Quantum trajectories of dissipative time-crystals

We explore the boundary time-crystal transition at the level of quantum trajectories which result from continuous monitoring. This Letter is motivated by recent experiments [G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, Nat. Phys. 19, 1345 (2023)10.1038/s41567-023-02064-w] realizing this many-body system and which allow one in principle to gain in situ information on its nonequilibrium dynamics. We find that the photon count signal as well as the homodyne current allow one to identify and characterize critical behavior at the time-crystal phase transition. In the time-crystal phase these quantities display persistent oscillations, resolvable in finite systems and in individual realizations. At the transition point the dynamics of the emission signals feature intermittent strong fluctuations, which can be understood through a simple nonlinear phase model. We furthermore show that the time-integrated homodyne current can serve as a useful dynamical order parameter. From this perspective the time crystal can be viewed as a state of matter in which different oscillation patterns coexist.