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Journal of Experimental and Theoretical Physics Letters

Bose-Einstein condensation of an ideal gas in a parabolic trap

В. А. Алексеев, В. В. Климов, D. D. Krylova

Published Oct 1, 1997• 0 citations
Publisher

Abstract

The temperature-dependent velocity distribution function is found for the case of Bose-Einstein condensation of a finite number of noninteracting atoms trapped in a three-dimensional anisotropic parabolic trap. It is shown that at a temperature T of the order of the condensation temperature T 0 the velocity distribution consists of an anisotropic part, reflecting the population of the ground state, and an isotropic part above the condensate.

DOI
10.1134/1.567552

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The temperature-dependent velocity distribution function is found for the case of Bose-Einstein condensation of a finite number of noninteracting atoms trapped in a three-dimensional anisotropic parabolic trap. It is shown that at a temperature T of the order of the condensation temperature T 0 the velocity distribution consists of an anisotropic part, reflecting the population of the ground state, and an isotropic part above the condensate.
Abstract By using the one-dimensional canonical partition function, we modelled an ideal gas-like form of a set of Bose-Einstein condensates confined by a three-dimensional anisotropic parabolic trap. The model itself was constructed by taking the eigenenergies of the one-dimensional Gross-Pitaevskii equation in the longitudinal direction and enabling the harmonic volume as the inverse cube of average geometric trapping frequency to substitute the real volume. In this paper, we showed that the condensates form an ideal gas represented by its equation of state and have similar mature to Einstein’s solid-like model with the corrections in both the low and high temperatures in the internal energy formulations.
Condensation of the weakly nonideal gas in an anisotropic parabolic trap was considered. The first order (with respect to the interaction parameter) correction Δ 0 = 0 ( T )— 0 0 ( T ) to the temperature dependence of the ground state population 0 0 ( T ) of the ideal gas atoms was found using thermodynamic perturbation theory. The correction found 0 proved to be much larger than the one calculated within the mean field theory [see Giorgini S, Pitaevskii L P, Stringari S Phys. Rev. A 54 R4633 (1996)] at temperatures below critical. This correction tends to zero as the temperature approaches the critical one; i.e., the interaction of the atoms does not lead to variation in the critical temperature. A simple analytic expression was found that describes the smooth variation of 0 0 ( T ) as it approaches zero in the vicinity of the critical temperature. A new criterion for applicability of the thermodynamic limit is given.
The anisotropic density distribution of Bose particles after expansion from a magnetic trap has been taken as an important signature of the formation of Bose-Einstein condensate. In this letter we show that even for a non-condensed trapped gas one can get an anisotropic distribution due to anisotropic collision rates along radial and axial directions. We demonstrate that the mean energies along those directions, different from the energy in the trap, have opposite evolution tendencies with the expansion. This difference is particularly important for the case where a large number of bosons is trapped. We also show that the velocity distribution of the non-condensed gas in the trap, assumed initially isotropic and Gaussian, becomes anisotropic and with linewidth dependent on the expansion direction.
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 study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.