Collective excitations of dissipative time crystals

We present a Floquet-theoretic description of atoms interacting periodically with a dissipative optical cavity. We derive an effective atom-only master equation, valid in the bad cavity regime.

Using this theory, we analyze the excitation spectrum of the atoms across the transition from a normal phase to a time-crystalline phase. We identify features in the excitation spectra, such as

mode softening when crossing a continuous equilibrium transition, that suggest a dynamical phase transition. We then analyze the excitation spectra when the periodic drive crosses a bistable regime and observe sudden jumps in the oscillation frequencies and relaxation rates. Finally, we discuss how these results can be detected experimentally by probing the cavity with an additional monochromatic drive. Our work provides important tools for analyzing the response of dynamical out-of-equilibrium phases.