Floquet Time Crystals Research Articles

Floquet time crystals as quantum sensors of ac fields

Floquet time crystals as quantum sensors of ac fields

Boundary discrete time crystals induced by topological superconductors in solvable spin chains

The Floquet time crystal, which breaks discrete time-translation symmetry, is an intriguing phenomenon in nonequilibrium systems. It is crucial to understand the rigidity and robustness of discrete time crystal (DTC) phases in a many-body system, and finding a precisely solvable model can pave a way for understanding of the DTC phase. Here, we propose and study a solvable spin chain model by mapping it to a Floquet superconductor through the Jordan-Wigner transformation. The phase diagrams of Floquet topological systems are characterized by topological invariants and tell the existence of anomalous edge states. The subharmonic oscillation, which is the typical signal of the DTC, can be generated from such edge states and protected by topology and solvability. We also examine the robustness of the DTC by adding symmetry-preserving and symmetry-breaking perturbations. Our results on the topologically protected DTC can provide a deep understanding of the DTC when generalized to other interacting or dissipative systems.

Topological [formula omitted] modes and beyond

This short Perspective article presents an overview of the discovery of topological $\pi$ modes as well as their physical significance in quantum computing and the understanding of an exotic phase of matter, i.e., the Floquet time crystal. The recent proposals of $2\pi/k$ modes as the generalizations of $\pi$ modes are further elucidated.

Clean two-dimensional Floquet time crystal

We consider the two-dimensional quantum Ising model, in absence of disorder, subject to periodic imperfect global spin flips. We show by a combination of exact diagonalization and tensor-network methods that the system can sustain a spontaneously broken discrete time-translation symmetry. Employing careful scaling analysis, we show the feasibility of a two-dimensional discrete time-crystal (DTC) pre-thermal phase. Despite an unbounded energy pumped into the system, in the high-frequency limit, a well-defined effective Hamiltonian controls a finite-temperature intermediate regime, wherein local time averages are described by thermal averages. As a consequence, the long-lived stability of the DTC relies on the existence of a long-range ordeblack phase at finite temperature. Interestingly, even for large deviations from the perfect spin flip, we observe a non-perturbative change in the decay rate of the order parameter, which is related to the long-lived stability of the magnetic domains in 2D.

Square-root Floquet topological phases and time crystals

Periodically driven (Floquet) phases are attractive due to their ability to host unique physical phenomena with no static counterparts. We propose a general approach in nontrivially devising a square-root version of existing Floquet phases, applicable both in noninteracting and in interacting setting. The resulting systems are found to yield richer physics that is otherwise absent in the original counterparts and is robust against parameter imperfection. These include the emergence of Floquet topological superconductors with arbitrarily many zero, $\ensuremath{\pi}$, and $\ensuremath{\pi}/2$ edge modes, as well as $4T$-period Floquet time crystals in disordered and disorder-free systems ($T$ being the driving period). Remarkably, our approach can be repeated indefinitely to obtain a ${2}^{n}\mathrm{th}$-root version of any periodically driven system, thus, allowing for the discovery and systematic construction of exotic Floquet phases.

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.

Observation of Photonic Topological Floquet Time Crystals

The two main research questions on time crystals are: What is a time crystal? Is there a material that is spontaneously crystalline in time? This study synthesizes a photonic material of topological Floquet time crystals and experimentally observes its indicative period‐ beating. A single‐particle picture is explicitly reconstructed of discrete time‐crystalline phase and is revealed using an appropriately‐designed photonic Floquet simulator the rigid period‐doubling as a signature of the breakage of the discrete time‐translational symmetry. Unlike the requirement of the many‐body localization, the photonic Floquet time crystal is derived from a newly defined single‐particle topological phase that can be extensively accessed by many pertinent nonequilibrium and periodically‐driven platforms. The observation will drive theoretical and technological interests toward condensed matter physics and topological photonics.

Time-periodic corner states from Floquet higher-order topology

The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. So far, all HOTI realisations have been based on static systems described by time-invariant Hamiltonians, without considering the time-variant situation. There is growing interest in Floquet systems, in which time-periodic driving can induce unconventional phenomena such as Floquet topological phases and time crystals. Recent theories have attempted to combine Floquet engineering and HOTIs, but there has been no experimental realisation so far. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along a spatial axis serving as an effective time-dependent drive. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection. This shows that space-time dynamics can induce anomalous higher-order topological phases unique to Floquet systems.

Critical properties of the prethermal Floquet time crystal

The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to\infty$, we determine the exponent $\nu$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\beta$ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent $\theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\nu, \beta, \theta$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector $\omega/(2 v)$, $v$ being the velocity of the quasiparticles, and $\omega$ being the drive frequency.

Nondispersing Wave Packets in Lattice Floquet Systems.

We show that in a one-dimensional translationally invariant tight binding chain, nondispersing wave packets can in general be realized as Floquet eigenstates-or linear combinations thereof-using a spatially inhomogeneous drive, which can be as simple as modulation on a single site. The recurrence time of these wave packets (their "round-trip" time) locks in at rational ratios sT/r of the driving period T, where s, r are coprime integers. Wave packets of different s/r can coexist under the same drive, yet travel at different speeds. They retain their spatial compactness either infinitely (s/r=1) or over a long time (s/r≠1). Discrete time translation symmetry is manifestly broken for s≠1, reminiscent of integer and fractional Floquet time crystals. We further demonstrate how to reverse engineer a drive protocol to reproduce a target Floquet micromotion, such as the free propagation of a wave packet, as if coming from a strictly linear energy spectrum. The variety of control schemes open up a new avenue for Floquet engineering in quantum information sciences.

Enhancing and controlling parametric instabilities in mechanical systems

We gain new fundamental insights on parametric instabilities that are at the heart of many physical phenomena from the dynamic buckling of slender structures in periodic compression to the emergence of Faraday waves or the spontaneous symmetry breaking in Floquet time crystals. Combining theoretical models and precision desktop experiments, we explain how to periodically vary the evolution function of a dynamical system to enhance and get control on parametric instabilities. We show on a proof of concept that is an electromagnetic pendulum: (i) how to observe extremely high orders of parametric resonance, even in the presence of dissipation, (ii) how to trigger and efficiently sustain the natural vibrations of an oscillator. The presented concepts being universal, they could offer new dynamical functionalities in various fields and at any scale, from actuation in soft robotics to vibrational motions in microelectromechanical resonators.

Critical properties of the Floquet time crystal within the Gaussian approximation

The periodically driven $O(N)$ model is studied near the critical line separating a disordered paramagnetic phase from a period doubled phase, the latter being an example of a Floquet time crystal. The time evolution of one-point and two-point correlation functions are obtained within the Gaussian approximation and perturbatively in the drive amplitude. The correlations are found to show not only period doubling, but also power-law decays at large spatial distances. These features are compared with the undriven $O(N)$ model, within the Gaussian approximation, in the vicinity of the paramagnetic-ferromagnetic critical point. The algebraic decays in space are found to be qualitatively different in the driven and the undriven cases. In particular, the spatiotemporal order of the Floquet time crystal leads to position-momentum and momentum-momentum correlation functions which are more long-ranged in the driven than in the undriven model. The light-cone dynamics associated with the correlation functions is also qualitatively different as the critical line of the Floquet time crystal shows a light cone with two distinct velocities, with the ratio of these two velocities scaling as the square-root of the dimensionless drive amplitude. The Floquet unitary, which describes the time evolution due to a complete cycle of the drive, is constructed for modes with small momenta compared to the drive frequency, but having a generic relationship with the square-root of the drive amplitude. At intermediate momenta, which are large compared to the square-root of the drive amplitude, the Floquet unitary is found to simply rotate the modes. On the other hand, at momenta which are small compared to the square-root of the drive amplitude, the Floquet unitary is found to primarily squeeze the modes, to an extent which increases upon increasing the wavelength of the modes, with a power-law dependence on it.

Homogeneous Floquet time crystal from weak ergodicity breaking

Recent works on observation of discrete time-crystalline signatures throw up major puzzles on the necessity of localization for stabilizing such out-of-equilibrium phases. Motivated by these studies, we delve into a clean interacting Floquet system, whose quasi-spectrum conforms to the ergodic Wigner-Dyson distribution, yet with an unexpectedly robust, long-lived time-crystalline dynamics in the absence of disorder or fine-tuning. We relate such behavior to a measure zero set of nonthermal Floquet eigenstates with long-range spatial correlations, which coexist with otherwise thermal states at near-infinite temperature and develop a high overlap with a family of translationally invariant, symmetry-broken initial conditions. This resembles the notion of "dynamical scars" that remain robustly localized throughout a thermalizing Floquet spectrum with fractured structure. We dub such a long-lived discrete time crystal formed in partially nonergodic systems, "scarred discrete time crystal" which is distinct by nature from those stabilized by either many-body localization or prethermalization mechanism.

Quantum frequency locking and downconversion in a driven qubit-cavity system

We study a periodically driven qubit coupled to a quantized cavity mode. Despite its apparent simplicity, this system supports a rich variety of exotic phenomena, such as topological frequency conversion as recently discovered in [Martin et al, PRX 7, 041008 (2017)]. Here we report on a qualitatively different phenomenon that occurs in this platform, where the cavity mode's oscillations lock their frequency to a rational fraction $r/q$ of the driving frequency $\Omega$. This phenomenon, which we term quantum frequency locking, is characterized by the emergence of $q$-tuplets of stationary (Floquet) states whose quasienergies are separated by $\Omega/q$, up to exponentially small corrections. The Wigner functions of these states are nearly identical, and exhibit highly-regular and symmetric structure in phase space. Similarly to Floquet time crystals, these states underlie discrete time-translation symmetry breaking in the model. We develop a semiclassical approach for analyzing and predicting quantum frequency locking in the model, and use it to identify the conditions under which it occurs.

Time-crystalline phases and period-doubling oscillations in one-dimensional Floquet topological insulators

In this work, we reported a ubiquitous presence of topological Floquet time crystal (TFTC) in one-dimensional periodically-driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry (DTS) breaking in our model require necessarily coexistence of anomalous topological invariants (0 modes and $\pi$ modes), instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between zero and pi modes can produce the period-doubling (2T, two cycles of the drive) that breaks the spontaneously, leading to the subharmonic response ($\omega/2$, half the drive frequency). The rigid period-oscillation is topologically-protected against perturbations due to both non-trivially opening of 0 and $\pi$-gaps in the quasienergy spectrum, thus, as a result, can be viewed as a specific "Rabi oscillation" between two Floquet eigenstates with certain quasienergy splitting $\pi/T$. Our modeling of the time-crystalline 'ground state' can be easily realized in experimental platforms such as topological photonics and ultracold fields. Also, our work can bring significant interests to explore topological phase transition in Floquet systems and to bridge the gap between Floquet topological insulators and photonics, and period-doubled time crystals.

Condensed matter physics in time crystals

Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.

Probing Floquet modes in a time periodic system with time defects using Faraday instability

Time crystals are systems whose properties are periodically modulated in time. Among these, Floquet time crystals exhibit momentum gaps in their band structures, analogous to energy gaps in spatial crystals. Recently, time defects with a -shift in the time modulation have been introduced theoretically as temporal analogues of spatial topological defects associated with localized edge modes. Here we perform experiments in a time periodic system using Faraday instability, a parametric excitation of a liquid bath by vertical shaking. Although time defects also trigger an exponentially decaying wave, we show that the analogy does not hold due to temporal causality and lack of energy conservation. However, these time defects provide an original way to explore momentum gaps and reveal their overdamped modes.

Prethermalization without Temperature

While a clean driven system generically absorbs energy until it reaches `infinite temperature', it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for non-trivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethermal regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localised cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al., [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.

Feedback-Induced Quantum Phase Transitions Using Weak Measurements.

We show that applying feedback and weak measurements to a quantum system induces phase transitions beyond the dissipative ones. Feedback enables controlling essentially quantum properties of the transition, i.e., its critical exponent, as it is driven by the fundamental quantum fluctuations due to measurement. Feedback provides the non-Markovianity and nonlinearity to the hybrid quantum-classical system, and enables simulating effects similar to spin-bath problems and Floquet time crystals with tunable long-range (long-memory) interactions.

Homogeneous Floquet time crystal protected by gauge invariance

We show that homogeneous lattice gauge theories can realize nonequilibrium quantum phases with long-range spatiotemporal order protected by gauge invariance instead of disorder. We study a kicked $\mathbb{Z}_2$-Higgs gauge theory and find that it breaks the discrete temporal symmetry by a period doubling. In a limit solvable by Jordan-Wigner analysis we extensively study the time-crystal properties for large systems and further find that the spatiotemporal order is robust under the addition of a solvability-breaking perturbation preserving the $\mathbb{Z}_2$ gauge symmetry. The protecting mechanism for the nonequilibrium order relies on the Hilbert space structure of lattice gauge theories, so that our results can be directly extended to other models with discrete gauge symmetries.