The concept of Floquet engineering is to subject a quantum system to time-periodic driving in such a way that it acquires interesting novel properties. It has been employed, for instance, for the realization of artificial magnetic fluxes in optical lattices and, typically, it is based on two approximations. First, the driving frequency is assumed to be low enough to suppress resonant excitations to high-lying states above some energy gap separating a low energy subspace from excited states. Second, the driving frequency is still assumed to be large compared to the energy scales of the low-energy subspace, so that also resonant excitations within this space are negligible. Eventually, however, deviations from both approximations will lead to unwanted heating on a time scale $τ$. Using the example of a one-dimensional system of repulsively interacting bosons in a shaken optical lattice, we investigate the optimal frequency (window) that maximizes $τ$. As a main result, we find that, when increasing the lattice depth, $τ$ increases faster than the experimentally relevant time scale given by the tunneling time $\hbar/J$, so that Floquet heating becomes suppressed.
Floquet engineering is the concept of tailoring a system by a periodic drive. It has been very successful in opening new classes of Hamiltonians to the study with ultracold atoms in optical lattices, such as artificial gauge fields, topological band structures and density-dependent tunneling. Furthermore, driven systems provide new physics without static counterpart such as anomalous Floquet topological insulators. In this review article, we provide an overview of the exciting developments in the field and discuss the current challenges and perspectives.
Floquet engineering provides a toolbox for the realization of novel quantum phases without static counterparts, while conventionally the realization may rely on the manipulation of complex temporal evolution. Here we propose a systematic and high-precision scheme to realize unconventional Floquet topological phases by engineering local band structures in particular momentum subspace called band inversion surfaces (BISs). This scheme is based on a new bulk-boundary correspondence that for a class of generic $d$-dimensional periodically driven systems, the local topological structure formed in e…
Quantum simulators based on atoms or molecules often have long-range interactions due to dipolar or Coulomb interactions. We present a method based on Floquet engineering to turn a long-range interaction into a short-range one. By modulating a magnetic-field gradient with one or a few frequencies, one reshapes the interaction profile, such that the system behaves as if it only had nearest-neighbor interactions. Our approach works in both one and two dimensions and for both spin-1/2 and spin-1 systems. It does not require individual addressing, and is applicable to all experimental systems with…
Considering the example of superconducting circuits, we show how Floquet engineering can be combined with reservoir engineering for the controlled preparation of target states. Floquet engineering refers to the control of a quantum system by means of time-periodic forcing, typically in the high-frequency regime, so that the system is governed effectively by a time-independent Floquet Hamiltonian with novel interesting properties. Reservoir engineering, on the other hand, can be achieved in superconducting circuits by coupling a system of artificial atoms (or qubits) dispersively to pumped leak…
We experimentally realized a time-periodically modulated 1D lattice for ultracold atoms featuring a pair of linear bands, each associated with a Floquet winding number: a topological invariant. These bands are spin-momentum locked and almost perfectly linear everywhere in the Brillouin zone (BZ), making this system a near-ideal realization of the 1D Dirac Hamiltonian. We characterized the Floquet winding number using a form of quantum state tomography, covering the BZ and following the micromotion through one Floquet period. Lastly, we altered the modulation timing to lift the topological prot…
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…
The insertion of a local magnetic flux, as the one created by a thin solenoid, plays an important role in gedanken experiments of quantum Hall physics. By combining Floquet engineering of artificial magnetic fields with the ability of single-site addressing in quantum-gas microscopes, we propose a scheme for the realization of such local solenoid-type magnetic fields in optical lattices. We show that it can be employed to manipulate and probe elementary excitations of a topological Chern insulator. This includes quantized adiabatic charge pumping along tailored paths inside the bulk, as well a…
We study Floquet engineering of the tunnel coupling between a pair of one-dimensional bosonic quasi-condensates in a tilted double-well potential. By modulating the energy difference between the two wells, we re-establish tunnel coupling and precisely control its amplitude and phase. This allows us to initiate coherence between two initially uncorrelated Bose gases and prepare different initial states in the emerging sine-Gordon Hamiltonian. We fully characterize the Floquet system and study the dependence of both equilibrium properties and relaxation on the modulation.
We provide a pedagogical technical guide to many of the key theoretical tools and ideas that underlie work in the field of Floquet engineering. We hope that this document will serve as a useful resource for new researchers aiming to enter the field, as well as experienced researchers who wish to gain new insight into familiar or possibly unfamiliar methods. This guide was developed out of supplementary material as a companion to our recent review, "Band structure engineering and non-equilibrium dynamics in Floquet topological insulators," Nature Reviews Physics 2, 229 (2020). The primary focus…
Controlling interactions is the key element for quantum engineering of many-body systems. Using time-periodic driving, a naturally given many-body Hamiltonian of a closed quantum system can be transformed into an effective target Hamiltonian exhibiting vastly different dynamics. We demonstrate such Floquet engineering with a system of spins represented by Rydberg states in an ultracold atomic gas. Applying a sequence of spin manipulations, we change the symmetry properties of the effective Heisenberg XYZ Hamiltonian. As a consequence, the relaxation behavior of the total spin is drastically mo…
We consider an ensemble of ultracold bosonic atoms within a near-planar cavity, driven by a far detuned laser whose phase is modulated at a frequency comparable to the transverse cavity mode spacing. We show that a strong, dispersive atom-photon coupling can be reached for many transverse cavity modes at once. The resulting Floquet polaritons involve a superposition of a set of cavity modes with a density excitation of the atomic cloud. The mutual interactions between these modes lead to distinct avoided crossings between the polaritons. Increasing the laser drive intensity, a low-lying multim…
Floquet engineering is one of the most vigorous fields in periodically driven (Floquet) systems, with which we can control phases of matter usually by high-frequency drives. In this paper, with Floquet engineering by a combination of high-frequency drives and resonant drives, we propose a way to realize nontrivial topological phases protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry only in the presence of a $\mathbb{Z}_2$ symmetry, using a robust emergent $\mathbb{Z}_2$ symmetry induced by the resonant drives. Moreover, the symmetry protected topological (SPT) phases are switchable be…
Hamiltonians which are inaccessible in static systems can be engineered in periodically driven many-body systems, i.e., Floquet many-body systems. We propose to use interacting particles in a one-dimensional (1D) harmonic potential with periodic kicking to investigate two-dimensional (2D) topological and many-body physics. Depending on the driving parameters, the Floquet Hamiltonian of single kicked harmonic oscillator has various lattice structures in phase space. The noncommutative geometry of phase space gives rise to the topology of the system. We investigate the effective interactions of …
Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions and mobility edges in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force dresses the hopping amplitudes between lattice sites, yielding alternate transitions between localized, mobility edge and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to five representative models of non-Hermitian quasicrystals, obtain the conditions of photon-assisted localizat…
Proposal for realizing quantum-spin systems on a two-dimensional square lattice with Dzyaloshinskii-Moriya interaction by Floquet engineering using Rydberg atoms
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…
The dynamic engineering of band structures for ultracold atoms in optical lattices represents an innovative approach to understand and explore the fundamental principles of topological matter. In particular, the folded Floquet spectrum determines the associated band topology via band inversion. We experimentally and theoretically study two-frequency phase modulation to asymmetrically hybridize the lowest two bands of a one-dimensional lattice. Using quasi-degenerate perturbation theory in the extended Floquet space we derive an effective two-band model that quantitatively describes our setting…
Optical control and manipulation of cold atoms has become an important topic in condensed matter. Widely employed are optical lattice shaking experiments which allow the introduction of artificial gauge fields, the design of topological bandstructures, and more general probing of quantum critical phenomena. Here we develop new numerical methods to simulate these periodically driven systems by implementing lattice shaking directly. As a result we avoid the usual assumptions associated with a simplified picture based on Floquet dynamics. A demonstrable success of our approach is that it yields q…
We develop the Floquet-Magnus expansion for a classical equation of motion under a periodic drive that is applicable to both isolated and open systems. For classical systems, known approaches based on the Floquet theorem fail due to the nonlinearity and the stochasticity of their equations of motion (EOMs) in contrast to quantum ones. Here, employing their master equation, we successfully extend the Floquet methodology to classical EOMs to obtain their Floquet-Magnus expansions, thereby overcoming this difficulty. Our method has a wide range of application from classical to quantum as long as …
Floquet dynamics of a quantum system subject to periodic modulations of system parameters provide a powerful tool for engineering new quantum matter with exotic properties. While system dynamics are significantly altered, the periodic modulation itself is usually induced externally and independent of Floquet dynamics. Here we propose a new type of Floquet physics for a Bose-Einstein condensate (BEC) subject to a shaken lattice generated inside a cavity, where the shaken lattice and atomic Floquet bands are mutually dependent, resulting in self-adapted Floquet dynamics. In particular, the shake…
In this paper, we propose that a simple model, in which fermions coupled to a dynamical lattice gauge field, can be engineered via the Floquet approach. The model possess both a independent Maxwell term and the local $Z_{2}$ gauge symmetry. Our proposal relies on a species dependent optical lattice, and can be achieved in one, two or three dimension. By an unitary transformation, this model can be mapped into a non-interacting composite Fermion system with fluctuating back ground charge. With the help of this composite Fermion picture, two characteristic observations are predicted. One is the …
We report on the experimental implementation of tunable occupation-dependent tunneling in a Bose-Hubbard system of ultracold atoms via time-periodic modulation of the on-site interaction energy. The tunneling rate is inferred from a time-resolved measurement of the lattice site occupation after a quantum quench. We demonstrate coherent control of the tunneling dynamics in the correlated many-body system, including full suppression of tunneling as predicted within the framework of Floquet theory. We find that the tunneling rate explicitly depends on the atom number difference in neighboring lat…
Semimetals exhibiting nodal lines or nodal surfaces represent a novel class of topological states of matter. While conventional Weyl semimetals exhibit momentum-space Dirac monopoles, these more exotic semimetals can feature unusual topological defects that are analogous to extended monopoles. In this work, we describe a scheme by which nodal rings and nodal spheres can be realized in synthetic quantum matter through well-defined periodic-driving protocols. As a central result of our work, we characterize these nodal defects through the quantum metric, which is a gauge-invariant quantity assoc…
The presence of quantum scars, athermal eigenstates of a many-body Hamiltonian with finite energy density, leads to absence of ergodicity and long-time coherent dynamics in closed quantum systems starting from simple initial states. Such non-ergodic coherent dynamics, where the system does not explore its entire phase space, has been experimentally observed in a chain of ultracold Rydberg atoms. We show, via study of a periodically driven Rydberg chain, that the drive frequency acts as a tuning parameter for several reentrant transitions between ergodic and non-ergodic regimes. The former regi…
Floquet engineering or coherent time periodic driving of quantum systems has been successfully used to synthesize Hamiltonians with novel properties. In ultracold atomic systems, this has led to experimental realizations of artificial gauge fields, topological band structures, and observation of dynamical localization, to name just a few. Here we present a Floquet-based framework to stroboscopically engineer Hamiltonians with spatial features and periodicity below the diffraction limit of light used to create them by time-averaging over various configurations of a 1D optical Kronig-Penney (KP)…
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet t…
Experimental engineering of Floquet topological phases in a one-dimensional optical lattice
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；Beijing Aerospace Institute for Metrology and Measurement Technology（北京航天计量测试技术研究所）；Hefei National Laboratory（合肥国家实验室）；School of Electronics Engineering and Computer Science, Peking University（北京大学电子工程与计算机科学学院）；International Quantum Academy（深圳国际量子学院）
Excitation energy transfer (EET) is one of the most important processes in both natural and artificial chemical systems including, for example, photosynthetic complexes and organic solar cells. The EET rate, however, is strongly suppressed when there is a large difference in the excitation energy between the donor and acceptor molecules. Here, we demonstrate both analytically and numerically that the EET rate can be greatly enhanced by periodically modulating the excitation energy difference. The enhancement of EET by using this Floquet engineering, in which the system's Hamiltonian is made pe…
The realization of synthetic gauge fields has attracted a lot of attention recently in relation with periodically driven systems and the Floquet theory. In ultra-cold atom systems in optical lattices and photonic networks, this allows to simulate exotic phases of matter such as quantum Hall phases, anomalous quantum Hall phases and analogs of topological insulators. In this paper, we apply the Floquet theory to engineer anisotropic Haldane models on the honeycomb lattice and two-leg ladder systems. We show that these anisotropic Haldane models still possess a topologically non-trivial band str…
We investigate the topological properties of a resonantly shaken one-dimensional optical lattice system, where the lattice position is periodically driven with two harmonic frequencies to generate one- and two-photon couplings between the two lowest orbitals. In a two-band approximation, we numerically show that degenerate edge states appear under a certain driving condition and that the corresponding topological phase is protected by the chiral symmetry of the periodically driven system. The system's micromotion is characterized with oscillating Zak phases and we find that the Zak phases are …
Periodic driving of a quantum system can significantly alter its energy bands and even change the band topology, opening a completely new avenue for engineering novel quantum matter. Although important progress has been made recently in measuring topological properties of Floquet bands in different systems, direct experimental measurement of Floquet band dispersions and their topology change is still demanding. Here we directly measure Floquet band dispersions in a periodically driven spin-orbit coupled ultracold Fermi gas. Using spin injection radio-frequency spectroscopy, we observe that the…
