Stabilities and dynamics of a three-level extensions of the Dicke model

We consider an extension of the Dicke model which consists of a three-level system coupled to a cavity field. It can describe a Bose-Einstein-condensate coupled to an optical cavity and driven by a bichromatic laser pump. Although its closed-system phase diagram is rather typical to Dicke-type models, the introduction of cavity dissipation induces unusual stabilities and instabilities. For a wide range of parameters, the lowest-energy normal state becomes unstable towards a large collection of inverted states with higher energy, which indicates a finite density of states. This voluminous stability is connected to an emergent local U(1) symmetry in the system, and can be easily probed experimentally due to the sensitivity of final states of dynamical evolutions to the ramping protocols.