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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

Dynamical symmetry of nonstationary systems

I.A. Malkin, Vladimir I. Man’ko, D. A. Trifonov

Published Aug 1, 1971• 25 citations
Publisher
DOI
10.1007/bf02731519

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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.
Advancing quantum sensing tools for investigating atomic and nanoscale systems is crucial for the progress of quantum technologies. While many protocols use quantum probes to extract information from stationary or weakly coupled environments, challenges intensify at the atomic scale and the nanoscale, where the environment is inherently out of equilibrium and/or strongly coupled with the sensor. In this work, we demonstrate that time-reversal symmetry in the control dynamics of a quantum sensor is broken when the qubit sensor interacts with environments that are out of equilibrium (with nonstationary fluctuations), contain quantum non-Gaussian correlations, or exhibit intrinsic time-reversal symmetry breaking. Leveraging this phenomenon, we introduce a quantum sensing paradigm based on time-asymmetric dynamical control sequences, enabling the probing of the distance of an environment from equilibrium, its time-reversal symmetry-breaking behavior, or its quantum non-Gaussian nature. Additionally, we propose sensing strategies to distinguish between these different sources of symmetry breaking. We validate our approach through proof-of-principle experimental quantum simulations using solid-state nuclear magnetic resonance, where we drive the environment of a qubit sensor out of equilibrium and use our protocol to quantify the nonstationary characteristics of the generated states. Our findings highlight the potential of this framework for quantifying nonstationary behavior, designing tailored pump-probe experiments (including measurements of quantum information scrambling), and detecting quantum states that exhibit time-reversal symmetry breaking, time-crystal structures, or quantum non-Gaussian fluctuations. Overall, this work constitutes a step forward in designing quantum devices for atomic scale and nanoscale sensing, broadening their applicability to complex and dynamic quantum environments.
In this paper, we report findings of nonstationary behavior observed in a fully connected qubit network, utilizing a random unitary evolution model in open quantum system theory. The environmental effect is reflected in the partial swap interaction between pairs of qubits with a certain probability. Our study begins with a simple Ising-type Hamiltonian and through many iterations of random unitary evolution, a nonstationary oscillatory state may arise, which encodes certain memory of the initial state. The nontrivial periodic motion of some local observables is indicative of a continuous time crystal phase. We also explore the extension of our study to other types of Hamiltonians and demonstrate that this nonstationary behavior is widespread in our model due to the generalized dynamical symmetry. Remarkably, both theoretical and numerical analysis support the robustness of the constructed time crystal phase to most types of noise. Our research provides a perspective for constructing the time-crystal phase in an open system model.