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Physical Review A

Extended Gross-Pitaevskii description for the dynamics of thermal dilute Bose gases

Michiel Wouters

Published Sep 10, 2014• 9 citations
Publisher

Abstract

We introduce a technique to simulate the dynamics of a thermal dilute Bose gas. With the method of the nonequilibrium statistical operator, a correction to the Gross-Pitaevskii equation (GPE) is derived. This modification of the GPE describes spontaneous scattering processes. It regularizes the ultraviolet catastrophe and captures the dynamics of the particles at all energies, including the tail of the momentum distribution. We apply our model to the simulation of an evaporative cooling quench of a two-dimensional Bose gas.

DOI
10.1103/physreva.90.033611

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We introduce a technique to simulate the dynamics of a thermal dilute Bose gas. With the method of the nonequilibrium statistical operator, a correction to the Gross-Pitaevskii equation (GPE) is derived. This modification of the GPE describes spontaneous scattering processes. It regularizes the ultraviolet catastrophe and captures the dynamics of the particles at all energies, including the tail of the momentum distribution. We apply our model to the simulation of an evaporative cooling quench of a two-dimensional Bose gas.
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We develop an approximate formalism suitable for performing simulations of the thermal dynamics of interacting Bose gases. The method is based on the observation that when the lowest-energy modes of the Bose field operator are highly occupied, they may be treated classically to a good approximation. We derive a finite-temperature Gross-Pitaevskii equation (FTGPE) for these modes which is coupled to an effective reservoir described by quantum kinetic theory. We discuss each of the terms that arise in this GPE, and their relevance to experimental systems. We then describe a simpler projected GPE that may be useful in simulating thermal Bose condensates. This classical method could be applied to other Bose fields.
A straightforward numerical technique, based on the Gross-Pitaevskii equation, is used to generate a self-consistent description of thermally excited states of a dilute boson gas. The process of evaporative cooling is then modeled by following the time evolution of the system using the same equation. It is shown that the subsequent rethermalization of the thermally excited state produces a cooler coherent condensate. Other results presented show that trapping vortex states with the ground state may be possible in a two-dimensional experimental environment.
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.
In experiments, Bose-Einstein condensates are prepared by cooling a dilute Bose gas in a trap. After the phase transition has been reached, the trap is switched off and the evolution of the condensate observed. The evolution is macroscopically described by the Gross-Pitaevskii equation. On the microscopic level, the dynamics of Bose gases are described by the NNN-body Schrödinger equation. We review our article [BdS12] in which we construct a class of initial data in Fock space which are energetically close to the ground state and prove that their evolution approximately follows the Gross-Pitaevskii equation. The key idea is to model two-particle correlations with a Bogoliubov transformation.