The truncated Wigner method for Bose-condensed gases: limits of validity and applications

Abstract

We study the truncated Wigner method applied to a weakly interacting spinless Bose condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ(r) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve 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, 2629-2644] and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three dimensional spatially homogeneous Bose condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev-Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross-Pitaevskii equation, thermalises to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy ǫmax of the Bogoliubov modes in the simulation does not exceed a few kBT .

Atomic quantum simulation of a three-dimensional U(1) gauge-Higgs model

“…However, we still expect that some qualitative feature can be captured by the Gross-Pitaevskii approach, as described in our previous studies [26,47] and later discussion. Quantum fluctuations can be included in the truncated Wigner approximation [48], which is obtained by taking into account quantum fluctuations around the classical path up to the second-order. The truncated Wigner approximation consists of (i) deriving an equation of motion of the average value of quantum operator, which is just the GPE itself; (ii) solving GPE for a given initial condition; and (iii) averaging over the solutions of GPE with different initial conditions with a certain weight.…”

Classical stochastic measurement trajectories: Bosonic atomic gases in an optical cavity and quantum measurement backaction

“…Multimode Wigner representations have been of great utility in studies of bosonic ultracold gases in closed systems [54][55][56][57][58][59][60] and can naturally include dissipation in open systems [51,53], as has been investigated in the context of three-body losses [60,61]. We derive below the equation of motion for the ensemble-averaged quasiprobability distribution, before showing in the subsequent section how individual classical measurement trajectories emerge from a mathematical correspondence to stochastic differential equations (SDEs).…”

“…The derivation of a Fokker-Planck equation is reminiscent of dropping the triple-derivative terms that arise from the s-wave interactions in the truncated Wigner approximation [54][55][56][57][58][59][60]63] for a closed bosonic atomic system. While we have used a discrete spatial basis argument here, similar arguments for truncating the interparticle interactions have been made using spectral basis decompositions [60,63].…”

Fock-state dynamics in Raman photoassociation of Bose-Einstein condensates

“…Here we adapt an approximation developed by Gardiner et al [27], which allows us to approximately represent Fock states without having to deal with negative pseudoprobabilities. As we investigate only the dynamics of the mean fields rather than quantum correlations, we will stochastically integrate the appropriate equations in the truncated Wigner representation [10,28,29], which we expect to give reliable results for the numbers of particles involved. For the parameters used in Ref.…”