We study the dynamics of atoms interacting periodically with a dissipative optical cavity and employ Floquet theory to analyze their low-frequency behavior. By means of an effective atom-only master equation, valid in the bad cavity regime, we characterize the excitation spectrum of the atoms across the transition from a normal phase to a time-crystalline phase where the atoms undergo stable oscillations. We identify features in the complex excitation spectra when crossing second and first order transitions where the order parameter changes continuously or abruptly. Finally, we discuss how these results can be detected experimentally by probing the cavity with an additional drive. Our work provides important tools for analyzing the response of dynamical out-of-equilibrium phases.
We study the dynamics of atoms interacting periodically with a dissipative optical cavity and employ Floquet theory to analyze their low-frequency behavior. By means of an effective atom-only master equation, valid in the bad cavity regime, we characterize the excitation spectrum of the atoms across the transition from a normal phase to a time-crystalline phase where the atoms undergo stable oscillations. We identify features in the complex excitation spectra when crossing second and first order transitions where the order parameter changes continuously or abruptly. Finally, we discuss how these results can be detected experimentally by probing the cavity with an additional drive. Our work provides important tools for analyzing the response of dynamical out-of-equilibrium phases.
We demonstrate the emergence of a time crystal of atoms in a high-finesse optical cavity driven by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representation for the driven-dissipative quantum dynamics, we confirm the rigidity and persistence of the time crystalline phase. We identify experimentally relevant parameter regimes for which the time crystal phase is long lived, and map out the dynamical phase diagram. We compare and contrast the incommensurate time crystal with the commensurate Dicke time crystal in the amplitude-modulated case.
Discrete time crystals are a recently proposed and experimentally observed out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic dynamics of a local observable repeats itself at an integer multiple of the driving period. We address this issue in a driven-dissipative setup, focusing on the modulated open Dicke model, which can be implemented by cavity or circuit QED systems. In the thermodynamic limit, we employ semiclassical approaches and find rich dynamical phases on top of the discrete time-crystalline order. In a deep quantum regime with few qubits, we find clear signatures of a transient discrete time-crystalline behavior, which is absent in the isolated counterpart. We establish a phenomenology of dissipative discrete time crystals by generalizing the Landau theory of phase transitions to Floquet open systems.
We consider continuous time-crystalline phases in dissipative many-body systems of atoms in cavities, focusing on the role of short-range interatomic interactions. First, we show that the latter can alter the nature of the time crystal by changing the type of the underlying critical bifurcation. Second, we characterize the heating mechanism and dynamics resulting from the short-range interactions and demonstrate that they make the time crystal inherently metastable. We argue this is generic for the broader class of dissipative time crystals in atom-cavity systems whenever the cavity loss rate is comparable to the atomic recoil energy. We observe that such a scenario for heating has several similarities to the one proposed for preheating in the early universe, where the oscillating coherent inflation field decays into a cascade of exponentially growing fluctuations. By extending approaches for dissipative dynamical systems to our many-body problem, we obtain analytical predictions for the parameters describing the phase transition and the heating rate inside the time-crystalline phase. We underpin and extend the analytical predictions of the heating rates with numerical simulations, which also show that the metastable regime exists when the inherent stochastic nature is taken into account.
We study the quantum dynamics of many-body arrays of two-level atoms in a driven cavity subject to collective decay and interactions mediated by the cavity field. We work in the bad cavity limit accessible, for example, using long-lived electronic clock states of alkaline-earth atoms, for which the bare atomic linewidth is much less than the cavity linewidth. In the absence of interactions, our system reduces to previously studied models of collective fluorescence. We show that while interactions do not qualitatively change the steady-state properties they lead to a drastic change in the dynamical properties. We find that, for some intervals of driving strengths, the system shows two very distinct types of transient behaviors that depend on the initial state of the system. In particular, there is a parameter regime where the system features oscillatory dynamics with a period of oscillation that becomes much shorter than the duration of the overall transient dynamics as the atom number increases. We use both mean-field and exact numerical calculations of the quantum system to investigate the dynamics.
Continuous time crystals represent a novel state in many-body systems that can self-organize into timeperiodic oscillations without external periodic driving. Recent experiments have achieved such systems in dissipative solid-state materials, where persistent temporal order is autonomously sustained. A decisive characteristic of time crystals is their robustness, meaning that despite various disturbances, including fluctuations in internal parameters and external noise, they can still maintain rhythmic behavior, which has scientific value and echnological application potential. Although previous studies have shown that specific experimental parameters have robustness, thare is a lack of a systematic framework for quantifying and predicting their resilience to disturbances, and the underlying physics of this robustnessis still not fully understood. The key unresolved problems include how nonlinear interactions and feedback mechanisms contribute to stability, and what the critical thresholds are for parameter variations beyond which temporal order collapses. This work addresses these gaps by systematically analyzing how internal parameters and external influences affect the oscillation period and overall stability. Internally, the dynamics are determined by dipole-dipole interactions and atomic transition strengths, which define the temporal symmetry breaking that occurs in the system. Externally, the response of the system is controlled by the strength of the optical driving field and the energy dissipation rate. A key finding is the determination of an intrinsic feedback mechanism for a dynamically stabile time crystal. This mechanism plays a role in restoring force, correcting deviations caused by minor disturbances, and maintaining the coherence of oscillatory phase. Moreover, the system displays nonlinear dynamical behavior, characterized by two different states: one is stable oscillation continuing under moderate disturbance, and the other is stronger disturbance causing dynamical phase transition, resulting in switching between disordered or dynamically unstable state and stable state. These results provide a comprehensive understanding of the various behaviors observed in continuous time crystals and lay an important theoretical foundation for utilizing their unique properties in advanced applications such as quantum information processing and precision metrology.
Spontaneous breaking of a continuous symmetry produces a massless Goldstone mode that can evolve across a degenerate manifold at zero energy cost. Goldstone modes have been identified primarily through excitation spectra, mode softening or collective oscillations. However, their time-domain transport under intrinsic fluctuations and dissipation has remained largely unexplored. Here we directly track the stochastic transport of a Goldstone mode in a self-organized atomic crystal inside an optical ring cavity. The ring cavity maps the order-parameter phase onto the real-space position of the emergent crystal. Without any external perturbation, fundamental photon-scattering recoil drives the collective transport, while cavity dissipation generates friction. We monitor individual trajectories of the atoms and their self-generated optical lattice by measuring the cavity output phase. We find that the diffusion constant decreases as 1/N, indicating that all atoms move collectively as a rigid object rather than independently. By tuning the Langevin driving force and cavity-mediated damping, we show that the normalized diffusion constant collapses onto a single universal curve. This work extends the study of continuous symmetry breaking from excitation-frequency measurements to real-time tracking of transport, and opens routes for studying non-equilibrium collective transport, phonon dynamics, and defect formation in driven-dissipative quantum matter.