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arXiv:2303.07725 · Physical Review A

Entangled time-crystal phase in an open quantum light-matter system

Robert Mattes, Igor Lesanovsky, Federico Carollo

Published Dec 18, 2023• 29 citations
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Abstract

Time crystals are nonequilibrium many-body phases in which the state of the system dynamically approaches a limit cycle. While these phases have recently been the focus of intensive research, it is still far from clear whether they can host quantum correlations. In fact, mostly classical correlations have been observed so far, and time crystals appear to be effectively classical high-entropy phases. Here, we consider the nonequilibrium behavior of an open quantum light-matter system, realizable in current experiments, which maps to a paradigmatic time-crystal model after an adiabatic elimination of the light field. The system displays a bistable regime with coexistent time-crystal and stationary phases, terminating at a tricritical point from which a second-order phase transition line departs. While light and matter are uncorrelated in the stationary phase, the time-crystal phase features bipartite correlations of both quantum and classical nature. Our work unveils that time-crystal phases in collective open quantum systems can sustain quantum correlations, including entanglement, and are thus more than effectively classical many-body phases.

DOI
10.1103/physreva.108.062216

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Time crystals are nonequilibrium many-body phases in which the state of the system dynamically approaches a limit cycle. While these phases have recently been the focus of intensive research, it is still far from clear whether they can host quantum correlations. In fact, mostly classical correlations have been observed so far, and time crystals appear to be effectively classical high-entropy phases. Here, we consider the nonequilibrium behavior of an open quantum light-matter system, realizable in current experiments, which maps to a paradigmatic time-crystal model after an adiabatic elimination of the light field. The system displays a bistable regime with coexistent time-crystal and stationary phases, terminating at a tricritical point from which a second-order phase transition line departs. While light and matter are uncorrelated in the stationary phase, the time-crystal phase features bipartite correlations of both quantum and classical nature. Our work unveils that time-crystal phases in collective open quantum systems can sustain quantum correlations, including entanglement, and are thus more than effectively classical many-body phases.
The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of nonstationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical nonlinear and nonstationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differ from those of stationary ones.
Abstract Open many-body quantum systems can exhibit intriguing nonequilibrium phases of matter, such as time crystals. In these phases, the state of the system spontaneously breaks the time-translation symmetry of the dynamical generator, which typically manifests through persistent oscillations of an order parameter. A paradigmatic model displaying such a symmetry breaking is the boundary time crystal (BTC), which has been extensively analyzed experimentally and theoretically. Despite the broad interest in these nonequilibrium phases, their thermodynamics and their fluctuating behavior remain largely unexplored, in particular for the case of coupled time crystals. In this work, we consider two interacting BTCs and derive a consistent interpretation of their thermodynamic behavior. We fully characterize their average dynamics and the behavior of their quantum fluctuations, which allows us to demonstrate the presence of quantum and classical correlations in both the stationary and the time-crystal phases displayed by the system. We furthermore exploit our theoretical derivation to explore possible applications of time crystals as quantum batteries, demonstrating their ability to efficiently store energy.
Time crystals are nonequilibrium phases of matter characterized by the emergence of temporal ordering, in which an interacting many-body system develops robust structure in its time evolution that is not trivially dictated by the external driving or environment. While related phenomena have long been studied in classical nonlinear systems, their realization in entangled quantum matter represents a distinct frontier. The theoretical understanding of discrete time crystals has substantially advanced, yet recent experiments using modern quantum devices and quantum processors reveal regimes beyond established paradigms. These developments call for an extended classification of time-crystalline phases according to both their stabilization mechanisms and their physical character, including discrete and continuous, closed and open, critical, topological, quasiperiodic, and controlled realizations. We review recent implementations of time crystals on quantum platforms and propose such a classification framework, identifying promising directions for the discovery of novel time-crystalline phases of matter.
Abstract Time-translation symmetry breaking is a mechanism for the emergence of non-stationary many-body phases, so-called time-crystals, in Markovian open quantum systems. Dynamical aspects of time-crystals have been extensively explored over the recent years. However, much less is known about their thermodynamic properties, also due to the intrinsic nonequilibrium nature of these phases. Here, we consider the paradigmatic boundary time-crystal system, in a finite-temperature environment, and demonstrate the persistence of the time-crystalline phase at any temperature. Furthermore, we analyze thermodynamic aspects of the model investigating, in particular, heat currents, power exchange and irreversible entropy production. Our work sheds light on the thermodynamic cost of sustaining nonequilibrium time-crystalline phases and provides a framework for characterizing time-crystals as possible resources for, e.g. quantum sensing. Our results may be verified in experiments, for example with trapped ions or superconducting circuits, since we connect thermodynamic quantities with mean value and covariance of collective (magnetization) operators.
In this paper we study genuine multipartite correlations (GMC's) in a boundary time crystal (BTC). Boundary time crystals are nonequilibrium quantum phases of matter in contact to an environment, for which a macroscopic fraction of the many-body system breaks the time translation symmetry. We analyze both (i) the structure (orders) of GMC's among the subsystems, as well as (ii) their build-up dynamics for an initially uncorrelated state. We find that, in the thermodynamic limit (and only in such a limit), multipartite correlations of all orders grow indefinitely in time in the BTC phase, further displaying a persistent oscillatory behavior around their mean growth. The orders of the correlations show a power-law decaying hierarchy among its kkk partitions. Moreover, in the long-time limit the GMC's are shown extensive with the system size, contrasting to the subextensive scaling in the non time crystals (ferromagnetic) phase of the model. We also discuss the classical and quantum nature of these correlations with basis on multipartite entanglement witnesses, specifically, the analysis of the quantum Fisher information (QFI). Both GMC and QFI are able to capture and distinguish the different phases of the model. Our paper highlights the genuine many-body properties of these peculiar nonequilibrium phases of matter.
Nonequilibrium phases of quantum matter featuring time crystalline eigenstate order have been realized recently on noisy intermediate-scale quantum (NISQ) devices. While ideal quantum time crystals exhibit collective subharmonic oscillations and spatiotemporal long-range order persisting for infinite times, the decoherence time of current NISQ devices sets a natural limit to the survival of these phases, restricting their observation to a shallow quantum circuit. Here we propose a time-periodic scheme that leverages quantum-classical feedback protocols in subregions of the system to enhance a time crystal signal significantly exceeding the decoherence time of the device. As a case of study, we demonstrate the survival of the many-body localized discrete time crystal phase in the one-dimensional periodically kicked Ising model, accounting for decoherence of the system with an environment. Based on classical simulation of quantum circuit realizations we find that this approach is suitable for implementation on existing quantum hardware and presents a prospective path to simulate complex quantum many-body dynamics that transcend the low depth limit of current digital quantum computers. Published by the American Physical Society 2024