The truncated Wigner method for Bose-condensed gases: limits of validity andapplications11Presented at the Theory of Quantum Gases and Quantum Coherence:The Salerno BEC Workshop, 3–5 June 2001. (A compilation of articles from thisworkshop was published in J. Phys. B: At. Mol. Opt. Phys. 34 (2001) issue23.)

Abstract

We study the truncated Wigner method applied to a weakly interactingspinless Bose-condensed gas which is perturbed away from thermalequilibrium by a time-dependent external potential. The principleof the method is to generate an ensemble of classical fields ψ(r)which samples the Wigner quasi-distribution function of the initial thermalequilibrium density operator of the gas, and then to evolve each classicalfield with the Gross–Pitaevskii equation. In the first part of the paper weimprove the sampling technique over our previous work (Sinatra et al2000J. Mod. Opt. 47 2629–44) and we test its accuracy against the exactlysolvable model of the ideal Bose gas. In the second part of the paper weinvestigate the conditions of validity of the truncated Wigner method. Forshort evolution times it is known that the time-dependent Bogoliubovapproximation is valid for almost pure condensates. The requirement that thetruncated Wigner method reproduces the Bogoliubov prediction leads to theconstraint that the number of field modes in the Wigner simulation must besmaller than the number of particles in the gas. For longer evolution timesthe nonlinear dynamics of the noncondensed modes of the field plays animportant role. To demonstrate this we analyse the case of a three-dimensionalspatially homogeneous Bose-condensed gas and we test the ability of thetruncated Wigner method to correctly reproduce the Beliaev–Landau dampingof an excitation of the condensate. We have identified the mechanismwhich limits the validity of the truncated Wigner method: the initialensemble of classical fields, driven by the time-dependent Gross–Pitaevskiiequation, thermalizes to a classical field distribution at a temperature Tclasswhich is larger than the initial temperatureT of the quantum gas.When Tclasssignificantly exceeds Ta spurious damping is observed in the Wigner simulation. This leads to the secondvalidity condition for the truncated Wigner method, Tclass − T ≪ T,which requires that the maximum energy ϵmaxof the Bogoliubov modes in the simulation does not exceed a few kB T.