Gapless Finite- T Theory of Collective Modes of a Trapped Gas

We present predictions for the frequencies of collective modes of trapped Bose-condensed ${}^{87}\mathrm{Rb}$ atoms at finite temperature. Our treatment includes a self-consistent treatment of the effects upon the mean field from finite- $T$ excitations and the anomalous average. This is the first gapless [A. Griffin, Phys. Rev. B 53, 9341 (1996)] calculation of this type for a trapped Bose-Einstein condensed gas. The corrections quantitatively account for the downward shift in the $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ excitation frequencies observed in recent experiments as the critical temperature is approached.