Dynamical quantum phase transitions (DQPTs) are characterized by nonanalytic behaviors of physical observables as functions of time. When a system is subject to time-periodic modulations, the nonanalytic signatures of its observables could recur periodically in time, leading to the phenomena of Floquet DQPTs. In this work, we systematically explore Floquet DQPTs in a class of periodically quenched one-dimensional system with chiral symmetry. By tuning the strength of quench, we find multiple Floquet DQPTs within a single driving period, with more DQPTs being observed when the system is initialized in Floquet states with larger topological invariants. Each Floquet DQPT is further accompanied by the quantized jump of a dynamical topological order parameter, whose values remain quantized in time if the underlying Floquet system is prepared in a gapped topological phase. The theory is demonstrated in a piecewise quenched lattice model, which possesses rich Floquet topological phases and is readily realizable in quantum simulators like the nitrogen-vacancy center in diamonds. Our discoveries thus open a new perspective for the Floquet engineering of DQPTs and the dynamical detection of topological phase transitions in Floquet systems.
The Floquet eigenvalue problem is analyzed for periodically driven Friedrichs models on discrete and continuous space. In the high-frequency regime, there exists a Floquet bound state consistent with the Floquet-Magnus expansion in the discrete Friedrichs model, while it is not the case in the continuous model. In the latter case, however, the bound state predicted by the Floquet-Magnus expansion appears as a metastable state whose lifetime diverges in the limit of large frequencies. We obtain the lifetime by evaluating the imaginary part of the quasi-energy of the Floquet resonant state. In t…
The Floquet-Magnus expansion is a useful tool to calculate an effective Hamiltonian for periodically driven systems. In this study, we investigate the convergence of the expansion for a one-body nonlinear system in a continuous space, a driven anharmonic oscillator. In this model, all eigenstates of the time evolution operator are found to be localized in energy space, and the expectation value of the energy is bounded from above. We first propose a general procedure to estimate the radius of convergence of the Floquet-Magnus expansion for periodically driven systems with an unbounded energy s…
An integrable model subjected to a periodic driving gives rise generally to a non-integrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb--Liniger model in presence of a linear potential with a periodic time--dependent strength is instead integrable and its quasi-energies can be determined using the Bethe ansatz approach. We discuss various aspects of the dynamics of the system at stroboscopic times and we also propose a possible experimental realisation of the periodically driven tilting in terms of a shaken rotated ring potential.
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 study the localization-delocalization transition of Floquet eigenstates in a driven fermionic chain with an incommensurate Aubry-André potential and a hopping amplitude which is varied periodically in time. Our analysis shows the presence of a mobility edge separating single-particle delocalized states from localized and multifractal states in the Floquet spectrum. Such a mobility edge does not have any counterpart in the static Aubry-André model and exists for a range of drive frequencies near the critical frequency at which the transition occurs. The presence of the mobility edge is shown…
By means of a Floquet analysis, we study the quantum dynamics of a fully connected Lipkin-Ising ferromagnet in a periodically driven transverse field showing that thermalization in the steady state is intimately connected to properties of the $N\to \infty$ classical Hamiltonian dynamics. When the dynamics is ergodic, the Floquet spectrum obeys a Wigner-Dyson statistics and the system satisfies the eigenstate thermalization hypothesis (ETH): Independently of the initial state, local observables relax to the $T=\infty$ thermal value, and Floquet states are delocalized in the Hilbert space. On th…
The violation of the Floquet version of eigenstate thermalization hypothesis is systematically discussed with realistic Hamiltonians. Our model is based on the PXP type interactions without disorder. We exactly prove the existence of many-body scar states in the Floquet eigenstates, by showing the explicit expressions of the wave functions. Using the underlying physical mechanism, various driven Hamiltonians with Floquet-scar states can be systematically engineered.
Floquet theory provides rigorous foundations for the theory of periodically driven quantum systems. In the case of non-periodic driving, however, the situation is not so well understood. Here, we provide a critical review of the theoretical framework developed for quasi-periodically driven quantum systems. Although the theoretical footing is still under development, we argue that quasi-periodically driven quantum systems can be treated with generalizations of Floquet theory in suitable parameter regimes. Moreover, we provide a generalization of the Floquet-Magnus expansion and argue that quasi…
Floquet engineering, modulating quantum systems in a time periodic way, lies at the central part for realizing novel topological dynamical states. Thanks to the Floquet engineering, various new realms on experimentally simulating topological materials have emerged. Conventional Floquet engineering, however, only applies to time periodic non-dissipative Hermitian systems, and for the quantum systems in reality, non-Hermitian process with dissipation usually occurs. So far, it remains unclear how to characterize topological phases of periodically driven non-Hermitian systems via the frequency sp…
A primer on the Floquet theory of periodically time-dependent quantum systems is provided, and it is shown how to apply this framework for computing the quasienergy band structure governing the dynamics of ultracold atoms in driven optical cosine lattices. Such systems are viewed here as spatially and temporally periodic structures living in an extended Hilbert space, giving rise to spatio-temporal Bloch waves whose dispersion relations can be manipulated at will by exploiting ac-Stark shifts and multiphoton resonances. The elements required for numerical calculations are introduced in a tutor…
Floquet higher order topological insulators (FHOTIs) are a novel topological phase that can occur in periodically driven lattices. An appropriate experimental platform to realize FHOTIs has not yet been identified. We introduce a periodically-driven bipartite (two-band) system that hosts FHOTI phases, and predict that this lattice can be realized in experimentally-realistic optical waveguide arrays, similar to those previously used to study anomalous Floquet insulators. The model exhibits interesting phase transitions from first-order to second-order topological matter by tuning a coupling str…
We investigate the analytical solution of a two-level system subject to a monochromatical, linearly polarized external field that was published a couple of years ago. In particular, we derive an explicit expression for the quasienergy. Moreover, we calculate the time evolution of a typical two-level system over a full period by evaluating series solutions of the confluent Heun equation. This is possible without invoking the connection problem of this equation since the complete time evolution of the system under consideration can be reduced to that of the first quarter-period.
We study the quantum version of a tilting and flashing Hamiltonian ratchets, consisting of a periodic potential and a time-periodic driving field. The system dynamics is governed by a Floquet evolution matrix bearing the symmetry of the corresponding Hamiltonian. The dc-current appears due to the desymmetrization of Floquet eigenstates, which become transporting when all the relevant symmetries are violated. Those eigenstates which mostly contribute to a directed transport reside in phase space regions corresponding to classical resonances. Quantum dynamics leads to the dependence of the avera…
We present a general formulation of Floquet states of periodically time-dependent open Markovian quasi-free fermionic many-body systems in terms of a discrete Lyapunov equation. Illustrating the technique, we analyze periodically kicekd XY spin 1/2 chain which is coupled to a pair of Lindblad reservoirs at its ends. A complex phase diagram is reported with re-entrant phases of long range and exponentially decaying spin-spin correlations as some of the system's parameters are varied. The structure of phase diagram is reproduced in terms of counting non-trivial stationary points of Floquet quasi…
We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory. The final results are equivalent to those obtained within a different approach [Phys.\ Rev.\ A {\bf 68}, 013820 (2003), Phys.\ Rev.\ X {\bf 4}, 031027 (2014)] and can also be related to the Floquet-Magnus expansion [J.\ Phys.\ A {\bf 34}, 3379 (2000)]. We discuss that the dependence on the d…
Electrons in a lattice exhibit time-periodic motion, known as Bloch oscillation, when subject to an additional static electric field. Here we show that a corresponding dynamics can occur upon replacing the spatially periodic potential by a time-periodic driving: Floquet oscillations of charge carriers in a spatially homogeneous system. The time lattice of the driving gives rise to Floquet bands that take on the role of the usual Bloch bands. For two different drivings (harmonic driving and periodic kicking through pulses) of systems with linear dispersion we demonstrate the existence of such o…
We theoretically report an analog of photon-assisted tunneling (PAT) in a periodically driven lattice array without a static biased potential by studying a three-channel waveguide system. This analog of PAT can be achieved by only periodically modulating the top waveguide and adjusting the distance between the bottom and its adjacent waveguide. It is numerically shown that the PAT resonances also exist in the five-channel waveguide system and probably exist in the waveguide arrays with other odd numbers of waveguides, but they will become weak as the number of waveguides increases. With origin…
Periodically driven non-Hermitian systems could possess exotic nonequilibrium phases with unique topological, dynamical and transport properties. In this work, we introduce an experimentally realizable two-leg ladder model subjecting to both time-periodic quenches and non-Hermitian effects, which belongs to an extended CII symmetry class. Due to the interplay between drivings and nonreciprocity, rich non-Hermitian Floquet topological phases emerge in the system, with each of them been characterized by a pair of even-integer topological invariants $(w_{0},w_π)\in2\mathbb{Z}\times2\mathbb{Z}$. U…
Anholonomies in the parametric dependences of the eigenvalues and the eigenvectors of Floquet operators that describe unit time evolutions of periodically driven systems, e.g., kicked rotors, are studied. First, an example of the anholonomies induced by a periodically pulsed rank-1 perturbation is given. As a function of the strength of the perturbation, the perturbed Floquet operator of the quantum map and its spectrum are shown to have a period. However, we show examples where each eigenvalue does not obey the periodicity of the perturbed Floquet operator and exhibits an anholonomy. Furtherm…
This work explores a fundamental dynamical structure for a wide range of many-body quantum systems under periodic driving. Generically, in the thermodynamic limit, such systems are known to heat up to infinite temperature states after infinite-time evolution, irrespective of dynamical details. In the present study, instead of considering infinitely long-time scale, we aim to provide a framework to understand the long but finite time behavior, namely the transient dynamics. In the analysis, we focus on the Floquet-Magnus (FM) expansion that gives a formal expression of the effective Hamiltonian…
Quantum computers are a leading platform for the simulation of many-body physics. This task has been recently facilitated by the possibility to program directly the time-dependent pulses sent to the computer. Here, we use this feature to simulate quantum lattice models with long-range hopping. Our approach is based on an exact mapping between periodically driven quantum systems and one-dimensional lattices in the synthetic Floquet direction. By engineering a periodic drive with a power-law spectrum, we simulate a lattice with long-range hopping, whose decay exponent is freely tunable. We propo…
We propose a general framework to solve tight binding models in D dimensional lattices driven by ac electric fields. Our method is valid for arbitrary driving regimes and allows to obtain effective Hamiltonians for different external fields configurations. We establish an equivalence with time independent lattices in D+1 dimensions, and analyze their topological properties. Further, we demonstrate that non-adiabaticity drives a transition from topological invariants defined in D+1 to D dimensions. Our approach provides a theoretical framework to analyze ac driven systems, with potential applic…
In this work, we present a new approach to disordered, periodically driven (Floquet) quantum many-body systems based on flow equations. Specifically, we introduce a continuous unitary flow of Floquet operators in an extended Hilbert space, whose fixed point is both diagonal and time-independent, allowing us to directly obtain the Floquet modes. We first apply this method to a periodically driven Anderson insulator, for which it is exact, and then extend it to driven many-body localized systems within a truncated flow equation ansatz. In particular we compute the emergent Floquet local integral…
We study the Floquet edge states in arrays of periodically curved optical waveguides described by the modulated Su-Schrieffer-Heeger model. Beyond the bulk-edge correspondence, our study explores the interplay between band topology and periodic modulations. By analysing the quasi-energy spectra and Zak phase, we reveal that, although topological and non-topological edge states can exist for the same parameters, \emph{they can not appear in the same spectral gap}. In the high-frequency limit, we find analytically all boundaries between the different phases and study the coexistence of topologic…
Laser technology has developed and accelerated photo-induced nonequilibrium physics from both scientific and engineering viewpoints. The Floquet engineering, i.e., controlling material properties and functionalities by time-periodic drives, is a forefront of quantum physics of light-matter interaction, but limited to ideal dissipationless systems. For the Floquet engineering extended to a variety of materials, it is vital to understand the quantum states emerging in a balance of the periodic drive and energy dissipation. Here we derive the general description for nonequilibrium steady states (…
We experimentally observe Floquet Raman transitions in the weakly driven solid state spin system of nitrogen-vacancy center in diamond. The periodically driven spin system simulates a two-band Wannier-Stark ladder model, and allows us to observe coherent spin state transfer arising from Raman transition mediated by Floquet synthetic levels. It also leads to the prediction of analog photon-assisted Floquet Raman transition and dynamical localisation in a driven two-level quantum system. The demonstrated rich Floquet dynamics offers new capabilities to achieve effective Floquet coherent control …
The periodically driven quantum Ising chain has recently attracted a large attention in the context of Floquet engineering. In addition to the common paramagnet and ferromagnet, this driven model can give rise to new topological phases. In this work we systematically explore its quantum phase diagram, by examining the properties of its Floquet ground state. We specifically focus on driving protocols with time-reversal invariant points, and demonstrate the existence of an infinite number of distinct phases. These phases are separated by second-order quantum phase transitions, accompanied by con…
Driving a quantum system periodically in time can profoundly alter its long-time correlations and give rise to exotic quantum states of matter. The complexity of the combination of many-body correlations and dynamic manipulations has the potential to uncover a whole field of new phenomena, but the theoretical and numerical understanding becomes extremely difficult. We now propose a promising numerical method by generalizing the density matrix renormalization group to a superposition of Fourier components of periodically driven many-body systems using Floquet theory. With this method we can stu…
We discuss the universal nature of relaxation in isolated many-body quantum systems subjected to global and strong periodic driving. Our rigorous Floquet analysis shows that the energy of the system remains almost constant up to an exponentially long time in frequency for arbitrary initial states and that an effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion is a quasi-conserved quantity for long time scale. These two general properties lead to intriguing classification on the initial stage of relaxation, one of which is similar to the prethermalization phenomenon i…
Detection of weak electromagnetic waves and hypothetical particles aided by quantum amplification is important for fundamental physics and applications. However, demonstrations of quantum amplification are still limited; in particular, the physics of quantum amplification is not fully explored in periodically driven (Floquet) systems, which are generally defined by time-periodic Hamiltonians and enable observation of many exotic quantum phenomena such as time crystals. Here we investigate the magnetic-field signal amplification by periodically driven $^{129}$Xe spins and observe signal amplifi…
We give a systematic review of the adiabatic theorem and the leading non-adiabatic corrections in periodically-driven (Floquet) systems. These corrections have a two-fold origin: (i) conventional ones originating from the gradually changing Floquet Hamiltonian and (ii) corrections originating from changing the micro-motion operator. These corrections conspire to give a Hall-type linear response for non-stroboscopic (time-averaged) observables allowing one to measure the Berry curvature and the Chern number related to the Floquet Hamiltonian, thus extending these concepts to periodically-driven…
Periodically driven quantum systems, known as Floquet systems, have been a focus of non-equilibrium physics in recent years, thanks to their rich dynamics. Not only time-periodic systems exhibit symmetries similar to those in spatially periodic systems, but they also display novel behavior due to symmetry breaking. Characterizing such dynamical symmetries is crucial, but the task is often challenging, due to limited driving strength and the lack of an experimentally accessible characterization protocol. Here, we show how to characterize dynamical symmetries including parity, rotation, and part…
Floquet theory combined with the generalized Van Vleck nearly degenerate perturbation theory, has been widely employed for studying various two-level systems that are driven by external fields via the time-dependent longitudinal (i.e., diagonal) couplings. However, three-level systems strongly driven by the time-dependent transverse (i.e., off-diagonal) couplings have rarely been investigated, due to the breakdown of the traditional rotating wave approximation. Meanwhile, the conventional perturbation theory is not directly applicable, since a small parameter for the perturbed part is no longe…
The invention of the maser stimulated many revolutionary technologies such as lasers and atomic clocks. Despite enormous progress, the realizations of masers are still confined to a limited variety of systems, in particular, the physics of masers remains unexplored in periodically driven (Floquet) systems, which are generally defined by time-periodic Hamiltonians and enable to observe many exotic phenomena such as time crystals. Here we investigate the Floquet system of periodically driven 129Xe gas under damping feedback, and surprisingly observe a multi-mode maser that oscillates at frequenc…
Much having learned about Floquet dynamics of pseudospin-$1/2$ system namely, graphene, we here address the stroboscopic properties of a periodically kicked {three-band fermionic system such as $α$-T$_3$ lattice. This particular model provides an interpolation between graphene and dice lattice via the continuous tuning of the parameter $α$ from 0 to 1.} In the case of dice lattice ($α=1$), we reveal that one can, in principle, engineer various types of low energy dispersions around some specific points in the Brillouin zone by tuning the kicking parameter in the Hamiltonian along a particular …
We study the dynamics of isolated interacting spin chains that are periodically driven by sudden quenches. Using full exact diagonalization of finite chains, we show that these systems exhibit three distinct regimes. For short driving periods, the Floquet Hamiltonian is well approximated by the time-averaged Hamiltonian, while for long periods the evolution operator exhibits properties of random matrices of a Circular Ensemble (CE). In-between, there is a crossover regime. Based on a finite-size scaling analysis and analytic arguments we argue that, for thermodynamically large systems and non-…
The dynamics of qubits coupled to a harmonic oscillator with time-periodic coupling is investigated in the framework of Floquet theory. This system can be used to model nonadiabatic phenomena that require a periodic modulation of the qubit/oscillator coupling. The case of a single qubit coupled to a resonator populated with $n= 0,1$ photons is explicitly treated. The time-dependent Schrödinger equation describing the system's dynamics is solved within the Floquet formalism and a perturbative approach in the time- and Laplace-domain. Good quantitative agreement is found between the analytical a…
Floquet states of periodically driven systems could exhibit rich topological properties. Many of them are absent in their static counterparts. One such example is the chiral edge states in anomalous Floquet topological insulators, whose description requires a new topological invariant and a novel type of bulk-edge correspondence. In this work, we propose a prototypical quenched lattice model, whose two Floquet bands could exchange their Chern numbers periodically and alternatively via touching at quasienergies 0 and $π$ under the change of a single system parameter. This process in principle a…
In this paper, we study the existence of Floquet topological insulators for PT symmetric non-Hermitian Hamiltonians. We consider an array of waveguide in 1D with periodically changing non-Hermitian potential and predict the existence of Floquet topological insulators in the system. We also extend the concept of Floquet topological phase to a two dimensional non-Hermitian system.
Floquet topological matter has emerged as one exciting platform to explore rich physics and game-changing applications of topological phases. As one remarkable and recently discovered feature of Floquet symmetry protected topological (SPT) phases, in principle a simple periodically driven system can host an arbitrary number of topological protected zero edge modes and pi edge modes, with Majorana zero modes and Majorana pi modes as examples protected by the particle-hole symmetry. This work advocates a new route to holonomic quantum computation by exploiting the co-existence of many Floquet SP…
The out-of-equilibrium dynamics of interaction quenched finite ultracold bosonic ensembles in periodically driven one-dimensional optical lattices is investigated. It is shown that periodic driving enforces the bosons in the outer wells of the finite lattice to exhibit out-of-phase dipole-like modes, while in the central well the atomic cloud experiences a local breathing mode. The dynamical behavior is investigated with varying driving frequency, revealing a resonant-like behavior of the intra-well dynamics. An interaction quench in the periodically driven lattice gives rise to admixtures of …
We present a systematic construction of effective Hamiltonians of periodically driven quantum systems. Because of an equivalence between the time dependence of a Hamiltonian and an interaction in its Floquet operator, flow equations, that permit to decouple interacting quantum systems, allow us to identify time-independent Hamiltonians for driven systems. With this approach, we explain the experimentally observed deviation of expected suppression of tunneling in ultracold atoms.
