Many-body quantum optics with decaying atomic spin states: (gamma, kappa) Dicke model

We provide a theory for quantum-optical realizations of the open Dicke model with internal, atomic spin states subject to spontaneous emission with rate gamma. This introduces a second decay channel for excitations to irreversibly dissipate into the environment, in addition to the photon loss with rate., which is composed of individual atomic decay processes and a collective atomic decay mechanism. The strength of the latter is determined by the cavity geometry. We compute the mean-field nonequilibrium steady states for spin and photon observables in the long-time limit, t -> infinity. Although gamma does not conserve the total angular momentum of the spin array, we argue that our solution is exact in the thermodynamic limit for the number of atoms N -> infinity. In light of recent and upcoming experiments realizing superradiant phase transitions using internal atomic states with pinned atoms in optical lattices, our work lays the foundation for the pursuit of a new class of open quantum magnets coupled to quantum light