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arXiv:cond-mat/0507492 · Physical Review A

Attractive ultracold bosons in a necklace optical lattice

Pierfrancesco Buonsante, Vittorio Penna, A. Vezzani

Published Oct 31, 2005• 49 citations
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Abstract

We study the ground-state properties of the Bose-Hubbard model with attractive interactions on an $M$-site one-dimensional periodic---necklacelike---lattice, whose experimental realization in terms of ultracold atoms is promised by a recently proposed optical trapping scheme, as well as by the control over the atomic interactions and tunneling amplitudes granted by well-established optical techniques. We compare the properties of the quantum model to a semiclassical picture based on a number-conserving $\mathrm{su}(M)$ coherent state, which results in a set of modified discrete nonlinear Schr\"odinger equations. We show that, owing to the presence of a correction factor ensuing from number conservation, the ground-state solution to these equations provides a remarkably satisfactory description of its quantum counterpart not only---as expected---in the weak-interaction, superfluid regime, but even in the deeply quantum regime of large interactions and possibly small populations. In particular, we show that in this regime, the delocalized, Schr\"odinger-cat-like quantum ground state can be seen as a coherent quantum superposition of the localized, symmetry-breaking ground state of the variational approach. We also show that, depending on the hopping to interaction ratio, three regimes can be recognized both in the semiclassical and quantum picture of the system.

DOI
10.1103/physreva.72.043620

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•Jan 1, 2010•0 citations
We study the ground-state properties of the Bose-Hubbard model with attractive interactions on an MMM-site one-dimensional periodic---necklacelike---lattice, whose experimental realization in terms of ultracold atoms is promised by a recently proposed optical trapping scheme, as well as by the control over the atomic interactions and tunneling amplitudes granted by well-established optical techniques. We compare the properties of the quantum model to a semiclassical picture based on a number-conserving su(M)\mathrm{su}(M)su(M) coherent state, which results in a set of modified discrete nonlinear Schr"odinger equations. We show that, owing to the presence of a correction factor ensuing from number conservation, the ground-state solution to these equations provides a remarkably satisfactory description of its quantum counterpart not only---as expected---in the weak-interaction, superfluid regime, but even in the deeply quantum regime of large interactions and possibly small populations. In particular, we show that in this regime, the delocalized, Schr"odinger-cat-like quantum ground state can be seen as a coherent quantum superposition of the localized, symmetry-breaking ground state of the variational approach. We also show that, depending on the hopping to interaction ratio, three regimes can be recognized both in the semiclassical and quantum picture of the system.
We consider a one-dimensional Bose–Hubbard model describing attractive ultracold bosons trapped in an M-site ring optical lattice within a variational su ( M )-coherent-state approach where the attractive-boson dynamics is reformulated in terms of (modified) discrete nonlinear Schrödinger equations. The delocalized, exact ground state of this model can be shown to be well represented as a Schrödinger-cat-like state by coherently superimposing the space-localized ground states obtained by our variational approach. In this paper we focus on two aspects of such space-localized (semiclassical) ground states. First, we prove that the (space-like) su ( M ) coherent states involved in our variational approach can be recast as momentum-like su ( M ) coherent states. Based on this property we show that the momentum-space boson distribution of Schrödinger-cat-like states exhibits unexpected features. Secondly, we study the low-energy modes of the discrete nonlinear Schrödinger equations showing that, for sufficiently small lattices, the ground state of the system exhibits a transition controlled by the model parameters from a single-pulse (localized) soliton mode to the superfluid (delocalized) mode. The non-trivial dependence of this effect on the lattice size is also discussed.
We consider the Bose-Hubbard model describing attractive bosonic particles hopping across the sites of a translation-invariant lattice and compare the relevant ground-state properties with those of the corresponding symmetry-breaking semiclassical nonlinear theory. The introduction of a suitable measure allows us to highlight many correspondences between the nonlinear theory and the inherently linear quantum theory, characterized by the well-known self-trapping phenomenon. In particular we demonstrate that the localization properties and bifurcation pattern of the semiclassical ground state can be clearly recognized at the quantum level. Our analysis highlights a finite-number effect.
We study the ground state of the attractive one-dimensional Bose-Hubbard model, and in particular the nature of the crossover between the weak interaction and strong interaction regimes for finite system sizes. Indicator properties such as the gap between the ground and first excited energy levels, and the incremental ground-state wave function overlaps are used to locate different regimes. Using mean-field theory we predict that there are two distinct crossovers connected to spontaneous symmetry breaking of the ground state. The first crossover arises in an analysis valid for large LLL with finite NNN, where LLL is the number of lattice sites and NNN is the total particle number. An alternative approach valid for large NNN with finite LLL yields a second crossover. For small system sizes we numerically investigate the model and observe that there are signatures of both crossovers. We compare with exact results from Bethe ansatz methods in several limiting cases to explore the validity for these numerical and mean-field schemes. The results indicate that for finite attractive systems there are generically three ground-state phases of the model.
We study the ground-state properties of ultracold bosons in an optical lattice in the regime of strong interactions. The system is described by a non-standard Bose–Hubbard model with both occupation-dependent tunneling and on-site interaction. We find that for sufficiently strong coupling, the system features a phase transition from a Mott insulator with one particle per site to a superfluid of spatially extended particle pairs living on top of the Mott background—instead of the usual transition to a superfluid of single particles/holes. Increasing the interaction further, a superfluid of particle pairs localized on a single site (rather than being extended) on top of the Mott background appears. This happens at the same interaction strength where the Mott-insulator phase with two particles per site is destroyed completely by particle–hole fluctuations for arbitrarily small tunneling. In another regime, characterized by weak interaction but high occupation numbers, we observe a dynamical instability in the superfluid excitation spectrum. The new ground state is a superfluid, forming a two-dimensional (2D) slab, localized along one spatial direction that is chosen spontaneously.
We study the quantum ground-state phases of the one-dimensional disordered Bose-Hubbard model with attractive interactions, realized by a chain of superconducting transmon qubits or cold atoms. We map the phase diagram using perturbation theory and exact diagonalization. Compared to the repulsive Bose-Hubbard model, the quantum ground-state behavior is dramatically different. At strong disorder of the on-site energies, all the bosons localize into the vicinity of a single site, contrary to the Bose glass behavior of the repulsive model. At weak disorder, depending on hopping, the ground state is either superfluid or a WWW state, which is a multisite and multiparticle entangled superposition of states where all the bosons occupy a single site. We show that the robustness of the WWW phase against disorder diminishes as the total number of bosons increases.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.