  Dicke model predicts a quantum phase transition from normal to superradiant
phases for a two-level atomic ensemble coupled with an optical cavity at zero
temperature. In a recent pioneer experiment [Nature 464, 1301 (2010)], such a
phase transition has been observed using a Bose-Einstein condensate (BEC) in an
optical cavity. Compared with the original Dicke model, the experimental system
features finite temperature and strong atom-photon nonlinear interaction in
BEC. In this Letter, we develop a finite temperature theory for the Dicke phase
transition of a BEC in an optical cavity, taking into account the atom-photon
nonlinear interaction. In addition to explaining the experimentally observed
transition from normal to superradiant phases at finite-temperature, we point
it out that a new phase, the coexistence of normal and superradient phases, was
also observed in the experiment. We show rich finite temperature phase diagrams
existing in the experimental system by tuning various experimental parameters.
We find that the specific heat of the BEC can serve as a powerful tool for
probing various phases.
