Collective Excitations of Dissipative Time Crystals

We investigate the dynamical response across phase transitions in dissipative time crystals. Using Floquet theory, we characterize the excitation spectra associated with the transition from a normal to a time-crystalline phase. Our analysis reveals two distinct types of dynamical phase transitions, distinguished by the behavior of an order parameter: a continuous (second-order) transition, where the order parameter evolves smoothly; and a discontinuous (first-order) transition, where it changes abruptly. The excitation spectra provide clear signatures of these transitions: the continuous transition is accompanied by the closing of a complex gap, whereas the discontinuous transition exhibits the coexistence of two excitation branches. Focusing on a concrete model—cold atoms strongly coupled to an optical cavity—we demonstrate how these excitations can be experimentally probed by driving the cavity with a longitudinal field. This framework not only clarifies the nature of out-of-equilibrium phases but also opens a route toward experimentally probing the universal properties of time-crystalline order.