The public and scientists constantly have different perspectives. While on a time crystal, they stand in line and ask: What is a time crystal? Show me a material that is spontaneously crystalline in time? This study synthesizes a photonic material of Floquet time crystals and experimentally observes its indicative period-2T beating. We explicitly reconstruct a discrete time-crystalline ground state and reveal using an appropriately-designed photonic Floquet simulator the rigid period-doubling as a signature of the spontaneous breakage of the discrete time-translational symmetry. Unlike the result of the exquisite many-body interaction, the photonic time crystal is derived from a single-particle topological phase that can be extensively accessed by many pertinent nonequilibrium and periodically-driven platforms. Our observation will drive theoretical and technological interests toward condensed matter physics and topological photonics, and demystify time crystals for the non-scientific public.
We propose a class of photonic Floquet topological insulators based on staggered helical lattices and an efficient numerical method for calculating their Floquet bandstructure. The lattices support anomalous Floquet topological insulator phases with vanishing Chern number and tunable topological transitions. At the critical point of the topological transition, the bandstructure hosts a single unpaired Dirac cone, which yields a variety of unusual transport effects: a discrete analogue of conical diffraction, weak antilocalization not limited by intervalley scattering, and suppression of Anders…
The Lieb Lattice exhibits intriguing properties that are of general interest in both the fundamental physics and practical applications. Here, we investigate the topological Landau-Zener Bloch oscillation in a photonic Floquet Lieb lattice, where the dimerized helical waveguides is constructed to realize the synthetic spin-orbital interaction through the Floquet mechanism, rendering us to study the impacts of topological transition from trivial gaps to non-trivial ones. The compact localized states of flat bands supported by the local symmetry of Lieb lattice will be associated with other band…
Edge modes in topological insulators are known to be robust against defects. We investigate if this also holds true when the defect is not static, but varies in time. We study the influence of defects with time-dependent coupling on the robustness of the transport along the edge in a Floquet system of helically curved waveguides. Waveguide arrays are fabricated via direct laser writing in a negative tone photoresist. We find that single dynamic defects do not destroy the chiral edge current, even when the temporal modulation is strong. Quantitative numerical simulation of the intensity in the …
Recent progresses on Floquet topological phases have shed new light on time-dependant quantum systems, among which one-dimensional (1D) Floquet systems have been under extensive theoretical research. However, an unambiguous experimental observation of these 1D Floquet topological phases has still been lacking. Here, by periodically bending ultrathin metallic arrays of coupled corrugated waveguides, a photonic Floquet simulator was well designed and successfully fabricated to simulate the periodically driven Su-Schrieffer-Heeger model. Intriguingly, under moderate driven frequencies, we first e…
In this letter we analyze an optical Fabry-Pérot resonator as a time-periodic driving of the (2D) optical field repeatedly traversing the resonator, uncovering that resonator twist produces a synthetic magnetic field applied to the light within the resonator, while mirror aberrations produce relativistic dynamics, anharmonic trapping, and spacetime curvature. We develop a Floquet formalism to compute the effective Hamiltonian for the 2D field, generalizing the idea that the intra-cavity optical field corresponds to an ensemble of non-interacting, massive, harmonically trapped particles. This w…
Phase-controlled transport of Floquet-driven compact chiral photonic states
Institute of Physics, University of Rostock（罗斯托克大学物理研究所）；Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile（智利大学理学院数学物理系）；Millennium Institute for Research in Optics–MIRO（光学研究千年研究所）
Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the "inherently dynamical" Floquet crystals. The DSI and quotient winding data, as we…
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 $π$ 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…
We demonstrate the emergence of a non-equilibrium superradiant phase in the dissipative Rabi-Dicke model. This phase is characterized by a photonic steady state that oscillates with a frequency close to the cavity frequency, in contrast to the constant photonic steady state of the equilibrium superradiant phase in the Dicke model. We relate this superradiant phase to the population inversion of Floquet states by introducing a Schwinger representation of the driven two-level systems in the cavity. This inversion is depleted near Floquet energies that are resonant with the cavity frequency to su…
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 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, $π$, and $π/2$ edge modes, as well as $4T$-period…
Although fragile topology has been intensely studied in static crystals in terms of Wannier obstruction, it is not clear how to generalize the concept to dynamical systems. In this work, we generalize the concept of fragile topology, and provide a definition of fragile topology for noninteracting Floquet crystals, which we refer to as dynamical fragile topology. In contrast to the static fragile topology defined by Wannier obstruction, dynamical fragile topology is defined for the nontrivial quantum dynamics characterized by the obstruction to static limits (OTSL). Specifically, the OTSL of a …
Dynamical Casimir effect of the optomechanical cavity interacting with one-dimensional photonic crystal is theoretically investigated in terms of the complex spectral analysis of Floquet-Liouvillian in the symplectic-Floquet space. The quantum vacuum fluctuation of the intra-cavity mode is parametrically amplified by a periodic motion of the mirror boundary, and the amplified photons are spontaneously emitted to the photonic band. We have derived the non-Hermitian effective Floquet-Liouvillian from the total system Liouvillian with the use of the Brillouin-Wigner-Feshbach projection method in …
Quantum walks are processes that model dynamics in coherent systems. Their experimental implementations proved key to unveil novel phenomena in Floquet topological insulators. Here we realize a photonic quantum walk in the presence of a synthetic gauge field, which mimics the action of an electric field on a charged particle. By tuning the energy gaps between the two quasi-energy bands, we investigate intriguing system dynamics characterized by the interplay between Bloch oscillations and Landau-Zener transitions. When both gaps at quasi-energy values 0 and $π$ are vanishingly small, the Floqu…
We theoretically study the dynamical Casimir effect (DCE), i.e., parametric amplification of a quantum vacuum, in an optomechanical cavity interacting with a photonic crystal, which is considered to be an ideal system to study the microscopic dissipation effect on the DCE. Starting from a total Hamiltonian including the photonic band system as well as the optomechanical cavity, we have derived an effective Floquet-Liouvillian by applying the Floquet method and Brillouin-Wigner-Feshbach projection method. The microscopic dissipation effect is rigorously taken into account in terms of the energy…
We demonstrate how to control the spectra and current flow of Dirac electrons in both a graphene sheet and a topological insulator by applying either two linearly polarized laser fields with frequencies $ω$ and $2ω$ or a monochromatic (one-frequency) laser field together with a spatially periodic static potential(graphene/TI superlattice). Using the Floquet theory and the resonance approximation, we show that a Dirac point in the electron spectrum can be split into several Dirac points whose relative location in momentum space can be efficiently manipulated by changing the characteristics of t…
We analytically identify a new class of quantum scars protected by spatiotemporal translation symmetries, dubbed Floquet-Bloch scars. They distinguish from previous (quasi-)static scars by a rigid spectral pairing only possible in Floquet systems, where strong interaction and drivings equalize the quasienergy corrections to all scars and maintain their spectral spacings against generic bilinear perturbations. Scars then enforce the spatial localization and rigid discrete time crystal (DTC) oscillations as verified numerically in a trimerized kagome lattice model relevant to recent cold atom ex…
We revisit the Floquet-Bloch eigenstates of a one-dimensional electron gas in the presence of the periodic Kronig-Penny potential and an oscillating electric field. Considering the appropriate boundary conditions for the wave function and its derivative, we derive the determining equations for the Floquet-Bloch eigenstates, which are represented by a single-infinite matrix rather than a double-infinite matrix needed for a generic potential. We numerically solve these equations, showing that there appear anticrossings at the crossing points of the different Floquet bands as well as the band gap…
Floquet driven systems represent an extremely interesting arena to study out-of-equilibrium phenomena. For instance, they provide realizations of discrete time crystals, where the discrete time translation symmetry of the periodic Hamiltonian is spontaneously broken by a subharmonic response of the system. However, the continuous presence of an external periodic driving is required within the current Floquet paradigm. We propose here the concept of spontaneous many-body Floquet state. This is a state that, in the absence of external periodic driving, self-oscillates like in the presence of a p…
In this work we discuss the existence of time-translation symmetry breaking in a kicked infinite-range-interacting clean spin system described by the Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under perturbations of the kicking protocol, its existence being intimately linked to the underlying $\mathbb{Z}_2$ symmetry breaking of the time-independent model. We show that the model being infinite-range and having an extensive amount of symmetry breaking eigenstates is essential for having the time-crystal behaviour. In particular we discuss the properties of the Floquet spectr…
We define what it means for time translation symmetry to be spontaneously broken in a quantum system, and show with analytical arguments and numerical simulations that this occurs in a large class of many-body-localized driven systems with discrete time-translation symmetry.
Floquet higher-order topological insulators and superconductors (HOTI/SCs) with an order-two space-time symmetry or antisymmetry are classified. This is achieved by considering unitary loops, whose nontrivial topology leads to the anomalous Floquet topological phases, subject to a space-time symmetry/antisymmetry. By mapping these unitary loops to static Hamiltonians with an order-two crystalline symmetry/antisymmetry, one is able to obtain the $K$ groups for the unitary loops and thus complete the classification of Floquet HOTI/SCs. Interestingly, we found that for every order-two nontrivial …
We analyse quasi-periodically driven quantum systems that can be mapped exactly to periodically driven ones and find Floquet Time Spirals in analogy with spatially incommensurate spiral magnetic states. Generalising the mechanism to many-body systems we discover that a form of discrete time-translation symmetry breaking can also occur in quasi-periodically driven systems. We construct a discrete time quasi-crystal stabilised by many-body localisation. Crucially, it persists also under perturbations that break the equivalence with periodic systems. As such it provides evidence of a stable quasi…
We construct a class of period-$n$-tupling discrete time crystals based on $\mathbb{Z}_n$ clock variables, for all the integers $n$. We consider two classes of systems where this phenomenology occurs, disordered models with short-range interactions and fully connected models. In the case of short-range models we provide a complete classification of time-crystal phases for generic $n$. For the specific cases of $n=3$ and $n=4$ we study in details the dynamics by means of exact diagonalisation. In both cases, through an extensive analysis of the Floquet spectrum, we are able to fully map the pha…
Photonic hyper-crystals combine the most interesting features of hyperbolic metamaterials and photonic crystals. Since the dispersion law of extraordinary photons in hyperbolic metamaterials does not exhibit the usual diffraction limit, photonic hyper-crystals exhibit light localization on deep subwavelength scales, leading to considerable enhancement of nonlinear photon-photon interaction. Therefore, similar to their conventional photonic crystal counterparts, nonlinear photonic hyper-crystals appear to be very promising in optical limiting and optical computing applications. Nonlinear optics…
We present a heuristic mathematical model of the relation between the geometry of a photonic crystal waveguide and the Purcell enhancement factor at a particular wavelength of interest. We use this model to propose approaches to the design of a photonic crystal waveguide maximizing the Purcell enhancement at a target wavelength. Numerical simulations indicate that the proposed structures exhibit robustness to fabrication defects introduced into photonic crystal geometry.
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-doubl…
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 mech…
We investigate the conditions under which periodically driven quantum systems subject to dissipation exhibit a stable subharmonic response. Noting that coupling to a bath introduces not only cooling but also noise, we point out that a system subject to the latter for the entire cycle tends to lose coherence of the subharmonic oscillations, and thereby the long-time temporal symmetry breaking. We provide an example of a short-ranged two-dimensional system which does not suffer from this and therefore displays persistent subharmonic oscillations stabilised by the dissipation. We also show that t…
A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Floquet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchers in the field to mostly focus on many-body localized systems. Some works have however shown that the standard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises the question whether an homogeneous discrete time crystal is possible in which the originally expected heating is evaded. Studying both a localized and an homog…
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…
We demonstrate that the prethermal regime of periodically driven (Floquet), classical many-body systems can host nonequilibrium phases of matter. In particular, we show that there exists an effective Hamiltonian that captures the dynamics of ensembles of classical trajectories despite the breakdown of this description at the single trajectory level. In addition, we prove that the effective Hamiltonian can host emergent symmetries protected by the discrete time-translation symmetry of the drive. The spontaneous breaking of such an emergent symmetry leads to a subharmonic response, characteristi…
We propose a method for entangling a system of two-level atoms in photonic crystals. The atoms are assumed to move in void regions of a photonic crystal. The interaction between the atoms is mediated either via a defect mode or via resonant dipole-dipole interaction. We show that these interactions can produce pure entangled atomic states. We analyze the problem with parameters typical for currently existing photonic crystals and Rydberg atoms. We show that the atoms can emerge from photonic crystals in entangled states. Depending on the linear dimensions of the crystal and on their velocity o…
We show that fractional decay may be observable in experiments using quantum dots and photonic crystals with parameters that are currently achievable. We focus on the case of inverse opal photonic crystals and locate the position in the crystal where the effect is most pronounced. Furthermore, we quantify the influence of absorptive loss and show that it is a limiting but not prohibitive effect.
