Temperature dependent properties of the dynamics of a two-level system interacting with a radiation field

A system of two-level atoms interacting with one mode of a quantized radiation field is considered by the quantum field theory. The existence of a phase transition to the spontaneous coherent state (SCS) is shown. This state arises from the Bose-Einstein condensation of the photons of the radiation field. The spectrum of the collective excitations of the system is found for all temperatures. In the disordered phase there are two modes of transverse collective excitations when the sizes of an atomic sample are much less than the wavelength and three modes of excitations when the sizes of an atomic sample can be compared with the wavelength of the radiation field. One of the modes is a soft mode provided that the threshold condition is fulfilled. In the SCS there are six transverse collective modes. One of them has zero frequency and the frequencies of other modes increase with decreasing temperature. Moreover, in the SCS three longitudinal collective excitations arise. The photon Green's function has a pole at the origin in the condensed phase which shows that the radiation field is in a coherent state.