Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties of these driven settings. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven honeycomb lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.
We describe an experimental setup for imaging topologically protected Floquet edge states using ultracold bosons in an optical lattice. Our setup involves a deep two dimensional optical lattice with a time dependent superlattice that modulates the hopping between neighboring sites. The finite waist of the superlattice beam yields regions with different topological numbers. One can observe chiral edge states by imaging the real-space density of a bosonic packet launched from the boundary between two topologically distinct regions.
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…
Spatiotemporal Non-Hermitian Skin Effect with Floquet-Engineered Ultracold Atoms
Quantum Institute for Light and Atoms, School of Physics, East China Normal University（华东师范大学物理学院光与原子量子研究所）；School of Physics and Astronomy, and Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学物理与天文学院及李政道研究所）；Shanghai Branch, Hefei National Laboratory（合肥国家实验室上海分部）；Shanghai Research Center for Quantum Sciences（上海量子科学研究与创新中心）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）
A gas of ultracold interacting quantum degenerate Fermions is considered in a three dimensional optical lattice which is externally modulated in the fre- quency and the amplitude. This theoretical study utilizes the Keldysh formalism to account for the system being out of thermodynamical equilibrium. A dynamical mean field theory, extended to non-equilibrium, is presented to calculate characteristic quantities such as the local density of states and the non-equilibrium distribution function. A dynamics Franz-Keldysh splitting is found which accounts for the non-equilibrium modification of the …
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…
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…
Topological band structures can be designed by subjecting lattice systems to time-periodic modulations, as was recently demonstrated in cold atoms and photonic crystals. However, changing the topological nature of Floquet Bloch bands from trivial to non-trivial, by progressively launching the time-modulation, is necessarily accompanied with gap-closing processes: this has important consequences for the loading of particles into a target Floquet band with non-trivial topology, and hence, on the subsequent measurements. In this work, we analyse how such loading sequences can be optimized in view…
Employing the external degrees of freedom of atoms as synthetic dimensions renders easy and new accesses to quantum engineering and quantum simulation. As a recent development, ultracold atoms suffering from two-photon Bragg transitions can be diffracted into a series of discrete momentum states to form a momentum lattice. Here we provide a detailed analysis on such a system, and, as a concrete example, report the observation of robust helical Floquet channels, by introducing periodic driving sequences. The robustness of these channels against perturbations is confirmed, as a test for their to…
Open physical systems with balanced loss and gain, described by non-Hermitian parity-time ($\mathcal{PT}$) reflection symmetric Hamiltonians, exhibit a transition which could engenders modes that exponentially decay or grow with time and thus spontaneously breaks the $\mathcal{PT}$-symmetry. Such $\mathcal{PT}$-symmetry breaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of $\mathcal{PT}$-symmetry breaking transitions by engineering time-periodic dissipation and coupling, whi…
Probing disorder-driven topological phase transitions via topological edge modes with ultracold atoms in Floquet-engineered honeycomb lattices
Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Fakultät für Physik, Ludwig-Maximilians-Universität（慕尼黑大学物理学院）
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 …
In systems of ultracold atoms, pairwise interactions are resonantly enhanced by the application of an oscillating magnetic field that is parallel to the spin-quantization axis of the atoms. The resonance occurs when the frequency of the applied field is precisely tuned near the transition frequency between the scattering atoms and a diatomic molecule. The resulting cross section can be made more than two orders of magnitude larger than the cross section in the absence of the oscillating field. The low momentum resonance properties have a universal description that is independent of the atomic …
Quantum simulation has the potential to investigate gauge theories in strongly-interacting regimes, which are up to now inaccessible through conventional numerical techniques. Here, we take a first step in this direction by implementing a Floquet-based method for studying $\mathbb{Z}_2$ lattice gauge theories using two-component ultracold atoms in a double-well potential. For resonant periodic driving at the on-site interaction strength and an appropriate choice of the modulation parameters, the effective Floquet Hamiltonian exhibits $\mathbb{Z}_2$ symmetry. We study the dynamics of the system…
We present a brief overview of the phases and dynamics of ultracold bosons in an optical lattice in the presence of a tilt. We begin with a brief summary of the possible experimental setup for generating the tilt. This is followed by a discussion of the effective low-energy model for these systems and its equilibrium phases. We also chart the relation of this model to the recently studied system of ultracold Rydberg atoms. Next, we discuss the non-equilibrium dynamics of this model for quench, ramp and periodic protocols with emphasis on the periodic drive which can be understood in terms of a…
We analyze time-of-flight absorption images obtained with dilute Bose-Einstein con-densates released from shaken optical lattices, both theoretically and experimentally. We argue that weakly interacting, ultracold quantum gases in kilohertz-driven optical potentials constitute equilibrium systems characterized by a steady-state distri-bution of Floquet-state occupation numbers. Our experimental results consistently indicate that a driven ultracold Bose gas tends to occupy a single Floquet state, just as it occupies a single energy eigenstate when there is no forcing. When the driving amplitude…
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…
We investigate the out-of-equilibrium properties of a system of interacting bosons in a ring lattice. We present a Floquet driving that induces clockwise (counterclockwise) circulation of the particles among the odd (even) sites of the ring which can be mapped to a fully connected model of clocks of two counter-rotating species. The clock-like motion of the particles is at the core of a period-$n$ discrete time crystal where $L=2n$ is the number of lattice sites. In presence of a "staircase-like" on-site potential, we report the emergence of a second characteristic timescale in addition to the…
We introduce a Floquet spinor Bose-Einstein condensate induced by a periodically driven quadratic Zeeman coupling whose frequency is larger than any other energy scales. By examining a spin-1 system available in ultracold atomic gases, we demonstrate that such an external driving field has great effect on the condensate through emergence of a unique spin-exchange interaction. We uncover that the ferromagnetic condensate has several unconventional stationary states and thus exhibits rich continuous phase transitions. On the other hand, the antiferromagnetic condensate is found to possess a nont…
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.
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 response of ultracold atoms to a weak force in the presence of a temporally strongly modulated optical lattice potential. It is experimentally demonstrated that the strong ac-driving allows for a tailoring of the mobility of a dilute atomic Bose-Einstein condensate with the atoms moving ballistically either along or against the direction of the applied force. Our results are in agreement with a theoretical analysis of the Floquet spectrum of a model system, thus revealing the existence of diabatic Floquet bands in the atom's band spectra and highlighting their role in the non-equi…
We present a quantitative, near-term experimental blueprint for the quantum simulation of topological insulators using lattice-trapped ultracold polar molecules. In particular, we focus on the so-called Hopf insulator, which represents a three-dimensional topological state of matter existing outside the conventional tenfold way and crystalline-symmetry-based classifications of topological insulators. Its topology is protected by a \emph{linking number} invariant, which necessitates long-range spin-orbit coupled hoppings for its realization. While these ingredients have so far precluded its rea…
The atomic Quantum Kicked Rotor is an outstanding "quantum simulator" for the exploration of transport in disordered quantum systems. Here we study experimentally the phase-shifted quantum kicked rotor, which we show to display properties close to an ideal disordered quantum system, opening new windows into the study of Anderson physics.
The concept of valence bond resonance plays a fundamental role in the theory of the chemical bond and is believed to lie at the heart of many-body quantum physical phenomena. Here we show direct experimental evidence of a time-resolved valence bond quantum resonance with ultracold bosonic atoms in an optical lattice. By means of a superlattice structure we create a three-dimensional array of independent four-site plaquettes, which we can fully control and manipulate in parallel. Moreover, we show how small-scale plaquette resonating valence bond states with s- and d-wave symmetry can be create…
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…
We consider the angular momentum of a harmonically trapped, noninteracting Fermi gas subject to either rotation or to an artificial gauge field. The angular momentum of the gas is shown to display oscillations as a function of the particle number or chemical potential. This phenomenon is analogous to the de Haas - van Alphen oscillations of the magnetization in the solid-state context. However, key differences exist between the solid-state and ultracold atomic gases that we point out and analyze. We explore the dependence of the visibility of these oscillations on the physical parameters and p…
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.
We study the stroboscopic and non-stroboscopic dynamics in the Floquet realization of the Harper-Hofstadter Hamiltonian. We show that the former produces the evolution expected in the high-frequency limit only for observables which commute with the operator to which the driving protocol couples. On the contrary, non-stroboscopic dynamics is capable of capturing the evolution governed by the Floquet Hamiltonian of any observable associated with the effective high-frequency model. We provide exact numerical simulations for the dynamics of the density operator following a quantum cyclotron orbit …
We propose the implementation of a quantum heat pump with ultracold atoms. It is based on two periodically driven coherently coupled quantum dots using ultracold atoms. Each dot possesses two relevant quantum states and is coupled to a fermionic reservoir. The working principle is based on energy-selective driving-induced resonant tunneling processes, where a particle that tunnels from one dot to the other either absorbs or emits the energy quantum $\hbarω$ associated with the driving frequency, depending on its energy. We characterize the device using Floquet theory and compare simple analyti…
We have realized a hybrid optomechanical system by coupling ultracold atoms to a micromechanical membrane. The atoms are trapped in an optical lattice, which is formed by retro-reflection of a laser beam from the membrane surface. In this setup, the lattice laser light mediates an optomechanical coupling between membrane vibrations and atomic center-of-mass motion. We observe both the effect of the membrane vibrations onto the atoms as well as the backaction of the atomic motion onto the membrane. By coupling the membrane to laser-cooled atoms, we engineer the dissipation rate of the membrane.…
We propose a simple scheme for realizing the physics of 1D anyons with ultracold bosonic atoms in an optical lattice. It relies on lattice-shaking-induced resonant tunneling against potential off-sets created by a combination of a lattice tilt and strong on-site interactions. No lasers additional to those used for the creation of the optical lattice are required. We also discuss experimental signatures of the continuous interpolation between bosons and fermions when the statistical angle $θ$ is varied from 0 to $π$. Since the anyonic momentum distribution is not measurable in the bosonic repre…
Floquet modulation has been widely used in optical lattices for coherent control of quantum gases, in particular for synthesizing artificial gauge fields and simulating topological matters. However, such modulation induces heating which can overwhelm the signal of quantum dynamics in ultracold atoms. Here we report that the thermal motion, instead of being a noise source, provides a new control knob in Floquet-modulated superradiance lattices, which are momentum-space tight-binding lattices of collectively excited states of atoms. The Doppler shifts combined with Floquet modulation provide eff…
We demonstrate the experimental implementation of an optical lattice that allows for the generation of large homogeneous and tunable artificial magnetic fields with ultracold atoms. Using laser-assisted tunneling in a tilted optical potential we engineer spatially dependent complex tunneling amplitudes. Thereby atoms hopping in the lattice accumulate a phase shift equivalent to the Aharonov-Bohm phase of charged particles in a magnetic field. We determine the local distribution of fluxes through the observation of cyclotron orbits of the atoms on lattice plaquettes, showing that the system is …
Spin-orbit coupling (SOC) is central to many physical phenomena, including fine structures of atomic spectra and quantum topological matters. Whereas SOC is in general fixed in a physical system, atom-laser interaction provides physicists a unique means to create and control synthetic SOC for ultracold atoms \cite{Dalibard}. Though significant experimental progresses have been made, a bottleneck in current studies is the lack of a two-dimensional (2D) synthetic SOC, which is crucial for realizing high-dimensional topological matters. Here, we report the experimental realization of 2D SOC in ul…
Time crystals, a phase showing spontaneous breaking of time-translation symmetry, has been an intriguing subject for systems far away from equilibrium. Recent experiments found such a phase both in the presence and absence of localization, while in theories localization by disorder is usually assumed a priori. In this work, we point out that time crystals can generally exist in systems without disorder. A series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle. Robust time crystalline orders are found in the strongly in…
We experimentally realize the spin-tensor momentum coupling (STMC) using the three ground Zeeman states coupled by three Raman laser beams in ultracold atomic system of $^{40}$K Fermi atoms. This new type of STMC consists of two bright-state bands as a regular spin-orbit coupled spin-1/2 system and one dark-state middle band. Using radio-frequency spin-injection spectroscopy, we investigate the energy band of STMC. It is demonstrated that the middle state is a dark state in the STMC system. The realized energy band of STMC may open the door for further exploring exotic quantum matters.
The Weyl semimetals [1-6] are three-dimensional (3D) gapless topological phases with Weyl cones in the bulk band, and host massless quasiparticles known as Weyl fermions which were theorized by Hermann Weyl in the last twenties [7]. The lattice theory constrains that Weyl cones must come in pairs, with the minimal number of cones being two. The semimetal with only two Weyl cones is an ideal Weyl semimetal (IWSM) which is the optimal platform to explore broad Weyl physics but hard to engineer in solids. Here, we report the experimental realization of the IWSM band by synthesising for the first …
We propose and study a quantum pump which emulates a traditional paddlewheel, that can be implemented with ultracold atoms in waveguides. We use wavepacket propagation to study its single-mode dynamics, which also determines its multimode current for mesoscopic setups. Energy flow with or without particle transport is possible. The spectrum reveals unusual features such as nonuniform Floquet side-bands and counter-intuitive scattering. Explanations are found by examining the scattering dynamics comparatively using quantum, classical and semiclassical pictures, indicating a rich system and expe…
Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the reali…
We report on the experimental realization of ultracold $^{174}{\rm Yb}$-$^{7}{\rm Li}$ (Boson-Boson) and $^{173}{\rm Yb}$-$^{7}{\rm Li}$ (Fermion-Boson) mixtures. They are loaded into three dimensional (3D) or one dimensional (1D) optical lattices that are species-selectively deep for the heavy Ytterbium (Yb) and shallow for the light bosonic Lithium (Li) component, realizing novel mixed dimensional systems. In the 1D optical lattice the band structure of $^{173}{\rm Yb}$ is reconstructed in the presence of $^{7}{\rm Li}$. Spectroscopic measurements of the $^{174}{\rm Yb}$-$^{7}{\rm Li}$ mixtu…
Topological states of interacting many-body systems are at the focus of current research due to the exotic properties of their elementary excitations. In this paper we suggest a realistic experimental setup for the realization of a simple version of such a phase. We show how delta-interacting bosons hopping on the links of a one-dimensional (1D) ladder can be used to simulate the thin-torus-limit of the two-dimensional (2D) Hofstadter-Hubbard model at one-quarter magnetic flux per plaquette. Bosons can be confined to ladders by optical superlattices, and synthetic magnetic fields can be realiz…
Ultracold atomic gases in two dimensions tuned close to a p-wave Feshbach resonance were expected to exhibit topological superfluidity, but these were found to be experimentally unstable. We show that one can induce a topological Floquet superfluid if weakly interacting atoms are brought suddenly close ("quenched") to such a resonance, in the time before the instability kicks in. The resulting superfluid possesses Majorana edge modes, yet differs from a conventional Floquet system as it is not driven externally. Instead, the periodic modulation is self-generated by the dynamics.
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…
Much recent experimental effort has focused on the realization of exotic quantum states and dynamics predicted to occur in periodically driven systems. But how robust are the sought-after features, such as Floquet topological surface states, against unavoidable imperfections in the periodic driving? In this work, we address this question in a broader context and study the dynamics of quantum systems subject to noise with periodically recurring statistics. We show that the stroboscopic time evolution of such systems is described by a noise-averaged Floquet superoperator. The eigenvectors and -v…
Floquet driven systems represent an extremely interesting arena to study out-of-equilibrium phenomena. For instance, they provide realizations of discrete time crystals, where the discrete time translation symmetry of the periodic Hamiltonian is spontaneously broken by a subharmonic response of the system. However, the continuous presence of an external periodic driving is required within the current Floquet paradigm. We propose here the concept of spontaneous many-body Floquet state. This is a state that, in the absence of external periodic driving, self-oscillates like in the presence of a p…
