The bulk-boundary correspondence, a fundamental principle relating the topological invariants of the bulk to the presence of edge states, is modified in periodically driven systems. Conventional bulk topological invariants are insufficient to predict the existence of topological edge modes in such systems. Although ultracold atoms provide excellent settings for clean realizations of Floquet protocols, the observation of real-space edge modes has so far remained elusive. Here we demonstrate an experimental protocol for realizing chiral edge modes in optical lattices through the periodic modulation of the tunnelling rate between neighbouring sites. In particular, we show how to efficiently prepare particles in edge modes in three distinct Floquet topological regimes in a periodically driven honeycomb lattice. Controlling the height and amplitude of the potential step, we characterize the emergence of edge modes and the dependence of their group velocity on the sharpness of the potential step. Our direct observation of topological edge modes provides a tool to study topological phases of matter in the presence of disorder and interactions, where conventional bulk observables are not applicable. The observation of edge modes in topological systems is challenging because precise control over the sample and occupied states is required. An experiment with atoms in a driven lattice now shows how edge modes with programmable potentials can be realized.

The impact of weak disorder and its spatial correlation on the topology of a Floquet system is not well understood so far. In this study, we investigate a model closely related to a two-dimensional Floquet system that has been realized in experiments. In the absence of disorder, we determine the phase diagram and identify an exotic phase characterized by edge states with alternating chirality in adjacent gaps. When weak disorder is introduced, we examine the disorder -averaged Bott index and analyze why the anomalous Floquet topological insulator is favored by both uncorrelated and correlated disorder, with the latter having a stronger effect. For a system with a ring -shaped gap in the energy spectrum, the Born approximation fails to explain the topological phase transition, unlike for a system with a pointlike gap.

The study of geometrically frustrated many -body quantum systems is of central importance to uncover novel quantum mechanical effects. We design a scheme where ultracold bosons trapped in a one-dimensional state -dependent optical lattice are modeled by a frustrated Bose -Hubbard Hamiltonian. A derivation of the Hamiltonian parameters based on Cesium atoms, further show large tunability of contact and nearestneighbor interactions. For pure contact repulsion, we discover the presence of two phases peculiar to frustrated quantum magnets: the bond -order -wave insulator with broken inversion symmetry and a chiral superfluid. When the nearest -neighbor repulsion becomes sizable, a further density -wave insulator with broken translational symmetry can appear. We show that the phase transition between the two spontaneously symmetry -broken phases is continuous, thus representing a one-dimensional deconfined quantum critical point not captured by the Landau-Ginzburg-Wilson symmetry -breaking paradigm. Our results provide a solid ground to unveil the novel quantum physics induced by the interplay of nonlocal interactions, geometrical frustration, and quantum fluctuations.

Optical box traps offer new possibilities for quantum-gas experiments. Building on their exquisite spatial and temporal control, we propose to engineer system-reservoir configurations using box traps, in view of preparing and manipulating topological atomic states in optical lattices. First, we consider the injection of particles from the reservoir to the system: this scenario is shown to be particularly well suited to activating energy-selective chiral edge currents, but also to prepare fractional Chern insulating ground states. Then, we devise a practical evaporative-cooling scheme to effectively cool down atomic gases into topological ground states. Our open-system approach to optical-lattice settings provides a new path for the investigation of ultracold quantum matter, including strongly correlated and topological phases.

The anomalous Floquet Anderson insulator (AFAI) has been theoretically predicted in stepwise periodically driven models, but its stability under more general driving protocols has not been determined. We show that adding disorder to the anomalous Floquet topological insulator realized with a continuous driving protocol in the experiment by Wintersperger et al. [Nat. Phys. 16, 1058 (2020)], supports an AFAI phase, where, for a range of disorder strengths, all the time averaged bulk states become localized, while the pumped charge in a Laughlin pump setup remains quantized.

We present measurements of three distinctive state -dependent wavelengths for the 1 S 0 - 3 P 0 clock transition in 174 Yb atoms. Specifically, we determine two magic wavelengths at 652.281(21) and 542 . 50205(19) THz, where the differential light shift on the 1 S 0 - 3 P 0 clock transition vanishes, and one tune -out wavelength at 541 . 8325(5) THz, where the polarizability of the 1 S 0 ground state exhibits a zero crossing. The two magic wavelengths are identified by spectroscopically interrogating cold 174 Yb atoms on the clock transition in a one-dimensional optical lattice. The ground -state tune -out wavelength is determined via a parametric heating scheme. With a simple empirical model, we then extrapolate the ground- and excited -state polarizability over a broad range of wavelengths in the visible spectrum.

The sine-Gordon model emerges as a low-energy theory in a plethora of quantum many-body systems. Here, we theoretically investigate tunnel-coupled Bose-Hubbard chains with strong repulsive interactions as a realization of the sine-Gordon model deep in the quantum regime. We propose protocols for quantum gas microscopes of ultracold atoms to prepare and analyze solitons, which are the fundamental topological excitations of the emergent sine-Gordon theory. With numerical simulations based on matrix product states, we characterize the preparation and detection protocols and discuss the experimental requirements.

Quantum gas microscopy is an in-situ imaging technique used to investigate many-body phenomena in cold-atom quantum simulators and can provide resolution at the single-particle level,. however, limiting factors, such as short lattice constants and finite signal-to-noise ratios, weaken image resolution. Here, the authors develop an algorithm based on unsupervised deep learning that can reconstruct the occupation of an optical lattice of Cs atoms from fluorescence images with high fidelity. In quantum gas microscopy experiments, reconstructing the site-resolved lattice occupation with high fidelity is essential for the accurate extraction of physical observables. For short interatomic separations and limited signal-to-noise ratio, this task becomes increasingly challenging. Common methods rapidly decline in performance as the lattice spacing is decreased below half the imaging resolution. Here, we present an algorithm based on deep convolutional neural networks to reconstruct the site-resolved lattice occupation with high fidelity. The algorithm can be directly trained in an unsupervised fashion with experimental fluorescence images and allows for a fast reconstruction of large images containing several thousand lattice sites. We benchmark its performance using a quantum gas microscope with cesium atoms that utilizes short-spaced optical lattices with lattice constant 383.5 nm and a typical Rayleigh resolution of 850 nm. We obtain promising reconstruction fidelities ≳ 96% across all fillings based on a statistical analysis. We anticipate this algorithm to enable novel experiments with shorter lattice spacing, boost the readout fidelity and speed of lower-resolution imaging systems, and furthermore find application in related experiments such as trapped ions.

We propose a scheme to implement Kitaev's honeycomb model with cold atoms, based on a periodic (Floquet) drive, in view of realizing and probing non-Abelian chiral spin liquids using quantum simulators. We derive the effective Hamiltonian to leading order in the inverse-frequency expansion, and show that the drive opens up a topological gap in the spectrum without mixing the effective Majorana and vortex degrees of freedom. We address the challenge of probing the physics of Majorana fermions, while having access only to the original composite spin degrees of freedom. Specifically, we propose to detect the properties of the chiral spin liquid phase using gap spectroscopy and edge quenches in the presence of the Floquet drive. The resulting chiral edge signal, which relates to the thermal Hall effect associated with neutral Majorana currents, is found to be robust for realistically prepared states. By combining strong interactions with Floquet engineering, our work paves the way for future studies of non-Abelian excitations and quantized thermal transport using quantum simulators.

We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utiliz-ing alkaline-earth(-like) atoms in state-dependent optical lattices. The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions. We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species. Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory. This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theo-ries that-while pivotal for a successful implementation-have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics. We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope 173Yb, addressing via simulations all key sources of imperfections, and pro-vide concrete parameter estimates for relevant energy scales in both one-and two-dimensional settings.

Intriguingly, quantum many-body systems may defy thermalization even without disorder. One example is so-called fragmented models, where the many-body Hilbert space fragments into dynamically disconnected subspaces that are not determined by the global symmetries of the model. In this Letter we demonstrate that the tilted one-dimensional Fermi-Hubbard model naturally realizes distinct effective Hamiltonians that are expected to support nonergodic behavior due to fragmentation, even at resonances between the tilt energy and the Hubbard on site interaction. We find that the effective description captures the observed dynamics in experimentally accessible parameter ranges of moderate tilt values. Specifically, we observe a pronounced dependence of the relaxation dynamics on the initial doublon fraction, which directly reveals the microscopic processes of the fragmented model. Our results pave the way for future studies of nonergodic behavior in higher dimensions.

Thouless pumping provides one of the simplest manifestations of topology in quantum systems and has attracted a lot of recent interest, both theoretically and experimentally. Since the seminal works by David Thouless and Qian Niu in 1983 and 1984, it has been argued that the quantization of the pumped charge is robust against weak disorder, but a clear characterization of the localization properties of the relevant states, and the breakdown of quantized transport in the presence of interaction or out of the adiabatic approximation, has long been debated. Thouless pumping is also the first example of a topological phase emerging in a periodically driven system. Driven systems can exhibit exotic topological phases without any static analogue and have been the subject of many recent proposals both in fermionic and in bosonic systems. Recent experimental studies have been performed in diverse platforms ranging from cold atoms to photonics and condensed-matter systems. This Review serves as a basis to understand the robustness of the topology of slowly driven systems and also highlights the rich properties of topological pumps and their diverse range of applications. Examples include systems with synthetic dimensions or work towards understanding higher-order topological phases, which underline the relevance of topological pumping for the fast-growing field of topological quantum matter.

The postulate of gauge invariance in nature does not lend itself directly to implementations of lattice gauge theories in modern setups of quantum synthetic matter. Unavoidable gauge-breaking errors in such devices require gauge invariance to be enforced for faithful quantum simulation of gauge-theory physics. This poses major experimental challenges, in large part due to the complexity of the gauge-symmetry generators. Here, we show that gauge invariance can be reliably stabilized by employing simplified local pseudogenerators designed such that within the physical sector they act identically to the actual local generator. Dynamically, they give rise to emergent exact gauge theories up to time scales polynomial and even exponential in the protection strength. This obviates the need for implementing often complex multibody full gauge symmetries, thereby further reducing experimental overhead in physical realizations. We showcase our method in the Z(2) lattice gauge theory, and discuss experimental considerations for its realization in modern ultracold-atom setups.

The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems, quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here, we show that quantized Thouless pumps connecting C-4-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically nontrivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multipole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas's theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.

Transporting cold atoms between distant sections of a vacuum system is a central ingredient in many quantum simulation experiments, in particular in setups, where large optical access and precise control over magnetic fields is needed. In this work, we demonstrate optical transport of cold cesium atoms over a total transfer distance of about 43 cm in less than 30 ms. The high speed is facilitated by a moving lattice, which is generated via the interference of a Bessel and a Gaussian laser beam. We transport about 3 x 10(6) atoms at a temperature of a few microkelvins with a transport efficiency of about 75%. We provide a detailed study of the transport efficiency for different accelerations and lattice depths and find that the transport efficiency is mainly limited by a fast initial loss most likely due to the sudden onset of the acceleration and the potential depth along the direction of gravity. To highlight the suitability of the optical-transport setup for quantum simulation experiments, we demonstrate the generation of a pure Bose-Einstein condensate with about 2 x 10(4) atoms. We find a robust final atom number within 2% over a duration of 2.5 h with a standard deviation of <5% between individual experimental realizations.

The central idea of this review k to consider quantum field theory models relevant for particle physics and replace the fermionic matter in these models by a bosonic one. This is mostly motivated by the fact that bosons are more 'accessible' and easier to manipulate for experimentalists, but this 'substitution' also leads to new physics and novel phenomena. It allows us to gain new information about among other things confinement and the dynamics of the deconfinement transition. We will thus consider bosons in dynamical lattices corresponding to the bosonic Schwinger or Z(2) Bose-Hubbard models. Another central idea of this review concerns atomic simulators of paradigmatic models of particle physics theory such as the ereuti-Hubbard ladder, or Cross-Nevey-Wilson and Wilson-Hubbard models. This article k not a general review of the rapidly growing field-it reviews activities related to quantum simulations for lattice field theories performed by the Quantum Optics Theory group at ICFO and their collaborators from 19 institutions all over the world. Finally, we will briefly describe our efforts to design experimentally friendly simulators of these and other models relevant for particle physics. This article is part of the theme issue 'Quantum technologies in particle physics'.

Quantum simulators have made a remarkable progress towards exploring the dynamics of many-body systems, many of which offer a formidable challenge to both theoretical and numerical methods. While state-of-the-art quantum simulators are, in principle, able to simulate quantum dynamics well outside the domain of classical computers, they are noisy and limited in the variability of the initial state of the dynamics and the observables that can be measured. Despite these limitations, here we show that such a quantum simulator can be used to in effect solve for the dynamics of a many-body system. We develop an efficient numerical technique that facilitates classical simulations in regimes not accessible to exact calculations or other established numerical techniques. The method is based on approximations that are well suited to describe localized one-dimensional Fermi-Hubbard systems. Since this new method does not have an error estimate and the approximations do not hold in general, we use a neutral-atom Fermi-Hubbard quantum simulator with L-exp similar or equal to 290 lattice sites to benchmark its performance in terms of accuracy and convergence for evolution times up to 700 tunneling times. We then use these approximations in order to derive a simple prediction of the behavior of interacting Bloch oscillations for spin-imbalanced Fermi-Hubbard systems, which we show to be in quantitative agreement with experimental results. Finally, we demonstrate that the convergence of our method is the slowest when the entanglement depth developed in the many-body system we consider is neither too small nor too large. This represents a promising regime for near-term applications of quantum simulators.

Kitaev's toric code is an exactly solvable model with Z(2)-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental realization remains an open challenge. Here, we propose a building block for Z(2) lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically nontrivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we outline experimental signatures of the ground-state wave function and introduce a minimal braiding protocol. Detecting a p-phase shift between Ramsey fringes in this protocol reveals the anyonic excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators.

The thermalization of isolated quantum many-body systems is deeply related to fundamental questions of quantum information theory. While integrable or many-body localized systems display non-ergodic behavior due to extensively many conserved quantities, recent theoretical studies have identified a rich variety of more exotic phenomena in between these two extreme limits. The tilted one-dimensional Fermi-Hubbard model, which is readily accessible in experiments with ultracold atoms, emerged as an intriguing playground to study non-ergodic behavior in a clean disorder-free system. While non-ergodic behavior was established theoretically in certain limiting cases, there is no complete understanding of the complex thermalization properties of this model. In this work, we experimentally study the relaxation of an initial charge-density wave and find a remarkably long-lived initial-state memory over a wide range of parameters. Our observations are well reproduced by numerical simulations of a clean system. Using analytical calculations we further provide a detailed microscopic understanding of this behavior, which can be attributed to emergent kinetic constraints. It was predicted that complex thermalizing behaviour can arise in many-body systems in the absence of disorder. Here, the authors observe non-ergodic dynamics in a tilted optical lattice that is distinct from previously studied regimes, and propose a microscopic mechanism that is due to emergent kinetic constrains.

Topological states of matter, such as fractional quantum Hall states, are an active field of research due to their exotic excitations. In particular, ultracold atoms in optical lattices provide a highly controllable and adaptable platform to study such new types of quantum matter. However, finding a clear route to realize non-Abelian quantum Hall states in these systems remains challenging. Here we use the density-matrix renormalization-group (DMRG) method to study the Hofstadter-Bose-Hubbard model at filling factor v = 1 and find strong indications that at alpha = 1/6 magnetic flux quanta per plaquette the ground state is a lattice analog of the continuum non-Abelian Pfaffian. We study the on-site correlations of the ground state, which indicate its paired nature at v = 1, and find an incompressible state characterized by a charge gap in the bulk. We argue that the emergence of a charge density wave on thin cylinders and the behavior of the two- and three-particle correlation functions at short distances provide evidence for the state being closely related to the continuum Pfaffian. The signatures discussed in this letter are accessible in current cold atom experiments and we show that the Pfaffian-like state is readily realizable in few-body systems using adiabatic preparation schemes.

Understanding the nature of confinement, as well as its relation with the spontaneous breaking of chiral symmetry, remains one of the long-standing questions in high-energy physics. The difficulty of this task stems from the limitations of current analytical and numerical techniques to address nonperturbative phenomena in non-Abelian gauge theories. In this work, we show how similar phenomena emerge in simpler models, and how these can be further investigated using state-of-the-art cold-atom quantum simulators. More specifically, we introduce the rotor Jackiw-Rebbi model, a (1 + 1)-dimensional quantum field theory where interactions between Dirac fermions are mediated by quantum rotors. Starting from a mixture of ultracold atoms in an optical lattice, we show how this quantum field theory emerges in the long-wavelength limit. For a wide and experimentally relevant parameter regime, the Dirac fermions acquire a dynamical mass via the spontaneous breakdown of chiral symmetry. We study the effect of both quantum and thermal fluctuations, and show how they lead to the phenomenon of chiral symmetry restoration. Moreover, we uncover a confinement-deconfinement quantum phase transition, where mesonlike fermions fractionalize into quarklike quasiparticles bound to topological solitons of the rotor field. The proliferation of these solitons at finite chemical potentials again serves to restore the chiral symmetry, yielding a clear analogy with the quark-gluon plasma in quantum chromodynamics, where the restored symmetry coexists with the deconfined fractional charges. Our results indicate how the interplay between these phenomena could be analyzed in more detail in realistic atomic experiments.

Standard topological invariants commonly used in static systems are not enough to fully capture the topological properties of Floquet systems. In a periodically driven quantum gas, chiral edge modes emerge despite all Chern numbers being equal to zero. 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 are two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers being 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.

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 interparticle 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 instabilities in weakly interacting Bose-Einstein condensates in strongly driven optical lattices through momentum-resolved measurements, in line with theoretical predictions. Parametric instabilities can trigger the destruction of weakly interacting Bose-Einstein condensates through the rapid growth of collective excitations, in particular in systems with weak harmonic confinement transverse to the lattice axis. Understanding the onset of parametric instabilities in driven quantum matter is crucial for determining optimal conditions for the engineering of modulation-induced many-body systems.

Quantum simulation has the potential to investigate gauge theories in strongly interacting regimes, which are currently inaccessible through conventional numerical techniques. Here, we take a first step in this direction by implementing a Floquet-based method for studying Z(2) I 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 Z(2) I symmetry. We study the dynamics of the system for different initial states and critically contrast the observed evolution with a theoretical analysis of the full time-dependent Hamiltonian of the periodically driven lattice model. We reveal challenges that arise due to symmetry-breaking terms and outline potential pathways to overcome these limitations. Our results provide important insights for future studies of lattice gauge theories based on Floquet techniques.

From the standard model of particle physics to strongly correlated electrons, various physical settings are formulated in terms of matter coupled to gauge fields. Quantum simulations based on ultracold atoms in optical lattices provide a promising avenue to study these complex systems and unravel the underlying many-body physics. Here, we demonstrate how quantized dynamical gauge fields can be created in mixtures of ultracold atoms in optical lattices, using a combination of coherent lattice modulation with strong interactions. Specifically, we propose implementation of Z(2) lattice gauge theories coupled to matter, reminiscent of theories previously introduced in high-temperature superconductivity. We discuss a range of settings from zero-dimensional toy models to ladders featuring transitions in the gauge sector to extended two-dimensional systems. Mastering lattice gauge theories in optical lattices constitutes a new route toward the realization of strongly correlated systems, with properties dictated by an interplay of dynamical matter and gauge fields.

We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half filling, resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counterpropagating (protected) edge modes with different velocities. In the strong-coupling limit, the energy spectrum shows flat bands. Therefore we build a perturbation theory and address further the bulk-edge correspondence. We also show the occurrence of a topological insulating phase with Chern number one when only the lowest band is filled. We generalize the effect to Haldane bilayer systems with asymmetric Semenoff masses. Moreover, we propose an alternative definition of the topological invariant on the Bloch sphere.

We experimentally study many-body localization (MBL) with ultracold atoms in a weak onedimensional quasiperiodic potential, which in the noninteracting limit exhibits an intermediate phase that is characterized by a mobility edge. We measure the time evolution of an initial charge density wave after a quench and analyze the corresponding relaxation exponents. We find clear signatures of MBL when the corresponding noninteracting model is deep in the localized phase. We also critically compare and contrast our results with those from a tight-binding Aubry-Andre model, which does not exhibit a singleparticle intermediate phase, in order to identify signatures of a potential many-body intermediate phase.