Floquet Systems Research Articles

Real-time simulation of light-driven spin chains on quantum computers

In this work, we study the real-time evolution of periodically driven (Floquet) systems on a quantum computer using IBM quantum devices. We consider a driven Landau-Zener model and compute the transition probability between the Floquet steady states as a function of time. We find that for this simple one-qubit model, Floquet states can develop in real time, as indicated by the transition probability between Floquet states. Next, we model light-driven spin chains and compute the time-dependent antiferromagnetic order parameter. We consider models arising from light coupling to the underlying electrons as well as those arising from light coupling to phonons. For the two-spin chains, the quantum devices yield time evolutions that match the effective Floquet Hamiltonian evolution for both models once readout error mitigation is included. For three-spin chains, zero-noise extrapolation yields a time dependence that follows the effective Floquet time evolution. Therefore, the current IBM quantum devices can provide information on the dynamics of small Floquet systems arising from light drives once error mitigation procedures are implemented.

Circular Rosenzweig-Porter random matrix ensemble

The Rosenzweig-Porter random matrix ensemble serves as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition in static systems. We propose a unitary (circular) analogue of this ensemble, which similarly captures the phenomenology of many-body localization in periodically driven (Floquet) systems. We define this ensemble as the outcome of a Dyson Brownian motion process. We show numerical evidence that this ensemble shares some key statistical properties with the Rosenzweig-Porter ensemble for both the eigenvalues and the eigenstates.

Coarse-Grained Entanglement and Operator Growth in Anomalous Dynamics.

In two-dimensional Floquet systems, many-body localized dynamics in the bulk may give rise to a chaotic evolution at the one-dimensional edges that is characterized by a nonzero chiral topological index. Such anomalous dynamics is qualitatively different from local-Hamiltonian evolution. Here we show how the presence of a nonzero index affects entanglement generation and the spreading of local operators, focusing on the coarse-grained description of generic systems. We tackle this problem by analyzing exactly solvable models of random quantum cellular automata (QCA) that generalize random circuits. We find that a nonzero index leads to asymmetric butterfly velocities with different diffusive broadening of the light cones and to a modification of the order relations between the butterfly and entanglement velocities. We propose that these results can be understood via a generalization of the recently introduced entanglement membrane theory, by allowing for a spacetime entropy current, which in the case of a generic QCA is fixed by the index. We work out the implications of this current on the entanglement "membrane tension" and show that the results for random QCA are recovered by identifying the topological index with a background velocity for the coarse-grained entanglement dynamics.

Quantum phase transitions and a disorder-based filter in a Floquet system

Two-dimensional periodically-driven topological insulators have been shown to exhibit numerous topological phases, including ones which have no static analog, such as anomalous Floquet topological phases. We study a two dimensional model of spinless fermions on a honeycomb lattice with periodic driving. We show that this model exhibits a rich mixture of weak and strong topological phases, which we identify by computing their scattering matrix invariants. Further, we do an in-depth analysis of these topological phases in the presence of spatial disorder and show the relative robustness of these phases against imperfections. Making use of this robustness against spatial disorder, we propose a filter which allows the passage of only edge states, and which can be realized using existing experimental techniques.

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.

Topological micromotion of Floquet quantum systems

The Floquet Hamiltonian has often been used to describe a time-periodic system. Nevertheless, because the Floquet Hamiltonian depends on a micro-motion parameter, the Floquet Hamiltonian with a fixed micro-motion parameter cannot faithfully represent a driven system, which manifests as the anomalous edge states. Here we show that an accurate description of a Floquet system requires a set of Hamiltonian exhausting all values of the micro-motion parameter, and this micro-motion parameter can be viewed as an extra synthetic dimension of the system. Therefore, we show that a $d$-dimensional Floquet system can be described by a $d+1$-dimensional static Hamiltonian, and the advantage of this representation is that the periodic boundary condition is automatically imposed along the extra-dimension, which enables a straightforward definition of topological invariants. The topological invariant in the $d+1$-dimensional system can ensure a $d-1$-dimensional edge state of the $d$-dimensional Floquet system. Here we show two examples where the topological invariant is a three-dimensional Hopf invariant. We highlight that our scheme of classifying Floquet topology on the micro-motion space is different from the previous classification of Floquet topology on the time space.

Floquet prethermal phase protected by U(1) symmetry on a superconducting quantum processor

Periodically driven systems, or Floquet systems, exhibit many novel dynamics and interesting out-of-equilibrium phases of matter. Those phases arising with the quantum systems' symmetries, such as global U(1) symmetry, can even show dynamical stability with symmetry protection. Here we experimentally demonstrate a U(1) symmetry-protected prethermal phase, via performing a digital-analog quantum simulation on a superconducting quantum processor. The dynamical stability of this phase is revealed by its robustness against external perturbations. We also find that the site-averaged square of the two-spin correlation function in this phase is stabilized by the interaction between the spins. Our work reveals a promising prospect in discovering emergent quantum dynamical phases with digital-analog quantum simulators.

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.

Floquet Edge Multicolor Solitons

AbstractTopological insulators are unique physical structures that are insulators in their bulk, but support currents at their edges which can be unidirectional and topologically protected from scattering on disorder and inhomogeneities. Photonic topological insulators can be crafted in materials that exhibit a strong nonlinear response, thus opening the door to the exploration of the interplay between nonlinearity and topological effects. Among the fascinating new phenomena arising from this interplay is the formation of topological edge solitons—hybrid asymmetric states localized across and along the interface due to different physical mechanisms. Such solitons have so far been studied only in materials with Kerr‐type, or cubic, nonlinearity. Here, the first example of the topological edge soliton supported by parametric interactions in nonlinear media is presented. Such solitons exist in Floquet topological insulators realized in arrays of helical waveguides made of a phase‐matchable material. Floquet edge solitons bifurcate from topological edge states in the spectrum of the fundamental frequency wave and remain localized over propagation distances drastically exceeding the helix period, while travelling along the edge of the structure. A theory of such states is developed. It is shown that multicolor solitons in a Floquet system exists in the vicinity of (formally infinite) set of linear resonances determined by the Floquet phase matching conditions. Away from resonance, soliton envelopes can be described by a period‐averaged single nonlinear Schrödinger equation with an effective cubic nonlinear coefficient whose magnitude and sign depend on the overall phase‐mismatch between the fundamental frequency and second harmonic waves. Such total phase‐mismatch includes the intrinsic mismatch and the geometrically‐induced mismatch introduced by the array, and its value reveals one of the genuine effects exhibited by the Floquet quadratic solitons. The results open fundamental new prospects for the exploration of a range of parametric frequency‐mixing phenomena in photonic Floquet quadratic nonlinear media.

Following Floquet states in high-dimensional Hilbert spaces

An iterative algorithm is established which enables one to compute individual Floquet states even for many-body systems with high-dimensional Hilbert spaces that are not accessible to commonly employed conventional methods. A strategy is proposed for following a Floquet state in response to small changes of a given system's Hamiltonian. The scheme is applied to a periodically driven Bose-Hubbard chain, verifying the possibility of pseudoadiabatic Floquet state following. In particular, it is demonstrated that a driving-induced Mott insulatorlike target Floquet state can be populated with high efficiency if the driving amplitude is turned on smoothly but not too slowly. We conclude that the algorithm constitutes a powerful tool for the future investigation of many-body Floquet systems.

Single-shot measurement of the photonic band structure in a fiber-based Floquet-Bloch lattice

Floquet-Bloch lattices are systems in which wave packets are subjet to periodic modulations both in time and space, showing rich dynamics. While this type of lattice is difficult to implement in solid-state physics, optical systems have provided excellent platforms to probe their physics: among other effects, they have revealed genuine phenomena such as the anomalous Floquet topological insulator and the funnelling of light into localised interface modes. Despite the crucial importance of the band dispersion in the photon dynamics and the topological properties of the lattice, the direct experimental measurement of the Floquet-Bloch bands has remained elusive. Here we report the direct measurement of the Floquet-Bloch bands of a photonic lattice with a single shot method. We use a system of two coupled fibre rings that implements a time-multiplexed Floquet-Bloch lattice. By Fourier transforming the impulse response of the lattice we obtain the band structure together with an accurate characterization of the lattice eigenmodes, i. e. the amplitudes and the phases of the Floquet-Bloch eigenvectors over the entire Brillouin zone. Our results open promising perspectives for the observation of topological effects in the linear and nonlinear regime in Floquet systems.

Fate of topological edge states in disordered periodically driven nonlinear systems

We explore topological edge states in periodically driven nonlinear systems. Based on a self-consistency method adjusted to periodically driven systems, we obtain stationary states associated with topological phases unique to Floquet systems. In addition, we study the linear stability of these edge states and reveal that Floquet stationary edge states experience a sort of transition between two regions I and II, in which lifetimes of these edge states are extremely long and short, respectively. We characterize the transitions in lifetimes by Krein signatures or equivalently the pseudo-Hermiticity breaking, and clarify that the transitions between regions I and II are signified by collisions of edge-dominant eigenstates of Floquet operators for fluctuations. We also analyze the effects of random potentials and clarify that lifetimes of various stationary edge states are equalized due to the randomness-induced mixing of edge- and bulk-dominant eigenstates. This intriguing phenomenon originating from a competition between the nonlinearity and randomness results in that random potentials prolong lifetimes in the region II and vice versa in the region I. These changes of lifetimes induced by nonlinear and/or random effects should be detectable in experiments by preparing initial states akin to the edge states.

Nielsen-Ninomiya Theorem with Bulk Topology: Duality in Floquet and Non-Hermitian Systems.

The Nielsen-Ninomiya theorem is a fundamental theorem on the realization of chiral fermions in static lattice systems in high-energy and condensed matter physics. Here we extend the theorem in dynamical systems, which include the original Nielsen-Ninomiya theorem in the static limit. In contrast to the original theorem, which is a no-go theorem for bulk chiral fermions, the new theorem permits them due to bulk topology intrinsic to dynamical systems. The theorem is based on duality enabling a unified treatment of periodically driven systems and non-Hermitian ones. We also present the extended theorem for nonchiral gapless fermions protected by symmetry. Finally, as an application of our theorem and duality, we predict a new type of chiral magnetic effect-the non-Hermitian chiral magnetic skin effect.

Trapping light in a Floquet topological photonic insulator by Floquet defect mode resonance

Floquet topological photonic insulators characterized by periodically varying Hamiltonians are known to exhibit much richer topological behaviors than static systems. In a Floquet insulator, the phase evolution of the Floquet–Bloch modes plays a crucial role in determining its topological behaviors. Here, we show that by perturbing the driving sequence, it is possible to manipulate the cyclic phase change in the system over each evolution period to induce self-interference of a bulk mode, leading to a resonance effect, which can be regarded as a Floquet counterpart of defect-mode resonance in static lattices. This Floquet Defect Mode Resonance (FDMR) is cavity-less since it does not require physical boundaries; its spatial localization pattern is, instead, determined by the driving sequence and is found to be different in topologically trivial and nontrivial lattices. We demonstrated excitation of FDMRs by edge modes in a Floquet octagon lattice on silicon-on-insulator, achieving extrinsic quality factors greater than 104. Imaging of the scattered light pattern directly revealed the hopping sequence of the Floquet system and confirmed the spatial localization of FDMR in a bulk-mode loop. The new Floquet topological resonator could find various applications in lasers, optical filters and switches, nonlinear cavity optics, and quantum optics.

Gauging anomalous unitary operators

Boundary theories of static bulk topological phases of matter are obstructed in the sense that they cannot be realized on their own as isolated systems. The obstruction can be quantified/characterized by quantum anomalies, in particular when there is a global symmetry. Similarly, topological Floquet evolutions can realize obstructed unitary operators at their boundaries. In this paper, we discuss the characterization of such obstructions by using quantum anomalies. As a particular example, we discuss time-reversal symmetric boundary unitary operators in one and two spatial dimensions, where the anomaly emerges as we gauge the so-called Kubo-Martin-Schwinger (KMS) symmetry. We also discuss mixed anomalies between particle number conserving U(1) symmetry and discrete symmetries, such as $C$ and $\mathit{CP}$, for unitary operators in odd spatial dimensions that can be realized at the boundaries of topological Floquet systems in even spatial dimensions.

Fermi's golden rule for heating in strongly driven Floquet systems

We study heating dynamics in isolated quantum many-body systems driven periodically at high frequency and large amplitude. Combining the high-frequency expansion for the Floquet Hamiltonian with Fermi's golden rule (FGR), we develop a master equation termed the Floquet FGR. Unlike the conventional one, the Floquet FGR correctly describes heating dynamics, including the prethermalization regime, even for strong drives, under which the Floquet Hamiltonian is significantly dressed, and nontrivial Floquet engineering is present. The Floquet FGR depends on system size only weakly, enabling us to analyze the thermodynamic limit with small-system calculations. Our results also indicate that, during heating, the system approximately stays in the thermal state for the Floquet Hamiltonian with a gradually rising temperature.

Dynamical symmetry indicators for Floquet crystals

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 well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.

Selection rules for breaking selection rules

Floquet systems often exhibit dynamical symmetries (DS) that govern the time-dependent dynamics and result in selection rules. When a DS is broken, selection rule deviations are expected. Typically, information about the symmetry-breaking perturbation/phase and the time-dependent dynamics can be extracted from these deviations, hence they are regarded as a background free gauge of symmetry breaking. However, to date, DS breaking & selection rule deviations are not described by a general approach, thus there is no universal insight about the interplay between selection rule deviations, the symmetry breaking perturbation, and the broken DS. Here we consider DS breaking in Floquet systems from a general standpoint, formulating a general theory that analytically connects the symmetry-broken and fully symmetric systems. Using an external laser (of arbitrary frequency and polarization), as a model DS breaking perturbation, we discover that the broken symmetry systematically imposes selection rules on the symmetry-broken system, which physically manifest as scaling laws of selection rule deviations. We term these rules ‘selection rules for breaking selection rules’—a new concept in physics. We numerically validate the analytical theory in the context of high harmonic generation. Our discovery is a general feature of nonlinear wave-mixing phenomena, and we expect it to apply to any Floquet system (classical & quantum) and to any DS breaking mechanism (either by intrinsic or extrinsic elements of the system).

Observation of Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems.

Periodically driven (Floquet) quantum systems have recently been a focus of nonequilibrium physics by virtue of their rich dynamics. Time-periodic systems not only exhibit symmetries that resemble those in spatially periodic systems, but also display novel behavior that arises from symmetry breaking. Characterization of such dynamical symmetries is crucial, but often challenging due to limited driving strength and lack of an experimentally accessible characterization technique. Here, we show how to reveal dynamical symmetries, namely, parity, rotation, and particle-hole symmetries, by observing symmetry-induced Floquet selection rules. Notably, we exploit modulated driving to reach the strong light-matter coupling regime, and we introduce a protocol to experimentally extract the transition matrix elements between Floquet states from the system coherent evolution. By using nitrogen-vacancy centers in diamond as an experimental test bed, we execute our protocol to observe symmetry-protected dark states and dark bands, and coherent destruction of tunneling. Our work shows how one can exploit the quantum control toolkit to study dynamical symmetries that arise in the topological phases of strongly driven Floquet systems.

Floquet chiral hinge modes and their interplay with Weyl physics in a three-dimensional lattice

We demonstrate that a three dimensional time-periodically driven (Floquet) lattice can exhibit chiral hinge states and describe their interplay with Weyl physics. A peculiar type of the hinge states are enforced by the repeated boundary reflections with lateral Goos-H\"{a}nchen like shifts occurring at the second-order boundaries of our system. Such chiral hinge modes coexist in a wide range of parameters regimes with Fermi arc surface states connecting a pair of Weyl points in a two-band model. We find numerically that these modes still preserve their locality along the hinge and their chiral nature in the presence of local defects and other parameter changes. We trace the robustness of such chiral hinge modes to special band structure unique in a Floquet system allowing all the eigenstates to be localized in quasi-one-dimensional regions parallel to each other when open hinge boundaries are introduced. The implementation of a model featuring both the second-order Floquet skin effect and the Weyl physics is straightforward with ultracold atoms in optical superlattices.