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arXiv:cond-mat/9910081 · Physical Review A

Collisional relaxation in diffuse clouds of trapped bosons

G. M. Kavoulakis, C. J. Pethick, H. Smith

Published Apr 6, 2000• 30 citations
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Abstract

The damping of collective modes and the relaxation of temperature anisotropies in a trapped Bose gas is determined at temperatures above the Bose-Einstein condensation temperature in the collisionless regime. We demonstrate for both cases how the effects of collisions may be treated perturbatively and calculate relaxation rates from a variational principle. Our results are compared with experiment and with previous theoretical calculations.

DOI
10.1103/physreva.61.053603

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The damping of collective modes and the relaxation of temperature anisotropies in a trapped Bose gas is determined at temperatures above the Bose-Einstein condensation temperature in the collisionless regime. We demonstrate for both cases how the effects of collisions may be treated perturbatively and calculate relaxation rates from a variational principle. Our results are compared with experiment and with previous theoretical calculations.
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Recently it was shown that the Landau-Khalatnikov two-fluid hydrodynamics describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. We use these equations to discuss the low frequency hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and damping of collective modes. A new feature is our use of frequency-dependent transport coefficients, which produce a natural cutoff by eliminating the collisionless low-density tail of the thermal cloud. Above the superfluid transition, our expression for the damping in trapped inhomogeneous gases is analogous to the result first obtained by Landau and Lifshitz for uniform classical fluids. We also use the moment method to discuss the crossover from the collisionless to the hydrodynamic region. Recent data for the monopole-quadrupole mode in the hydrodynamic region of a trapped gas of metastable 4He{}^{4}\mathrm{He}4He is discussed. We also present calculations for the damping of the analogous m=0m=0m=0 monopole-quadrupole condensate mode in the superfluid phase.
Explicit expressions for all the transport coefficients have recently been found for a trapped Bose-condensed gas at finite temperatures. These transport coefficients are used to define the characteristic relaxation times, which determine the crossover between the mean-field collisionless and the two-fluid hydrodynamic regime. These relaxation times are evaluated as a function of the position in the trap potential. We show that all the relaxation times are dominated by the collisions between the condensate and the noncondensate atoms, and are much smaller than the standard classical collision time used in most of the current literature. The study [Stamper-Kurn et al., Phys. Rev. Lett. 81, 500 (1998)] of the collective modes at finite temperature is shown to have been well within the two-fluid hydrodynamic regime.
We study, in the framework of the Boltzmann-Bloch-Nordheim equation (BBN), the kinetic properties of a boson gas above the Bose-Einstein transition temperature Tc.{T}_{c}.Tc​. The BBN equation is solved numerically within an algorithm that we test with exact analytical results for the collision rate of a homogeneous system in thermal equilibrium. In the classical regime $(T\ensuremath{\gg}{T}_{c}),$ the relaxation time of a quadrupolar deformation in momentum space is proportional to the mean free collision time ${\ensuremath{\tau}}_{\mathrm{relax}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{coll}}\ensuremath{\propto}{T}^{\ensuremath{-}1/2}.$ Approaching the critical temperature, quantum statistic effects become dominant, and the collision rate increases dramatically. Nevertheless, this does not affect the relaxation properties of the gas that depend only on the spontaneous collision term in BBN. The relaxation time becomes ${\ensuremath{\tau}}_{\mathrm{relax}}\ensuremath{\propto}(T\ensuremath{-}{T}_{c}{)}^{\ensuremath{-}1/2},$ exhibiting a critical slowing down. These phenomena can be experimentally confirmed looking at the damping properties of collective motions induced on trapped atoms. The possibility to observe a transition from collisionless (zero-sound) to hydrodynamic (such as first and second-sound) is finally discussed.
This article is devoted to the study of two-dimensional Bose gases harmonically confined. We first summarize their equilibrium properties. For such a gas above the critical temperature, we also derive the frequencies and the damping of the collective oscillations and we investigate its expansion after releasing of the trap. The method is well suited to study the collisional effects taking place in the system and in particular to discuss the crossover between the hydrodynamic and the collisionless regimes. We establish the link between the relaxation times relevant for the damping of the collective oscillations and for the time-of-flight expansion. We also evaluate the collision rate and its relationship with the relaxation time.