Theory for Dissipative Time Crystals in Coupled Parametric Oscillators.

Discrete time crystals are novel phases of matter that break the discrete time translational symmetry of a periodically driven system. In this Letter, we propose a classical system of weakly nonlinear parametrically driven coupled oscillators as a test bed to understand these phases. Such a system of parametric oscillators can be used to model period-doubling instabilities of Josephson junction arrays as well as semiconductor lasers. To show that this instability leads to a discrete time crystal we first show that a certain limit of the system is close to Langevin dynamics in a symmetry breaking potential. We numerically show that this phase exists even in the presence of Ising symmetry breaking using a Glauber dynamics approximation. We then use a field theoretic argument to show that these results are robust to other approximations including the semiclassical limit when applied to dissipative quantum systems.