We develop and study an atom-only description of the Dicke model with time-periodic couplings between atoms and a dissipative cavity mode. The cavity mode is eliminated, giving rise to effective atom-atom interactions and dissipation. We use this effective description to analyze the dynamics of the atoms that undergo a transition to a dynamical superradiant phase with macroscopic coherences in the atomic medium and the light field. Using Floquet theory in combination with the atom-only description we provide a precise determination of the phase boundaries and of the dynamical response of the atoms. From this we can predict the existence of dissipative time crystals that show a subharmonic response with respect to the driving frequency. We show that the atom-only theory can describe the relaxation into such a dissipative time crystal and that the damping rate can be understood in terms of a cooling mechanism.

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• Cavity quantum electrodynamics
• Cooling & trapping
• Dissipative dynamics
• Open quantum systems
• Parametric resonance
• Time crystals
• Floquet systems
• Dicke model
• Lindblad equation
• Master equation