Abstract Non-equilibrium phase transitions are a vibrant frontier in physics, driving breakthroughs across diverse disciplines in recent years. Thermal spin ensembles, a captivating many-body quantum system, have harnessed their collective strengths to set remarkable measurement benchmarks in equilibrium. However, their non-equilibrium dynamics remain unexplored, with nonlinear interactions among constituents offering a rich playground for phase transition studies. Here, we employ nonlinear magnetic feedback to achieve real-time observation and control of phase transitions in a driven-dissipative spin ensemble, unveiling dynamic shifts between limit cycles, subharmonics, and chaos. This work not only deepens our grasp of critical phenomena in non-equilibrium systems but also paves the way for chaos-based sensing and information processing technologies, bridging fundamental physics with transformative applications.
Abstract Non-equilibrium phase transitions are a vibrant frontier in physics, driving breakthroughs across diverse disciplines in recent years. Thermal spin ensembles, a captivating many-body quantum system, have harnessed their collective strengths to set remarkable measurement benchmarks in equilibrium. However, their non-equilibrium dynamics remain unexplored, with nonlinear interactions among constituents offering a rich playground for phase transition studies. Here, we employ nonlinear magnetic feedback to achieve real-time observation and control of phase transitions in a driven-dissipative spin ensemble, unveiling dynamic shifts between limit cycles, subharmonics, and chaos. This work not only deepens our grasp of critical phenomena in non-equilibrium systems but also paves the way for chaos-based sensing and information processing technologies, bridging fundamental physics with transformative applications.
Spontaneous symmetry breaking plays a pivotal role in physics ranging from the emergence of elementary particles to the phase transitions of matter. The spontaneous breaking of continuous time translation symmetry leads to a novel state of matter named continuous time crystal (CTC). It exhibits periodic oscillation without the need for periodic driving, and the relative phases for repetitively realized oscillations are random. However, the mechanism behind the spontaneous symmetry breaking in CTCs, particularly the random phases, remains elusive. Here we propose and experimentally realize two types of CTCs based on distinct mechanisms: manifold topology and near-chaotic motion. We observe both types of CTCs in thermal atomic ensembles by artificially synthesizing spin-spin nonlinear interactions through a measurement-feedback scheme. Our work provides general recipes for the realization of CTCs, and paves the way for exploring CTCs in various systems.
Nonlinear spin systems exhibit rich and exotic dynamical phenomena, offering promising applications ranging from spin masers and time crystals to precision measurement. Recent theoretical work [T. Wang et al., Commun. Phys. 8, 41 (2025)] predicted intriguing nonlinear dynamical phases arising from inhomogeneous magnetic fields and feedback interactions. However, experimental exploration of these predictions remains lacking. Here, we report the observation of nonlinear spin dynamics in dual-bias magnetic fields with dual-cell alkali-metal atomic gases and present three representative stable dynamical behaviors of limit cycles, quasi-periodic orbits, and chaos. Additionally, we probe the nonlinear phase transitions between these phases by varying the feedback gain and the difference of dual-bias magnetic fields. Furthermore, we demonstrate the robustness of the limit cycle and quasi-periodic orbit against the noise of magnetic fields. Our findings establish a versatile platform for exploring complex spin dynamics and open new avenues for the realization of multimode spin masers, time crystals and quasi-crystals, and high-precision magnetometers.
We study the application of a new method for simulating nonlinear dynamics of many-body spin systems using quantum measurement and feedback [Mu\~noz-Arias et al., Phys. Rev. Lett. 124, 110503 (2020)] to a broad class of many-body models known as p-spin Hamiltonians, which describe Ising-like models on a completely connected graph with p-body interactions. The method simulates the desired mean-field dynamics in the thermodynamic limit by combining nonprojective measurements of a component of the collective spin with a global rotation conditioned on the measurement outcome. We apply this protocol to simulate the dynamics of the p-spin Hamiltonians and demonstrate how different aspects of criticality in the mean-field regime are readily accessible with our protocol. We study applications including properties of dynamical phase transitions and the emergence of spontaneous symmetry breaking in the adiabatic dynamics of the collective spin for different values of the parameter p. We also demonstrate how this method can be employed to study the quantum-to-classical transition in the dynamics continuously as a function of system size.
A strongly interacting driven spin system can display a variety of interesting many- body quantum phenomena due to the interplay of interaction, disorder and coherent driving. Using a dense electronic spin ensemble of atom-like impurities in diamond, we study non-equilibrium quantum many-body dynamics involving ∼10^6 spins. Specifically, we describe: (1) investigation of thermalization dynamics of a closed quantum system in the presence of long-range interactions and on-site disorder, (2) realization of new non-equilibrium phases of matter, such as discrete time crystals, and (3) probing emergent phase transitions in a driven many-body system. In addition, the high-density spin ensembles in the solid state also show great promise for practical sensing applications. As a practical application of these studies, we demonstrate high sensitivity magnetic-field sensing with fault-tolerant pulsed control. These studies demonstrate that dense electronic spin ensembles in the solid state can be a powerful tool to simulate quantum many-body dynamics, explore new coherent non-equilibrium phenomena and to detect external signals with superior sensitivity and high spectral and spatial resolution.
Non-equilibrium quantum phase transitions (NEQPTs) represent a frontier in condensed matter physics and quantum statistical mechanics, characterized by abrupt changes in the steady-state properties of open quantum systems or by critical dynamics in closed systems far from thermal equilibrium. Unlike their equilibrium counterparts, NEQPTs are profoundly influenced by driving forces, dissipation, and the specific dynamics of quantum quenches. Understanding these complex phenomena requires theoretical frameworks that can accurately capture both strong correlations and real-time evolution in the presence of environmental interactions. Traditional theoretical tools, such as purely variational methods or purely perturbative expansions, often fall short when dealing with the intricate interplay of coherent and incoherent processes characteristic of NEQPTs. Variational approaches, while adept at capturing ground state properties and short-time dynamics within a chosen ansatz, struggle with long-time thermalization and the inclusion of subtle quantum and thermal fluctuations beyond the variational manifold. Conversely, perturbative methods, particularly those based on the Keldysh formalism, can systematically include interactions and environmental effects but typically rely on a small expansion parameter, limiting their applicability to weakly interacting or weakly driven regimes. This paper proposes a novel "Variational-Perturbative Unification" framework designed to overcome these limitations. The core idea is to combine the strengths of both approaches: using a variational ansatz to capture the dominant non-equilibrium many-body state or dynamics, and then systematically incorporating quantum and thermal fluctuations, as well as dissipation, through a perturbative expansion around this optimized variational state. This hybrid methodology aims to provide a robust and accurate description of NEQPTs, offering insights into critical exponents, universality classes, and the dynamic pathways to steady states. We explore the theoretical underpinnings of this unified approach, discuss its potential applications to driven-dissipative systems and quantum quenches, and highlight how it can reveal novel non-equilibrium phenomena.
During recent years the interest to dynamics of quantum systems has grown considerably. Quantum many body systems out of equilibrium often manifest behavior, different from the one predicted by standard statistical mechanics and thermodynamics in equilibrium. Since the dynamics of a many body quantum system typically involve many excited eigenstates, with a non-thermal distribution, the time evolution of such a system provides an unique way for investigation of non-equilibrium quantum statistical mechanics. Last decade such new subjects like quantum quenches, thermalization, pre-thermalization, equilibration, generalized Gibbs ensemble, etc. are among the most attractive topics of investigation in modern quantum physics. One of the most interesting themes in the study of dynamics of quantum many-body systems out of equilibrium is connected with the recently proposed important concept of dynamical quantum phase transitions. During the last few years a great progress has been achieved in studying of those singularities in the time dependence of characteristics of quantum mechanical systems, in particular, in understanding how the quantum critical points of equilibrium thermodynamics affect their dynamical properties. Dynamical quantum phase transitions reveal universality, scaling, connection to the topology, and many other interesting features. Here we review the recent achievements of this quickly developing part of low temperature quantum physics. The study of dynamical quantum phase transitions is especially important in context of their connection to the problem of the modern theory of quantum information, where namely non-equilibrium dynamics of many-body quantum system plays the major role.