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The nonlinear semiclassical dynamics of the unbalanced, open Dicke model

In recent years there have been significant advances in the study of many-body interactions between atoms and light confined to optical cavities. One model which has received widespread attention of late is the Dicke model, which under certain conditions exhibits a quantum phase transition to a state in which the atoms collectively emit light into the cavity mode, known as superradiance. We consider a generalization of this model that features independently controllable strengths of the co- and counter-rotating terms of the interaction Hamiltonian. We study this system in the semiclassical (mean field) limit, i.e., neglecting the role of quantum fluctuations. Under this approximation, the model is described by a set of nonlinear differential equations, which determine the system's semiclassical evolution. By taking a dynamical systems approach, we perform a comprehensive analysis of these equations to reveal an abundance of novel and complex dynamics. Examples of the novel phenomena that we observe are the emergence of superradiant oscillations arising due to Hopf bifurcations, and the appearance of a pair of chaotic attractors arising from period-doubling cascades, followed by their collision to form a single, larger chaotic attractor via a sequence of infinitely many homoclinic bifurcations. Moreover, we find that a flip of the collective spin can result in the sudden emergence of chaotic dynamics. Overall, we provide a comprehensive roadmap of the possible dynamics that arise in the unbalanced, open Dicke model in the form of a phase diagram in the plane of the two interaction strengths. Hence, we lay out the foundations to make further advances in the study of the fingerprint of semiclassical chaos when considering the master equation of the unbalanced Dicke model, that is, the possibility of studying a manifestation of quantum chaos in a specific, experimentally realizable system.