Abstract We investigate the nonlinear dynamics of two coupled driven–dissipative Rydberg atomic chains, each of which formed a previously reported continuous time crystal phase. By tuning the inter-chain interaction, the two time-crystalline subsystems with distinct intrinsic oscillation frequencies will naturally mix, and support luxuriant stable and collective oscillatory behavior. We identify a dynamical phase characterized by quasi-periodic oscillations with multiple incommensurate frequencies, corresponding to a continuous-time quasi-crystalline order. In addition, we identify a phase transition from a time quasi-crystalline phase to a time-crystalline phase at sufficiently strong inter-chain coupling. This transition can be understood as the evolution from partially synchronized oscillations between the two chains to fully synchronized dynamics. From a mathematical perspective, we confirm that the transition is governed by either a saddle-node or a torus bifurcation, depending on the separation between the two original oscillation frequencies. These theoretical results provide guidance for exploring complex periodic dynamics in Rydberg atomic platforms.
Abstract We investigate the nonlinear dynamics of two coupled driven–dissipative Rydberg atomic chains, each of which formed a previously reported continuous time crystal phase. By tuning the inter-chain interaction, the two time-crystalline subsystems with distinct intrinsic oscillation frequencies will naturally mix, and support luxuriant stable and collective oscillatory behavior. We identify a dynamical phase characterized by quasi-periodic oscillations with multiple incommensurate frequencies, corresponding to a continuous-time quasi-crystalline order. In addition, we identify a phase transition from a time quasi-crystalline phase to a time-crystalline phase at sufficiently strong inter-chain coupling. This transition can be understood as the evolution from partially synchronized oscillations between the two chains to fully synchronized dynamics. From a mathematical perspective, we confirm that the transition is governed by either a saddle-node or a torus bifurcation, depending on the separation between the two original oscillation frequencies. These theoretical results provide guidance for exploring complex periodic dynamics in Rydberg atomic platforms.
We report an oscillatory form in the two coupled dissipative Rydberg atomic chains by modulating its spacing. Such oscillation has $\ensuremath{\pi}$-phase difference between two neighboring sites, which distinguishes it from antiferromagnetic-type synchronization in the previous studies. Theoretically, we find a phase with two coexisting types of continuous time crystals and recognize that the transition belongs to Hopf and pitchfork bifurcation. Furthermore, we generalize the conclusion to multiple chains and verify the uniqueness of the synchronization mode. We also discuss its experimental feasibility.
Abstract Discrete time quasi-crystals are non-equilibrium phenomena with quasi-periodic order in time domain, an extension of the discrete time-crystal phase. As a natural platform to explore the non-equilibrium phases, Rydberg atomic arrays have demonstrated the quantum simulation of the discrete-time crystal phase, associated with quantum many-body scar state. However, the existence of discrete time quasi-crystal in this platform has yet to be conceived. Here, we propose a method to generate the discrete time quasi-crystal behavior by coupling two discrete time-crystals, where two external driving frequencies have the maximum incommensurability. Upon the characterization of the phase diagram and the robustness, we explore the entanglement entropy between two subsystems. We find that the emergence of the aperiodic response appears at an appropriate driving frequency, which could be well interpreted by the model with two coupled oscillators. Our method thus offers the possibilities to explore the phases of matters in the quantum simulator.
We report a new oscillatory form in the two coupled dissipative Rydberg atomic chains by modulating its spacing. Such oscillation has π-phase difference between two neighboring sites, which distinguishes itself from antiferromagnetic-type synchronization in the previous studies. Theoretically, we find a phase with coexisting two types of continuous time crystals, and recognize that the transition belongs to Hopf and pitchfork bifurcation. Furthermore, we generalize the conclusion to multiple chains and verify the uniqueness of the new synchronization mode. We also discuss its experimental feasibility.
Driven nonlinear oscillators constitute a universal paradigm for understanding synchronization, frequency pulling, and frequency comb formation in nonequilibrium systems. Here, we realize such an emergent nonlinear oscillator in strongly interacting cesium Rydberg vapor, where coherent optical excitation, dissipation, and long-range interactions give rise to a driven-dissipative time crystal phase with intrinsic oscillation frequencies. Applying a radio-frequency (RF) field allows controlled tuning of the intrinsic oscillation frequency. Under RF heterodyne conditions, we observe intermodulation, frequency pulling, and, at strong drive, the emergence of a comb-like spectrum in the atomic coherence. We quantitatively capture these observations using a four-level mean-field model and demonstrate a classical analogue with a driven Van der Pol oscillator. Our results establish interacting Rydberg ensembles as a tunable platform for exploring nonequilibrium time-crystalline order, nonlinear synchronization, and frequency comb generation in many-body atomic systems.
Time crystals, as temporal analogs of space crystals, manifest as stable and periodic behavior that breaks time translation symmetry. In an open quantum system, many-body interaction subjected to dissipation allows one to develop the time crystalline order in an unprecedented way, as refer to dissipative time crystals. Here we report the observation of multiple time crystals in the continuously driven-dissipative and strongly interacting Rydberg thermal gases, in which continuous time crystals, sub-harmonic time crystals and high-harmonic time crystals are observed in the same system by manipulating the Rydberg excitation. Our work provides new ways to explore the non-equilibrium phases of matter in open systems. Such time crystals with persistent oscillation rooted in emergent quantum correlations, may emerge as a ubiquitous tool in quantum metrology, for instance, continuous sensing and parameter estimation surpassing the standard quantum limit. The authors observed multiple time crystals in the continuously driven-dissipative and strongly interacting Rydberg thermal gases. This discovery may benefit the field of quantum metrology, such as continuous sensing, potentially surpassing the standard quantum limit, and time crystalline order as a frequency standard.
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.