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arXiv:1708.01472 · Physical Review Letters

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

Zongping Gong, Ryusuke Hamazaki, Masahito Ueda

Published Jan 25, 2018• 240 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.120.040404

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Abstract Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, an exotic new phase of matter. We present observations characteristic of discrete time crystalline order in a driven system of paramagnetic P-donor impurities in isotopically enriched 28Si cooled below 10 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of discrete time crystalline order. This signal has a finite lifetime of ∼100 Floquet periods, but this effect is long-lived relative to coherent spin–spin interaction timescales, lasting ∼104 times longer. We present simulations of the system based on the paradigmatic central spin model and show good agreement with experiment. We investigate the role of dissipation and interactions within this model, and show that both are capable of giving rise to discrete time crystal-like behaviour.

New Journal of Physics•Aug 1, 2020•37 citations
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.
Discrete time crystals (DTC) have been demonstrated experimentally in several different quantum systems in the past few years. Spin couplings and cavity losses have been shown to play crucial roles for realizing DTC order in open many-body systems out of equilibrium. Recently, it has been proposed that eternal and transient DTC can be present with an open Floquet setup in the thermodynamic limit and in the deep quantum regime with few qubits, respectively. In this work, we consider the effects of spin damping and spin dephasing on the DTC order in spin-optomechanical and open cavity systems in which the spins can be all-to-all coupled. In the thermodynamic limit, it is shown that the existence of dephasing can destroy the coherence of the system and finally lead the system to its trivial steady state. Without dephasing, eternal DTC is displayed in the weak damping regime, which may be destroyed by increasing the all-to-all spin coupling or the spin damping. By contrast, the all-to-all coupling is constructive to the DTC in the moderate damping regime. We also focus on a model which can be experimentally realized by a suspended hexagonal boron nitride (hBN) membrane with a few spin color centers under microwave drive and Floquet magnetic field. Signatures of transient DTC behavior are demonstrated in both weak and moderate dissipation regimes without spin dephasing. Relevant experimental parameters are also discussed for realizing transient DTC order in such an hBN optomechanical system.
Periodically driven quantum systems host a range of non-equilibrium phenomena which are unrealizable at equilibrium. Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, a putative quantum phase of matter. We present the observation of discrete time crystalline order in a driven system of paramagnetic PPP -donor impurities in isotopically enriched 28Si^{28}Si28Si cooled below 101010 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of DTC order. We propose a theoretical model based on the paradigmatic central spin model which is in good agreement with experimental observations, and investigate the role of dissipation in the stabilisation of the DTC. Both experiment and theory indicate that the order in this system is primarily a dissipative effect, and which persists in the presence of spin-spin interactions. We present a theoretical phase diagram as a function of interactions and dissipation for the central spin model which is consistent with the experiments. This opens up questions about the interplay of coherent interaction and dissipation for time-translation symmetry breaking in many-body Floquet systems.
Discrete time crystals (DTCs) represent a nonequilibrium phase of matter characterized by spontaneousbreaking of discrete time-translational symmetry under periodic driving. While Floquet topological phases offer apromising route to stabilizing DTC behavior, their persistence in realistic open quantum systems remains an openproblem. In this work, we numerically investigate a periodically driven one-dimensional Ising chain locally coupledto a dissipative two-level system (TLS) defect, motivated by microscopic defects commonly observed in superconductingcircuit-QED devices. Using Lindblad master-equation simulations implemented in QuTiP, we show that weaklydissipative TLS defects can dynamically synchronize with the Floquet edge mode and inherit its subharmonic oscillations.As the defect coupling and dissipation strength increase, the system undergoes a crossover toward decoherence andFloquet thermalization, accompanied by the suppression of edge-localized period doubling. By combining quasienergyspectroscopy, boundary-condition analysis, rigidity diagnostics, and entropy evolution, we characterize the stabilitylimits of Floquet topological time-crystalline phases in realistic noisy quantum hardware.
Abstract Periodically driven quantum systems manifest various non-equilibrium features which are absent at equilibrium. For example, discrete time-translation symmetry can be broken in periodically driven quantum systems leading to an exotic phase of matter, called discrete time crystal (DTC). For open quantum systems, previous studies showed that DTC can be found only when there exists a meta-stable state in the undriven system. However, by investigating the simplest Bose–Hubbard model with dissipation and time periodically tunneling, we find in this paper that a 2T DTC can appear even when the meta-stable state is absent in the undriven system. This observation extends the understanding of DTC and shed more light on the physics behind the DTC. Besides, by the detailed analysis of simplest two-sites model, we show further that the two-sites model can be used as basic building blocks to construct large rings in which a nT DTC might appear. These results might find applications into engineering exotic phases in driven open quantum systems.
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.
Abstract Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, an exotic new phase of matter. We present observations characteristic of discrete time crystalline order in a driven system of paramagnetic P-donor impurities in isotopically enriched 28Si cooled below 10 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of discrete time crystalline order. This signal has a finite lifetime of ∼100 Floquet periods, but this effect is long-lived relative to coherent spin–spin interaction timescales, lasting ∼104 times longer. We present simulations of the system based on the paradigmatic central spin model and show good agreement with experiment. We investigate the role of dissipation and interactions within this model, and show that both are capable of giving rise to discrete time crystal-like behaviour.