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Physical Review A

Mott-insulator shells in the three-dimensional Bose-Hubbard model with harmonic confinement

Makoto Yamashita, Michael W. Jack

Published Feb 6, 2009• 8 citations
Publisher

Abstract

Ultracold atom experiments offer unprecedented potential for the quantitative comparison of quantum many-body effects with theory. This comparison is hindered by the lack of numerical methods capable of dealing with these large inhomogeneous three-dimensional systems. In this paper we demonstrate the applicability of a highly efficient numerical method based on the Gutzwiller approximation for calculating the ground state of the Bose-Hubbard model in the regimes of the recent cold atom experiments. The numerical method is applied to the experimental regime employed by Campbell et al. [Science 314, 281 (2006)], where Mott-insulator shells were directly imaged via atomic clock shifts. We calculate several quantities that are closely related to the current experimental measurement methods: average number distributions, momentum distributions, two-photon spectra, and three-body recombination loss rates. Our results shed light on a number of features of the experimental observations and clarify both the crossover behavior around the superfluid--Mott-insulator transition point and the experimentally observed three-body decay rate of the Mott-insulator shells.

DOI
10.1103/physreva.79.023609

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Ultracold atom experiments offer unprecedented potential for the quantitative comparison of quantum many-body effects with theory. This comparison is hindered by the lack of numerical methods capable of dealing with these large inhomogeneous three-dimensional systems. In this paper we demonstrate the applicability of a highly efficient numerical method based on the Gutzwiller approximation for calculating the ground state of the Bose-Hubbard model in the regimes of the recent cold atom experiments. The numerical method is applied to the experimental regime employed by Campbell et al. [Science 314, 281 (2006)], where Mott-insulator shells were directly imaged via atomic clock shifts. We calculate several quantities that are closely related to the current experimental measurement methods: average number distributions, momentum distributions, two-photon spectra, and three-body recombination loss rates. Our results shed light on a number of features of the experimental observations and clarify both the crossover behavior around the superfluid--Mott-insulator transition point and the experimentally observed three-body decay rate of the Mott-insulator shells.
The Bose-Hubbard model describes the physics of a system of bosonic ultracold atoms in an optical lattice, in which a phase transition is present between a superfluid phase and a Mott insulator one. \nThe exact solution of this Hamiltonian is only feasible to find the ground-state of small systems, while other techniques (as mean-field schemes or quantum Monte Carlo) are necessary to study systems of larger size. \n \nAs a first application, we study the trapped model - relevant for the comparison with current experiments - through an inhomogeneous mean-field scheme. We describe some signatures of the phase crossover between superfluid and Mott insulator. In particular, the visibility of the quasimomentum distribution shows some kinks as a function of the lattice depth; we describe these features and we link them with the ones observed in other works in the literature. \n \nAs a second application, we use quantum Monte Carlo techniques to study the one-dimensional Bose-Hubbard model with long-range interactions and we focus on the appearance of the Haldane insulating phase, distinguishable from the Mott one through the presence of non-local hidden order. \nNon-local correlation functions are also used to describe the difference between the superfluid phase and the Mott insulator one.
We present an extensive study of Mott insulator (MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with harmonic traps. For this we apply the inhomogeneous mean-field theory developed by Sheshadri et al. [Phys. Rev. Lett. 75, 4075 (1995)]. Our results for the BH model with one type of spinless bosons agree quantitatively with quantum Monte Carlo simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also extend our inhomogeneous mean-field theory to study BH models with harmonic traps and (a) two species of bosons or (b) spin-1 bosons. With two species of bosons, we obtain rich phase diagrams with a variety of SF and MI phases and associated shells when we include a quadratic confining potential. For the spin-1 BH model, we show, in a representative case, that the system can display alternating shells of polar SF and MI phases, and we make interesting predictions for experiments in such systems.
We study the ground-state properties of spin-1 bosonic atoms trapped in a one-dimensional optical lattice with harmonic confinement. We have developed a highly efficient numerical method based on the Gutzwiller approximation to clarify the inhomogeneous magnetic phases induced by the harmonic confining potential. A quantum phase transition from a superfluid to a Mott insulator is examined for the respective cases with antiferromagnetic and ferromagnetic interactions. It is shown that the profile of local spins over the lattice sites changes as a function of the interaction strength forming the characteristic structures, which closely correlates with the average number of atoms at the sites. For antiferromagnetic cases, we find that there is a crossover from a superfluid polar condensate to Mott-insulator domains consisting of either a spin-singlet pair or a single spin at each site.
We present an experimental measurement of the ensemble averages of both the kinetic and interaction energies of the three-dimensional Bose-Hubbard model at finite temperature and various optical lattice depths across weakly to strongly interacting regimes, for an almost unit filling factor within single-band tight-binding approximation. The kinetic energy is obtained through Fourier transformation of a time-of-flight signal, and the interaction energy is measured using a newly developed atom-number-projection spectroscopy technique, by exploiting an ultranarrow optical transition of two-electron atoms. The obtained experimental results can be used as benchmarks for state-of-the-art numerical methods of quantum many-body theory. As an illustrative example, we compare the measured energies with numerical calculations involving the Gutzwiller and cluster-Gutzwiller approximations, assuming realistic trap potentials and particle numbers at nonzero entropy (finite temperature); we obtain good agreement without fitting parameters. We also discuss the possible application of this method to temperature estimations for atoms in optical lattices using the thermodynamic relation. This study offers a unique advantage of cold atom system for ``quantum simulators.''
Cold atoms in periodic potentials are versatile quantum systems for implementing simple models prevalent in condensed matter theory. Here we realize the 2D Bose-Hubbard model by loading a Bose-Einstein condensate into an optical lattice, and study the resulting Mott insulator. The measured momentum distributions agree quantitatively with theory (no adjustable parameters). In these systems, the Mott insulator forms in a spatially discrete shell structure which we probe by focusing on correlations in atom shot noise. These correlations show a marked dependence on the lattice depth, consistent with the changing size of the insulating shell expected from simple arguments.
The competition between short-range and cavity-mediated infinite-range interactions in a cavity-boson system leads to the existence of a superfluid phase and a Mott-insulator phase within the self-organized regime. In this work, we quantitatively compare the steady-state phase boundaries of this transition measured in experiments and simulated using the Multiconfigurational Time-Dependent Hartree Method for Indistinguishable Particles. To make the problem computationally feasible, we represent the full system by the exact many-body wave function of a two-dimensional four-well potential. We argue that the validity of this representation comes from the nature of both the cavity-atomic system and the Bose-Hubbard physics. Additionally, we show that the chosen representation only induces small systematic errors, and that the experimentally measured and theoretically predicted phase boundaries agree reasonably well. We thus demonstrate a new approach for the quantitative numerical modeling for the physics of the superfluid--Mott-insulator phase boundary.