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Journal of Physics B Atomic Molecular and Optical Physics

A Hartree - Fock study of a Bose condensed gas

Patrik Öhberg, Stig Stenholm

Published Jun 28, 1997• 22 citations
Publisher

Abstract

We present a numerical study of an inhomogeneous condensed Bose gas by solving the Hartree - Fock equations. The density profile of the Bose gas is solved with several different harmonic trapping potentials in one, two and three dimensions. We also present an investigation of the critical temperature given by the Hartree - Fock theory and discuss possible shortcomings of the theory to describe the transition region between an ordinary gas and a Bose condensed gas.

DOI
10.1088/0953-4075/30/12/005

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We present a numerical study of an inhomogeneous condensed Bose gas by solving the Hartree - Fock equations. The density profile of the Bose gas is solved with several different harmonic trapping potentials in one, two and three dimensions. We also present an investigation of the critical temperature given by the Hartree - Fock theory and discuss possible shortcomings of the theory to describe the transition region between an ordinary gas and a Bose condensed gas.
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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.
We calculate the pair correlation function of an interacting Bose gas in a harmonic trap directly via Path Integral Quantum Monte Carlo simulation for various temperatures and compare the numerical result with simple approximative treatments. Around the critical temperature of Bose-Einstein condensation, a description based on the Hartree-Fock approximation is found to be accurate. At low temperatures the Hartree-Fock approach fails and we use a local density approximation based on the Bogoliubov description for a homogeneous gas. This approximation agrees with the simulation results at low temperatures, where the contribution of the phonon-like modes affects the long range behavior of the correlation function. Further we discuss the relation between the pair correlation and quantities measured in recent experiments.
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 devote a particular attention to the role played by the repulsive interaction on the chemical potential of condensed atoms, the chemical potential of non-condensed atoms, the anomalous fraction and the heat specific capacity for Bose gas in harmonic trap. First, we point out that the generalized Hartree-Fock Bogoliubov approximations (GHFB) produces a change in the specific temperature via estimation experience, and we discuss these inadequacies. We calculate a behavior of condensate density in Thomas Fermi approximation, where the thermal cloud is not negligible. Finally, we compare our results with various recent numerical estimates and experience.
The density of states of a Bose-condensed gas confined in a harmonic trap is investigated. The predictions of Bogoliubov theory are compared with those of Hartree-Fock theory and of the hydrodynamic model. We show that the Hartree-Fock scheme provides an excellent description of the excitation spectrum in a wide range of energy, revealing a major role played by single-particle excitations in these confined systems. The crossover from the hydrodynamic regime, holding at low energies, to the independent-particle regime is explicitly explored by studying the frequency of the surface mode as a function of their angular momentum. The applicability of the semiclassical approximation for the excited states is also discussed. We show that the semiclassical approach provides simple and accurate formulas for the density of states and the quantum depletion of the condensate.