Phase Transitions in Some Generalized Dicke Models of Superradiance

The Dicke model of superradiance studied in a previous paper by Wang and Hioe is generalized (i) to include an interaction energy of the form $a^{†}{\sigma }^{+}+a{\sigma }^{-}$, where $a,a^{†}$ and ${\sigma }^{-},{\sigma }^{+}$ are the usual photon and spin operators, (ii) to include the spatial variation of the electromagnetic field, and (iii) to include the possibility of small classical motions of atoms from their equilibrium positions. The free energies of these generalized models are calculated exactly in the thermodynamic limit, and their phase-transition properties are presented.