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Frontiers in Optics

Time Crystals and Nonequilibrium Dissipative Phase Transitions Mediated by Squeezed Bath

Zhenghao Zhang, Qingtian Miao, G. S. Agarwal

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We investigate a driven collective-spin system coupled to a squeezed vacuum reservoir. Squeezing enhances the sharpness of the dissipative phase transition near the critical point and alters the Liouvillian spectrum in a non-monotonic way.
DOI
10.1364/fio.2025.jtu5a.22

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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.
The nonequilibrium dissipative phase transition, arising from the competition of cooperative behavior and coherent field driving, discovered in the 1970s by Narducci et al. [Phys. Rev. A18, 1571 (1978)] and Walls et al. [Prog. Theor. Phys. Suppl.64, 307 (1978)], 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.
Quantum synchronization (QS) in dissipative systems is often inferred from smooth phase locking, leaving open whether its breakdown constitutes a genuine nonequilibrium transition. Here we introduce a Liouvillian framework that classifies driven-dissipative dynamics by the structure of the undriven dissipative background and show that QS breaks down via a Hopf-type dynamical phase transition into a boundary time crystal (BTC). The character of this transition is determined by the background attractor: systems with a self-sustained oscillator (SSO) support robust non-resonant BTCs, whereas those with a polar fixed point (PFP) sustain BTCs only at resonance and lose them under detuning. We identify sharp dynamical and spectral signatures of the QS-BTC transition and thereby establish, within U(1)-symmetric collective-spin Lindbladians driven by a single coherent tone, a background-based allowed/forbidden criterion that unifies QS, its breakdown, and time-crystalline order within a single Liouvillian framework.
In this work, we study the driven-dissipative dynamics of a coherently-driven spin ensemble with a squeezed, superradiant decay. This decay consists of a sum of both raising and lowering collective spin operators with a tunable weight. The model presents different critical non-equilibrium phases with a gapless Liouvillian that are associated to particular symmetries and that give rise to distinct kinds of non-ergodic dynamics. In Ref. [1] we focus on the case of a strong-symmetry and use this model to introduce and discuss the effect of dissipative freezing, where, regardless of the system size, stochastic quantum trajectories initialized in a superposition of different symmetry sectors always select a single one of them and remain there for the rest of the evolution. Here, we deepen this analysis and study in more detail the other type of non-ergodic physics present in the model, namely, the emergence of non-stationary dynamics in the thermodynamic limit. We complete our description of squeezed superradiance by analysing its metrological properties in terms of spin squeezing and by analysing the features that each of these critical phases imprint on the light emitted by the system.
We investigate quantum thermometry in a driven--dissipative Kerr cavity coupled to a thermal reservoir. The system exhibits a finite-size precursor of a dissipative phase transition, characterized by a pronounced minimum of the Liouvillian gap and a sharp jump in the steady-state physical observables such as average photon number at the critical driving strength. We show that this regime leads to strong enhancement of the quantum Fisher information (QFI) for temperature estimation. Using an effective two-branch description, we show that the enhancement originates from temperature-induced redistribution of the weight factors in photon number distribution between low- and high-photon-number branches, which is described by an effective binary Fisher information. By optimizing the coherent drive, the enhanced response persists over an extended low-temperature, low-thermal-occupation regime and yields a favorable relative temperature-uncertainty bound. These results identify finite-size precursors of dissipative phase transitions in Kerr-cavity platforms as useful resources for tunable nonequilibrium quantum thermometry. We further show that the predicted thermometric enhancement is accessible in a parameter regime compatible with circuit quantum electrodynamics (circuit-QED) platforms.