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AIP conference proceedings

Phase Diagram and Visibility of Optically Trapped Bosons

Richard T. Scalettar

Published Jan 1, 2006• 0 citations
Publisher

Abstract

Over the last several years, the boson‐Hubbard Hamiltonian has been widely used as an effective model of the physics of ultra‐cold optically trapped atoms. In this paper we will review its equilibrium properties, in one dimension, both in the absence and in the presence of an external potential. Then we will describe some new results for the visibility V of the interference pattern obtained after the free expansion of a gas. We will show that the evolution of V exhibits kinks with increasing interaction strength U/t, replicating phenomena seen experimentally. The behavior of the density profiles ni correlates well with the features in V. These profiles reveal an unexpected ‘freezing’ — a range of interaction strengths in which they remain unchanged. Finally, we show that V exhibits an increase with U/t signaling when a central ni = 2 Mott region melts and coherence develops between two superfluid regions with local densities 1 < ni < 2.

DOI
10.1063/1.2178045

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Over the last several years, the boson‐Hubbard Hamiltonian has been widely used as an effective model of the physics of ultra‐cold optically trapped atoms. In this paper we will review its equilibrium properties, in one dimension, both in the absence and in the presence of an external potential. Then we will describe some new results for the visibility V of the interference pattern obtained after the free expansion of a gas. We will show that the evolution of V exhibits kinks with increasing interaction strength U/t, replicating phenomena seen experimentally. The behavior of the density profiles ni correlates well with the features in V. These profiles reveal an unexpected ‘freezing’ — a range of interaction strengths in which they remain unchanged. Finally, we show that V exhibits an increase with U/t signaling when a central ni = 2 Mott region melts and coherence develops between two superfluid regions with local densities 1 < ni < 2.
We study the phase coherence and visibility of trapped atomic condensates on one-dimensional optical lattices, by means of quantum Monte Carlo simulations. We obtain structures in the visibility similar to the kinks recently observed experimentally by Gerbier et al. [Phys. Rev. Lett. 95, 050404 (2005); 10.1103/PhysRevLett.95.050404cond-mat/0507087]. We examine these features in detail and offer a connection to the evolution of the density profiles as the depth of the lattice is increased. Our simulations reveal that, as the interaction strength U is increased, the evolution of superfluid and Mott-insulating domains stall for finite intervals of U. The density profiles do not change with increasing U. We show here that in one dimension the visibility provides unequivocal signatures of the melting of Mott domains with densities larger than 1.
We investigate the phase coherence properties of ultracold Bose gases in optical lattices, with special emphasis on the Mott insulating phase. We show that phase coherence on short length scales persists even deep in the insulating phase, preserving a finite visibility of the interference pattern observed after free expansion. This behavior can be attributed to a coherent admixture of particle-hole pairs to the perfect Mott state for small but finite tunneling. In addition, small but reproducible kinks are seen in the visibility, in a broad range of atom numbers. We interpret them as signatures for density redistribution in the shell structure of the trapped Mott insulator.
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 compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasiperiodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wavelength is incommensurate with the main lattice wavelength. After discussing the conditions under which the model can be realized experimentally, the study of the density vs the chemical potential curves for a nontrapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density-wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth V2{V}_{2}V2​ in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott-insulating phase is found at finite V2{V}_{2}V2​. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The latter feature can be investigated by looking at the expansion of the cloud after releasing the trap.
We study cold atomic gases with a contact interaction and confined into one-dimension. Crossing the confinement induced resonance the correlation between the bosons increases, and introduces an effective range for the interaction potential. Using the mapping onto the sine-Gordon model and a Hubbard model in the strongly interacting regime allows us to derive the phase diagram in the presence of an optical lattice. We find the appearance of a phase transition from a Luttinger liquid with algebraic correlations into a crystalline phase with a particle on every second lattice site.
We study cold atomic gases with a contact interaction and confined into one-dimension. Crossing the confinement induced resonance the correlation between the bosons increases, and introduces an effective range for the interaction potential. Using the mapping onto the sine-Gordon model and a Hubbard model in the strongly interacting regime allows us to derive the phase diagram in the presence of an optical lattice. We demonstrate the appearance of a phase transition from a Luttinger liquid with algebraic correlations into a crystalline phase with a particle on every second lattice site.