Time Crystal Phase Research Articles

Prolonging a discrete time crystal by quantum-classical feedback

Nonequilibrium phases of quantum matter featuring time crystalline eigenstate order have been realized recently on noisy intermediate-scale quantum (NISQ) devices. While ideal quantum time crystals exhibit collective subharmonic oscillations and spatiotemporal long-range order persisting for infinite times, the decoherence time of current NISQ devices sets a natural limit to the survival of these phases, restricting their observation to a shallow quantum circuit. Here we propose a time-periodic scheme that leverages quantum-classical feedback protocols in subregions of the system to enhance a time crystal signal significantly exceeding the decoherence time of the device. As a case of study, we demonstrate the survival of the many-body localized discrete time crystal phase in the one-dimensional periodically kicked Ising model, accounting for decoherence of the system with an environment. Based on classical simulation of quantum circuit realizations we find that this approach is suitable for implementation on existing quantum hardware and presents a prospective path to simulate complex quantum many-body dynamics that transcend the low depth limit of current digital quantum computers. Published by the American Physical Society 2024

Discrete time crystal phase of higher dimensional integrable models

This paper investigates the possibility of generating Floquet-time crystals in higher dimensions (d≥2) through the time-periodic driving of integrable free-fermionic models. The realization leads to rigid time-crystal phases that are ideally resistant to thermalization and decoherence. By utilizing spin-orbit coupling, we are able to realize a robust time-crystal phase that can be detected using novel techniques. Moreover, we discuss the significance of studying the highly persistent subharmonic responses and their implementation in a Kitaev spin liquid, which contributes to our understanding of time translational symmetry breaking and its practical implications.

Time Crystal in a Single-Mode Nonlinear Cavity.

Time crystal is a class of nonequilibrium phases with broken time-translational symmetry. Here, we demonstrate the time crystal in a single-mode nonlinear cavity. The time crystal originates from the self-oscillation induced by a linear gain and is stabilized by a nonlinear damping. We show in the time crystal phase there are sharp dissipative gap closing and pure imaginary eigenvalues of the Liouvillian spectrum in the thermodynamic limit. Dynamically, we observe a metastable regime with the emergence of quantum oscillation, followed by a dissipative evolution with a timescale much longer than the oscillating period. Moreover, we show there is a dissipative phase transition at the Hopf bifurcation, which can be characterized by the photon number fluctuation in the steady state. These results pave a new promising way for further experiments and deepen our understanding of time crystals.

Quantum metrology with boundary time crystals

Quantum sensing is one of the arenas that exemplifies the superiority of quantum technologies over their classical counterparts. Such superiority, however, can be diminished due to unavoidable noise and decoherence of the probe. Thus, metrological strategies to fight against or profit from decoherence are highly desirable. This is the case of certain types of decoherence-driven many-body systems supporting dissipative phase transitions, which might be helpful for sensing. Boundary time crystals are exotic dissipative phases of matter in which the time-translational symmetry is broken, and long-lasting oscillations emerge in open quantum systems at the thermodynamic limit. We show that the transition from a symmetry unbroken into a boundary time crystal phase, described by a second-order transition, reveals quantum-enhanced sensitivity quantified through quantum Fisher information. We also determine the critical exponents of the system and establish their relationship. Our scheme is indeed a demonstration of harnessing decoherence for achieving quantum-enhanced sensitivity. From a practical perspective, it has the advantage of being independent of initialization and can be captured by a simple measurement.

Fractal nature of high-order time crystal phases

Fractal nature of high-order time crystal phases

Emergence and Dynamical Stability of a Charge Time-Crystal in a Current-Carrying Quantum Dot Simulator.

Periodically driven open quantum systems that never thermalize exhibit a discrete time-crystal behavior, a nonequilibrium quantum phenomenon that has shown promise in quantum information processing applications. Measurements of time-crystallinity are currently limited to (magneto-) optical experiments in atom-cavity systems and spin-systems making it an indirect measurement. We theoretically show that time-crystallinity can be measured directly in the charge-current from a spin-less Hubbard ladder, which can be simulated on a quantum-dot array. We demonstrate that one can dynamically tune the system out and then back on a time-crystal phase, proving its robustness against external forcings. These findings motivate further theoretical and experimental efforts to simulate the time-crystal phenomena in current-carrying nanoscale systems.

Floquet time crystals in driven spin systems with all-to-all p -body interactions

We show the emergence of Floquet time crystal (FTC) phases in the Floquet dynamics of periodically driven $p$-spin models, which describe a collection of spin-1/2 particles with all-to-all $p$-body interactions. Given the mean-field nature of these models, we treat the problem exactly in the thermodynamic limit and show that, for a given $p$, these systems can host various robust time-crystalline responses with period $nT$, where $T$ is the period of the drive and $n$ an integer between 2 and $p$. In particular, the case of four-body interactions ($p=4$) gives rise to both a usual period-doubling crystal, and also a novel period-quadrupling phase. We develop a comprehensive framework to predict robust subharmonic response in classical area-preserving maps, and use this as a basis to predict the occurrence and characterize the stability of the resulting mean-field FTC phases in the quantum regime. Our analysis reveals that the robustness of the time-crystal behavior is reduced as their period increases, and establishes a connection between the emergence of time crystals, described by eigenstate ordering and robust subharmonic response, and the phenomenology of excited state and dynamical quantum phase transitions. Finally, for the models hosting two or more coexisting time crystal phases, we define protocols where the periodic subharmonic response of the system can be varied in time via the non-periodic modulation of an external control parameter.

Emergent parametric resonances and time-crystal phases in driven Bardeen-Cooper-Schrieffer systems

We study the out-of-equilibrium dynamics of a Bardeen-Cooper-Schrieffer condensate subject to a periodic drive. We demonstrate that the combined effect of drive and interactions results in emerging parametric resonances, analogous to a vertically driving pendulum. In particular, Arnold tongues appear when the driving frequency matches 2Δ0/n, with n being a natural number and Δ0 being the equilibrium gap parameter. Inside the Arnold tongues we find a commensurate time-crystal condensate which retains the U(1) symmetry breaking of the parent superfluid/superconducting phase and shows an additional time-translational symmetry breaking. Outside these tongues, the synchronized collective Higgs mode found in quench protocols is stabilized without the need of a strong perturbation. Our results are directly relevant to cold-atom and condensed-matter systems and do not require very long energy relaxation times to be observed.Received 28 July 2021Revised 15 October 2021Accepted 19 October 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.L042023Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDynamical phase transitionsExotic phases of matterNonequilibrium statistical mechanicsSuperconductivitySuperfluiditySynchronizations-wavePhysical SystemsFloquet systemsUltracold gasesTechniquesBCS theoryMethods in superconductivityCondensed Matter, Materials & Applied PhysicsNonlinear Dynamics

Anharmonic classical time crystals: A coresonance pattern formation mechanism

Driven many-body systems have been shown to exhibit discrete time crystal phases characterized by broken discrete time-translational symmetry. This has been achieved generally through a subharmonic response, in which the system undergoes one oscillation every other driving period. Here, we demonstrate that classical time crystals do not need to resonate in a subharmonic fashion but instead can also exhibit a continuously tunable anharmonic response to driving, which we show can emerge through a coresonance between modes in different branches of the dispersion relation in a parametrically driven medium. This response, characterized by a typically incommensurate ratio between the resonant frequencies and the driving frequency, is demonstrated by introducing a time crystal model consisting of an array of coupled pendula with alternating lengths. Importantly, the coresonance mechanism is the result of a bifurcation involving a fixed point and an invariant torus, with no intermediate limit cycles. This bifurcation thus gives rise to a many-body symmetry-breaking phenomenon directly connecting the symmetry-unbroken phase with a previously uncharacterized phase of matter, which we call an anharmonic time crystal phase. The mechanism is shown to generalize to driven media with any number of coupled fields and is expected to give rise to anharmonic responses in a range of weakly damped pattern-forming systems, with potential applications to the study of nonequilibrium phases, frequency conversion, and acoustic cloaking.

Bistability and time crystals in long-ranged directed percolation

Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections. Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata.

Dynamical Transitions and Critical Behavior between Discrete Time Crystal Phases.

In equilibrium physics, spontaneous symmetry breaking and elementary excitation are two concepts closely related with each other: the symmetry and its spontaneous breaking not only control the dynamics and spectrum of elementary excitations, but also determine their underlying structures. In this Letter, based on an exactly solvable model, we propose a phase ramping protocol to study an excitationlike behavior of a nonequilibrium quantum matter: a discrete time crystal phase with spontaneous temporal translational symmetry breaking. It is shown that slow ramping could induce a dynamical transition between two Z_{2} symmetry breaking time crystal phases in time domain, which can be considered as a temporal analog of the soliton excitation spatially sandwiched by two degenerate charge density wave states in polyacetylene. By tuning the ramping rate, we observe a critical value at which point the transition duration diverges, resembling the critical slowing down phenomenon in nonequilibrium statistic physics. We also discuss the effect of stochastic sequences of such phase ramping processes and its implication to the stability of the discrete time crystal phase against noisy perturbations.

Time crystals in the driven transverse field Ising model under quasiperiodic modulation

We investigate the transverse field Ising model subject to a two-step periodic driving protocol and quasiperiodic modulation of the Ising couplings. Analytical results on the phase boundaries associated with Majorana edge modes and numerical results on the localization of single-particle excitations are presented. The implication of a region with fully localized domain-wall-like excitations in the parameter space is eigenstate order and exact spectral pairing of Floquet eigenstates, based on which we conclude the existence of time crystals. We also examine various correlation functions of the time crystal phase numerically, in support of its existence.

Discrete time crystal in the gradient-field Heisenberg model

We show that time crystal phases, which are known to exist for disorder-based many-body localized systems, also appear in systems where localization is due to strong magnetic field gradients. Specifically, we study a finite Heisenberg spin chain in the presence of a gradient field, which can be realized experimentally in quantum dot systems using micromagnets or nuclear spin polarization. Our numerical simulations reveal time crystalline order over a broad range of realistic quantum dot parameters, as evidenced by the long-time preservation of spin expectation values and the asymptotic form of the mutual information. We also consider the undriven system and present several diagnostics for many-body localization that are complementary to those recently studied. Our results show that these non-ergodic phases should be realizable in modest-sized quantum dot spin arrays using only demonstrated experimental capabilities.

Time crystals in a shaken atom-cavity system

We demonstrate the emergence of a time crystal of atoms in a high-finesse optical cavity driven by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representation for the driven-dissipative quantum dynamics, we confirm the rigidity and persistence of the time crystalline phase. We identify experimentally relevant parameter regimes for which the time crystal phase is long-lived, and map out the dynamical phase diagram. We compare and contrast the incommensurate time crystal with the commensurate Dicke time crystal in the amplitude-modulated case.

Stabilization and manipulation of multispin states in quantum-dot time crystals with Heisenberg interactions

A discrete time crystal is a recently discovered non-equilibrium phase of matter that has been shown to exist in disordered, periodically driven Ising spin chains. In this phase, if the system is initially prepared in one of a certain class of pure multispin product states, it periodically returns to this state over very long time scales despite the presence of interactions, disorder, and pulse imperfections. Here, we show that this phase occurs in GaAs quantum dot spin arrays containing as few as three quantum dots, each confining one electron spin, for naturally occurring levels of nuclear spin noise and charge noise. Although the physical interaction in these arrays is a nearest-neighbor Heisenberg exchange interaction, we show that this can be effectively converted into an Ising interaction by applying additional pulses during each drive period. We show that by changing the rotation axis of these pulses, we can select the direction of the Ising interaction and, consequently, the quantization axis of the stabilized multispin states. Moreover, we demonstrate that it is possible to perform coherent rotations of the stabilized states while remaining in the time crystal phase. These findings open up the intriguing possibility of using time crystal phases to extend the lifetime of quantum states for information applications.