Session A14: Focus Session: Quantum Simulation of Condensed Matter Systems With Ultracold Atoms

Simulating Charged Particles in a Magnetic Field with Ultra-cold Atoms Using Light-induced Effective Gauge Fields

We experimentally study light-induced gauge potentials in a $^{87}$Rb Bose-Einstein condensate. Instead of rotating the trap, we prepare the atoms in a spatially-varying optically dressed state. The atomic spin state is dressed by a spatially varying two-photon Raman coupling between the three $F=1$ hyperfine ground states. The resulting effective magnetic field is equivalent to rotating the condensate (and transforming to the rotating frame), and thus generates vortices. The inter-vortex distance is given by $\sqrt{2\pi} l_B$. Using the technique, the minimum possible $l_B \approx\sqrt{R_{\rm TF} \lambda/8 \pi}$ is the magnetic length for a uniform field, $R_{\rm TF}$ is the condensate diameter, and $\lambda\approx805\ {\rm nm}$ is the optical wavelength. We prepare the condensate in the dressed state, whose projection onto internal states of various state-dependent Bragg momenta are well understood.   

Two-Dimensional Electron Gas with Cold Atoms in Non-Abelian Gauge Potentials

Motivated by the possibility of creating non-Abelian fields using cold atoms in optical lattices, we study two-dimensional electron gases in a lattice, subjected to such fields. In the continuum limit, the system characterized by a two-component ``magnetic flux" describes a harmonic oscillator existing in two different charge states (mimicking a particle-hole pair). A key feature of the non-Abelian system is a splitting of the Landau energy levels, which broaden into bands. These Landau bands result in a coarse-grained Hofstadter ``moth." Furthermore, the bands overlap, leading to effective relativistic effects. Similar features also characterize the corresponding 2D lattice problem when at least one of the components of the magnetic flux is an irrational number. Some unique aspects of the transport properties of the non-Abelian system are the possibility of inducing localization by varying the quasimomentum, and the absence of localization of certain zero-energy states exhibiting a linear dispersion relation. Furthermore, non-Abelian systems provide an interesting localization scenario where the localization transition is accompanied by a transition from relativistic to non-relativistic theory.   

Observing {\it Zitterbewegung} in Ultracold Atoms

We propose an optical lattice scheme which would permit the experimental observation of {\it Zitterbewegung} (ZB) with ultracold, neutral atoms. A four-level ``tripod" variant of the usual setup for stimulated Raman adiabatic passage (STIRAP) has been proposed for generating non-Abelian gauge fields. [1] Dirac-like Hamiltonians, which exhibit ZB, are simple examples of such non-Abelian gauge fields; we show how a variety of them can arise, and how ZB can be observed, in a tripod system. We predict that the ZB should occur at experimentally accessible frequencies and amplitudes. \newline [1] J. Ruseckas, G. Juzeli{\=u}nas, P. \"{O}hberg, M. Fleischhauer, {\it Physical Review Letters} {\bf 95}, 010404 (2005).   

Detecting the Bose glass in optical lattices

We theoretically propose a method for the unambiguous experimental detection of Bose-glass behavior in the central region of a system of bosons trapped in an optical lattice system, and to discriminate the Bose glass from more conventional Mott and band insulators. The method is based on probing the compressibility of the system in the trap center by gradually increasing the trap frequency. Straightforward measurements of the average particle density in the center of the trap and of the momentum distribution allow to detect the migration of particles from the wings into \emph{localized} states in the center of the system under trap squeezing. We discuss the application of the method to simple optical lattices, and to commensurate and incommensurate optical superlattices; we moreover discuss the potential of trap squeezing techniques to probe the density of states of exotic phases realized in optical lattices.   

Phase Coherence and Superfluid-Insulator Transition in a Disordered Bose-Einstein Condensate$^1$

We have studied both the transport and phase coherence properties of a Bose-Einstein condensate (BEC) of $^7$Li atoms subject to a disordered potential with tunable strength ($V_D$). The BEC is created in an elongated optical trap, while the disordered potential is produced by laser speckle. We probe transport of the disordered BEC by either slowly or abruptly offsetting the trap relative to the disordered potential. At high $V_D$, we observe pinning of the disordered BEC and suppression of its dipole excitation, consistent with the transition to an insulator. We use \textit{in-situ} imaging to detect density modulation, while time-of-flight (TOF) imaging is used to probe phase coherence. At moderate $V_D$, we observe small density fluctuations in the \textit{in-situ} images, and random but \textit{reproducible} interference patterns in the TOF images. This interference reflects phase coherence in the disordered BEC and is interpreted as \textit{speckle for matter waves}. At higher $V_D$, the TOF interference contrast diminishes while the \textit{in-situ} density fluctuations increase, signifying a fragmented ``granular" condensate with little phase coherence. $^1$Supported by NSF, ONR, NASA, Welch and Keck Foundations. $^2$Now at Purdue University.   

Effects of disorder on the interacting Fermi gases in one-dimensional optical lattices

Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to an harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a random potential on these ground-state phases. We employ density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder and harmonic trap are treated as external potentials. We find that disorder has two main effects: it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and it induces an anomaly in the compressibility at low density from quenching of percolation.   

Probing non local order parameters in highly correlated Bose insulators

Ground states of integer spin chains are known since the late 80's to sustain highly non local order described by infinite string operators of the spins. Such states defy the usual Landau theory description and can be considered simple prototypes of topological order. Recently we showed that spinless Bose insulators with nearest neighbor or longer range repulsion in one dimension can exhibit similar string order in terms of the boson density [1]. The tunability of cold atomic systems would allow much more flexibility in probing the non local order than spin systems do. For example the bosons can be tuned across a quantum phase transition between the exotic insulator, which we term Haldane insulator, and the usual Mott insulator. Investigating how the transition responds to external perturbations lends direct access to properties of the string order parameter. I will demonstrate this with several new results obtained from a field theoretic description of the phases and confirmed by numerical calculations using DMRG. Particularly revealing of the unusual character of the string order is the prediction that any external perturbation, which breaks the lattice inversion symmetry, would eliminate the distinction between the Haldane and Mott phases and allow a fully gapped adiabatic connection between them. This is remarkable given that neither phase involves spontaneous breaking of lattice inversion symmetry. We also predict that inter-chain tunneling destroys the direct phase transition between the two insulators by establishing an intermediate superfluid phase. Finally I will discuss how the new phases and phase transitions may be realized and probed in actual experiments with ultra cold atoms or polar molecules. \newline [1] E. G. Dalla Torre, E. Berg and E. Altman, Phys. Rev. Lett. 97, 260401 (2006)   

A Hexagonal Lattice Ion Trap for Quantum Simulation of Spin Models

Quantum simulations of one dimensional spin systems are being implemented by ions in linear Paul traps; however, a natural extension to two dimensional quantum spin simulations cannot be realized in the linear Paul trap geometry. Planar lattice traps not only offer the possibility for two dimensional simulations, but also hold two advantages over linear traps: first, neighboring ions in lattice traps have well defined and uniform spacings; second, quantum simulations with planar traps can be scaled up more easily than with linear traps. We develop a hexagonal lattice trap that allows vibrational coupling between ions due to their Coulomb repulsion, which is essential for effective spin-spin interaction. We present fabrication details, preliminary testing results, and a proposal for simulating geometrical spin frustration with three ions in a triangular configuration.   

Trapped-Ion Quantum Simulations of Spin Systems: From Two Qubits to Thousands

Due to the exponential growth of a quantum system's state-space with its size, the current technological limit for simulating the evolution of many-quantum-spin systems with classical computers (CC) is 36 spin-$\frac{1}{2}$ particles. While CC's cannot be scaled to meet the exponentially increased demand in computational resources, mapping the Hamiltonians of such problems onto that of a quantum simulator (QS) completely avoids this exponential scaling problem, allowing for efficient simulations of much larger systems. QS may be the first attainable application of quantum information processing, enabling exploration of parts of the phase space not accessible in the original system and possibly providing an exponential speedup of computations for even just a few tens of interacting qubits when compared to CC methods. Following the work of Porras and Cirac [Phys. Rev. Lett. 92, 207901-1 (2004)] we discuss the status of an experiment at Los Alamos for demonstrating a proof of principle QS of an Ising-like spin-spin interaction in a transverse magnetic field. We also discuss a novel architecture for microfabricated ion trap arrays geared toward enabling large scale QS and one-way quantum computing with potentially thousands of ions [arXiv:0711.0233].   

Quantitative determination of the Hubbard model phase diagram from optical lattice experiments: overcoming the singular nature of the thermodynamic limit

We propose an experiment to obtain the phase diagram of the fermionic Hubbard model, for any dimensionality, using cold atoms in optical lattices. It is based on measuring the total energy for a sequence of trap profiles. It combines finite-size scaling with an additional `finite-curvature scaling' necessary to reach the homogeneous limit. We illustrate its viability in the 1D case, simulating experimental data in the Bethe-Ansatz local density approximation. Including experimental errors, the filling corresponding to the Mott transition can be determined with better than 3 per cent accuracy. The main obstacle that our method overcomes is the singular nature of the thermodynamic limit of atom traps. We discuss this surprising phenomenon and describe a simpler experiment that could be used to demonstrate it.   

The p$_{x,y}$-orbital counterpart of graphene -- cold fermions in the honeycomb optical lattices

We study the ground states of cold atoms in the tight-binding bands built from p-orbitals on a two dimensional honeycomb optical lattice. The band structure includes two completely flat bands. Exact many-body ground states with on-site repulsion can be found at low particle densities, for both fermions and bosons. We find crystalline order at n=1/6 with a $\sqrt{3} \times \sqrt{3}$ structure breaking a number of discrete lattice symmetries. In fermionic systems, if the repulsion is strong enough, we find the bonding strength becomes \emph{dimerized} at n=1/2. Experimental signatures of crystalline order can be detected through the noise correlations in time of flight experiments.   

DMRG Studies for Strongly-Correlated Fermions on a Triangular Optical Lattice

Strongly-interacting fermions in a triangular lattice attract much attention because not only the interaction but also the geometrical frustration is expected to cause non-trivial behaviors. In solids, most of materials parameters, e.g., interaction strength between electrons (fermions), fermion density, and crystalline potential are almost fixed depending on the sample fabrication and our research area are restricted. In contrast, cold atom systems enable to study it systematically because some crucial parameters are precisely controllable. Thus, we expect that the cold Fermi atoms on a triangular lattice bring us information on both strongly-correlated and frustrated systems. In this study, we examine a system described by the triangular Hubbard model by using the parallel density-matrix renormalization group (DMRG). In addition, we also investigate effects of random potential made by speckle laser in the system. We evaluate the effects of frustration, strong interaction, and randomness simultaneously.   

Order parameter statistics at a quantum phase transition

Universality implies that at a second order phase transition the probability distribution of the order parameter takes a universal scaling form. This distribution is a natural way to characterize the quantum critical properties of ultracold atomic gases, since its histogram may be readily obtained by repeated `single-shot' measurements. In this work we obtain the exact scaling probability distribution for the simplest quantum critical point: that of the transverse field Ising model in 1D. Using a novel identity for the Ising model correlation functions, we map the problem to a particular case of the anisotropic Kondo model.