  The Dicke model is a staple of theoretical cavity Quantum Electrodynamics
(cavity QED), describing the interaction between an ensemble of atoms and a
single radiation mode of an optical cavity. It has been studied both quantum
mechanically and semiclassically for two-level atoms, and demonstrates a rich
variety of dynamics such as phase transitions, phase multistability, and chaos.
In this work we explore an open, spin-1 Dicke model with independent co- and
counter-rotating coupling terms as well as a quadratic Zeeman shift enabling
control over the atomic energy-level structure. We investigate the stability of
operator and moment equations under two approximations and show the system
undergoes phase transitions. To compliment these results, we relax the
aforementioned approximations and investigate the system semiclassically. We
show evidence of phase transitions to steady-state and oscillatory
superradiance in this semiclassical model, as well as the emergence of chaotic
dynamics. The varied and complex behaviours admitted by the model highlights
the need to more rigorously map its dynamics.
