The Hepp-Lieb-Dicke model is ubiquitous in cavity quantum electrodynamics, describing spin-cavity coupling which does not conserve excitation number. Coupling the closed spin-cavity system to an environment realizes the open Dicke model, and by tuning the structure of the environment or the system-environment coupling, interesting spin-only models can be engineered. In this work, we focus on a variation of the multi-species open Dicke model which realizes mediated nonreciprocal interactions between the spin species and, consequently, a dynamical limit-cycle phase. In particular, we improve upon adiabatic elimination and, instead, employ a Redfield master equation in order to describe the effective dynamics of the spin-only system. We assess this approach at the mean-field level, comparing it both to adiabatic elimination and the full spin-cavity model, and find that the predictions are sensitive to the presence of single-particle incoherent decay. Additionally, we clarify the symmetries of the model and explore the dynamical limit-cycle phase in the case of explicit PT-symmetry breaking, finding a region of phase coexistence terminating at a codimension-two exceptional point. Lastly, we go beyond mean-field theory by exact numerical diagonalization of the master equation, appealing to permutation symmetry in order to increase the size of accessible systems. We find signatures of phase transitions even for small system sizes.
Peter B. Littlewood Joseph Jachinowski (Mon,) studied this question.