A dissipative time crystal with or without Z2 symmetry breaking.

We study an emergent semiclassical time crystal composed of two interacting driven-dissipative bosonic modes. The system has a discrete spatial symmetry which, depending on the strength of the drive, can be broken in the time-crystalline phase or it cannot. An exact semiclassical mean-field analysis, numerical simulations in the quantum regime, and the spectral analysis of the Liouvillian are combined to show the emergence of the time crystal and to prove the robustness of the oscillation period against quantum fluctuations. [ABSTRACT FROM AUTHOR]