Abstract. 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 .PACS numbers: 03.75.Fi, 67.40.Kh 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.

Superfluidity in coupled electron-hole sheets of bilayer graphene is predicted here to be multicomponent because of the conduction and valence bands. We investigate the superfluid crossover properties as functions of the tunable carrier densities and the tunable energy band gap E_{g}. For small band gaps there is a significant boost in the two superfluid gaps, but the interaction-driven excitations from the valence to the conduction band can weaken the superfluidity, even blocking the system from entering the Bose-Einstein condensate (BEC) regime at low densities. At a given larger density, a band gap E_{g}∼80-120  meV can carry the system into the strong-pairing multiband BCS-BEC crossover regime, the optimal range for realization of high-T_{c} superfluidity.

Superfluidity has recently been reported in double electron-hole bilayer graphene. The multiband nature of the bilayers is important because of the very small band gaps between conduction and valence bands. The long range nature of the superfluid pairing interaction means that screening must be fully taken into account. We have carried out a systematic mean-field investigation that includes (i) contributions to screening from both intraband and interband excitations, (ii) the lowenergy band structure of bilayer graphene with its small band gap and flattened Mexican hat-like low-energy bands, (iii) the large density of states at the bottom of the bands, (iv) electron-hole pairing in the multibands, and (v) electron-hole pair transfers between the conduction and valence band condensates. We find that the superfluidity strongly modifies the intraband contributions to the screening, but that the interband contributions are unaffected. Unexpectedly, the net effect of the screening is to suppress Josephson-like pair transfers and to confine the superfluid pairing entirely to the conduction band condensate even for very small band gaps, making the system behave similarly to a one-band superfluid.