We investigate the dynamics of coupled discrete time crystals (DTCs) in the coupled Dicke model or Dicke dimer (DD) with dissipation. By numerically simulating the mean-field dynamics of the system, we show that coupling two DTCs decreases their oscillation amplitude following a transient relaxation period. For sufficiently strong coupling, the coupled DTCs become unstable, resulting in the system relaxing towards the normal phase (NP). We demonstrate that the relaxation time diverges near the critical coupling strength, suggesting a continuous phase transition. Using the truncated Wigner approximation (TWA), we explore how the dynamics are affected by the introduction of quantum fluctuations. We also consider a scenario in which an NP is briefly coupled to a DTC, seeding the NP to transform into a DTC that is either synchronized or anti-synchronized relative to the DTC. We investigate the robustness of this phenomenon in the presence of noise. Using these results, we propose a scheme to perform NAND and NOR operations with three DTCs by sequentially coupling the input and output sites. We identify the parameter regimes in which the logic gate architecture remains valid, and investigate how the protocol is affected by the presence of quantum fluctuations.
We investigate the dynamics of coupled discrete time crystals (DTCs) in the coupled Dicke model or Dicke dimer (DD) with dissipation. By numerically simulating the mean-field dynamics of the system, we show that coupling two DTCs decreases their oscillation amplitude following a transient relaxation period. For sufficiently strong coupling, the coupled DTCs become unstable, resulting in the system relaxing towards the normal phase (NP). We demonstrate that the relaxation time diverges near the critical coupling strength, suggesting a continuous phase transition. Using the truncated Wigner approximation (TWA), we explore how the dynamics are affected by the introduction of quantum fluctuations. We also consider a scenario in which an NP is briefly coupled to a DTC, seeding the NP to transform into a DTC that is either synchronized or anti-synchronized relative to the DTC. We investigate the robustness of this phenomenon in the presence of noise. Using these results, we propose a scheme to perform NAND and NOR operations with three DTCs by sequentially coupling the input and output sites. We identify the parameter regimes in which the logic gate architecture remains valid, and investigate how the protocol is affected by the presence of quantum fluctuations.
We study discrete time crystals (DTCs) in periodically driven quantum systems, in the presence of non-Markovian dissipation. In contrast to DTCs observed in earlier works in the presence of Markovian dynamics, using the open Dicke model in presence of Jaynes-Cummings-like dissipation, we show that non-Markovian regime can be highly beneficial for stabilizing DTCs over a wide range of parameter values. This may be attributed to periodically varying dissipation rates even at long times in the case of non-Markovian dynamics. Further the Markovian and non-Markovian regimes show sharp distinctions for intermediate strengths of the dissipator coefficient, with a time-independent steady state in the Markovian regime being replaced by varied dynamical phases, including DTC order, in the non-Markovian regime. We also verify the robustness of the DTC phase in the non-Markovian regime by introducing errors both in the Hamiltonian as well as in the dissipation. Our study shows the possibility of using DTC as a probe for non-Markovian dynamics in periodically modulated open quantum systems, at long times.
We investigate the discrete time crystal (DTC) phase in a qubit ensemble, periodically driven by its interaction with either a photon or a transmon field, which is prone to dissipative leakage. We find this DTC to be robust against changes in detuning and anharmonicity of the field mode. Additionally, we study the system in the semiclassical limit, where mean-field approximations are valid, and demonstrate the utility of a suitable semiclassical Hamiltonian for this purpose. Intriguingly, we observe that the system exhibits a transient DTC even with only two qubits. We examine the dynamics of bipartite entanglement between the qubits and the field. Our findings show that the entanglement saturates to a steady value early in the dynamics, following a sudden peak. We find a strong positive correlation between this long-term entanglement value and the lifetime of the transient DTC, in a wide range of the parameter regime where the field is due to a lossy photon or a lossy transmon mode, with small detuning.
It has been shown that Discrete Time Crystal (DTC) behavior may arise in the periodically driven, open Dicke model in the thermodynamic limit (N→∞). In this work, we demonstrate that finite N non-mean-field fluctuations in this model can lead to decay rates and amplitudes of transient DTCs that display a non-trivial non-monotonic behavior with an increasing number of atoms. We show that the second-order cumulant expansion fails to capture the finite-size dynamics, while a stochastic version of the Truncated Wigner Approximation (TWA) is remarkably accurate, particularly in the large atom ensemble limit. Using the TWA and the Monte Carlo Wave Function (MCWF) approach, we identify three distinct dynamical regimes of transient DTC behavior in the finite-size Dicke model. We outline the mechanism behind the non-monotonic finite size dynamical decay of DTC order and discuss the implications for experiments performed on finite-sized atomic assemblies.
We demonstrate that the Kibble-Zurek mechanism (KZM) holds for open systems transitioning from a disordered phase to a discrete time crystal (DTC). Specifically, we observe the characteristic power-law scaling with quench time of the number of spatial defects and the transition delay measured from the time at which the system crosses the critical point. We show analytically that this universal behavior can be traced back to how systems that can be mapped onto a dissipative linear parametric oscillator (DLPO) satisfy the adiabatic-impulse (AI) approximation, evinced by the divergence of the relaxation time of the DLPO near a critical point. We verify our predictions in both the classical and quantum regimes by considering two systems: the Sine-Gordon model, which is a paradigmatic system for emulating classical DTCs; and the open Dicke lattice model, an array of spin-boson systems subject to quantum fluctuations. We establish a universality class for DTC formation in systems that can be mapped onto a DLPO and show that the classical and quantum models considered here belong to this class.
We describe a general and simple paradigm for discrete time crystals (DTCs), systems with a stable subharmonic response to an external driving field, in a classical thermal setting. We consider, specifically, an Ising model in two dimensions, as a prototypical system with a phase transition into stable phases distinguished by a local order parameter, driven by thermal dynamics and periodically kicked with a noisy protocol. By means of extensive numerical simulations for large sizes-allowed by the classical nature of our model-we show that the system features a true disorder-DTC order phase transition as a function of the noise strength, with a robust DTC phase extending over a wide parameter range. We demonstrate that, when the dynamics is observed stroboscopically, the phase transition to the DTC state appears to be in the equilibrium two-dimensional Ising universality class. However, we explicitly show that the DTC is a genuine nonequilibrium state. More generally, we speculate that systems with thermal phase transitions to multiple competing phases can give rise to DTCs when appropriately driven.
The simulation of non-equilibrium quantum many-body systems on noisy intermediate-scale quantum (NISQ) processors represents a central challenge and opportunity in modern condensed matter physics. Periodically driven (Floquet) systems can host exotic phases of matter with no equilibrium counterpart, most notably discrete time crystals (DTCs) characterized by spontaneous breaking of discrete time-translation symmetry and robust subharmonic responses. However, the realization of DTCs faces fundamental obstacles: driven many-body systems generically absorb energy and thermalize to an infinite-temperature state, destroying time-crystalline order. Two primary mechanisms have been identified to stabilize DTCs—many-body localization (MBL) through strong disorder, and Floquet prethermalization wherein high-frequency driving suppresses heating over exponentially long timescales. This work investigates the digital quantum simulation of prethermal DTCs on current superconducting quantum processors, with a focus on understanding and mitigating the impact of hardware noise. Recent experiments on 133-qubit heavy-hexagonal devices have demonstrated the realization of clean two-dimensional DTCs and incommensurately modulated DTCs (IM-DTCs) using kicked Ising models, observing period-doubling oscillations persisting for up to 100 Floquet cycles . Crucially, these simulations reveal that quantum noise, often perceived as detrimental, can in certain regimes stabilize novel prethermal DTC phases—a phenomenon identified as noise-induced DTCs supported by symmetry charge pumping or random gate fluctuations . The heavy-hexagonal lattice geometry, compared to square lattices, exhibits significantly slower quantum scrambling, thereby extending the prethermal plateau and enhancing DTC stability. We address two key challenges limiting the scalability of DTC simulations: limited qubit connectivity and environmental noise. By leveraging the universality of native gate sets, we demonstrate how to implement effective couplings among physically disconnected qubits at the cost of increased circuit depth, enabling the study of interaction-range effects on DTC stabilization—a phenomenon previously unobserved experimentally . Furthermore, we benchmark the performance of error mitigation protocols based on depolarising noise models, achieving excellent agreement between error-mitigated hardware data and tensor-network simulations in regimes where classical computations remain tractable . Our results highlight the complementary roles of quantum hardware and classical simulation: quantum processors provide access to regimes where entanglement growth outpaces tensor-network methods, while careful validation against classical benchmarks ensures the reliability of quantum outcomes. This study establishes a concrete pathway for using NISQ devices as experimental platforms for synthesizing and probing novel non-equilibrium quantum phases, advancing both our fundamental understanding of Floquet many-body dynamics and the practical capabilities of quantum simulation. The identification of noise as an active ingredient in phase stabilization suggests new directions for quantum engineering of spatiotemporal orders on near-term hardware.