Classical-Field Method for Time Dependent Bose-Einstein Condensed Gases

Abstract

We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas. DOI: 10.1103/PhysRevLett.87.210404 PACS numbers: 03.75.Fi, 05.10.Gg, 42.50. -p Since the first experimental demonstrations of BoseEinstein condensation in atomic gases [1], the role played by finite temperature effects in the physics of BoseEinstein condensed alkali gases has drawn increasing attention. For example, in thermal equilibrium, both the spatial density of the condensate atoms [2] and the distribution of number of particles in the condensate [3] are modified due to the presence of the thermal atoms. Similarly, we must take into account the noncondensed atoms in order to explain time dependent phenomena such as the damping and frequency shifts of collective modes [4,5] or the evolution of the recently created vortices [6], where the dissipation of the condensate motion is provided by the noncondensed atoms [7].The widely used Gross-Pitaevskii equation, the nonlinear Schrödinger equation for the condensate wave function, is not able to describe these effects since it neglects the interaction between the condensate and the noncondensed atoms [8]. One possibility to go beyond the GrossPitaevskii equation is to use Bogoliubov theory, which is a perturbative method valid for a small noncondensed fraction. The Bogoliubov method can be applied to thermal equilibrium but also to time dependent situations: in a U(1) symmetry breaking point of view, to zeroth order one solves the time dependent Gross-Pitaevskii equation for the condensate field c 0 ͑ r, t͒, to first order one linearizes the quantum field equations around the classical field c 0 ͑ r, t͒ to get the dynamics of noncondensed particles, and to second order one includes the backaction of noncondensed particles on the condensate. However, in this way one can predict only small corrections to the Gross-Pitaevskii equation. Additionally, if the number of noncondensed particles increases during the evolution of the system, the Bogoliubov approach is valid only for short times [9]. Another existing approach is the mean field HartreeFock-Bogoliubov approximation. This approach is known however to present consistency problems and is still the object of research [10].In this Letter we propose an alternative method to study the time evolution of Bose condensed gases perturbed from an initial thermal equilibrium, based on the classical field approximation in the Wigner representation, the so-called truncated Wigner approximation. The classical field approximation amounts to evolving a set of initially randomly distributed atomic fields w...

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