The truncated Wigner or classical field method for Bose condensed gases: limits of validity and applications
Abstract
We study the truncated Wigner method applied to the the time evolution of a weakly interacting Bose condensed gas perturbed away from thermal equilibrium. The method generates an ensemble of classical fields which samples the Wigner quasi-distribution function of the initial thermal density operator of the gas, and then evolves each classical field with the Gross-Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work [Jour. of Mod. Opt. 47, 2629 (2000)] and we test its accuracy against the exactly solvable ideal Bose gas model. In the second part of the paper we investigate the validity conditions of the classical field approximation. For short times the time-dependent Bogoliubov approximation is valid for almost pure condensates. We use this fact to test the truncated Wigner method: this leads to the constraint that the number of field modes in the Wigner simulation should be smaller than the number of particles in the gas. For longer times the nonlinear dynamics of the noncondensed modes is important. To show this we analyse the case of a 3D homogeneous Bose condensed gas and we test the ability of the method to reproduce the Beliaev-Landau damping of a collective excitation. We find that the ensemble of classical fields thermalises to a classical field distribution at a temperature T_class larger than the initial temperature T. When T_class >> T a spurious damping is observed. This leads to the second validity condition, T_class-T << T, which requires that the maximum energy of the Bogoliubov modes in the simulation does not exceed a few times k_B T.Keywords
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