourier modes is performed, the resulting truncated GPE (TGPE) can also describe the correct thermal behavior of a Bose gas, as long as all relevant modes are highly occupied [M.J. Davis, S.A. Morgan, K. Burnett, Simulations of Bose fields at finite temperature, Phys. Rev. Lett. 87 (16) (2001) 160402]. We review some of our groupʼs recent GPE- and TGPE-based numerical studies of superfluid dynamics and BEC stability. The relations with experiments are discussed.
We show that the projected Gross-Pitaevskii equation (PGPE) can be mapped exactly onto Hamilton's equations of motion for classical position and momentum variables. Making use of this mapping, we adapt techniques developed in statistical mechanics to calculate the temperature and chemical potential of a classical Bose field in the microcanonical ensemble. We apply the method to simulations of the PGPE, which can be used to represent the highly occupied modes of Bose condensed gases at finite temperature. The method is rigorous, valid beyond the realms of perturbation theory, and agrees with an earlier method of temperature measurement for the same system. Using this method we show that the critical temperature for condensation in a homogeneous Bose gas on a lattice with a uv cutoff increases with the interaction strength. We discuss how to determine the temperature shift for the Bose gas in the continuum limit using this type of calculation, and obtain a result in agreement with more sophisticated Monte Carlo simulations. We also consider the behavior of the specific heat.
We introduce a time-dependent projected Gross-Pitaevskii equation to describe a partially condensed homogeneous Bose gas, and find that this equation will evolve randomized initial wave functions to equilibrium. We compare our numerical data to the predictions of a gapless, second order theory of Bose-Einstein condensation [S. A. Morgan, J. Phys. B 33, 3847 (2000)], and find that we can determine a temperature when the theory is valid. As the Gross-Pitaevskii equation is nonperturbative, we expect that it can describe the correct thermal behavior of a Bose gas as long as all relevant modes are highly occupied. Our method could be applied to other boson fields.
In this paper, we develop a numerical procedure for investigating the dynamics of trapped Bose gases based on the ZGN theory. The dynamical equations used consist of a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud. The former is solved using a fast Fourier transform split-operator technique while the Boltzmann equation is treated by means of N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. This scheme has been applied to a model equilibration problem, dipole oscillations in isotropic traps and scissors modes in anisotropic traps. In the case of the latter, the frequencies and damping rates of the condensate mode have been extracted from the simulations for a wide range of temperatures. Good agreement with recent experiments has been found.
We introduce a technique to simulate the dynamics of a thermal dilute Bose gas. With the method of the nonequilibrium statistical operator, a correction to the Gross-Pitaevskii equation (GPE) is derived. This modification of the GPE describes spontaneous scattering processes. It regularizes the ultraviolet catastrophe and captures the dynamics of the particles at all energies, including the tail of the momentum distribution. We apply our model to the simulation of an evaporative cooling quench of a two-dimensional Bose gas.