We investigate the long-time behavior of quantum N-level systems that are coupled to a Markovian environment and subject to periodic driving. As our main result, we obtain a general algebraic condition ensuring that all solutions of a periodic quantum master equation with Lindblad form approach a unique limit cycle. Quite intuitively, this criterion requires that the dissipative terms of the master equation connect all subspaces of the system Hilbert space during an arbitrarily small fraction of the cycle time. Our results provide a natural extension of Spohn's algebraic condition for the approach to equilibrium to systems with external driving.
Theoretical treatments of periodically-driven quantum thermal machines (PD-QTMs) are largely focused on the limit-cycle stage of operation characterized by a periodic state of the system. Yet, this regime is not immediately accessible for experimental verification. Here, we present a general thermodynamic framework that can handle the performance of PD-QTMs both before and during the limit-cycle stage of operation. It is achieved by observing that periodicity may break down at the ensemble average level, even in the limit-cycle phase. With this observation, and using conventional thermodynamic…
When the discrete time-translation symmetry of isolated, periodically driven systems is spontaneously broken, a new phase of matter can emerge. We review some recent developments on both the theoretical underpinnings and experimental realizations of time crystalline order. Particular attention is placed on delineating the key features of time crystals, which distinguish them from other oscillatory non-equilibrium phenomena.
Low-noise quantum frequency conversion in periodically-poled nonlinear crystals has proved challenging when the pump wavelength is shorter than the target wavelength. This is - at least in large part - a consequence of the parasitic spontaneous parametric down-conversion of pump photons, whose efficiency is increased by fabrication errors in the periodic poling. Here we characterise the poling quality of commercial periodically-poled bulk potassium titanyl phosphate (ppKTP) by measuring the sum-frequency generation (SFG) efficiency over a large phase mismatch range from 0 to more than 400$π$. …
Periodically-driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics. Hence, it is of interest to understand what constraints such dynamics must satisfy. We derive a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the recurrence probability. We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters. To illustrate the relevance of these results for modern quantum systems we d…
We study the coherent dynamics of a quantum many-body system subject to a time-periodic driving. We argue that in many cases, destructive interference in time makes most of the quantum averages time-periodic, after an initial transient. We discuss in detail the case of a quantum Ising chain periodically driven across the critical point, finding that, as a result of quantum coherence, the system never reaches an infinite temperature state. Floquet resonance effects are moreover observed in the frequency dependence of the various observables, which display a sequence of well-defined peaks or dip…
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…
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 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…
We discuss dynamics of periodically-driven open quantum systems. The time evolution of the quantum state is described by the quantum master equation and the form of the dissipator is chosen so that the instantaneous stationary state is given by the Gibbs distribution. We find that the correlation between the population part and the coherence part of the density operator is induced by an adiabatic gauge potential. Although the introduction of the counterdiabatic term eliminates the correlation, additional correlations prevent a convergence to the Gibbs distribution. We study the performance of …
We present an effective Hamiltonian theory available for some quasi-periodically driven quantum systems which does not need the knowledge of the Fourier frequencies of the control signal. It could also be available for some chaotically driven quantum systems. It is based on the Koopman approach which generalizes the Floquet approach used with periodically driven systems. We show the properties of the quasi-energy states (eigenvectors of the effective Hamiltonian) as quasi-recurrent states of the quantum system.
We investigate the quantum recurrence phenomena in periodically driven systems. We calculate the classical period and the quantum recurrence time and develop their interdependence. We further predict the behavior of the recurrence phenomena for the power law potentials.
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…
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…
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.
We present theoretical study of revival phenomena for a wave packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The classical motion, revival, and super-revival time scale are derived exactly for wave packet excited in the one dimension box potential.
We present a short derivation and discussion of the master equation for an open quantum system weakly coupled to a heat bath and then its generalization to the case of with periodic external driving based on the Floquet theory. Further, a single heat bath is replaced by several ones. We present also the definition of heat currents which satisfies the second law of thermodynamics and apply the general results to a simple model of periodically modulated qubit.
We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similar as for undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but invididually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at highest computational cost.
Periodically driven dynamics of open quantum systems is very interesting because typically non-equilibrium steady state is reached, which is characterized by a non-vanishing current. In this work, we study time discrete and periodically driven dynamics experimentally for a single photon that its coupled to its environment. We develop a comprehensive theory which explains the experimental observations and offers an analytical characterization of the non-equilibrium steady states of the system. We demonstrate that the periodic driving and the properties of the environment can be engineered in su…
We develop a flow renormalization approach for periodically-driven quantum systems, which reveals prethermal dynamical regimes and associated timescales via direct correspondence between real time and flow time behavior. In this formalism, the dynamical problem is recast in terms of coupling constants of the theory flowing towards an attractive fixed point that represents the thermal Floquet Hamiltonian at long times, while narrowly avoiding a series of unstable fixed points which determine distinct prethermal regimes at intermediate times. We study a class of relevant perturbations that trigg…
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…
Quantum revivals are investigated for the dynamics of an atom in a driven gravitational cavity. It is demonstrated that the external driving field influences the revival time significantly. Analytical expressions are presented which are based on second order perturbation theory and semiclassical secular theory. These analytical results explain the dependence of the revival time on the characteristic parameters of the problem quantitatively in a simple way. They are in excellent agreement with numerical results.
We propose a practical protocol to generate and observe a non-Abelian geometric phase using a periodically driven Raman process in the hyperfne ground state manifold of atoms in a dilute ultracold gas. Our analysis is based upon recent developments and application of Floquet theory to periodically driven quantum systems. The simulation results show the non-Abelian gauge transformation and the non-commuting property of the SU(2) transformation operators. Based on these results, we propose a possible experimental implementation with an ultracold dilute Bose gas.
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…
We calculate the quantum revival time for a wave-packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The dependence of the revival time on various parameters of the driven system is shown analytically. As an example of application of our approach, we compare the analytically obtained values of the revival time for various modulation strengths with the numerically computed ones in the case of a driven gravitational cavity. We show that they are in very good agreement.
We demonstrate that the counting statistics of currents in periodically driven ergodic stochastic systems can show sharp changes of some of its properties in response to continuous changes of the driving protocol. To describe this effect, we introduce a new topological phase factor in evolution of the moment generating function which is akin to the topological geometric phase in evolution of a periodically driven quantum mechanical system with time-reversal symmetry. This phase leads to the prediction of a sign change for the difference of the probabilities to find even and odd number particle…
Using a perturbative solution for a periodically driven two-level quantum system, we show how to obtain phase factors for both a two-level quantum system and two two-level quantum systems non-interacting and interacting. The method is easily implemented by numerical routines and presents the advantage of being stable for long-time periods. We furthermore explore the possibility of implementing a quantum phase gate using the perturbative solution.
Quantum phases of matter have many relevant applications in quantum computation and quantum information processing. Current experimental feasibilities in diverse platforms allow us to couple two or more subsystems in different phases. In this letter, we investigate the situation where one couples two domains of a periodically-driven spin chain where one of them is ergodic while the other is fully localized. By combining tools of both graph and Floquet theory, we show that the localized domain remains stable for strong disorder, but as this disorder decreases the localized domain becomes ergodi…
We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adia…
A random matrix theory approach is applied in order to analyze the localization properties of local spectral density for a generic system of coupled quantum states with strong static imperfection in the unperturbed energy levels. The system is excited by an external periodic field, the temporal profile of which is close to monochromatic one. The shape of local spectral density is shown to be well described by the contour obtained from a relevant model of periodically driven two-states system with irreversible losses to an external thermal bath. The shape width and the inverse participation rat…
Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a new type of DTC in a periodically driven surface code, the subharmonic signature of which is only observable with respect to some nonlocal order parameter. By modifying the commonly used metrics for characterizing ordinary DTCs, i.e., the spectral functions, spin-glass order parameters, and two-point correlators, we numerically investigate the different phases of the proposed system. Specifically, depending on the system parameters and boundary conditions,…
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.
Using a dynamical model relevant to cold-atom experiments, we show that long-lasting exponential spreading of wave packets in momentum space is possible. Numerical results are explained via a pseudo-classical map, both qualitatively and quantitatively. Possible applications of our findings are also briefly discussed.
I develop the solution to the problem of an electron confined in a composite quadratic well subject to a simple, external periodic force. The method of solution illustrates several of the basic techniques useful in formally solving the one-dimensional, time-dependent Schrodinger equation. One of the aims of this exercise is to see how far is it possible to push analytics, before plowing into numerical methods. I hope this presentation of the problem may result useful to others seeking to gain additional experience in the details of solving the time-dependent Schrodinger equation in one space d…
We develop a realistic protocol to observe a robust topological dynamics of two-particle bound states in a lattice model with on-site interactions and suitably designed time-dependent hoppings. This Floquet scheme can be realistically implemented on existing digital quantum computer platforms. Marked differences from the topological single-particle dynamics of two independent particles and clear signatures of the entanglement between the two constituent particles are highlighted.
We theoretically report the emergence of $Z_4$ parafermion edge modes in a periodically driven spinful superconducting chain with modest fermionic Hubbard interaction. These parafermion edge modes represent $\pm π/(2T)$ quasienergy excitations ($T$ being the driving period), which have no static counterpart and arise from the interplay between interaction effect and periodic driving. At special parameter values, these exotic quasiparticles can be analytically and exactly derived. Strong numerical evidence of their robustness against variations in parameter values and spatial disorder is furthe…
We demonstrate that a site-dependent driving of a periodic potential allows for the controlled manipulation of a quantum particle on length scales of the lattice spacing. Specifically we observe for distinct driving frequencies a near depletion of certain sites which is explained by a resonant mixing of the involved Floquet-Bloch modes occurring at these frequencies. Our results could be exploited as a scheme for a site-selective loading of e.g. ultracold atoms into an optical lattices.
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…
We study the evolution of bosons in a periodically driven optical lattice during a slow change of the driving amplitude. Both the regime of high frequency and low frequency driving are investigated. In the low frequency regime, resonant absorption of energy is observed. In the high frequency regime, the dynamics is compared to a system with an effective Hamiltonian in which the atoms are `dressed' by the driving field. This `dressing' can dramatically change the amplitude and sign of the effective tunneling. A particular focus of this study is the investigation of the time-scales necessary for…
We investigate multi-"photon" interband excitation processes in an optical lattice that is driven periodically in time by a modulation of the lattice depth. Assuming the system to be prepared in the lowest band, we compute the excitation spectrum numerically. Moreover, we estimate the effective coupling parameters for resonant interband excitation processes analytically, employing degenerate perturbation theory in Floquet space. We find that below a threshold driving strength, interband excitations are suppressed exponentially with respect to the inverse driving frequency. For sufficiently low…
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…
Periodically-driven quantum systems are currently explored in view of realizing novel many-body phases of matter. This approach is particularly promising in gases of ultracold atoms, where sophisticated shaking protocols can be realized and inter-particle interactions are well controlled. The combination of interactions and time-periodic driving, however, often leads to uncontrollable heating and instabilities, potentially preventing practical applications of Floquet-engineering in large many-body quantum systems. In this work, we experimentally identify the existence of parametric instabiliti…
An array of single Rydberg atoms driven by a frequency modulated light field is studied. The periodic modulation effectively modifies the Rabi coupling, leading to unprecedented dynamics in the presence of Rydberg-Rydberg interactions. They include state dependent population trapping, the Rydberg blockade for small and anti-blockades at large interaction strengths. Interestingly, the Schrieffer-wolf transformation reveals a fundamental process in Rydberg gases, correlated Rabi oscillations, arising from the long-range interactions, provides an alternative depiction for Rydberg blockade and it …
We study a Fermionic chain with nearest-neighbor hopping and density-density interactions, where the nearest-neighbor interaction term is driven periodically. We show that such a driven chain exhibits prethermal strong Hilbert space fragmentation (HSF) in the high drive amplitude regime at specific drive frequencies $ω_m^{\ast}$. This constitutes the first realization of HSF for out-of-equilibrium systems. We obtain analytic expressions of $ω_m^{\ast}$ using a Floquet perturbation theory and provide exact numerical computation of entanglement entropy, equal-time correlation functions, and the …
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…
We propose a realization of a synthetic Random Flux Model in a two-dimensional optical lattice. Starting from Bose-Hubbard Hamiltonian for two atom species we show how to use fast-periodic modulation of the system parameters to construct random gauge field. We investigate the transport properties of such a system and describe the impact of time-reversal symmetry breaking and correlations in disorder on Anderson localization length.
We study the classical dynamics of many interacting particles in a periodically driven one-dimensional (1D) system. We show that under the rotating wave approximation (RWA), a short-distance 1D interaction ($δ$ function or hard-core interaction), becomes a long-distance two-dimensional (2D) interaction which only depends on the distance in the phase space of the rotating frame. The RWA interaction describes the effect of the interaction on the slowly changing amplitude and phase of the oscillating particles, while the fast oscillations take on the role of a force carrier, which allows for inte…
Periodically driven systems can lead to a directed motion of particles. We investigate this ratchet effect for a bosonic Mott insulator where both a staggered hopping and a staggered local potential vary periodically in time. If driving frequencies are smaller than the interaction strength and the density of excitations is small, one obtains effectively a one-particle quantum ratchet describing the motion of doubly occupied sites (doublons) and empty sites (holes). Such a simple quantum machine can be used to manipulate the excitations of the Mott insulator. For suitably chosen parameters, for…
