Time crystals and nonequilibrium dissipative phase transitions mediated by squeezed bath

Nonequilibrium dissipative phase transition, arising from the competition of cooperative behavior and coherent field driving, discovered in the 1970s by Narducci et al. and Walls et al., has been found to exhibit time-crystal behavior when the driving field exceeds the cooperative decay rate. This was seen through the study of the eigenvalues of the Liouvillian superoperator that describes the joint effect of drive and cooperativity. The cooperative decay depends on the nature of the reservoir correlations. If the reservoir correlations have phase-sensitive behavior, then the eigenvalues of the Liouvillian will be different. We investigate the time-crystal behavior of the nonequilibrium dissipative phase transitions under the influence of a squeezed vacuum reservoir. We analyze the steady-state phase diagram as a function of the control parameter and demonstrate that increasing the squeezing strength sharpens the dissipative phase transition. Spectral analysis of the Liouvillian reveals gap closings and the emergence of purely imaginary eigenvalues in the thermodynamic limit, indicating the time-crystal phase. We find that the real parts of subleading eigenvalues exhibit nonmonotonic behavior with increasing squeezing, reflecting the sensitivity of relaxation dynamics to the reservoir properties. Time-domain simulations confirm that the oscillation frequencies correspond to the imaginary parts of the Liouvillian eigenvalues. We also present results on quantum fluctuations in the time-crystal phase. Our results call attention to the study of time crystals in models of cooperativity based on engineered environments.