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arXiv:cond-mat/9611061 · Physical Review A

Bose-Einstein condensation in anisotropic harmonic traps

Tor Haugset, Hårek Haugerud, Jens O. Andersen

Published Apr 1, 1997• 61 citations
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Abstract

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.

DOI
10.1103/physreva.55.2922

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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.
By defining the characteristic length, the boundary effects of Bose-Einstein condensation in a three-dimensional harmonic trap are investigated using the Euler-MacLaurin formula. Results show that the condensed fraction of particles reduces due to the finite-size effects and the effects of finite particle number; the corrections of the condensation fraction and the condensation temperature have, respectively, a maximum value due to the boundary effect, hence it is very effective to optimize the parameters of the harmonic traps for improving the condensation fraction and the condensation temperature. In the jump of heat capacity exist the boundary effects and the effects of finite particle number, and the jump of heat capacity has a minimum because the parameters of harmonic traps are selected to be reasonable. The equation of state is derived for a finite ideal Bose gas system in a three-dimensional harmonic trap; the anisotropy (or isotropy) of the pressure is determined by the anisotropy (or isotropy) of the frequency of the harmonic potential.
Bose-Einstein condensation (BEC) of an ideal gas is investigated for a finite number of particles. In three dimensions, we find a transition temperature which is lower than in the thermodynamic limit. Lowering the dimension increases the transition temperature and is therefore favorable for BEC. This is in contrast to the standard result obtained in the thermodynamic limit which states that BEC is not possible in, e.g., a one-dimensional (1D) harmonic potential. As a result, 1D atom traps, such as radially tightly confining magnetic traps or optical dipole traps, are promising for studying BEC. \textcopyright{} 1996 The American Physical Society.
The thermodynamic properties of an ideal q-boson gas trapped in an n-dimensional harmonic potential are studied, based on the distribution function of q-bosons. The critical temperature Tc,q of Bose-Einstein condensation (BEC) and the heat capacity C of the system are derived analytically. It is shown that for the q-boson gas trapped in a harmonic potential, BEC may occur in any dimension when q ≠ 1, the critical temperature is always higher than that of an ordinary Bose gas (q = 1), and the heat capacity is continuous at Tc,q for a one-dimensional system but discontinuous at Tc,q for a two- or multi-dimensional system.
We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
We consider a finite number of noninteracting bosons trapped in an isotropic three-dimensional harmonic oscillator. Using the grand canonical ensemble, we calculate the heat capacity of the system as a function of temperature, for several values of the number of particles. We find that a new definition of critical temperature is necessary for a finite number of trapped particles, and present a definition that is experimentally convenient and in good agreement with a recent definition given by W. Ketterle and N. J. van Druten [Phys. Rev. A 54, 656 (1996)].
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.