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Revue Nature et Technologie

Thermodynamics of a trapped Bose gas at low temperature : Departure from the Thomas-Fermi regime.

Abdelâali Boudjemâa, Mohamed Benarous

Published Mar 24, 2011• 0 citations
OpenAlex

Abstract

We apply the finite temperature Thomas-Fermi (TF) approximation by using the Time dependent Hartree-Fock-Bogoliubov equations. The exploration of these equations at finite temperature leads to the fact that all the thermodynamics quantities deviate from the standard values in the TF regime .

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We derive a description of the spatially inhomogeneous Bose-Einstein condensate, treating the system locally as a homogeneous Bose gas. This approach, similar to the Thomas-Fermi model for the inhomogeneous many-particle fermion system, is well suited to describe the atomic Bose-Einstein condensates that have recently been obtained experimentally through atomic trapping and cooling. In this paper, we confine our attention to the zero-temperature case, although the treatment can be generalized to finite temperatures, as we shall discuss elsewhere. Several features of this approach, which we shall call the Thomas-Fermi-Bogoliubov description, are very attractive. (i) It is simpler than the Hartree-Fock-Bogoliubov technique. We can obtain analytical results in the case of weakly interacting bosons for quantities such as the chemical potential, the local depletion, pairing, pressure, and density of states. (ii) The method provides an estimate for the error due to the inhomogeneity of the Bose-condensed system. This error is a local quantity so that the validity of the description for a given trap and a given number of trapped atoms can be tested as a function of position. We see, for example, that at the edge of the condensate the Thomas-Fermi-Bogoliubov theory always breaks down. (iii) The Thomas-Fermi-Bogoliubov description can be generalized to treat the statistical mechanics of the Bose gas at finite temperatures.
We present a detailed study of the temperature dependence of the condensate and noncondensate density profiles of a Bose-condensed gas in a parabolic trap. These quantities are calculated self-consistently using the Hartree-Fock-Bogoliubov equations within the Popov approximation. Below the Bose-Einstein transition the excitation frequencies have a relatively weak temperature dependence even though the condensate is strongly depleted. As the condensate density goes to zero through the transition, the excitation frequencies are strongly affected and approach the frequencies of a noninteracting trapped gas in the high temperature limit.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.
Abstract This chapter is devoted to the thermodynamic behaviour of an interacting trapped Bose gas. It developes the Hartree–Fock theory for nonuniform interacting Bose gases at low temperatures, and discusses the problem of the shift of the critical temperature caused by interactions and also the thermodynamic behaviour of the trapped gas below the critical temperature. The chapter outlines general procedures, which do not only hold for Bose gases, to determine the equation of state of uniform matter, starting from the knowledge of the density profile in the trap. Finally, the chapter discusses some properties of the collective oscillations of a trapped Bose gas at a finite temperature.
We analyse numerically the finite-temperature behaviour of a dilute trapped Bose gas containing a large number of atoms. The generalized Hartree–Fock–Bogoliubov approximation is used to take into account the mutual interactions between the condensate and the thermal cloud. We confront our results with recent experiments and literature and explain in particular the experimentally observed departure from the Thomas–Fermi regime for the condensate radius and the aspect ratio. This clearly illustrates the compression effect of the thermal cloud on the condensate.
We rely on a variational approach to derive a set of equations governing a trapped self-interacting Bose gas at finite temperature. In this work, we analyze the static situation both at zero and finite temperature in the Thomas-Fermi limit. We derive simple analytic expressions for the condensate properties at finite temperature. The noncondensate and anomalous density profiles are also analyzed in terms of the condensate fraction. The results are quite encouraging owing to the simplicity of the formalism.