We study the ground state properties of the Bose-Hubbard model with attractive interactions on a M -site one-dimensional periodic -necklace-like -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 ) coherent state, which results into a set of modified discrete nonlinear Schrödinger 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, Schrödinger-cat-like quantum ground state can be seen as a coherent quantum superposition of the localized, symmetry-breaking groundstate 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. PACS numbers: 03.75.Lm 05.30.Jp, 03.65.Sq 31.15.Pf I. OVERVIEWOwing to the impressive progress in experimental techniques, ultracold neutral atoms trapped in optical lattices are nowadays widely recognized as a versatile toolbox bringing into reality ideal models of condensed matter physics [1]. A prominent example in this respect is no doubt the Bose-Hubbard model, originally introduced to sketch the physics of superfluid helium in porous media [2] and subsequently shown to be realizable in terms of optically trapped ultracold Bosonic atoms [3,4]. This model, describing interacting Bosonic particles hopping across the M sites of a discrete structure, is characterized by a Hamiltonian of the forma † m and n m = a † m a m are on-site Bosonic operators, U measures the strength of the (on-site) bosonboson interaction, T is the hopping amplitude across neighboring sites and J is the so-called adjacency matrix, describing the lattice topology. Its generic entry, J mm ′ , equals 1 if the sites m and m ′ are adjacent, and 0 otherwise. Parameters U and T are directly related to well defined experimental quantities, i.e. the scattering length of the Bosonic atoms and the intensity of the laser beams giving rise to the optical lattice, respectively [3]. The possibility of tuning the lattice strength over a wide range of values played a fundamental role in the experimental observation [4] of the superfluid-insulator quantum phase transition predicted for Hamiltonian (1) in the case of repulsive interaction [2]. Further aspects of versatility of optically trapped ultracold atoms lie in the possibil-ity of tuning the atomic scattering length, and hence the boson-boson interaction, via Fe...
The dynamics of the three coupled bosonic wells (trimer) containing N bosons is investigated within a standard (mean-field) semiclassical picture based on the coherent-state method. Various periodic solutions (configured as π-like, dimerlike and vortex states) representing collective modes are obtained analitically when the fixed points of trimer dynamics are identified on the N =const submanifold in the phase space. Hyperbolic, maximum and minimum points are recognized in the fixed-point set by studying the Hessian signature of the trimer Hamiltonian.The system dynamics in the neighbourhood of periodic orbits (associated to fixed points) is studied via numeric integration of trimer motion equations thus revealing a diffused chaotic behavior (not excluding the presence of regular orbits), macroscopic effects of population-inversion and selftrapping. In particular, the behavior of orbits with initial conditions close to the dimerlike periodic orbits shows how the self-trapping effect of dimerlike integrable subregimes is destroyed by the presence of chaos.
Quantum dynamics of the Bose-Hubbard Model is investigated through a semiclassical hamiltonian picture provided by the Time-Dependent Variational Principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ll 1$. The phase diagram is in good agreement with Quantum Monte Carlo results and third order strong coupling perturbative expansion.Comment: 4 pages Revtex, 2 figures .eps, to be published in PR
We show that a site-dependent mean-field approach captures the quantum phases of the disordered Bose-Hubbard model commonly adopted to describe ultracold bosons in random optical lattice potentials. The different phases, namely superfluid, Mott insulator, Bose-glass and -- at finite temperature -- normal fluid, are characterized by means of the superfluid and condensate fractions, and compressibility of the system. We point out that both the boundaries of the Mott lobes and the nature of the phase surrounding them are related to the spectral features of a purely off-diagonal non-interacting Anderson model. We compare our results to other works.Comment: 5 pages, 4 figures, added some references and reformulated some sentence
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