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Physica Scripta

Bose–Einstein Condensation in Harmonic Oscillator Potentials

Yi Xue-Xi, Haijun Wang, Chang-Pu Sun

Published Mar 1, 1998• 6 citations
Publisher

Abstract

Bose–Einstein Condensation (BEC) of both the ideal and the weakly inter-acting cold alkali gases confined by anisotropic harmonic oscillator poten-tials is investigated in this paper. It is shown that the transition point is shifted towards lower temperatures because of the trapped potentials, and the specific heat below the transition point is no longer proportional to T 3/2 . Expressions of modified condensation temperature, internal energy and the specific heat are derived for the ideal trapped gases explicitly. Moreover, the effect of interactions among the atoms on the transition temperature is also given and discussed.

DOI
10.1088/0031-8949/57/3/002

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•Mar 1, 2005•0 citations
Bose–Einstein Condensation (BEC) of both the ideal and the weakly inter-acting cold alkali gases confined by anisotropic harmonic oscillator poten-tials is investigated in this paper. It is shown that the transition point is shifted towards lower temperatures because of the trapped potentials, and the specific heat below the transition point is no longer proportional to T 3/2 . Expressions of modified condensation temperature, internal energy and the specific heat are derived for the ideal trapped gases explicitly. Moreover, the effect of interactions among the atoms on the transition temperature is also given and discussed.
A simple and direct approach to handle summation is presented. With this approach, we analytically investigate Bose–Einstein condensation of ideal Bose gas trapped in an isotropic harmonic oscillator potential. We get the accurate expression of which is very close to (0.43% larger than) the experimental data. We find the curve of internal energy of the system vs. temperature has a turning point which marks the beginning of a condensation. We also find that there exists specific heat jump at the transition temperature, no matter whether the system is macroscopic or finite. This phenomenon could be a manifestation of a phase transition in finite systems.
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.
A simple and direct approach to handle summation is presented. With this approach, we analytically investigate Bose-Einstein condensation of ideal Bose gas trapped in an isotropic harmonic oscillator potential. We get the accurate expression of Tc which is very close to (0.43% larger than) the experimental data. We find the curve of internal energy of the system vs. temperature has a turning point which marks the beginning of a condensation. We also find that there exists specific heat jump at the transition temperature, no matter whether the system is macroscopic or finite. This phenomenon could be a manifestation of a phase transition in finite systems.
We present an analysis of Bose-Einstein condensation for a system of non-interacting spin-0 particles in a harmonic oscillator confining potential trap. We discuss why a confined system of particles differs both qualitatively and quantitatively from an identical system which is not confined. One crucial difference is that a confined system is not characterized by a critical temperature in the same way as an unconfined system such as the free boson gas. We present the results of both a numerical and analytic analysis of the problem of Bose-Einstein condensation in a general anisotropic harmonic oscillator confining potential.
The Bose-Einstein condensation of a dilute gas of rubidium-87 atoms was achieved by cooling a small number of atoms in a magnetic trap. The effective potential of the trap is to lowest order harmonic and under these conditions we estimate the critical temperature and the heat capacity of an ideal gas of Bose particles. Previously this has been estimated within the semiclassical approximation. However, because of the small number of atoms, the resulting TcT_cTc​ deviates slightly from our result obtained by exact numerical calculations. For the gap of the heat capacity at the transition point the deviation is 20%20\%20%.
One of the most interesting properties of boson gases is that under special conditions, there is a possibility of a phase transition, in a critical temperature below which all bosons condensate into the ground state. This phenomenon is called Bose – Einstein Condensation (BEC). In this paper, we investigate BEC in a harmonic oscillator trap. We conclude that, in contrast to a free boson gas, there is no critical temperature for phase transition in a harmonic oscillator trap. However , by numerical and analytical calculation, it is possible to obtain a temperature at which the heat capacity is maximum. We call this the critical temperature . Possible explanation for all these features will be explained in this paper.