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arXiv:cond-mat/0410496 · Physical Review A

Projected Gross-Pitaevskii equation for harmonically confined Bose gases at finite temperature

P. B. Blakie, Matthew J. Davis

Published Dec 7, 2005• 106 citations
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Abstract

We extend the projected Gross-Pitaevskii equation formalism of Davis et al. [Phys. Rev. Lett. 87, 160402 (2001)] to the experimentally relevant case of thermal Bose gases in harmonic potentials and outline a robust and accurate numerical scheme that can efficiently simulate this system. We apply this method to investigate the equilibrium properties of the harmonically trapped three-dimensional projected Gross-Pitaevskii equation at finite temperature and consider the dependence of condensate fraction, position, and momentum distributions and density fluctuations on temperature. We apply the scheme to simulate an evaporative cooling process in which the preferential removal of high-energy particles leads to the growth of a Bose-Einstein condensate. We show that a condensate fraction can be inferred during the dynamics even in this nonequilibrium situation.

DOI
10.1103/physreva.72.063608

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We extend the projected Gross-Pitaevskii equation formalism of Davis et al. [Phys. Rev. Lett. 87, 160402 (2001)] to the experimentally relevant case of thermal Bose gases in harmonic potentials and outline a robust and accurate numerical scheme that can efficiently simulate this system. We apply this method to investigate the equilibrium properties of the harmonically trapped three-dimensional projected Gross-Pitaevskii equation at finite temperature and consider the dependence of condensate fraction, position, and momentum distributions and density fluctuations on temperature. We apply the scheme to simulate an evaporative cooling process in which the preferential removal of high-energy particles leads to the growth of a Bose-Einstein condensate. We show that a condensate fraction can be inferred during the dynamics even in this nonequilibrium situation.
Abstract We have devised a simulation method for studying the evaporative cooling process of trapped Bose-Einstein condensates at finite temperatures, leveraging the Zaremba-Nikuni-Griffin (ZNG) formalism. This framework incorporates the generalized Gross-Pitaevskii (GP) equation and a semiclassical kinetic equation governing the behavior of the thermal cloud. Our approach builds upon the numerical techniques pioneered by Jackson and Zaremba, which involve simulating the kinetic equation using test particles. A pivotal aspect of our method lies in emulating the evaporative cooling process by selectively removing test particles with high energy. Through extensive simulations, we demonstrate the efficacy of our approach in accurately describing condensate growth during evaporative cooling. Furthermore, we extend our simulations to investigate the formation of vortex lattices during evaporative cooling in the presence of a rotating thermal cloud. This comprehensive analysis provides valuable insights into the intricate dynamics of trapped Bose-Einstein condensates undergoing evaporative cooling processes.
We have achieved a full implementation of the stochastic projected Gross-Pitaevskii equation for a three-dimensional trapped Bose gas at finite temperature. Our work advances previous applications of this theory, which have only included growth processes, by implementing number-conserving scattering processes. We evaluate an analytic expression for the coefficient of the scattering term and compare it to that of the growth term in the experimental regime, showing that the two coefficients are comparable in size. We give an overview of the numerical implementation of the deterministic and stochastic terms for the scattering process, and use simulations of a condensate excited into a large amplitude breathing mode oscillation to characterize the importance of scattering and growth processes in an experimentally accessible regime. We find that in such nonequilibrium regimes the scattering can dominate over the growth, leading to qualitatively different system dynamics. In particular, the scattering causes the system to rapidly reach thermal equilibrium without greatly depleting the condensate, suggesting that it provides a highly coherent energy-transfer mechanism.
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.
We develop a formalism based on the projected Gross-Pitaevskii equation to simulate the finite-temperature collective mode experiments of Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. We examine the m=0m=0m=0 and m=2m=2m=2 quadrupolar modes on the temperature range 0.51Tc−−0.83Tc0.51{T}_{c}--0.83{T}_{c}0.51Tc​−−0.83Tc​ and calculate the frequencies of, and phase between, the condensate and noncondensate modes, and the condensate mode damping rate. This study is the first quantitative comparison of the projected Gross-Pitaevskii equation to experimental results in a dynamical regime.
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.
We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature Tc 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.