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arXiv:2601.20186

A general interpretation of nonlinear connected time crystals: quantum self-sustaining combined with quantum synchronization

Song-hai Li, Najmeh Es'haqi-Sani, Xingli Li, Wenlin Li

quant-phSubmitted Jan 28, 2026• Announced Jan 29, 2026• 0 citations
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Abstract

Although classical nonlinear dynamics suggests that sufficiently strong nonlinearity can sustain oscillations, quantization of such model typically yields a time-independent steady state that respects time-translation symmetry and thus precludes time-crystal behavior. We identify dephasing as the primary mechanism enforcing this symmetry, which can be suppressed by intercomponent phase correlations. Consequently, a sufficient condition for realizing a continuous time crystal is a nonlinear quantum self-sustaining system exhibiting quantum synchronization among its constituents. As a concrete example, we demonstrate spontaneous oscillations in a synchronized array of van der Pol oscillators, corroborated by both semiclassical dynamics and the quantum Liouville spectrum. These results reduce the identification of time crystals in many-body systems to the evaluation of only two-body correlations and provide a framework for classifying uncorrelated time crystals as trivial.

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Although classical nonlinear dynamics suggests that sufficiently strong nonlinearity can sustain oscillations, quantization of such model typically yields a time-independent steady state that respects time-translation symmetry and thus precludes time-crystal behavior. We identify dephasing as the primary mechanism enforcing this symmetry, which can be suppressed by intercomponent phase correlations. Consequently, a sufficient condition for realizing a continuous time crystal is a nonlinear quantum self-sustaining system exhibiting quantum synchronization among its constituents. As a concrete example, we demonstrate spontaneous oscillations in a synchronized array of van der Pol oscillators, corroborated by both semiclassical dynamics and the quantum Liouville spectrum. These results reduce the identification of time crystals in many-body systems to the evaluation of only two-body correlations and provide a framework for classifying uncorrelated time crystals as trivial.
Continuous time crystals (CTCs) are characterized by sustained oscillations that break the time-translation symmetry. Since the ruling out of equilibrium CTCs by no-go theorems, the emergence of such dynamical phases has been observed in various driven-dissipative quantum platforms. The current understanding of CTCs is mainly based on mean-field theories, which fail to address the problem of whether the continuous time-translation symmetry can be broken in noisy, spatially extended systems absent in all-to-all couplings. Here, we propose a CTC realized in a quantum contact model through self-organized bistability. The CTCs stem from the interplay between collective dissipation induced by the first-order absorbing phase transitions and slow constant driving provided by an incoherent pump. The stability of such oscillatory phases in finite dimensions under the action of intrinsic quantum fluctuations is scrutinized by the functional renormalization group method and numerical simulations. Occurring at the edge of many-body synchronization, the CTC phase exhibits an inherent period and amplitude with a coherence time linearly diverging with system size, thus also constituting a boundary time crystal. Our results serve as a solid route towards self-protected CTCs in strongly interacting open systems. Published by the American Physical Society 2024
While a generic open quantum system decays to its steady state, continuous time crystals (CTCs) develop spontaneous oscillation and never converge to a stationary state. Just as crystals develop correlations in space, CTCs do so in time. Here, we introduce a continuous quasi time crystals (CQTC). Despite being characterized by the presence of nondecaying oscillations, this phase does not retain its long-range order, making it time analogous to quasicrystal structures. We investigate the emergence of this phase in a minimal model, made of two coupled collective spin subsystems, each developing a CTC phase upon the action of a strong enough drive. The addition of a coupling enables the emergence of different synchronized phases, where both subsystems oscillate at the same frequency. In the transition between different CTC orders, the system develops chaotic dynamics with aperiodic oscillations. These chaotic features differ from those of closed quantum systems, as the dynamics is not characterized by a unitary evolution. We investigate the connection between chaos and this quasicrystalline phase using mean-field techniques, and we confirm these results by including quantum fluctuations at the lowest order.
We implement nonlinear anharmonic interaction in the coupled van der Pol oscillators to investigate the quantum synchronization behavior of the systems. We study the quantum synchronization in two oscillator models, coupled quantum van der Pol oscillators and anharmonic self-oscillators. We demonstrate that the considered systems exhibit a high-order synchronization through coupling in both classical and quantum domains. We show that, due to the anharmonicity of the nonlinear interaction between the oscillators, the system exhibits phonon blockade in the phase-locking regime, which is a pure nonclassical effect and has not been observed in the classical domain. We also demonstrate that, for coupled anharmonic oscillators, the system shows a multiple resonance phase-locking behavior due to nonlinear interaction. We point out that the synchronization blockade arises due to strong anticorrelation between the oscillators, which leads to phonon antibunching in the same parametric regime. In the anharmonic oscillator case we illustrate the simultaneous occurrence of bunching and antibunching effects as a consequence of simultaneous negative and positive correlation between the anharmonic oscillators. We examine the aforementioned characteristic features in the frequency entrainment of the oscillators using a power spectrum where one can observe normal mode splitting and the Mollow triplet in the strong coupling regime. Finally, we propose a possible experimental realization for the considered system in the trapped ion and optomechanical setting.
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.
Time crystals are classified into discrete time crystals and continuous time crystals based on whether they spontaneously break time-translation symmetry. Continuous-time crystals do not require external driving. By introducing AdS/CFT duality to time crystals, we derive their thermodynamic limit and find that in strongly correlated many-body systems such as a 3D optical lattice(ions or tweezer in supplemental materials), cooperative many-body tunneling enables time crystals to oscillate spontaneously. In strongly correlated quantum systems driven by many-body cooperative tunneling, we discover a universal scaling law governing the time-crystalline phase transition at a critical temperature.
Continuous time crystals, i.e., nonequilibrium phases with a spontaneously broken continuous time-translational symmetry, have been studied and recently observed in the long time dynamics of open quantum systems. Here, we investigate a lattice of interacting three-level particles and find two distinct time-crystal phases that cannot be described within mean-field theory. Remarkably, one of them emerges only in the presence of correlations, upon accounting for beyond-mean-field effects. Our findings extend explorations of continuous time-translational symmetry breaking in dissipative systems beyond the classical phenomenology of periodic orbits in a low-dimensional nonlinear system. The proposed model applies directly to the laser-driven dynamics of interacting Rydberg states in neutral-atom arrays and suggests that the predicted time-crystal phases are observable in such experiments.