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 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
L with finite
N, where
L is the number of lattice sites and
N is the total particle number. An alternative approach valid for large
N with finite
L 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.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with
F the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of
F and in this range the deviation of
W from the Bogoliubov value is small. There is, however, a transition point
FT, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by
FT. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
Properties of bosonic atoms in small systems with a periodic quasi one-dimensional circular toroidal lattice potential subjected to rotation are examined by performing exact diagonalization in a truncated many body space. The expansion of the many-body Hamiltonian is considered in terms of the first band Bloch functions, and no assumption regarding restriction to nearest-neighbor hopping (tight-binding approximation) is involved. A finite size version of the zero temperature phase diagrams of Fisher et al. \cite{Fisher} is obtained and the results, in remarkable quantitative correspondence with the results available for larger systems, discussed. Ground state properties relating to superfluidity are examined in the context of two-fluid phenomenology. The basic tool, consisting of the intrinsic inertia associated with small rotation angular velocities in the lab frame, is used to obtain ground state `superfluid fractions' numerically. They are analytically associated with one-body, uniform solenoidal currents in the case of the adopted geometry. These currents are in general incoherent superpositions of contributions from each eigenstate of the associated reduced one-body densities, with the corresponding occupation numbers as weights. Full coherence occurs therefore only when only one eigenstate is occupied by all bosons. The obtained numerical values for the superfluid fractions remain small throughout the parameter region corresponding to the `Mott insulator to superfluid' transition, and saturate at unity only as the lattice is completely smoothed out.
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
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 spectrum and stationary states in a ring-shaped lattice potential in the context of ultracold atoms with attractive interatomic interactions. We determine analytical solutions in the absence of a lattice by mapping them to those for repulsive interactions, and then we numerically follow the transformation of those solutions as the lattice is introduced and strengthened. Several features emerge that are specific to negative nonlinearity, which include soliton branches detaching to create new ground states, gaps opening up at the bottom of the primary spectral branch, and multiple splitting and rejoining of some branches. We correlate the spectral features with the behavior of the density and phase of the corresponding eigenstates, and track them along branches and as various system parameters change. We find that the phase is sensitive to how a specific point in the spectrum is approached, particularly relevant at certain persistent gaps in the spectrum. The symmetry and stability properties are generally found to be opposite of that found for repulsive interactions.