Collective excitations of Bose-Einstein condensed gases at finite temperatures

We have applied the Popov version of the Hartree-Fock-Bogoliubov (HFB) approximation to calculate the finite-temperature excitation spectrum of a Bose-Einstein condensate (BEC) of ${}^{87}\mathrm{Rb}$ atoms. For lower values of the temperature, we find excellent agreement with recently published experimental data for the JILA time-averaged orbiting potential trap. In contrast to recent comparison of the results of HFB-Popov theory with experimental condensate fractions and specific heats, there is disagreement of the theoretical and recent experimental results near the BEC phase transition temperature.