Site-resolved imaging of fermionic ^{6}Li in an optical lattice

In Situ Imaging of a Single-Atom Wave Packet in Continuous Space

We report on the imaging of the in situ spatial distribution of deterministically prepared single-atom wave packets as they expand in a plane, finding excellent agreement with the scaling dynamics predicted by the Schrödinger equation. Our measurement provides a direct and quantitative observation of the textbook free expansion of a one-particle Gaussian wave packet, which we believe has no equivalent in the existing literature. Second, we utilize these expanding wave packets as a benchmark to develop a protocol for the controlled projection of a spatially extended wave function from continuous space onto the sites of a deep optical lattice and subsequent single-atom imaging using quantum gas microscopy techniques. By probing the square modulus of the wave function for various lattice ramp-up times, we show how to obtain a near-perfect projection onto lattice sites. Establishing this protocol represents a crucial prerequisite to the realization of a quantum gas microscope for continuum physics. The method demonstrated here for imaging a wave packet whose initial extent greatly exceeds the pinning lattice spacing, is designed to be applicable to the many-body wave function of interacting systems in continuous space, promising a direct access to their microscopic properties, including spatial correlation functions up to high order and large distances.

Formation of individual stripes in a mixed-dimensional cold-atom Fermi-Hubbard system

The relation between d-wave superconductivity and stripes is fundamental to the understanding of ordered phases in high-temperature cuprate superconductors1-6. These phases can be strongly influenced by anisotropic couplings, leading to higher critical temperatures, as emphasized by the recent discovery of superconductivity in nickelates7-10. Quantum simulators with ultracold atoms provide a versatile platform to engineer such couplings and to observe emergent structures in real space with single-particle resolution. Here we show, to our knowledge, the first signatures of individual stripes in a cold-atom Fermi-Hubbard quantum simulator using mixed-dimensional (mixD) settings. Increasing the energy scale of hole-hole attraction to the spin exchange energy, we access the interesting crossover temperature regime in which stripes begin to form11. We observe extended, attractive correlations between hole dopants and find an increased probability of forming larger structures akin to individual stripes. In the spin sector, we study correlation functions up to the third order and find results consistent with stripe formation. These observations are interpreted as a precursor to the stripe phase, which is characterized by interleaved charge and spin density wave ordering with fluctuating lines of dopants separating domains of opposite antiferromagnetic order12-14.

Corner Charge Fluctuation as an Observable for Quantum Geometry and Entanglement in Two-Dimensional Insulators

Measuring bipartite fluctuations of a conserved charge, such as the particle number, is a powerful approach to understanding quantum systems. When the measured region has sharp corners, the bipartite fluctuation receives an additional contribution known to exhibit a universal angle dependence in 2D isotropic and uniform systems. Here we establish that, for generic lattice systems of interacting particles, the corner charge fluctuation is directly related to quantum geometry. We first provide a practical scheme to isolate the corner contribution on lattices and analytically prove that its angle dependence in the "small-angle limit" measures exclusively the integrated quantum metric. A model of a compact obstructed atomic insulator is introduced to illustrate this effect analytically, while numerical verification for various Chern insulator models further demonstrate the experimental relevance of the corner charge fluctuation in a finite-size quantum simulator as a probe of quantum geometry. Last but not least, for free fermions, we unveil an intimate connection between quantum geometry and quantum information through the lens of corner entanglement entropies.

Dynamical phase transition in the occupation fraction statistics for noncrossing Brownian particles

We consider a system of N noncrossing Brownian particles in one dimension. We find the exact rate function that describes the long-time large deviation statistics of their occupation fraction in a finite interval in space. Remarkably, we find that, for any general N≥2, the system undergoes N-1 dynamical phase transitions of second order. The N-1 transitions are the boundaries of N phases that correspond to different numbers of particles which are in the vicinity of the interval throughout the dynamics. We achieve this by mapping the problem to that of finding the ground-state energy for N noninteracting spinless fermions in a square-well potential. The phases correspond to different numbers of single-body bound states for the quantum problem. We also study the process conditioned on a given occupation fraction and the large-N limiting behavior.

Universal Kardar-Parisi-Zhang Dynamics in Integrable Quantum Systems

Although the Bethe ansatz solution of the spin-1/2 Heisenberg model dates back nearly a century, the anomalous nature of its high-temperature transport dynamics has only recently been uncovered. Indeed, numerical and experimental observations have demonstrated that spin transport in this paradigmatic model falls into the Kardar-Parisi-Zhang (KPZ) universality class. This has inspired the significantly stronger conjecture that KPZ dynamics, in fact, occur in all integrable spin chains with non-Abelian symmetry. Here, we provide extensive numerical evidence affirming this conjecture. Moreover, we observe that KPZ transport is even more generic, arising in both supersymmetric and periodically driven models. Motivated by recent advances in the realization of SU(N)-symmetric spin models in alkaline-earth-based optical lattice experiments, we propose and analyze a protocol to directly investigate the KPZ scaling function in such systems.

Canonical Density Matrices from Eigenstates of Mixed Systems

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy.

Quantum gas microscope assisted with T-shape vacuum viewports

A quantum gas microscope plays an important role in cold-atom experiments, which provides a high-resolution imaging of the spatial distributions of cold atoms. Here we design, build and calibrate an integrated microscope for quantum gases with all the optical components fixed outside the vacuum chamber. It provides large numerical aperture (NA) of 0.75, as well as good optical access from side for atom loading in cold-atom experiments due to long working distance (7 mm fused silica+6 mm vacuum) of the microscope objective. We make a special design of the vacuum viewport with a T-shape window, to suppress the window flatness distortion introduced by the metal-glass binding process, and protect the high-resolution imaging from distortions due to unflattened window. The achieved Strehl ratio is 0.9204 using scanning-near-field microscopy (SNOM) fiber coupling incoherent light as point light source.

Realizing the symmetry-protected Haldane phase in Fermi-Hubbard ladders

Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain^1,2. Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter^3,4. More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement^5–7. Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.

A ladder-like arrangement of an ultracold gas of lithium atoms trapped in an optical lattice enables the observation of a symmetry-protected topological phase.

Integral fluctuation theorem and generalized Clausius inequality for microcanonical and pure states

Fluctuation theorems are cornerstones of modern statistical mechanics and their standard derivations routinely rely on the crucial assumption of a canonical equilibrium state. Yet rigorous derivations of certain fluctuation theorems for microcanonical states and pure energy eigenstates in isolated quantum systems are still lacking and constitute a major challenge to theory. In this work we tackle this challenge and present such a derivation of an integral fluctuation theorem (IFT) by invoking two central and physically natural conditions, i.e., the so-called "stiffness" and "smoothness" of transition probabilities. Our analytical arguments are additionally substantiated by numerical simulations for archetypal many-body quantum systems, including integrable as well as nonintegrable models of interacting spins and hard-core bosons on a lattice. These simulations strongly suggest that "stiffness" and "smoothness" are indeed of vital importance for the validity of the IFT for microcanonical and pure states. Our work contrasts with recent approaches to the IFT based on Lieb-Robinson speeds and the eigenstate thermalization hypothesis.

Gap Statistics for Confined Particles with Power-Law Interactions

We consider the N particle classical Riesz gas confined in a one-dimensional external harmonic potential with power-law interaction of the form 1/r^{k}, where r is the separation between particles. As special limits it contains several systems such as Dyson's log-gas (k→0^{+}), the Calogero-Moser model (k=2), the 1D one-component plasma (k=-1), and the hard-rod gas (k→∞). Despite its growing importance, only large-N field theory and average density profile are known for general k. In this Letter, we study the fluctuations in the system by looking at the statistics of the gap between successive particles. This quantity is analogous to the well-known level-spacing statistics which is ubiquitous in several branches of physics. We show that the variance goes as N^{-b_{k}} and we find the k dependence of b_{k} via direct Monte Carlo simulations. We provide supporting arguments based on microscopic Hessian calculation and a quadratic field theory approach. We compute the gap distribution and study its system size scaling. Except in the range -1<k<0, we find scaling for all k>-2 with both Gaussian and non-Gaussian scaling forms.

Eigenstate fluctuation theorem in the short- and long-time regimes

The canonical ensemble plays a crucial role in statistical mechanics in and out of equilibrium. For example, the standard derivation of the fluctuation theorem relies on the assumption that the initial state of the heat bath is the canonical ensemble. On the other hand, the recent progress in the foundation of statistical mechanics has revealed that a thermal equilibrium state is not necessarily described by the canonical ensemble but can be a quantum pure state or even a single energy eigenstate, as formulated by the eigenstate thermalization hypothesis (ETH). Then a question raised is how these two pictures, the canonical ensemble and a single energy eigenstate as a thermal equilibrium state, are compatible in the fluctuation theorem. In this paper, we theoretically and numerically show that the fluctuation theorem holds in both of the long- and short-time regimes, even when the initial state of the bath is a single energy eigenstate of a many-body system. Our proof of the fluctuation theorem in the long-time regime is based on the ETH, while it was previously shown in the short-time regime on the basis of the Lieb-Robinson bound and the ETH [Phys. Rev. Lett. 119, 100601 (2017)0031-900710.1103/PhysRevLett.119.100601]. The proofs for these time regimes are theoretically independent and complementary, implying the fluctuation theorem in the entire time domain. We also perform a systematic numerical simulation of hard-core bosons by exact diagonalization and verify the fluctuation theorem in both of the time regimes by focusing on the finite-size scaling. Our results contribute to the understanding of the mechanism that the fluctuation theorem emerges from unitary dynamics of quantum many-body systems and can be tested by experiments with, e.g., ultracold atoms.

Collective P-Wave Orbital Dynamics of Ultracold Fermions

We consider the nonequilibrium orbital dynamics of spin-polarized ultracold fermions in the first excited band of an optical lattice. A specific lattice depth and filling configuration is designed to allow the p_{x} and p_{y} excited orbital degrees of freedom to act as a pseudospin. Starting from the full Hamiltonian for p-wave interactions in a periodic potential, we derive an extended Hubbard-type model that describes the anisotropic lattice dynamics of the excited orbitals at low energy. We then show how dispersion engineering can provide a viable route to realizing collective behavior driven by p-wave interactions. In particular, Bragg dressing and lattice depth can reduce single-particle dispersion rates, such that a collective many-body gap is opened with only moderate Feshbach enhancement of p-wave interactions. Physical insight into the emergent gap-protected collective dynamics is gained by projecting the Hamiltonian into the Dicke manifold, yielding a one-axis twisting model for the orbital pseudospin that can be probed using conventional Ramsey-style interferometry. Experimentally realistic protocols to prepare and measure the many-body dynamics are discussed, including the effects of band relaxation, particle loss, spin-orbit coupling, and doping.

Long distance optical transport of ultracold atoms: A compact setup using a Moiré lens

We present a compact and robust setup to optically transport ultracold atoms over long distances. Using a focus-tunable moiré lens that has recently appeared in the market, we demonstrate transport of up to a distance of 465 mm. A transfer efficiency of 70% is achieved with a negligible temperature change at 11 μK. With its high thermal stability and low astigmatism, the moiré lens is superior to fluid-based varifocal lenses. It is much more compact and stable than a lens mounted on a linear translation stage, allowing for simplified experimental setups.

Counting statistics for noninteracting fermions in a d-dimensional potential

We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.

Dominant Fifth-Order Correlations in Doped Quantum Antiferromagnets

Traditionally, one- and two-point correlation functions are used to characterize many-body systems. In strongly correlated quantum materials, such as the doped 2D Fermi-Hubbard system, these may no longer be sufficient, because higher-order correlations are crucial to understanding the character of the many-body system and can be numerically dominant. Experimentally, such higher-order correlations have recently become accessible in ultracold atom systems. Here, we reveal strong non-Gaussian correlations in doped quantum antiferromagnets and show that higher-order correlations dominate over lower-order terms. We study a single mobile hole in the t-J model using the density matrix renormalization group and reveal genuine fifth-order correlations which are directly related to the mobility of the dopant. We contrast our results to predictions using models based on doped quantum spin liquids which feature significantly reduced higher-order correlations. Our predictions can be tested at the lowest currently accessible temperatures in quantum simulators of the 2D Fermi-Hubbard model. Finally, we propose to experimentally study the same fifth-order spin-charge correlations as a function of doping. This will help to reveal the microscopic nature of charge carriers in the most debated regime of the Hubbard model, relevant for understanding high-T_{c} superconductivity.

Versatile objectives with NA = 0.55 and NA = 0.78 for cold-atom experiments

We present two sets of versatile high-numerical-apeture objectives suitable for various cold-atom experiments. The objectives are assembled entirely by the commercial on-shelf singlets. The two objectives are initially optimized at working wavelength of 852 nm with a standard 5-mm silica optical flat window. They have numerical apertures of NA=0.55 and NA=0.78, working distances of 23 and 12.8 mm, diffraction-limited fields of view of 98 and 15 μm, and spatial resolutions of 0.94 and 0.67 μm, respectively. These performances are simulated by the ray-tracing software and experimentally confirmed by imaging line patterns and a point-like emitter on a resolution chart. The two objectives can be further reoptimized at any single wavelengths from ultraviolet to near infrared and for various optical flat window with different thickness by only tuning one of lens spacing. The two objectives provide convenient and flexible options to observe and address individual atoms in single atom arrays or optical lattices for various cold-atom experiments.

Doublon-Hole Correlations and Fluctuation Thermometry in a Fermi-Hubbard Gas

We report on the single atom and single site-resolved detection of the total density in a cold atom realization of the 2D Fermi-Hubbard model. Fluorescence imaging of doublons is achieved by splitting each lattice site into a double well, thereby separating atom pairs. Full density readout yields a direct measurement of the equation of state, including direct thermometry via the fluctuation-dissipation theorem. Site-resolved density correlations reveal the Pauli hole at low filling, and strong doublon-hole correlations near half filling. These are shown to account for the difference between local and nonlocal density fluctuations in the Mott insulator. Our technique enables the study of atom-resolved charge transport in the Fermi-Hubbard model, the site-resolved observation of molecules, and the creation of bilayer Fermi-Hubbard systems.

Spinless Fulde-Ferrell superfluid in Haldane model with nearest-neighbor interaction

Fulde-Ferrell (FF) state is an exotic superconducting or superfluid state, characterized by Cooper pairs with a finite center-of-mass momentum, and it has been intensively investigated in spinful systems, but the spin population imbalance is usually necessary. Here we propose a model to realize and modulate the spinless FF superfluid, namely, Haldane model with nearest-neighbor interaction. The FF state appears when the space-inversion and time-reversal symmetries are broken simultaneously, and there exists the first order phase transition. A higher Berezinskii-Kosterlitz-Thouless transition temperature is obtained at larger absolute value of Peierls phase and smaller energy offset of the sublattices. Our proposal provides a feasible option for experimental researchers to detect the FF superfluid.

High-resolution imaging of Rydberg atoms in optical lattices using an aspheric-lens objective in vacuum

We present a high-resolution, simple, and versatile system for imaging ultracold Rydberg atoms in optical lattices. The imaging objective is a single aspheric lens [with a working distance of 20.6 mm and a numerical aperture (NA) of 0.51] placed inside the vacuum chamber. Adopting a large-working-distance lens leaves room for electrodes and electrostatic shields to control electric fields around Rydberg atoms. With this setup, we achieve a Rayleigh resolution of 1.10 μm or 1.41λ (λ = 780 nm), limited by the NA of the aspheric lens. For systems of highly excited Rydberg states with blockade radii greater than a few μm, the resolution achieved is sufficient for studying many physical processes of interest.

Noncommuting conserved charges in quantum many-body thermalization

In statistical mechanics, a small system exchanges conserved quantities-heat, particles, electric charge, etc.-with a bath. The small system thermalizes to the canonical ensemble or the grand canonical ensemble, etc., depending on the quantities. The conserved quantities are represented by operators usually assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small system's long-time state was dubbed "the non-Abelian thermal state (NATS)." We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which forms the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to near the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

Dynamical Sign Reversal of Magnetic Correlations in Dissipative Hubbard Models

In quantum magnetism, the virtual exchange of particles mediates an interaction between spins. Here, we show that an inelastic Hubbard interaction fundamentally alters the magnetism of the Hubbard model due to dissipation in spin-exchange processes, leading to sign reversal of magnetic correlations in dissipative quantum dynamics. This mechanism is applicable to both fermionic and bosonic Mott insulators, and can naturally be realized with ultracold atoms undergoing two-body inelastic collisions. The dynamical reversal of magnetic correlations can be detected by using a double-well optical lattice or quantum-gas microscopy, the latter of which facilitates the detection of the magnetic correlations in one-dimensional systems because of spin-charge separation. Our results open a new avenue toward controlling quantum magnetism by dissipation.

String patterns in the doped Hubbard model

Understanding strongly correlated quantum many-body states is one of the most difficult challenges in modern physics. For example, there remain fundamental open questions on the phase diagram of the Hubbard model, which describes strongly correlated electrons in solids. In this work, we realize the Hubbard Hamiltonian and search for specific patterns within the individual images of many realizations of strongly correlated ultracold fermions in an optical lattice. Upon doping a cold-atom antiferromagnet, we find consistency with geometric strings, entities that may explain the relationship between hole motion and spin order, in both pattern-based and conventional observables. Our results demonstrate the potential for pattern recognition to provide key insights into cold-atom quantum many-body systems.

Ultrafast dynamics of strongly correlated fermions-nonequilibrium Green functions and selfenergy approximations

This article presents an overview on recent progress in the theory of nonequilibrium Green functions (NEGF). We discuss applications of NEGF simulations to describe the femtosecond dynamics of various finite fermionic systems following an excitation out of equilibrium. This includes the expansion dynamics of ultracold atoms in optical lattices following a confinement quench and the excitation of strongly correlated electrons in a solid by the impact of a charged particle. NEGF, presently, are the only ab initio quantum approach that is able to study the dynamics of correlations for long times in two and three dimensions. However, until recently, NEGF simulations have mostly been performed with rather simple selfenergy approximations such as the second-order Born approximation (SOA). While they correctly capture the qualitative trends of the relaxation towards equilibrium, the reliability and accuracy of these NEGF simulations has remained open, for a long time. Here we report on recent tests of NEGF simulations for finite lattice systems against exact-diagonalization and density-matrix-renormalization-group benchmark data. The results confirm the high accuracy and predictive capability of NEGF simulations-provided selfenergies are used that go beyond the SOA and adequately include strong correlation and dynamical-screening effects. With an extended arsenal of selfenergies that can be used effectively, the NEGF approach has the potential of becoming a powerful simulation tool with broad areas of new applications including strongly correlated solids and ultracold atoms. The present review aims at making such applications possible. To this end we present a selfcontained introduction to the theory of NEGF and give an overview on recent numerical applications to compute the ultrafast relaxation dynamics of correlated fermions. In the second part we give a detailed introduction to selfenergies beyond the SOA. Important examples are the third-order approximation, the [Formula: see text] approximation, the T-matrix approximation and the fluctuating-exchange approximation. We give a comprehensive summary of the explicit selfenergy expressions for a variety of systems of practical relevance, starting from the most general expressions (general basis) and the Feynman diagrams, and including also the important cases of diagonal basis sets, the Hubbard model and the differences occuring for bosons and fermions. With these details, and information on the computational effort and scaling with the basis size and propagation duration, readers will be able to choose the proper basis set and straightforwardly implement and apply advanced selfenergy approximations to a broad class of systems.

An investigation of pre-crystalline order, ruling out Pauli crystals and introducing Pauli anti-crystals

Fluid states of matter can locally exhibit characteristics of the onset of crystalline order. Traditionally this has been theoretically investigated using multipoint correlation functions. However new measurement techniques now allow multiparticle configurations of cold atomic systems to be observed directly. This has led to a search for new techniques to characterize the configurations that are likely to be observed. One of these techniques is the configuration density (CD), which has been used to argue for the formation of “Pauli crystals” by non-interacting electrons in e.g. a harmonic trap. We show here that such Pauli crystals do not exist, but that other other interesting spatial structures can occur in the form of an “anti-Crystal”, where the fermions preferentially avoid a lattice of positions surrounding any given fermion. Further, we show that configuration densities must be treated with great care as naive application can lead to the identification of crystalline structures which are artifacts of the method and of no physical significance. We analyze the failure of the CD and suggest methods that might be more suitable for characterizing multiparticle correlations which may signal the onset of crystalline order. In particular, we introduce neighbour counting statistics (NCS), which is the full counting statistics of the particle number in a neighborhood of a given particle. We test this on two dimensional systems with emerging triangular and square crystal structures.

Quantum Computation Toolbox for Decoherence-Free Qubits Using Multi-Band Alkali Atoms

Protocols for designing and manipulating qubits with ultracold alkali atoms in 3D optical lattices are introduced. These qubits are formed from two-atom spin superposition states that create a decoherence-free subspace immune to stray magnetic fields, dramatically improving coherence times while still enjoying the single-site addressability and Feshbach resonance control of state-of-the-art alkali atom systems. The protocol requires no continuous driving or spin-dependent potentials, and instead relies upon the population of a higher motional band to realize naturally tunable in-site exchange and cross-site superexchange interactions. As a proof-of-principle example of their utility for entanglement generation for quantum computation, it is shown that the cross-site superexchange interactions can be used to engineer 1D cluster states. Explicit protocols for experimental preparation and manipulation of the qubits are also discussed, as well as methods for measuring more complex quantities such as out-of-time-ordered correlation functions (OTOCs).

Quantum Adiabatic Doping with Incommensurate Optical Lattices

Quantum simulations of Fermi-Hubbard models have been attracting considerable effort in the optical lattice research, with the ultracold antiferromagnetic atomic phase reached at half filling in recent years. An unresolved issue is to dope the system while maintaining the low thermal entropy. Here we propose to achieve the low temperature phase of the doped Fermi-Hubbard model using incommensurate optical lattices through adiabatic quantum evolution. In this theoretical proposal, we find that one major problem about the adiabatic doping is atomic localization in the incommensurate lattice, potentially causing an exponential slowing down of the adiabatic procedure. We study both one- and two-dimensional incommensurate optical lattices, and find that the localization prevents efficient adiabatic doping in the strong lattice regime for both cases. With density matrix renormalization group calculation, we further show that the slowing down problem in one dimension can be circumvented by considering interaction induced many-body delocalization, which is experimentally feasible using Feshbach resonance techniques. This protocol is expected to be efficient as well in two dimensions where the localization phenomenon is less stable.

High-power, frequency-doubled Nd:GdVO4 laser for use in lithium cold atom experiments

We report on an 888 nm-pumped Nd:GdVO₄ ring laser operational over a wavelength range from 1340.3 nm to 1342.1 nm with a maximum output power of 7.4 W at 1341.2 nm and a beam quality parameter M²<1.1. To our knowledge this is the highest single-longitudinal-mode power obtained with a Nd:GdVO₄ crystal laser. We use a commercial frequency-doubling cavity to achieve 1.2 W at 671.0 nm and 4.0 W at 670.6 nm for use in lithium cold atom experiments. Respectively, these wavelengths are approximately resonant with and 250 GHz blue-detuned from the lithium D-lines. Thus, this source provides ample power for laser cooling of lithium atoms while also offering substantial power for experiments requiring light 10's to 100's of GHz blue-detuned from the primary lithium transitions.

Exact Localized and Ballistic Eigenstates in Disordered Chaotic Spin Ladders and the Fermi-Hubbard Model

We demonstrate the existence of exact atypical many-body eigenstates in a class of disordered, interacting one-dimensional quantum systems that includes the Fermi-Hubbard model as a special case. These atypical eigenstates, which generically have finite energy density and are exponentially many in number, are populated by noninteracting excitations. They can exhibit Anderson localization with area-law eigenstate entanglement or, surprisingly, ballistic transport at any disorder strength. These properties differ strikingly from those of typical eigenstates nearby in energy, which we show give rise to diffusive transport as expected in a chaotic quantum system. We discuss how to observe these atypical eigenstates in cold-atom experiments realizing the Fermi-Hubbard model, and comment on the robustness of their properties.

An adaptable two-lens high-resolution objective for single-site resolved imaging of atoms in optical lattices

In this paper, we present a high-resolution, simple, and versatile imaging system for single-site resolved imaging of atoms in optical lattices. The system, which relies on an adaptable infinite conjugate two-lens design, has a numerical aperture of 0.52, which can in the ideal case be further extended to 0.57. It is optimized for imaging on the sodium D₂-line but allows us to tune the objective's diffraction limited performance between 400 nm and 1000 nm by changing the distance between the two lenses. Furthermore, the objective is designed to be integrated into a typical atomic physics vacuum apparatus where the operating distance can be large (>20 mm) and diffraction limited performance still needs to be achieved when imaging through thick vacuum windows (6 mm to 10 mm). Imaging gold nanoparticles, using a wavelength of 589 nm which corresponds to the D₂-line of sodium atoms, we measure diffraction limited performance and a resolution corresponding to an Airy radius of less than 0.7 µm, enabling potential single-site resolution in the commonly used 532 nm optical lattice spacing.

Cluster State Generation with Spin-Orbit Coupled Fermionic Atoms in Optical Lattices

Measurement-based quantum computation, an alternative paradigm for quantum information processing, uses simple measurements on qubits prepared in cluster states, a class of multiparty entangled states with useful properties. Here we propose and analyze a scheme that takes advantage of the interplay between spin-orbit coupling and superexchange interactions, in the presence of a coherent drive, to deterministically generate macroscopic arrays of cluster states in fermionic alkaline earth atoms trapped in three dimensional (3D) optical lattices. The scheme dynamically generates cluster states without the need of engineered transport, and is robust in the presence of holes, a typical imperfection in cold atom Mott insulators. The protocol is of particular relevance for the new generation of 3D optical lattice clocks with coherence times >10  s, 2 orders of magnitude larger than the cluster state generation time. We propose the use of collective measurements and time reversal of the Hamiltonian to benchmark the underlying Ising model dynamics and the generated many-body correlations.

Conductivity Spectrum of Ultracold Atoms in an Optical Lattice

We measure the conductivity of neutral fermions in a cubic optical lattice. Using in situ fluorescence microscopy, we observe the alternating current resultant from a single-frequency uniform force applied by displacement of a weak harmonic trapping potential. In the linear response regime, a neutral-particle analog of Ohm's law gives the conductivity as the ratio of total current to force. For various lattice depths, temperatures, interaction strengths, and fillings, we measure both real and imaginary conductivity, up to a frequency sufficient to capture the transport dynamics within the lowest band. The spectral width of the real conductivity reveals the current dissipation rate in the lattice, and the integrated spectral weight is related to thermodynamic properties of the system through a sum rule. The global conductivity decreases with increased band-averaged effective mass, which at high temperatures approaches a T-linear regime. Relaxation of current is observed to require a finite lattice depth, which breaks Galilean invariance and enables damping through collisions between fermions.

Narrow-Line Cooling and Imaging of Ytterbium Atoms in an Optical Tweezer Array

Engineering controllable, strongly interacting many-body quantum systems is at the frontier of quantum simulation and quantum information processing. Arrays of laser-cooled neutral atoms in optical tweezers have emerged as a promising platform because of their flexibility and the potential for strong interactions via Rydberg states. Existing neutral atom array experiments utilize alkali atoms, but alkaline-earth atoms offer many advantages in terms of coherence and control, and also open the door to new applications in precision measurement and time keeping. In this Letter, we present a technique to trap individual alkaline-earth-like ytterbium (Yb) atoms in optical tweezer arrays. The narrow ^{1}S_{0}-^{3}P_{1} intercombination line is used for both cooling and imaging in a magic-wavelength optical tweezer at 532 nm. The low Doppler temperature allows for imaging near the saturation intensity, resulting in a very high atom detection fidelity. We demonstrate the imaging fidelity concretely by observing rare (<1 in 10^{4} images) spontaneous quantum jumps into and out of a metastable state. We also demonstrate stochastic loading of atoms into a two-dimensional, 144-site tweezer array. This platform will enable advances in quantum information processing, quantum simulation, and precision measurement. The demonstrated narrow-line Doppler imaging may also be applied in tweezer arrays or quantum gas microscopes using other atoms with similar transitions, such as erbium and dysprosium.

Site-Specific and Coherent Manipulation of Individual Qubits in a 1D Optical Lattice with a 532-nm Site Separation

We demonstrate gate operations on a single qubit at a specific site without perturbing the coherence of an adjacent qubit in a 1D optical lattice when the site separation is only 532 nm. Three types of spin rotations are performed on the target qubit with fidelities between 0.88±0.05 and 0.99±0.01, whereas the superposition state of the adjacent one is preserved with fidelities between 0.93±0.04 and 0.97±0.04. The qubit is realized by a pair of Zeeman-sensitive ground hyperfine states of a ^{7}Li atom, and each site is identified by its resonance frequency in a magnetic field gradient of 1.6  G/cm. We achieve the site-specific resolving power in the frequency domain by using magic polarization for the lattice beam that allows a Fourier-limited transition linewidth as well as by highly stabilizing the lattice parameters and the ambient conditions. We also discuss a two-atom entanglement scheme using a blockade by cold collisional shifts in a 1D superlattice, for which a coherent manipulation of individual qubits is a prerequisite.

Implementation of a stable, high-power optical lattice for quantum gas microscopy

We describe the design and implementation of a stable high-power 1064 nm laser system to generate optical lattices for experiments with ultracold quantum gases. The system is based on a low-noise laser amplified by an array of four heavily modified, high-power fiber amplifiers. The beam intensity is stabilized and controlled with a nonlinear feedback loop. Using real-time monitoring of the resulting optical lattice, we find the stability of the lattice site positions to be well below the lattice spacing over the course of hours. The position of the harmonic trap produced by the Gaussian envelope of the lattice beams is stable to about one lattice spacing and the long-term (six-month) relative root-mean-square stability of the lattice spacing itself is 0.5%.

Measuring the Single-Particle Density Matrix for Fermions and Hard-Core Bosons in an Optical Lattice

Ultracold atoms in optical lattices provide clean, tunable, and well-isolated realizations of paradigmatic quantum lattice models. With the recent advent of quantum-gas microscopes, they now also offer the possibility to measure the occupations of individual lattice sites. What, however, has not yet been achieved is to measure those elements of the single-particle density matrix, which are off- diagonal in the occupation basis. Here, we propose a scheme to access these basic quantities both for fermions as well as hard-core bosons and investigate its accuracy and feasibility. The scheme relies on the engineering of a large effective tunnel coupling between distant lattice sites and a protocol that is based on measuring site occupations after two subsequent quenches.

30 W, sub-kHz frequency-locked laser at 532 nm

We report on the realization of a high-power, ultranarrow-linewidth, and frequency-locked 532 nm laser system. The laser system consists of single-pass and intra-cavity second harmonic generation of a continuous-wave Ytterbium doped fiber laser at 1064 nm in the nonlinear crystal of periodically poled lithium niobate and lithium triborate, respectively. With 47 W infrared input, 30 W green laser is generated through the type I critical phase matching in the intracavity lithium triborate crystal. The laser linewidth is measured to be on the order of sub-kHz, which is achieved by simultaneously locking the single-pass frequency doubling output onto the iodine absorption line R69 (36-1) at 532 nm. Furthermore, the phase locking between the laser system and another slave 1064 nm laser is demonstrated with relative frequency tunability being up to 10 GHz. Our results completely satisfy the requirements of 532 nm laser for quantum simulation with ultracold atoms.

Spin transport in a Mott insulator of ultracold fermions

Strongly correlated materials are expected to feature unconventional transport properties, such that charge, spin, and heat conduction are potentially independent probes of the dynamics. In contrast to charge transport, the measurement of spin transport in such materials is highly challenging. We observed spin conduction and diffusion in a system of ultracold fermionic atoms that realizes the half-filled Fermi-Hubbard model. For strong interactions, spin diffusion is driven by super-exchange and doublon-hole–assisted tunneling, and strongly violates the quantum limit of charge diffusion. The technique developed in this work can be extended to finite doping, which can shed light on the complex interplay between spin and charge in the Hubbard model.

Λ-Enhanced Imaging of Molecules in an Optical Trap

We report on nondestructive imaging of optically trapped calcium monofluoride molecules using in situ Λ-enhanced gray molasses cooling. 200 times more fluorescence is obtained compared to destructive on-resonance imaging, and the trapped molecules remain at a temperature of 20  μK. The achieved number of scattered photons makes possible nondestructive single-shot detection of single molecules with high fidelity.

Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps

We compute the joint statistics of the momenta p_{i} of N noninteracting fermions in a trap, near the Fermi edge, with a particular focus on the largest one p_{max}. For a 1D harmonic trap, momenta and positions play a symmetric role, and hence the joint statistics of momenta are identical to that of the positions. In particular, p_{max}, as x_{max}, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e., for "flat traps" near their minimum, with V(x)∼x^{2n} and n>1. These are based on generalizations of the Airy kernel that we obtain explicitly. The fluctuations of p_{max} are governed by new universal distributions determined from the nth member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.

Floquet Engineering of Optical Solenoids and Quantized Charge Pumping along Tailored Paths in Two-Dimensional Chern Insulators

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 as the controlled population of edge modes.

Quantum State Engineering of a Hubbard System with Ultracold Fermions

Accessing new regimes in quantum simulation requires the development of new techniques for quantum state preparation. We demonstrate the quantum state engineering of a strongly correlated many-body state of the two-component repulsive Fermi-Hubbard model on a square lattice. Our scheme makes use of an ultralow entropy doublon band insulator created through entropy redistribution. After isolating the band insulator, we change the underlying potential to expand it into a half-filled system. The final many-body state realized shows strong antiferromagnetic correlations and a temperature below the exchange energy. We observe an increase in entropy, which we find is likely caused by the many-body physics in the last step of the scheme. This technique is promising for low-temperature studies of cold-atom-based lattice models.

Wavelength-limited optical accordion

We demonstrate a method to create dynamic optical lattices with lattice constant tunable down to the optical wavelength limit. The periodicity of 1D lattice is to be adjusted by rotating the incoming direction of one of the two interfering laser beams with its fiber port. The relative phase between the stationary and rotating lasers are stabilized with a heterodyne phase-lock loop (Ma et al, Opt. Lett. 19, 1777, 1994), by reflecting part of the rotating laser beam back from a cylindrical mirror near the experiment. Our preliminary demonstration shows tuning of lattice constant λ2sinθ/2, limited by our imaging resolution, between θ = 3° and 20°, with stable and tunable phase. The results can be extended to achieve lattice constant tuning range from ∼ 10λ down to λ/2. We discuss extension of the demonstrated scheme for improved vibration suppression, and for lattice utilizing broadband lasers. Finally we propose a 2D accordion lattice design for quantum gas experiments.

Full-Counting Many-Particle Dynamics: Nonlocal and Chiral Propagation of Correlations

The ability to measure single quanta allows the complete characterization of small quantum systems known as full-counting statistics. Quantum gas microscopy enables one to observe many-body systems at the single-atom precision. We extend the idea of full-counting statistics to nonequilibrium open many-particle dynamics and apply it to discuss the quench dynamics. By way of illustration, we consider an exactly solvable model to demonstrate the emergence of unique phenomena such as nonlocal and chiral propagation of correlations, leading to a concomitant oscillatory entanglement growth. We find that correlations can propagate beyond the conventional maximal speed, known as the Lieb-Robinson bound, at the cost of probabilistic nature of quantum measurement. These features become most prominent at the real-to-complex spectrum transition point of an underlying parity-time-symmetric effective non-Hermitian Hamiltonian. A possible experimental situation with quantum gas microscopy is discussed.

The Fulde-Ferrell-Larkin-Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review

We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.

Quantum simulations with ultracold atoms in optical lattices

Quantum simulation, a subdiscipline of quantum computation, can provide valuable insight into difficult quantum problems in physics or chemistry. Ultracold atoms in optical lattices represent an ideal platform for simulations of quantum many-body problems. Within this setting, quantum gas microscopes enable single atom observation and manipulation in large samples. Ultracold atom-based quantum simulators have already been used to probe quantum magnetism, to realize and detect topological quantum matter, and to study quantum systems with controlled long-range interactions. Experiments on many-body systems out of equilibrium have also provided results in regimes unavailable to the most advanced supercomputers. We review recent experimental progress in this field and comment on future directions.

Λ-enhanced grey molasses on the D2 transition of Rubidium-87 atoms

Laser cooling based on dark states, i.e. states decoupled from light, has proven to be effective to increase the phase-space density of cold trapped atoms. Dark-states cooling requires open atomic transitions, in contrast to the ordinary laser cooling used for example in magneto-optical traps (MOTs), which operate on closed atomic transitions. For alkali atoms, dark-states cooling is therefore commonly operated on the D₁ transition nS_1/2 → nP_1/2. We show that, for ⁸⁷Rb, thanks to the large hyperfine structure separations the use of this transition is not strictly necessary and that “quasi-dark state” cooling is efficient also on the D₂ line, 5S_1/2 → 5P_3/2. We report temperatures as low as (4.0 ± 0.3) μK and an increase of almost an order of magnitude in the phase space density with respect to ordinary laser sub-Doppler cooling.

An energy-resolved atomic scanning probe

We propose a method to probe the local density of states (LDOS) of atomic systems that provides both spatial and energy resolution. The method combines atomic and tunneling techniques to supply a simple, yet quantitative and operational, definition of the LDOS for both interacting and non-interacting systems: It is the rate at which particles can be siphoned from the system of interest by a narrow energy band of non-interacting states contacted locally to the many-body system of interest. Ultracold atoms in optical lattices are a natural platform for implementing this broad concept to visualize the energy and spatial dependence of the atom density in interacting, inhomogeneous lattices. This includes models of strongly correlated condensed matter systems, as well as ones with non-trivial topologies.

Ultracold Atoms in a Square Lattice with Spin-Orbit Coupling: Charge Order, Superfluidity, and Topological Signatures

We present an ab initio, numerically exact study of attractive fermions in square lattices with Rashba spin-orbit coupling. The ground state of this system is a supersolid, with coexisting charge and superfluid order. The superfluid is composed of both singlet and triplet pairs induced by spin-orbit coupling. We perform large-scale calculations using the auxiliary-field quantum Monte Carlo method to provide the first full, quantitative description of the charge, spin, and pairing properties of the system. In addition to characterizing the exotic physics, our results will serve as essential high-accuracy benchmarks for the intense theoretical and especially experimental efforts in ultracold atoms to realize and understand an expanding variety of quantum Hall and topological superconductor systems.

On the observability of Pauli crystals in experiments with ultracold trapped Fermi gases

The best known manifestation of the Fermi-Dirac statistics is the Pauli exclusion principle: no two identical fermions can occupy the same one-particle state. This principle enforces high-order correlations in systems of many identical fermions and is responsible for a particular geometric arrangement of trapped particles even when all mutual interactions are absent. These geometric structures, called Pauli crystals, are predicted for a system of N identical atoms trapped in a harmonic potential. They emerge as the most frequent configurations in a collection of single-shot pictures of the system. Here we study how fragile Pauli crystals are when realistic experimental limitations are taken into account. The influence of the number of single-shots pictures available to analysis, thermal fluctuations and finite efficiency of detection are considered. The role of these sources of noise on the possibility of experimental observation of Pauli crystals is shown and conditions necessary for the detection of the geometrical arrangements of particles are identified.

Fast Nondestructive Parallel Readout of Neutral Atom Registers in Optical Potentials

We demonstrate the parallel and nondestructive readout of the hyperfine state for optically trapped ^{87}Rb atoms. The scheme is based on state-selective fluorescence imaging and achieves detection fidelities >98% within 10 ms, while keeping 99% of the atoms trapped. For the readout of dense arrays of neutral atoms in optical lattices, where the fluorescence images of neighboring atoms overlap, we apply a novel image analysis technique using Bayesian inference to determine the internal state of multiple atoms. Our method is scalable to large neutral atom registers relevant for future quantum information processing tasks requiring fast and nondestructive readout and can also be used for the simultaneous readout of quantum information stored in internal qubit states and in the atoms' positions.