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arXiv:2303.13334 · Physical review. B./Physical review. B

Bridging closed and dissipative discrete time crystals in spin systems with infinite-range interactions

Jayson G. Cosme, Jim Skulte, Ludwig Mathey

Published Jul 6, 2023• 14 citations
arXivHTMLPDF

Abstract

We elucidate the role that the dissipation in a bosonic channel plays in the prevalence and stability of time crystals (TCs) in a periodically driven spin-boson system described by the Dicke model. Here, the bosons are represented by photons, and they mediate the infinite-range interactions between the spin systems. For strong dissipation, we study the dynamics using an effective atom-only description and the closed Lipkin-Meshkov-Glick model. By mapping out the phase diagrams for varying dissipation strengths, ranging from zero to infinitely strong, we demonstrate that the area in the phase diagram, where a TC exists, grows with the dissipation strength but only up to an optimal point, beyond which most of the TCs become unstable. We find TCs in both closed-system and dissipative regimes, but dissipative TCs are shown to be more robust against random noise in the drive and are only weakly affected by the choice of initial state. We present the finite-sized behavior and the scaling of the lifetime of the TCs with respect to the number of spins and the interaction strength within a fully quantum mechanical description.

DOI
10.1103/physrevb.108.024302

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We elucidate the role that the dissipation in a bosonic channel plays in the prevalence and stability of time crystals (TCs) in a periodically driven spin-boson system described by the Dicke model. Here, the bosons are represented by photons, and they mediate the infinite-range interactions between the spin systems. For strong dissipation, we study the dynamics using an effective atom-only description and the closed Lipkin-Meshkov-Glick model. By mapping out the phase diagrams for varying dissipation strengths, ranging from zero to infinitely strong, we demonstrate that the area in the phase diagram, where a TC exists, grows with the dissipation strength but only up to an optimal point, beyond which most of the TCs become unstable. We find TCs in both closed-system and dissipative regimes, but dissipative TCs are shown to be more robust against random noise in the drive and are only weakly affected by the choice of initial state. We present the finite-sized behavior and the scaling of the lifetime of the TCs with respect to the number of spins and the interaction strength within a fully quantum mechanical description.
Abstract The Dicke model—a paradigmatic example of superradiance in quantum optics—describes an ensemble of atoms which are collectively coupled to a leaky cavity mode. As a result of the cooperative nature of these interactions, the system’s dynamics is captured by the behavior of a single mean-field, collective spin. In this mean-field limit, it has recently been shown that the interplay between photon losses and periodic driving of light–matter coupling can lead to time-crystalline-like behavior of the collective spin (Gong et al 2018 Phys. Rev. Lett. 120 040404). In this work, we investigate whether such a Dicke time crystal (TC) is stable to perturbations that explicitly break the mean-field solvability of the conventional Dicke model. In particular, we consider the addition of short-range interactions between the atoms which breaks the collective coupling and leads to complex many-body dynamics. In this context, the interplay between periodic driving, dissipation and interactions yields a rich set of dynamical responses, including long-lived and metastable Dicke-TCs, where losses can cool down the many-body heating resulting from the continuous pump of energy from the periodic drive. Specifically, when the additional short-range interactions are ferromagnetic, we observe time crystalline behavior at non-perturbative values of the coupling strength, suggesting the possible existence of stable dynamical order in a driven-dissipative quantum many-body system. These findings illustrate the rich nature of novel dynamical responses with many-body character in quantum optics platforms.
We establish a link between metastability and a discrete time-crystalline phase in a periodically driven open quantum system. The mechanism we highlight requires neither the system to display any microscopic symmetry nor the presence of disorder, but relies instead on the emergence of a metastable regime. We investigate this in detail in an open quantum spin system, which is a canonical model for the exploration of collective phenomena in strongly interacting dissipative Rydberg gases. Here, a semiclassical approach reveals the emergence of a robust discrete time-crystalline phase in the thermodynamic limit in which metastability, dissipation, and interparticle interactions play a crucial role. We perform numerical simulations in order to investigate the dependence on the range of interactions, from all to all to short ranged, and the scaling with system size of the lifetime of the time crystal.
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 pinpoint the conditions necessary for discrete time crystal (DTC) formation in fully connected spin-cavity systems from the perspective of parametric resonance by mapping these systems onto oscillatorlike models. We elucidate the role of nonlinearity and dissipation by mapping the periodically driven open Dicke model onto effective linear and nonlinear oscillator models, while we analyze the effect of global symmetry breaking using the Lipkin-Meshkov-Glick model with tunable anisotropy. We show that the system's nonlinearity restrains the dynamics from becoming unbounded when driven resonantly. On the other hand, dissipation keeps the oscillation amplitude of the period-doubling instability fixed, which is a key feature of DTCs. The presence of global symmetry breaking in the absence of driving is found to be crucial in the parametric resonant activation of period-doubling response. We provide analytic predictions for the resonant frequencies and amplitudes leading to DTC formation for both systems using their respective oscillator models.
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 develop a semiclassical Langevin approach to study finite-size effects in dissipative time-crystalline spin systems. For a spin-1/21/21/2 model with power-law Lindblad operators, we find finite-size oscillations exponentially decaying in time, and their decay rate decreases algebraically with system size in the long-range regime. The scaling exponent of this decay time provides a direct diagnostic of the robustness of the time-crystalline, and we find that this robustness extends beyond the range of power-law dissipation exponents where the system dynamics is mean-field in the thermodynamic limit. We also apply our approach to a spin-one model with local dissipation and long-range Hamiltonian interactions, formulating it in terms of Gell-Mann variables. In this case, finite-size stability is quantified through the deviation time from the mean-field behavior and the behavior of the dominant Fourier peak. We find that both quantities scale as a power law with the system size -- marking thereby a time-crystal behavior -- when the exponent of the power-law interaction is below a certain threshold, that turns out to be in agreement with previous cumulant-expansion findings. Our results show that semiclassical Langevin dynamics provides a useful finite-size probe of dissipative time crystals in regimes beyond exact Lindblad simulations.