  We show a dissipative phase transition in a driven nonlinear quantum
oscillator in which a discrete time-translation symmetry is broken either
continuously or discretely. The corresponding regimes display either continuous
or discrete time crystals, which we analyze numerically and analytically beyond
the classical limit addressing observable dynamics, Liouvillian spectral
features, and quantum fluctuations. Via an effective semiclassical description,
we show that phase diffusion dominates when the symmetry is broken
continuously, which manifests as a band of eigenmodes with a lifetime growing
linearly with the mean-field excitation number. Instead, in the discrete
symmetry broken phase, the leading fluctuation process corresponds to quantum
activation with a single mode that has an exponentially growing lifetime.
Interestingly, the transition between these two regimes manifests itself
already in the quantum regime as a spectral singularity, namely as an
exceptional point mediating between phase diffusion and quantum activation.
Finally, we discuss this transition between different time-crystal orders in
the context of synchronization phenomena.

Quantum Physics;Statistical Mechanics