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arXiv:cond-mat/9609234 · Physical Review A

Variational Thomas-Fermi theory of a nonuniform Bose condensate at zero temperature

Eddy Timmermans, Paolo Tommasini, Kerson Huang

Published May 1, 1997• 74 citations

Abstract

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.

DOI: 10.1103/physreva.55.3645

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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 derive and discuss the temperature dependence of the condensate and noncondensate density profile of a Bose-Einstein condensate gas with Feshbach resonance in a parabolic trap. These quantities are calculated self-consistently using the generalized Hartree-Fock-Bogoliubov (HFB) equations within the Bogoliubov approximation. At zero temperature, the HFB equation can be solved by means of a variation method that gives the low-excitation spectrum. Moreover, within the two-body collision theory, we estimate the relationship between the number of atoms in the condensate and the external magnetic field B . The result is in good agreement with data from recent experiments.

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.

Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order ${R}^{\ensuremath{-}4}\mathrm{ln}R$ and ${R}^{\ensuremath{-}4},$ where R is the number-dependent dimensionless condensate radius in units of the oscillator length $\sqrt{\ensuremath{\Elzxh}/m{\ensuremath{\omega}}_{0}}.$ The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order ${R}^{\ensuremath{-}4}.$

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.

We analyze the static TDHFB equations in the Thomas-Fermi limit for a gas of bosons in a harmonic trap . These equations naturally generalize the Gross-Pitaevskii equation. We first build a simple enough method that allows f or the determination of the various density profile s. At zero temperature, we obtain familiar expressions for the chemical pot ential and the condensate radius. The standard Thomas-Fermi profile for the condensate density is also recovered. For finit e temperatures and above the transition, we derive analytical expressions for the condensate radius, the chemical potential, the number of condensed atoms and the depletion as functions of the temperature. We observe that the condensate radius and the column density are surprisingly very slow f unctions of the temperature. Furthermore, the non-condensed density, although being quite uniform inside the trap, gro ws rapidly with the temperature. These facts imply therefore that the c ondensed atoms are very strongly attached and exhib it a certain robustness against ''decondensation''. Moreover, the transitio n to the non condensed phase seems to be much more controlled by the thermal cloud which rapidly grows from the borders toward the centre of the trap.

We derive and discuss the equations of motion for the condensate and its fluctuations for a dilute, weakly interacting Bose gas in an external potential within the self-consistent Hartree-Fock-Bogoliubov (HFB) approximation. Account is taken of the depletion of the condensate and the anomalous Bose correlations, which are important at finite temperatures. We give a critical analysis of the self-consistent HFB approximation in terms of the Hohenberg-Martin classification of approximations (conserving vs gapless) and point out that the Popov approximation to the full HFB gives a gapless single-particle spectrum at all temperatures. The Beliaev second-order approximation is discussed as the spectrum generated by functional differentiation of the HFB single-particle Green's function. We emphasize that the problem of determining the excitation spectrum of a Bose-condensed gas (homogeneous or inhomogeneous) is difficult because of the need to satisfy several different constraints.