Dynamical phases of a Bose-Einstein condensate in a bad optical cavity at optomechanical resonance

We study the emergence of dynamical phases of a Bose-Einstein condensate that is optomechanically coupled to a dissipative cavity mode and transversally driven by a laser. We focus on the regime close to the optomechanical resonance, where the atoms' refractive index shifts the cavity into resonance, assuming fast cavity relaxation. We derive an effective model for the atomic motion, where the cavity degrees of freedom are eliminated using perturbation theory in the atom-cavity coupling and benchmark its predictions using numerical simulations based on the full model. Away from the optomechanical resonance, perturbation theory in the lowest order (adiabatic elimination) reliably describes the dynamics and predicts chaotic phases with unstable oscillations. Interestingly, the dynamics close to the optomechanical resonance are qualitatively captured only by including the corrections to next order (nonadiabatic corrections). In this regime we find limit-cycle phases that describe stable oscillations of the density with a well-defined frequency. We further show that such limit-cycle solutions are metastable configurations of the adiabatic model. Our work sheds light on the mechanisms that are required to observe dynamical phases and predict their existence in atom-cavity systems where a substantial timescale separation is present.