Thermodynamic Properties of a Trapped Interacting Bose Gas

A Bose gas in an external potential is studied by means of the local density approximation. Analytical results are derived for the thermodynamic properties of an ideal Bose gas in a generic power-law trapping potential, and their dependence on the mutual interaction of atoms in the case of a non-ideal Bose gas. PACS numbers: 03.75.Fi, 32.80.Pj Typeset using REVTEX 1 One of the most striking consequences of quantum statistics is the condensation of an ideal Bose-Einstein gas where the zero momentum state can become macroscopically occupied at a sufficiently low temperature [1]. Some of the features of the transition to a superfluid phase exhibited by the Bose fluid He is often interpreted essentially as a result of this Bose-Einstein condensation (BEC), the strongest degeneracy effect of a Boson system [2]. For many years, it was considered hopeless to experimentally observe BEC in an atomic gas with weak interactions. With the development of techniques to trap and cool atoms, BEC was recently observed directly in dilute atomic vapors [3–5]. The new experimental achievements have stimulated great interest in the theoretical study of inhomogeneous Bose gases. The thermodynamic properties of trapped atomic Bose gases undergoing BEC can be altered by the spatially varying trapping potential. The interaction between atoms may have a significant effect on the thermodynamic properties. There have been several investigations analyzing the dependence of the critical temperature on the trapping potential and weak interaction in the Bose gas [6–8]. The thermodynamic properties of Bose gases in an external potential have also been discussed in Ref. [6], but no analytic results have been given. Here we shall derive some analytical expressions for the thermodynamic properties of a trapped non-interacting and interacting Bose gas under the local density approximation. Let us first consider the case of an ideal Bose gas trapped in an external potential. When the energy level spacing due to the trapping potential is much smaller than the thermal energy kT = β, the local density approximation is adequate [9–11]. Space may then be divided into small cells, and in each cell we may consider the trapping potential V (r) to be constant. With a trivial modification to indicate r dependence explicitly we may directly extend the formula of the grand potential, which can be found in any textbook on statistical mechanics, e.g. Ref. [1], to write the local grand potential above the critical temperature Tc as Ω̃BE(r) = − kT λ g5/2(ζ) (1)