Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Time crystals and nonequilibrium dissipative phase transitions mediated by squeezed bath

Zhenghao Zhang, Qingtian Miao, G. S. Agarwal

Published Aug 25, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1103/nhdh-nvmz

Related papers

Time crystals and nonequilibrium dissipative phase transitions mediated by squeezed bath

Zhenghao Zhang, Qingtian Miao, G. S. Agarwal

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.

arXiv:2507.09072•Jul 11, 2025•0 citations

Time Crystals and Nonequilibrium Dissipative Phase Transitions Mediated by Squeezed Bath

Zhenghao Zhang, Qingtian Miao, G. S. Agarwal

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.

Frontiers in Optics•Jan 1, 2025•0 citations

Superoperator structures and no-go theorems for dissipative quantum phase transitions

Thomas Barthel, Yikang Zhang

In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.

arXiv:2012.05505•Physical Review A•May 26, 2022•10 citations

Dissipative time crystal in an asymmetric nonlinear photonic dimer

Kilian Seibold, R. Rota, Vincenzo Savona

We investigate the behavior of two coupled nonlinear photonic cavities, in the presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean-field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly nonclassical properties and its dynamics exhibits the long-lived Josephson-like oscillations typical of dissipative time crystals, as indicated by the presence of purely imaginary eigenvalues in the spectrum of the Liouvillian superoperator in the thermodynamic limit.

arXiv:1910.03499•Physical Review A•Mar 30, 2020•80 citations

Quantum sensing in the vicinity of dissipative phase transition

Anonymous, G. S. Agarwal

Recently, there has been considerable interest in studying quantum sensing near the critical points of a system evolving unitarily and undergoing a phase transition. However, there are many quantum systems in which dissipation is very important. The quantum phase transition in such systems results as a result of competition between nonlinearity, drive, and dissipation. These quantum phase transitions belong to the class of nonequilibrium phase transitions. In this work, we present a detailed study of quantum sensing of the system parameters near the critical points of a dissipative phase transition. We illustrate the key results by considering a coherently driven nonlinear Kerr medium interacting with environment. The first-order quantum phase transition is characterized by the closing of the gap in eigenvalues of the underlying Liouvillian. The steady-state sensing away from the critical point has standard quantum limit scaling with respect to the effective system size, whereas Heisenberg scaling is found at the critical point. The latter is subject to a critical slowing down. We have also examined sensing of system parameters using the metastable states and find better sensing performance at intermediate timescales, i.e., when the system has not yet settled in steady state. Detailed results on the scaling behavior in different time domains are given.

Physical Review Research•Aug 11, 2026•0 citations

Emergent continuous time crystal in dissipative quantum spin system without driving

Shu Yang, Zeqing Wang, Libin Fu +1 more

Time crystals are a nonequilibrium phase of matter that extend fundamental spontaneous symmetry breaking into the temporal dimension, typically requiring external driving for their realization. Here, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system without external coherent or incoherent driving. Through numerical analysis of spin dynamics, we identify nonstationary steady states, some of which are limit cycles with persistent periodic oscillations, while others exhibit chaotic, aperiodic behavior. The emergence of limit cycle steady states breaks the continuous time-translation symmetry of this time-independent many-body system, classifying them as continuous time crystals. We further validate these oscillatory behaviors by testing their stability against local perturbations and assess the robustness of the emergent continuous time crystals by introducing isotropic Gaussian white noise. This work provides insights into the intricate interplay between the dissipation and spin interaction, and opens possibilities for realizing dissipation-induced, heating-immune time crystals. Time crystals represent a nonequilibrium phase of matter without an equilibrium counterpart. The authors investigate the nonequilibrium steadystate phase diagram of a two-dimensional dissipative Heisenberg model in the absence of driving, and identify a continuous time crystalline phase exhibiting persistent oscillations in time.

arXiv:2403.08476•Communications Physics•Mar 24, 2025•12 citations
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.
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.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
We investigate the behavior of two coupled nonlinear photonic cavities, in the presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean-field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly nonclassical properties and its dynamics exhibits the long-lived Josephson-like oscillations typical of dissipative time crystals, as indicated by the presence of purely imaginary eigenvalues in the spectrum of the Liouvillian superoperator in the thermodynamic limit.
Recently, there has been considerable interest in studying quantum sensing near the critical points of a system evolving unitarily and undergoing a phase transition. However, there are many quantum systems in which dissipation is very important. The quantum phase transition in such systems results as a result of competition between nonlinearity, drive, and dissipation. These quantum phase transitions belong to the class of nonequilibrium phase transitions. In this work, we present a detailed study of quantum sensing of the system parameters near the critical points of a dissipative phase transition. We illustrate the key results by considering a coherently driven nonlinear Kerr medium interacting with environment. The first-order quantum phase transition is characterized by the closing of the gap in eigenvalues of the underlying Liouvillian. The steady-state sensing away from the critical point has standard quantum limit scaling with respect to the effective system size, whereas Heisenberg scaling is found at the critical point. The latter is subject to a critical slowing down. We have also examined sensing of system parameters using the metastable states and find better sensing performance at intermediate timescales, i.e., when the system has not yet settled in steady state. Detailed results on the scaling behavior in different time domains are given.
Time crystals are a nonequilibrium phase of matter that extend fundamental spontaneous symmetry breaking into the temporal dimension, typically requiring external driving for their realization. Here, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system without external coherent or incoherent driving. Through numerical analysis of spin dynamics, we identify nonstationary steady states, some of which are limit cycles with persistent periodic oscillations, while others exhibit chaotic, aperiodic behavior. The emergence of limit cycle steady states breaks the continuous time-translation symmetry of this time-independent many-body system, classifying them as continuous time crystals. We further validate these oscillatory behaviors by testing their stability against local perturbations and assess the robustness of the emergent continuous time crystals by introducing isotropic Gaussian white noise. This work provides insights into the intricate interplay between the dissipation and spin interaction, and opens possibilities for realizing dissipation-induced, heating-immune time crystals. Time crystals represent a nonequilibrium phase of matter without an equilibrium counterpart. The authors investigate the nonequilibrium steadystate phase diagram of a two-dimensional dissipative Heisenberg model in the absence of driving, and identify a continuous time crystalline phase exhibiting persistent oscillations in time.