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Quantum systems with SU($N$) symmetry are paradigmatic settings for quantum many-body physics. They have been studied for the insights they provide into complex materials and their ability to stabilize exotic ground states. Ultracold alkaline-earth atoms were predicted to exhibit SU($N$) symmetry for $N=2I+1=1,2,\ldots,10$, where $I$ is the nuclear spin. Subsequent experiments have revealed rich many-body physics. However, alkaline-earth atoms realize this symmetry only for fermions with repulsive interactions. In this paper, we predict that ultracold molecules shielded from destructive collisions with static electric fields or microwaves exhibit SU($N$) symmetry, which holds because deviations of the s-wave scattering length from the spin-free values are only about 3\% for CaF with static-field shielding and are estimated to be even smaller for bialkali molecules. They open the door to $N$ as large as $32$ for bosons and $36$ for fermions. They offer important features unachievable with atoms, including bosonic systems and attractive interactions., Comment: 11 pages, 8 figures, 2 tables

The SAT problem is a prototypical NP-complete problem of fundamental importance in computational complexity theory with many applications in science and engineering; as such, it has long served as an essential benchmark for classical and quantum algorithms. This study shows numerical evidence for a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random sampling for finding all solutions to 3-SAT problems (All-SAT). G-QAOA is less resource-intensive and more adaptable for 3-SAT and Max-SAT than Grover's algorithm, and it surpasses conventional QAOA in its ability to sample all solutions. We show these benefits by classical simulations of many-round G-QAOA on thousands of random 3-SAT instances. We also observe G-QAOA advantages on the IonQ Aria quantum computer for small instances, finding that current hardware suffices to determine and sample all solutions. Interestingly, a single-angle-pair constraint that uses the same pair of angles at each G-QAOA round greatly reduces the classical computational overhead of optimizing the G-QAOA angles while preserving its quadratic speedup. We also find parameter clustering of the angles. The single-angle-pair protocol and parameter clustering significantly reduce obstacles to classical optimization of the G-QAOA angles.

Ultracold molecules confined in optical lattices or tweezer traps can be used to process quantum information and simulate the behaviour of many-body quantum systems. Molecules offer several advantages for these applications. They have a large set of stable states with strong transitions between them and long coherence times. They can be prepared in a chosen state with high fidelity, and the state populations can be measured efficiently. They have controllable long-range dipole-dipole interactions that can be used to entangle pairs of molecules and generate interesting many-body states. We review the advances that have been made and the challenges still to overcome, and describe the new ideas that will unlock the full potential of the field., Comment: 15 pages, 4 figures. Comments welcome

Two-level atoms coupled with single-mode cavity photons are predicted to exhibit a quantum phase transition when the coupling strength exceeds a critical value, entering a phase in which atomic polarization and photonic field are finite even at zero temperature and without external driving. However, this phenomenon, the superradiant phase transition (SRPT), is forbidden by a no-go theorem due to the existence of the diamagnetic term in the Hamiltonian. Here, we present spectroscopic evidence for a magnonic SRPT in ErFeO$_3$, where the role of the photonic mode (two-level atoms) in the photonic SRPT is played by an Fe$^{3+}$ magnon mode (Er$^{3+}$ spins). The absence of the diamagnetic term in the Fe$^{3+}$-Er$^{3+}$ exchange coupling ensures that the no-go theorem does not apply. Terahertz and gigahertz magnetospectroscopy experiments revealed the signatures of the SRPT -- a kink and a softening, respectively, of two spin-magnon hybridized modes at the critical point.

The physics of doped Mott insulators is at the heart of strongly correlated materials and is believed to constitute an essential ingredient for high-temperature superconductivity. In systems with higher SU(N) spin symmetries, even richer magnetic ground states appear at a filling of one particle per site compared to the case of SU(2) spins, but their fate upon doping remains largely unexplored. Here we address this question by studying a single hole in the SU(3) $t$-$J$ model, whose undoped ground state features long-range, diagonal spin stripes. By analyzing both ground state and dynamical properties utilizing the density matrix renormalization group, we establish the appearence of magnetic polarons consisting of chargons and flavor defects, whose dynamics is constrained to a single effective dimension along the ordered diagonal. We semi-analytically describe the system using geometric string theory, where paths of hole motion are the fundamental degrees of freedom. With recent advances in the realization and control of SU(N) Fermi-Hubbard models with ultracold atoms in optical lattices, our results can directly be observed in quantum gas microscopes with single-site resolution. Our work suggests the appearance of intricate ground states at finite doping constituted by emergent, coupled Luttinger liquids along diagonals, and is a first step towards exploring a wealth of physics in doped SU(N) Fermi-Hubbard models on various geometries., Comment: 5 + 5 pages

It is commonly believed that there are only two types of particle exchange statistics in quantum mechanics, fermions and bosons, with the exception of anyons in two dimension. In principle, a second exception known as parastatistics, which extends outside of two dimensions, has been considered but was believed to be physically equivalent to fermions and bosons. In this paper we show that nontrivial parastatistics inequivalent to either fermions or bosons can exist in physical systems. These new types of identical particles obey generalized exclusion principles, leading to exotic free-particle thermodynamics distinct from any system of free fermions and bosons. We formulate our theory by developing a second quantization of paraparticles, which naturally includes exactly solvable non-interacting theories, and incorporates physical constraints such as locality. We then construct a family of exactly solvable quantum spin models in one and two dimensions where free paraparticles emerge as quasiparticle excitations, and their exchange statistics can be physically observed and is notably distinct from fermions and bosons. This demonstrates the possibility of a new type of quasiparticle in condensed matter systems, and, more speculatively, the potential for previously unconsidered types of elementary particles., Comment: 27 pages, Mathematica codes accompanying this paper are available at https://github.com/lagrenge94/Mathematica-codes-for-parastatistics

Synthetic dimension platforms offer unique pathways for engineering quantum matter. We compute the phase diagram of a many-body system of ultracold atoms (or polar molecules) with a set of Rydberg states (or rotational states) as a synthetic dimension, where the particles are arranged in real space in optical microtrap arrays and interact via dipole-dipole exchange interaction. Using mean-field theory, we find three ordered phases - two are localized in the synthetic dimension, predicted as zero-temperature ground states in Refs. [Sci. Rep., 8, 1 (2018) and Phys. Rev. A 99, 013624 (2019)], and a delocalized phase. We characterize them by identifying the spontaneously broken discrete symmetries of the Hamiltonian. We also compute the phase diagram as a function of temperature and interaction strength, for both signs of the interaction. For system sizes with more than six synthetic sites and attractive interactions, we find that the thermal phase transitions can be first or second order, which leads to a tri-critical point on the phase boundary. By examining the dependence of the tri-critical point and other special points of the phase boundary on the synthetic dimension size, we shed light on the physics for thermodynamically large synthetic dimension., Comment: 9 pages, 3 figures

We use state-of-the-art numerical techniques to compute ground state correlations in the two-dimensional SU(3) Fermi Hubbard model at $1/3$-filling, modeling fermions with three possible spin flavors moving on a square lattice with an average of one particle per site. We find clear evidence of a quantum critical point separating a non-magnetic uniform metallic phase from a regime where long-range `spin' order is present. In particular, there are multiple successive transitions to states with regular, long-range alternation of the different flavors, whose symmetry changes as the interaction strength increases. In addition to the rich quantum magnetism, this important physical system allows one to study integer filling and the associated Mott transition disentangled from nesting, in contrast to the usual SU(2) model. Our results also provide a significant step towards the interpretation of present and future experiments on fermionic alkaline-earth atoms, and other realizations of SU($N$) physics.

We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice at $1/3$-filling using numerically exact determinant quantum Monte Carlo (DQMC) and numerical linked-cluster expansion (NLCE) techniques. We present the different regimes of the model in the $T-U$ plane, which are characterized by local and short-range correlations, and capture signatures of the metal-insulator transition and magnetic crossovers. These signatures are detected as the temperature scales characterizing the rise of the compressibility, and an interaction-dependent change in the sign of the diagonal spin-spin correlation function. The analysis of the compressibility estimates the location of the metal-insulator quantum critical point at $U_c/t \sim 6$, and provides a temperature scale for observing Mott physics at finite-$T$. Furthermore, from the analysis of the spin-spin correlation function we observe that for $U/t \gtrsim6$ and $T \sim J = 4t^2/U$ there is a development of a short-range two sublattice (2-SL) antiferromagnetic structure, as well as an emerging three sublattice (3-SL) antiferromagnetic structure as the temperature is lowered below $T/J \lesssim 0.57$. This crossover from 2-SL to 3-SL magnetic ordering agrees with Heisenberg limit predictions, and has observable effects on the density of on-site pairs. Finally, we describe how the features of the regimes in the $T$-$U$ plane can be explored with alkaline-earth-like atoms in optical lattices with currently-achieved experimental techniques and temperatures. The results discussed in this manuscript provide a starting point for the exploration of the SU(3) FHM upon doping.

Novel geometries can be created by coupling internal states of atoms or molecules to mimic movement in real-space, Comment: An early draft of Physics Today 76(4), 62 (2023). Please refer to the published paper for a significantly improved (edited and formatted) version. A free pdf is available at https://kaden.rice.edu/p2023-2.pdf

The experimental realization of Fermi-Hubbard tweezer arrays opens a new stage for engineering fermionic matter, where programmable lattice geometries and Hubbard model parameters are combined with single-site imaging. In order to use these versatile experimental Fermi-Hubbard models as quantum simulators, it is crucial to know the Hubbard parameters describing them. Here we develop methods to calculate the Hubbard model parameters of arbitrary two-dimensional lattice geometries: the tunneling $t$, on-site potential $V$, and interaction $U$, for multiple bands and for both fermions and bosons. We show several examples. One notable finding is that a finite array of equally strong and separated individual tweezer potentials actually sums to give a non-periodic total potential and thus spatially non-uniform Hubbard parameters. We demonstrate procedures to find trap configurations that equalize these parameters. More generally, these procedures solve the inverse problem of calculating Hubbard parameters: given desired Hubbard parameters, find trap configurations to realize them. These methods will be critical tools for using tunnel-coupled tweezer arrays., Comment: 14 pages, 8 figures

Ultracold polar molecules uniquely combine a rich structure of long-lived internal states with access to controllable long-range, anisotropic dipole-dipole interactions. In particular, the rotational states of polar molecules confined in optical tweezers or optical lattices may be used to encode interacting qubits for quantum computation or pseudo-spins for simulating quantum magnetism. As with all quantum platforms, the engineering of robust coherent superpositions of states is vital. However, for optically trapped molecules, the coherence time between rotational states is typically limited by inhomogeneous light shifts. Here we demonstrate a rotationally-magic optical trap for RbCs molecules that supports a Ramsey coherence time of 0.78(4) seconds in the absence of dipole-dipole interactions. This extends to >1.4 seconds at the 95% confidence level using a single spin-echo pulse. In our magic trap, dipolar interactions become the dominant mechanism by which Ramsey contrast is lost for superpositions that generate oscillating dipoles. By changing the states forming the superposition, we tune the effective dipole moment and show that the coherence time is inversely proportional to the strength of the dipolar interaction. Our work unlocks the full potential of the rotational degree of freedom in molecules for quantum computation and quantum simulation., Comment: 12 pages, 7 figures (main text and supplementary information combined)

Synthetic dimensions, wherein dynamics occurs in a set of internal states, have found great success in recent years in exploring topological effects in cold atoms and photonics. However, the phenomena thus far explored have largely been restricted to the non-interacting or weakly interacting regimes. Here, we extend the synthetic dimensions playbook to strongly interacting systems of Rydberg atoms prepared in optical tweezer arrays. We use precise control over driving microwave fields to introduce a tunable $U(1)$ flux in a four-site lattice of coupled Rydberg levels. We find highly coherent dynamics, in good agreement with theory. Single atoms show oscillatory dynamics controllable by the gauge field. Small arrays of interacting atoms exhibit behavior suggestive of the emergence of ergodic and arrested dynamics in the regimes of intermediate and strong interactions, respectively. These demonstrations pave the way for future explorations of strongly interacting dynamics and many-body phases in Rydberg synthetic lattices., Comment: 6 pages, 4 figures; 6 pages and 5 figures of Supplementary Material

We characterize the equation of state (EoS) of the SU($N>2$) Fermi-Hubbard Model (FHM) in a two-dimensional single-layer square optical lattice. We probe the density and the site occupation probabilities as functions of interaction strength and temperature for $N = 3, 4$ and 6. Our measurements are used as a benchmark for state-of-the-art numerical methods including determinantal quantum Monte Carlo (DQMC) and numerical linked cluster expansion (NLCE). By probing the density fluctuations, we compare temperatures determined in a model-independent way by fitting measurements to numerically calculated EoS results, making this a particularly interesting new step in the exploration and characterization of the SU($N$) FHM., Comment: 8 pages, 3 figures; Supplemental Material

We introduce a classical analog of quantum matter in ultracold molecule -- or Rydberg atom -- synthetic dimensions, extending the Potts model to include interactions J1 between atoms adjacent in both real and synthetic space and studying its finite temperature properties. For intermediate values of J1, the resulting phases and phase diagrams are similar to those of the clock and Villain models, in which three phases emerge. There exists a sheet phase analogous to that found in quantum synthetic dimension models between the high temperature disordered phase and the low temperature ferromagnetic phase. We also employ machine learning to uncover non-trivial features of the phase diagram using the learning by confusion approach. The key result there is that the method is able to discern several successive phase transitions., Comment: 12 pages, 10 figures

It has been proved that in gapped ground states of locally-interacting quantum systems, the effect of local perturbations decays exponentially with distance. However, in systems with power-law ($1/r^\alpha$) decaying interactions, no analogous statement has been shown, and there are serious mathematical obstacles to proving it with existing methods. In this paper we prove that when $\alpha$ exceeds the spatial dimension $D$, the effect of local perturbations on local properties a distance $r$ away is upper bounded by a power law $1/r^{\alpha_1}$ in gapped ground states, provided that the perturbations do not close the spectral gap. The power-law exponent $\alpha_1$ is tight if $\alpha>2D$ and interactions are two-body, where we have $\alpha_1=\alpha$. The proof is enabled by a method that avoids the use of quasiadiabatic continuation and incorporates techniques of complex analysis. This method also improves bounds on ground state correlation decay, even in short-range interacting systems. Our work generalizes the fundamental notion that local perturbations have local effects to power-law interacting systems, with broad implications for numerical simulations and experiments., Comment: 14 pages, 3 figures

We prepare high-filling two-component arrays of up to fifty fermionic atoms in optical tweezers, with the atoms in the ground motional state of each tweezer. Using a stroboscopic technique, we configure the arrays in various two-dimensional geometries with negligible Floquet heating. Full spin- and density-resolved readout of individual sites allows us to post-select near-zero entropy initial states for fermionic quantum simulation. We prepare a correlated state in a two-by-two tunnel-coupled Hubbard plaquette, demonstrating all the building blocks for realizing a programmable fermionic quantum simulator.

We study a three-dimensional (3D) classical Ising model that is exactly solvable when some coupling constants take certain imaginary values. The solution combines and generalizes the Onsager-Kaufman solution of the 2D Ising model and the solution of Kitaev's honeycomb model, leading to a three-parameter phase diagram with a third order phase transition between two distinct phases. Interestingly, the phases of this model are distinguished by topological features: the expectation value of a certain family of loop observables depend only on the topology of the loop (whether the loop is contractible), and are quantized at rational values that differ in the two phases. We show that a related exactly solvable 3D classical statistical model with real coupling constants also shows the topological features of one of these phases. Furthermore, even in the model with complex parameters, the partition function has some physical relevance, as it can be interpreted as the transition amplitude of a quantum dynamical process and may shed light on dynamical quantum phase transitions., Comment: 19 pages, 7 figures

We demonstrate the use of matrix product state (MPS) models for discriminating quantum data on quantum computers using holographic algorithms, focusing on classifying a translationally invariant quantum state based on $L$ qubits of quantum data extracted from it. We detail a process in which data from single-shot experimental measurements are used to optimize an isometric tensor network, the tensors are compiled into unitary quantum operations using greedy compilation heuristics, parameter optimization on the resulting quantum circuit model removes the post-selection requirements of the isometric tensor model, and the resulting quantum model is inferenced on either product state or entangled quantum data. We demonstrate our training and inference architecture on a synthetic dataset of six-site single-shot measurements from the bulk of a one-dimensional transverse field Ising model (TFIM) deep in its antiferromagnetic and paramagnetic phases. We experimentally evaluate models on Quantinuum's H1-2 trapped ion quantum computer using entangled input data modeled as translationally invariant, bond dimension 4 MPSs across the known quantum phase transition of the TFIM. Using linear regression on the experimental data near the transition point, we find predictions for the critical transverse field of $h=0.962$ and $0.994$ for tensor network discriminators of bond dimension $\chi=2$ and $\chi=4$, respectively. These predictions compare favorably with the known transition location of $h=1$ despite training on data far from the transition point. Our techniques identify families of short-depth variational quantum circuits in a data-driven and hardware-aware fashion and robust classical techniques to precondition the model parameters, and can be adapted beyond machine learning to myriad applications of tensor networks on quantum computers, such as quantum simulation and error correction., Comment: 19 pages, 9 figures. Abstract shortened compared to uploaded version to meet arXiv requirements

We calculate properties of dipolar interacting ultracold molecules or Rydberg atoms in a semi-synthetic three-dimensional configuration -- one synthetic dimension plus a two-dimensional real space optical lattice or periodic microtrap array -- using the stochastic Green function Quantum Monte Carlo method. Through a calculation of thermodynamic quantities and appropriate correlation functions, along with their finite size scalings, we show that there is a second order transition to a low temperature phase in which two-dimensional `sheets' form in the synthetic dimension of internal rotational or electronic states of the molecules or Rydberg atoms, respectively. Simulations for different values of the interaction $V$, which acts between atoms or molecules that are adjacent both in real and synthetic space, allow us to compute a phase diagram. We find a finite-temperature transition at sufficiently large $V$, as well as a quantum phase transition -- a critical value $V_c$ below which the transition temperature vanishes.

Combinatorial optimization problems on graphs have broad applications in science and engineering. The Quantum Approximate Optimization Algorithm (QAOA) is a method to solve these problems on a quantum computer by applying multiple rounds of variational circuits. However, there exist several challenges limiting the real-world applications of QAOA. In this paper, we demonstrate on a trapped-ion quantum computer that QAOA results improve with the number of rounds for multiple problems on several arbitrary graphs. We also demonstrate an advanced mixing Hamiltonian that allows sampling of all optimal solutions with predetermined weights. Our results are a step towards applying quantum algorithms to real-world problems., Comment: 8 pages, 5 figures

Ultracold Rydberg atom arrays are an emerging platform for quantum simulation and computing. However, decoherence in these systems remains incompletely understood. Recent experiments [Guardado-Sanchez et al. Phys. Rev. X 8, 021069 (2018)] observed strong decoherence in the quench and longitudinal-field-sweep dynamics of two-dimensional Ising models realized with Lithium-6 Rydberg atoms in optical lattices. This decoherence was conjectured to arise from spin-motion coupling. Here we show that spin-motion coupling indeed leads to decoherence in qualitative, and often quantitative, agreement with the experimental data, treating the difficult spin-motion coupled problem using the discrete truncated Wigner approximation method. We also show that this decoherence will be an important factor to account for in future experiments with Rydberg atoms in optical lattices and microtrap arrays, and discuss methods to mitigate the effect of motion, such as using heavier atoms or deeper traps.

The SU(2) symmetric Fermi-Hubbard model (FHM) plays an essential role in strongly correlated fermionic many-body systems. In the one particle per site and strongly interacting limit ${U/t \gg 1}$, it is effectively described by the Heisenberg Hamiltonian. In this limit, enlarging the spin and extending the typical SU(2) symmetry to SU($N$) has been predicted to give exotic phases of matter in the ground state, with a complicated dependence on $N$. This raises the question of what -- if any -- are the finite-temperature signatures of these phases, especially in the currently experimentally relevant regime near or above the superexchange energy. We explore this question for thermodynamic observables by numerically calculating the thermodynamics of the SU($N$) FHM in the two-dimensional square lattice near densities of one particle per site, using determinant Quantum Monte Carlo and Numerical Linked Cluster Expansion. Interestingly, we find that for temperatures above the superexchange energy, where the correlation length is short, the energy, number of on-site pairs, and kinetic energy are universal functions of $N$. Although the physics in the regime studied is well beyond what can be captured by low-order high-temperature series, we show that an analytic description of the scaling is possible in terms of only one- and two-site calculations., Comment: 22 pages, 15 figures

The scope of analog simulation in atomic, molecular, and optical systems has expanded greatly over the past decades. Recently, the idea of synthetic dimensions -- in which transport occurs in a space spanned by internal or motional states coupled by field-driven transitions -- has played a key role in this expansion. While approaches based on synthetic dimensions have led to rapid advances in single-particle Hamiltonian engineering, strong interaction effects have been conspicuously absent from most synthetic dimensions platforms. Here, in a lattice of coupled atomic momentum states, we show that atomic interactions result in large and qualitative changes to dynamics in the synthetic dimension. We explore how the interplay of nonlinear interactions and coherent tunneling enriches the dynamics of a one-band tight-binding model, giving rise to macroscopic self-trapping and phase-driven Josephson dynamics with a nonsinusoidal current-phase relationship, which can be viewed as stemming from a nonlinear band structure arising from interactions., Comment: 6 pages, 3 figures, 3 pages of supplementary material

Mott insulators are paradigms of strongly correlated physics, giving rise to phases of matter with novel and hard-to-explain properties. Extending the typical SU(2) symmetry of Mott insulators to SU($N$) is predicted to give exotic quantum magnetism at low temperatures, but understanding the effect of strong quantum fluctuations for large $N$ remains an open challenge. In this work, we experimentally observe nearest-neighbor spin correlations in the SU(6) Hubbard model realized by ytterbium atoms in optical lattices. We study one-dimensional, two-dimensional square, and three-dimensional cubic lattice geometries. The measured SU(6) spin correlations are dramatically enhanced compared to the SU(2) correlations, due to strong Pomeranchuk cooling. We also present numerical calculations based on exact diagonalization and determinantal quantum Monte Carlo. The experimental data for a one-dimensional lattice agree with theory, without any fitting parameters. The detailed comparison between theory and experiment allows us to infer from the measured correlations a lowest temperature of $\left[{0.096 \pm 0.054 \, \rm{(theory)} \pm 0.030 \, \rm{(experiment)}}\right]/k_{\rm B}$ times the tunneling amplitude. For two- and three-dimensional lattices, experiments reach entropies below where our calculations converge, highlighting the experiments as quantum simulations. These results open the door for the study of long-sought SU($N$) quantum magnetism., Comment: 13 pages, 8 figures

Finite-size error (FSE), the discrepancy between an observable in a finite system and in the thermodynamic limit, is ubiquitous in numerical simulations of quantum many body systems. Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing. Here we derive rigorous upper bounds on the FSE of local observables in real time quantum dynamics simulations initialized from a product state. In $d$-dimensional locally interacting systems with a finite local Hilbert space, our bound implies $ |\langle \hat{S}(t)\rangle_L-\langle \hat{S}(t)\rangle_\infty|\leq C(2v t/L)^{cL-\mu}$, with $v$, $C$, $c$, $\mu $ constants independent of $L$ and $t$, which we compute explicitly. For periodic boundary conditions (PBC), the constant $c$ is twice as large as that for open boundary conditions (OBC), suggesting that PBC have smaller FSE than OBC at early times. The bound can be generalized to a large class of correlated initial states as well. As a byproduct, we prove that the FSE of local observables in ground state simulations decays exponentially with $L$, under a suitable spectral gap condition. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.

The realization of antiferromagnetic (AF) correlations in ultracold fermionic atoms on an optical lattice is a significant achievement. Experiments have been carried out in one, two, and three dimensions, and have also studied anisotropic configurations with stronger tunneling in some lattice directions. Such anisotropy is relevant to the physics of cuprate superconductors and other strongly correlated materials. Moreover, this anisotropy might be harnessed to enhance AF order. Here we numerically investigate, using Determinant Quantum Monte Carlo, a simple realization of anisotropy in the 3D Hubbard model in which the tunneling between planes, $t_\perp$, is unequal to the intraplane tunneling $t$. This model interpolates between the three-dimensional isotropic ($t_\perp = t$) and two-dimensional ($t_\perp =0$) systems. We show that at fixed interaction strength to tunneling ratio ($U/t$), anisotropy can enhance the magnetic structure factor relative to both 2D and 3D results. However, this enhancement occurs at interaction strengths below those for which the N\'eel temperature $T_{\rm N\acute{e}el}$ is largest, in such a way that the structure factor cannot be made to exceed its value in isotropic 3D systems at the optimal $U/t$. We characterize the 2D-3D crossover in terms of the magnetic structure factor, real space spin correlations, number of doubly-occupied sites, and thermodynamic observables. An interesting implication of our results stems from the entropy's dependence on anisotropy. As the system evolves from 3D to 2D, the entropy at a fixed temperature increases. Correspondingly, at fixed entropy, the temperature will decrease going from 3D to 2D. This suggests a cooling protocol in which the dimensionality is adiabatically changed from 3D to 2D., Comment: 13 pages, 13 figures

We develop a numerical linked cluster expansion (NLCE) method that can be applied directly to inhomogeneous systems, for example Hamiltonians with disorder and dynamics initiated from inhomogeneous initial states. We demonstrate the method by calculating dynamics for single-spin expectations and spin correlations in two-dimensional spin models on a square lattice, starting from a checkerboard state. We show that NLCE can give moderate to dramatic improvement over an exact diagonalization of comparable computational cost, and that the advantage in computational resources grows exponentially as the size of the clusters included grows. Although the method applies to any type of NLCE, our explicit benchmarks use the rectangle expansion. Besides showing the capability to treat inhomogeneous systems, these benchmarks demonstrate the rectangle expansion's utility out of equilibrium., Comment: 9 pages, 6 figures

Motivated by a recent experiment [Revelle et al. Phys. Rev. Lett. 117, 235301 (2016)] that characterized the one- to three-dimensional crossover in a spin-imbalanced ultracold gas of $^6$Li atoms trapped in a two-dimensional array of tunnel-coupled tubes, we calculate the phase diagram for this system using Hartree-Fock Bogoliubov-de Gennes mean-field theory, and compare the results with experimental data. Mean-field theory predicts fully spin-polarized normal, partially spin-polarized normal, spin-polarized superfluid, and spin-balanced superfluid phases in a homogeneous system. We use the local density approximation to obtain density profiles of the gas in a harmonic trap. We compare these calculations with experimental measurements in Revelle {\em et al.} as well as previously unpublished data. Our calculations qualitatively agree with experimentally-measured densities and coordinates of the phase boundaries in the trap, and quantitatively agree with experimental measurements at moderate-to-large polarizations. Our calculations also reproduce the experimentally-observed universal scaling of the phase boundaries for different scattering lengths at a fixed value of scaled inter-tube tunneling. However, our calculations have quantitative differences with experimental measurements at low polarization, and fail to capture important features of the one- to three-dimensional crossover observed in experiments. These suggest the important role of physics beyond-mean-field theory in the experiments. We expect that our numerical results will aid future experiments in narrowing the search for the FFLO phase., Comment: 13 pages, 8 figures. Minor revisions; Conclusions unchanged

We calculate the collective modes of ultracold trapped alkaline-earth fermionic atoms, which possess an SU($N$) symmetry of the nuclear spin degree of freedom, and a controllable $N$, with $N$ as large as $10$. We calculate the breathing and quadrupole modes of two-dimensional and three-dimensional harmonically trapped gases in the normal phase. We particularly concentrate on two-dimensional gases, where the shift is more accessible experimentally, and the physics has special features. We present results as a function of temperature, interaction strength, density, and $N$. We include calculations across the collisionless to hydrodynamic crossover. We assume the gas is interacting weakly, such that it can be described by a Boltzmann-Vlasov equation that includes both mean-field terms and the collision integral. We solve this with an approximate scaling ansatz, taking care in two-dimensions to preserve the scaling symmetry of the system. We predict the collective mode frequency shifts and damping, showing that these are measurable in experimentally relevant regimes. We expect these results to furnish powerful tools to characterize interactions and the state of alkaline-earth gases, as well as to lay the foundation for future work, for example on strongly interacting gases and SU($N$) spin modes., Comment: 11 pages, 4 figures (including appendices)

Quantum simulators are a promising technology on the spectrum of quantum devices from specialized quantum experiments to universal quantum computers. These quantum devices utilize entanglement and many-particle behaviors to explore and solve hard scientific, engineering, and computational problems. Rapid development over the last two decades has produced more than 300 quantum simulators in operation worldwide using a wide variety of experimental platforms. Recent advances in several physical architectures promise a golden age of quantum simulators ranging from highly optimized special purpose simulators to flexible programmable devices. These developments have enabled a convergence of ideas drawn from fundamental physics, computer science, and device engineering. They have strong potential to address problems of societal importance, ranging from understanding vital chemical processes, to enabling the design of new materials with enhanced performance, to solving complex computational problems. It is the position of the community, as represented by participants of the NSF workshop on "Programmable Quantum Simulators," that investment in a national quantum simulator program is a high priority in order to accelerate the progress in this field and to result in the first practical applications of quantum machines. Such a program should address two areas of emphasis: (1) support for creating quantum simulator prototypes usable by the broader scientific community, complementary to the present universal quantum computer effort in industry; and (2) support for fundamental research carried out by a blend of multi-investigator, multi-disciplinary collaborations with resources for quantum simulator software, hardware, and education., Comment: 41 pages -- references and acknowledgments added in v2

The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity. Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications. We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions. Most prominently, in systems with a large local Hilbert space dimension $D$, our method gives an LR velocity that grows much slower than previous bounds with $D$ as $D\to \infty$. For example, in the Heisenberg model with spin $S$, we find $v\leq$ const. compared to the previous $v\propto S$ which diverges at large $S$, and in multiorbital Hubbard models with $N$ orbitals, we find $v\propto \sqrt{N}$ instead of previous $v\propto N$, and similarly in the $N$-state truncated Bose-Hubbard model and Wen's quantum rotor model. Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the $d$-dimensional quantum Ising model and perturbed toric code models. Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time., Comment: 23 pages, 8 figures

Ground state counting plays an important role in several applications in science and engineering, from estimating residual entropy in physical systems, to bounding engineering reliability and solving combinatorial counting problems. While quantum algorithms such as adiabatic quantum optimization (AQO) and quantum approximate optimization (QAOA) can minimize Hamiltonians, they are inadequate for counting ground states. We modify AQO and QAOA to count the ground states of arbitrary classical spin Hamiltonians, including counting ground states with arbitrary nonnegative weights attached to them. As a concrete example, we show how our method can be used to count the weighted fraction of edge covers on graphs, with user-specified confidence on the relative error of the weighted count, in the asymptotic limit of large graphs. We find the asymptotic computational time complexity of our algorithms, via analytical predictions for AQO and numerical calculations for QAOA, and compare with the classical optimal Monte Carlo algorithm (OMCS), as well as a modified Grover's algorithm. We show that for large problem instances with small weights on the ground states, AQO does not have a quantum speedup over OMCS for a fixed error and confidence, but QAOA has a sub-quadratic speedup on a broad class of numerically simulated problems. Our work is an important step in approaching general ground-state counting problems beyond those that can be solved with Grover's algorithm. It offers algorithms that can employ noisy intermediate-scale quantum devices for solving ground state counting problems on small instances, which can help in identifying more problem classes with quantum speedups., Comment: 25 pages including bibliography and appendices; 5 figures

We introduce a model to study the collisions of two ultracold diatomic molecules in one dimension interacting via pairwise potentials. We present results for this system, and argue that it offers lessons for real molecular collisions in three dimensions. We analyze the distribution of the adiabatic potentials in the hyperspherical coordinate representation as well as the distribution of the four-body bound states in the adiabatic approximation (i.e. no coupling between adiabatic channels). It is found that while the adiabatic potential distribution transitions from chaotic to non-chaotic as the two molecules are separated, the four-body bound states show no visible chaos in the distribution of nearest-neighbor energy level spacing. We also study the effects of molecular properties, such as interaction strength, interaction range, and atomic mass, on the resonance density and degree of chaos in the adiabatic potentials. We numerically find that the dependence of the four-body bound state density on these parameters is captured by simple scaling laws, in agreement with previous analytic arguments, even though these arguments relied on uncontrolled approximations. This agreement suggests that similar scaling laws may also govern real molecular collisions in three dimensions., Comment: 8 pages, 7 figures

We investigate two-frequency photoassociation of a weakly bound molecular state, focusing on a regime where the ac Stark shift is comparable to the halo-state energy. In this "high-intensity" regime, we observe features absent in low-intensity two-frequency photoassociation. We experimentally measure the spectra of $^{86}$Sr atoms coupled to the least bound state of the $^{86}$Sr$_2$ ground electronic channel through an intermediate electronically excited molecular state. We compare the spectra to a simple three-level model that includes a two-frequency drive on each leg of the transition. With numerical solution of the time-dependent Schrodinger equation, we show that this model accurately captures (1) the existence of experimentally observed satellite peaks that arise from nonlinear processes, (2) the locations of the two-photon peak in the spectrum, including ac Stark shifts, and (3) in some cases, spectral lineshapes. To better understand these numerical results, we develop an approximate treatment of this model, based on Floquet and perturbation theory, that gives simple formulas that accurately capture the halo-state energies. We expect these expressions to be valuable tools to analyze and guide future two-frequency photoassociation experiments., Comment: 12 pages, 7 figures

We present two-photon photoassociation to the least-bound vibrational level of the X$^1\Sigma_g^+$ electronic ground state of the $^{86}$Sr$_2$ dimer and measure a binding energy of $E_b=-83.00(7)(20)$\,kHz. Because of the very small binding energy, this is a halo state corresponding to the scattering resonance for two $^{86}$Sr atoms at low temperature. The measured binding energy, combined with universal theory for a very weakly bound state on a potential that asymptotes to a van der Waals form, is used to determine an $s$-wave scattering length $a=810.6(12)$\,$a_0$, which is consistent with, but substantially more accurate than the previously determined $a=798(12)\,a_0$ found from mass-scaling and precision spectroscopy of other Sr isotopes. For the intermediate state, we use a bound level on the metastable $^1S_0-{^3P_1}$ potential. Large sensitivity of the dimer binding energy to light near-resonant with the bound-bound transition to the intermediate state suggests that $^{86}$Sr has great promise for manipulating atom interactions optically and probing naturally occurring Efimov states., Comment: 10 pages, 9 figures

We present a multichannel model for elastic interactions, comprised of an arbitrary number of coupled finite square-well potentials, and derive semi-analytic solutions for its scattering behavior. Despite the model's simplicity, it is flexible enough to include many coupled short-ranged resonances in the vicinity of the collision threshold, as is necessary to describe ongoing experiments in ultracold molecules and lanthanide atoms. We also introduce a simple, but physically realistic, statistical ensemble for parameters in this model. We compute the resulting probability distributions of nearest-neighbor resonance spacings and analyze them by fitting to the Brody distribution. We quantify the ability of alternative distribution functions, for resonance spacing and resonance number variance, to describe the crossover regime. The analysis demonstrates that the multichannel square-well model with the chosen ensemble of parameters naturally captures the crossover from integrable to chaotic scattering as a function of closed channel coupling strength., Comment: 11 pages, 8 figures

We compare the continuous and discrete truncated Wigner approximations of various spin models' dynamics to exact analytical and numerical solutions. We account for all components of spin-spin correlations on equal footing, facilitated by a recently introduced geometric correlation matrix visualization technique [R. Mukherjee {\em et al.}, Phys. Rev. A {\bf 97}, 043606 (2018)]. We find that at modestly short times, the dominant error in both approximations is to substantially suppress spin correlations along one direction., Comment: 19 pages, 19 figures; The first two authors contributed equally

The Fermi-Hubbard model describes ultracold fermions in an optical lattice and exhibits antiferromagnetic long-ranged order below the N\'{e}el temperature. However, reaching this temperature in the lab has remained an elusive goal. In other atomic systems, such as trapped ions, low temperatures have been successfully obtained by adiabatic demagnetization, in which a strong effective magnetic field is applied to a spin-polarized system, and the magnetic field is adiabatically reduced to zero. Unfortunately, applying this approach to the Fermi-Hubbard model encounters a fundamental obstacle: the $SU(2)$ symmetry introduces many level crossings that prevent the system from reaching the ground state, even in principle. However, by breaking the $SU(2)$ symmetry with a spin-dependent tunneling, we show that adiabatic demagnetization can achieve low temperature states. Using density matrix renormalization group (DMRG) calculations in one dimension, we numerically find that demagnetization protocols successfully reach low temperature states of a spin-anisotropic Hubbard model, and we discuss how to optimize this protocol for experimental viability. By subsequently ramping spin-dependent tunnelings to spin-independent tunnelings, we expect that our protocol can be employed to produce low-temperature states of the Fermi-Hubbard Model., Comment: References added

We use network analysis to describe and characterize an archetypal quantum system - an Ising spin chain in a transverse magnetic field. We analyze weighted networks for this quantum system, with link weights given by various measures of spin-spin correlations such as the von Neumann and Renyi mutual information, concurrence, and negativity. We analytically calculate the spin-spin correlations in the system at an arbitrary temperature by mapping the Ising spin chain to fermions, as well as numerically calculate the correlations in the ground state using matrix product state methods, and then analyze the resulting networks using a variety of network measures. We demonstrate that the network measures show some traits of complex networks already in this spin chain, arguably the simplest quantum many-body system. The network measures give insight into the phase diagram not easily captured by more typical quantities, such as the order parameter or correlation length. For example, the network structure varies with transverse field and temperature, and the structure in the quantum critical fan is different from the ordered and disordered phases., Comment: 10 pages, 10 figures + references

We study Read and Green's mean-field model of the spinless $p_x+ip_y$ superconductor [N.Read and D.Green, Phys. Rev. B 61, 10267 (2000)] at a special set of parameters where we find the analytic expressions for the topologically degenerate ground states and the Majorana modes, including in finite systems with edges and in the presence of an arbitrary number of vortices. The wavefunctions of these ground states are similar (but not always identical) to the Moore-Read Pfaffian states proposed for the $\nu=5/2$ fractional quantum Hall system, which are interpreted as the p-wave superconducting states of composite fermions. The similarity in the long-wavelength universal properties is expected from previous work, but at the special point studied herein the wavefunctions are exact even for short-range, non-universal properties. As an application of these results, we show how to obtain the non-Abelian statistics of the vortex Majorana modes by explicitly calculating the analytic continuation of the ground state wavefunctions when vortices are adiabatically exchanged, an approach different from the previous one based on universal arguments. Our results are also useful for constructing particle number-conserving (and interacting) Hamiltonians with exact projected mean-field states., Comment: 16 pages, 2 figures

We demonstrate that a numerical linked cluster expansion method is a powerful tool to calculate quantum dynamics. We calculate the dynamics of the magnetization and spin correlations in the two-dimensional transverse field Ising and XXZ models evolved from a product state. Such dynamics are directly probed in ongoing experiments in ultracold atoms, molecules, and ions. We show that a numerical linked cluster expansion gives dramatically more accurate results at short-to-moderate times than exact diagonalization, and simultaneously requires fewer computational resources. More specifically, the cluster expansion frequently produces more accurate results than an exact diagonalization calculation that would require $10^{5}$--$10^{10}$ more computational operations and memory., Comment: 6 pages, 4 figures

Synthetic dimensions alter one of the most fundamental properties in nature, the dimension of space. They allow, for example, a real three-dimensional system to act as effectively four-dimensional. Driven by such possibilities, synthetic dimensions have been engineered in ongoing experiments with ultracold matter. We show that rotational states of ultracold molecules can be used as synthetic dimensions extending to many - potentially hundreds of - synthetic lattice sites. Microwaves coupling rotational states drive fully controllable synthetic inter-site tunnelings, enabling, for example, topological band structures. Interactions leads to even richer behavior: when molecules are frozen in a real space lattice with uniform synthetic tunnelings, dipole interactions cause the molecules to aggregate to a narrow strip in the synthetic direction beyond a critical interaction strength, resulting in a quantum string or a membrane, with an emergent condensate that lives on this string or membrane. All these phases can be detected using measurements of rotational state populations., Comment: 5-page article + 4 figures + references; 7 pages + 4 figures in Supplement