Bose–Einstein condensation versus Dicke–Hepp–Lieb transition in an optical cavity

•Atoms inside a driven cavity can undergo two transitions: self-organization and BEC.•The phase diagram has four phases which coexist at a bi-critical point.•Atom–cavity coupling creates a dynamical lattice for the atoms.•Finite temperature can enhance the tendency towards self-organization.•We calculate the detailed spectrum of the polaritonic excitations.

We provide an exact solution for the interplay between Bose–Einstein condensation and the Dicke–Hepp–Lieb self-organization transition of an ideal Bose gas trapped inside a single-mode optical cavity and subject to a transverse laser drive. Based on an effective action approach, we determine the full phase diagram at arbitrary temperature, which features a bi-critical point where the transitions cross. We calculate the dynamically generated band structure of the atoms and the associated suppression of the critical temperature for Bose–Einstein condensation in the phase with a spontaneous periodic density modulation. Moreover, we determine the evolution of the polariton spectrum due to the coupling of the cavity photons and the atomic field near the self-organization transition, which is quite different above or below the Bose–Einstein condensation temperature. At low temperatures, the critical value of the Dicke–Hepp–Lieb transition decreases with temperature and thus thermal fluctuations can enhance the tendency to a periodic arrangement of the atoms.