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arXiv:1003.0426 · Physical Review A

Quantum signatures of the self-trapping transition in attractive lattice bosons

Pierfrancesco Buonsante, Vittorio Penna, A. Vezzani

Published Oct 19, 2010• 31 citations
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Abstract

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.

DOI
10.1103/physreva.82.043615

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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 illustrate how dynamical transitions in nonlinear semiclassical models can be recognized as phase transitions in the corresponding---inherently linear---quantum model, where, in a statistical-mechanics framework, the thermodynamic limit is realized by letting the particle population go to infinity at fixed size. We focus on lattice bosons described by the Bose-Hubbard (BH) model and discrete self-trapping (DST) equations at the quantum and semiclassical levels, respectively. After showing that the Gaussianity of the quantum ground states is broken at the phase transition, we evaluate finite-population effects by introducing a suitable scaling hypothesis; we work out the exact value of the critical exponents and provide numerical evidence confirming our hypothesis. Our analytical results rely on a general scheme obtained from a large-population expansion of the eigenvalue equation of the BH model. In this approach the DST equations resurface as solutions of the zeroth-order problem.
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 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 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.
An analytical insight into the symmetry-breaking mechanisms underlying the transition from Josephson to self-trapping regimes in Bose-Einstein condensates is presented. We obtain expressions for the ground-state properties of the system of a gas of attractive bosons modeled by a two-site Bose-Hubbard Hamiltonian with an external bias. Simple formulas are found relating the appearance of fragmentation in the condensate with the large quantum fluctuations of the population imbalance occurring in the transition from the Josephson to the self-trapped regime.
The dynamics of vvv bosons initially created on the same site of a finite-size lattice is analyzed according to a Bose version of the Hubbard model. For a boson number greater than 2, it is shown that the interplay between symmetry breaking and nonlinearity favors the occurrence of localized bound states. In a localized state, the vvv bosons are trapped close to each other and they behave as a single particle whose wave function is exponentially localized near a lattice side. Consequently, the creation of vvv bosons on a side site mainly excites a localized state so that the main part of the energy stays localized on the excited site over an infinite time scale.