Dissipation-induced instabilities of two-component Bose-Einstein condensates in optical cavities

We investigate the dynamics of a gas of ultra-cold spin-1 atoms inside an optical cavity, which is driven transversely by an external laser whose polarization is not aligned with that of the cavity field. By considering the atom population to be equally distributed between the +1 and -1 spin states, we obtain a two-component Dicke model with complex light-matter couplings. We study the effects of cavity losses on the system by computing the steady-state phase diagram and observe the emergence of dynamical instabilities in the form of limit cycles, induced by the interplay between coherent and dissipative processes. We characterize the physical mechanisms behind the unstable behavior and study the role of cavity fluctuations in the system.