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Quantum simulation provides important tools in studying strongly correlated many-body systems with controllable parameters. As a hybrid of two fundamental models in quantum optics and in condensed matter physics, the Rabi-Hubbard model demonstrates rich physics through the competition between local spin-boson interactions and long-range boson hopping. Here we report an experimental realization of the Rabi-Hubbard model using up to $16$ trapped ions and present a controlled study of its equilibrium properties and quantum dynamics. We observe the ground-state quantum phase transition by slowly quenching the coupling strength, and measure the quantum dynamical evolution in various parameter regimes. With the magnetization and the spin-spin correlation as probes, we verify the prediction of the model Hamiltonian by comparing theoretical results in small system sizes with experimental observations. For larger-size systems of $16$ ions and $16$ phonon modes, the effective Hilbert space dimension exceeds $2^{57}$, whose dynamics is intractable for classical supercomputers., 6 pages, 3 figures + 13 pages, 10 figures

Thermalization of isolated quantum systems is a long-standing fundamental problem where different mechanisms are proposed over time. We contribute to this discussion by classifying the diverse quench dynamical behaviours of spin-1 Bose-Einstein condensates, which includes well-defined quantum collapse and revivals, thermalization, and certain special cases. These special cases are either nonthermal equilibration with no revival but a collapse even though the system has finite degrees of freedom or no equilibration with no collapse and revival. Given that some integrable systems are already shown to demonstrate the weak form of eigenstate thermalization hypothesis (ETH), we determine the regions where ETH holds and fails in this integrable isolated quantum system. The reason behind both thermalizing and nonthermalizing behaviours in the same model under different initial conditions is linked to the discussion of `rare' nonthermal states existing in the spectrum. We also propose a method to predict the collapse and revival time scales and how they scale with the number of particles in the condensate. We use a sudden quench to drive the system to non-equilibrium and hence the theoretical predictions given in this paper can be probed in experiments., Comment: 14 pages, 16 figures

The local phonon modes in a Coulomb crystal of trapped ions can represent a Hubbard system of coupled bosons. We selectively prepare single excitations at each site and observe free hopping of a boson between sites, mediated by the long-range Coulomb interaction between ions. We then implement phonon blockades on targeted sites by driving a Jaynes-Cummings interaction on individually addressed ions to couple their internal spin to the local phonon mode. The resulting dressed states have energy splittings that can be tuned to suppress phonon hopping into the site. This new experimental approach opens up the possibility of realizing large-scale Hubbard systems from the bottom up with tunable interactions at the single-site level., 8 pages, 5 figures, 1 table. Supplementary materials included

We show that ultracold atoms can be controlled in multi-band optical lattices through spatially periodic Raman pulses for investigation of a class of strongly correlated physics related to the Kondo problem. The underlying dynamics of this system is described by a spin-dependent fermionic or bosonic Kondo-Hubbard lattice model even if we have only spin-independent atomic collision interaction. We solve the bosonic Kondo-Hubbard lattice model through a mean-field approximation, and the result shows a clear phase transition from the ferromagnetic superfluid to the Kondo-signet insulator at the integer filling., Comment: 4 pages, 2 figures

Three-dimensional (3D) topological insulators in general need to be protected by certain kinds of symmetries other than the presumed $U(1)$ charge conservation. A peculiar exception is the Hopf insulators which are 3D topological insulators characterized by an integer Hopf index. To demonstrate the existence and physical relevance of the Hopf insulators, we construct a class of tight-binding model Hamiltonians which realize all kinds of Hopf insulators with arbitrary integer Hopf index. These Hopf insulator phases have topologically protected surface states and we numerically demonstrate the robustness of these topologically protected states under general random perturbations without any symmetry other than the $U(1)$ charge conservation that is implicit in all kinds of topological insulators., 7 pages (including supplemental material), 4 figures

We propose a method to generate massive entanglement in a spinor Bose-Einstein condensate from an initial product state through an adiabatic sweep of the magnetic field across a quantum phase transition induced by competition between the spin-dependent collision interaction and the quadratic Zeeman effect. The generated many-body entanglement is characterized by the experimentally measurable entanglement depth in the proximity of the Dicke state. We show that the scheme is robust to practical noise and experimental imperfection and under realistic conditions it is possible to generate genuine entanglement for hundreds of atoms.

Spin squeezed states are a class of entangled states of spins that have practical applications to precision measurements. In recent years spin squeezing with one-axis twisting (OAT) has been demonstrated experimentally with spinor BECs with more than 10^3 atoms. Although the noise is below the standard quantum limit, the OAT scheme cannot reduce the noise down to the ultimate Heisenberg limit. Here we propose an experimentally feasible scheme based on optimized quantum control to greatly enhance the performance of OAT to approach the Heisenberg limit, requiring only an OAT Hamiltonian and the use of several coherent driving pulses. The scheme is robust against technical noise and can be readily implemented for spinor BECs or trapped ions with current technology., 4 pages, 6 figures

We study topological properties of the Bose-Hubbard model with repulsive interactions in a one-dimensional optical superlattice. We find that the Mott insulator states of the single-component (two-component) Bose-Hubbard model under fractional fillings are topological insulators characterized by a nonzero charge (or spin) Chern number with nontrivial edge states. For ultracold atomic experiments, we show that the topological Chern number can be detected through measuring the density profiles of the bosonic atoms in a harmonic trap., 5 pages, published version

We show anisotropy of the dipole interaction between magnetic atoms or polar molecules can stabilize new quantum phases in an optical lattice. Using a well controlled numerical method based on the tensor network algorithm, we calculate phase diagram of the resultant effective Hamiltonian in a two-dimensional square lattice - an anisotropic Hubbard model of hard-core bosons with attractive interaction in one direction and repulsive interaction in the other direction. Besides the conventional superfluid and the Mott insulator states, we find the striped and the checkerboard charge density wave states and the supersolid phase that interconnect the superfluid and the striped solid states. The transition to the supersolid phase has a mechanism different from the case of the soft-core Bose Hubbard model., Comment: 5 pages, 5 figures.

Fermions in an optical lattice near a wide Feshbach resonance are expected to be described by an effective Hamiltonian of the general Hubbard model with particle-assisted tunneling rates resulting from the strong atomic interaction [Phys. Rev. Lett. 95, 243202 (2005)]. Here, we propose a scheme to unambiguously test the predictions of this effective Hamiltonian through manipulation of ultracold atoms in an inhomogeneous optical superlattice. The structure of the low-energy Hilbert space as well as the particle assisted tunneling rates can be inferred from measurements of the time-of-flight images., 4 pages, 4 figures

We propose to utilize density distributions from a series of time-of-flight images of an expanding cloud to reconstruct single-particle correlation functions of trapped ultra-cold atoms. In particular, we show how this technique can be used to detect off-diagonal correlations of atoms in a quasi-one-dimensional trap, where both real- and momentum- space correlations are extracted at a quantitative level. The feasibility of this method is analyzed with specific examples, taking into account finite temporal and spatial resolutions in experiments., Comment: 7 pages, 4 figures

It is hard to stabilize the p-wave superfluid state of cold atomic gas in free space due to inelastic collisional losses. We consider the p-wave Feshbach resonance in an optical lattice, and show that it is possible to have a stable p-wave superfluid state where the multi-atom collisional loss is suppressed through the quantum Zeno effect. We derive the effective Hamiltonian for this system, and calculate its phase diagram in a one-dimensional optical lattice. The results show rich phase transitions between the p-wave superfluid state and different types of insulator states induced either by interaction or by dissipation., Comment: 5 pages, 5 figures

We teleport quantum information between two distant ytterbium ions trapped in different vacuum chambers separated by one meter. Full state tomography shows that the heralded probabilistic process employed has a fidelity of 90%.

We study through controlled numerical simulation the ground state properties of spin-polarized strongly interacting fermi gas in an anisotropic optical lattice, which is described by an effective one-dimensional general Hubbard model with particle correlated hopping rate. We show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type of state, while enhanced by a negative correlated hopping rate, can be completely suppressed by positive particle correlated hopping, yielding to an unusual magnetic phase even for particles with on-site attractive interaction We also find several different phase separation patterns for these atoms in an inhomogeneous harmonic trap, depending on the correlated hopping rate.

We discuss the experimental signal for the Bose-Einstein condensation of cold atoms in an optical lattice at finite temperature. Instead of using the visibility of the interference pattern via time-of-flight imaging, we show that the momentum-space density profile in the first Brillouin zone, in particular its bimodal distribution, provides an unambiguous signal for the Bose-Einstein condensation. We confirm this point with detailed calculations of the change in the atomic momentum distribution across the condensation phase transition, taking into account both the global trapping potential and the atomic interaction effects.

We present a numerical study on ground state properties of a one-dimensional (1D) general Hubbard model (GHM) with particle-assisted tunnelling rates and repulsive on-site interaction (positive-U), which describes fermionic atoms in an anisotropic optical lattice near a wide Feshbach resonance. For our calculation, we utilize the time evolving block decimation (TEBD) algorithm, which is an extension of the density matrix renormalization group and provides a well-controlled method for 1D systems. We show that the positive-U GHM, when hole-doped from half-filling, exhibits a phase with coexistence of quasi-long-range superfluid and charge-density-wave orders. This feature is different from the property of the conventional Hubbard model with positive-U, indicating the particle-assisted tunnelling mechanism in GHM brings in qualitatively new physics., updated with published version

We systematically study the BCS-BEC crossover and the quantum phase transition in ultracold {sup 6}Li and {sup 40}K atoms across a wide Feshbach resonance. The background scattering lengths for {sup 6}Li and {sup 40}K have opposite signs, which lead to very different behaviors for these two types of atoms. For {sup 40}K, both the two-body and the many-body calculations show that the system always has two branches of solutions: one corresponds to a deeply bound molecule state; and the other, the one accessed by the current experiments, corresponds to a weakly bound state with population always dominantly in the open channel. For {sup 6}Li, there is only a unique solution with the standard crossover from the weakly bound Cooper pairs to the deeply bound molecules as one sweeps the magnetic field through the crossover region. Because of this difference, for the experimentally accessible state of {sup 40}K, there is a quantum phase transition at zero temperature from the superfluid to the normal Fermi gas at the positive detuning of the magnetic field where the s-wave scattering length passes its zero point. For {sup 6}Li, however, the system changes continuously across the zero point of the scattering length. For both typesmore » of atoms, we also give a detailed comparison between the results from the two-channel and the single-channel model over the whole region of the magnetic field detuning.« less

We investigate the dynamic response of a highly degenerate ultracold Fermi gas to varying magnetic fields near the Feshbach resonance. Following the experimental practice, we calculate the dynamics of the excitation gap in response to the oscillating and linearly ramped magnetic fields, respectively. For oscillating magnetic fields, depending on the frequencies and the amplitudes of the oscillations, the response of the system shows either linear or rich nonlinear behavior. In addition, both the spectral studies through the linear-response theory and the time-domain simulations suggest the existence of a resonant frequency corresponding to the dissociation threshold of the fermion atom pairs. For linearly ramped magnetic fields, the response of the excitation gap shows damped oscillations, with the final value dependent on the rate of the field sweep. This work may find its applications in exploring the dynamics of the ultracold atom superfluidity near the BCS\char21{}Bose-Einstein-condensation crossover region.

We map out the detailed phase diagram of a trapped ultracold Fermi gas with population imbalance across a wide Feshbach resonance. We show that under the local density approximation, the properties of the atoms in any (anisotropic) harmonic traps are universally characterized by three dimensionless parameters: the normalized temperature, the dimensionless interaction strength, and the population imbalance. We then discuss the possible quantum phases in the trap, and quantitatively characterize their phase boundaries in various typical parameter regions., Comment: 9 pages, 4 figures

We propose a detection method for ultracold atoms which allows reconstruction of the full one-particle and two-particle correlation functions from the measurements. The method is based on Fourier sampling of the time-of-flight images through two consecutive impulsive Raman pulses. For applications of this method, we discuss a few examples, including detection of phase separation between superfluid and Mott insulators, various types of spin or superfluid orders, entanglement, exotic or fluctuating orders., Comment: 4 pages, 2 figures

We derive a set of dynamical mean-field equations for strongly interacting fermionic atoms in a potential trap across a Feshbach resonance. Our derivation is based on a variational ansatz, which generalizes the crossover wavefunction to the inhomogeneous case, and the assumption that the order parameter is slowly varying over the size of the Cooper pairs. The equations reduce to a generalized time-dependent Gross-Pitaevskii equation on the BEC side of the resonance. We discuss an iterative method to solve these mean-field equations, and present the solution for a harmonic trap as an illustrating example to self-consistently verify the approximations made in our derivation., Comment: replaced with the published version

Crystals of trapped atomic ions are among the most promising platforms for the quantum simulation of many-body quantum physics. Here, we describe recent developments in the simulation of quantum magnetic spin models with trapped 171 Yb + ions, and discuss the possibility of scaling the system to a level of complexity where classical computation becomes intractable.