Internal energy and condensate fraction of a trapped interacting Bose gas

We present a semiclassical two–fluid model for an interacting Bose gas confined in an anisotropic harmonic trap and solve it in the experimentally relevant region for a spin–polarized gas of 87 Rb atoms, obtaining the temperature dependence of the internal energy and of the condensate fraction. Our results are in agreement with recent experimental observations by Ensher et al. Bose–Einstein condensation (BEC) has recently been realized in dilute vapours of spin–polarized alkali atoms, using advanced techniques for cooling and trapping[1, 2, 3, 4, 5]. These condensates consist of several thousands to several million atoms confined in a well which is generated from nonuniform magnetic fields. The confining potential is accurately harmonic along the three Cartesian directions and has cylindrical symmetry in most experimental setups. The determination of thermodynamic properties such as the condensate fraction and the internal energy as functions of temperature is at present of primary interest in the study of these condensates[4, 5]. The nature of BEC is fundamentally affected by the presence of the confining potential[6] and finite size corrections are appreciable, leading for instance to a reduction in the critical temperature[7, 8, 9, 10]. Interaction effects are very small in the normal phase but become significant with the condensation–induced density increase. The correction to the transition temperature due to interactions has been recently computed by Giorgini et al [11]. The temperature dependence of the condensate fraction was recently measured[5] for a sample of around 40000 87 Rb atoms, the observed lowering in transition temperature being in agreement with theoretical predictions within experimental resolution. In the same work the internal energy was measured during ballistic expansion and found to be significantly higher in the BEC phase than predicted by the ideal–gas model. While the increase is easily understood as a consequence of the interatomic repulsions, a quantitative estimate is still lacking. In this work we present a two–fluid mean–field model which is able to explain the above–mentioned effects, giving results in agreement with experiment for both the condensate fraction and the internal energy as functions of temperature.