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 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 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 investigated the spatial phase separation of the two components forming a bosonic mixture distributed in a four-well lattice with a ring geometry. We studied the ground state of this system, described by means of a binary Bose–Hubbard Hamiltonian, by implementing a well-known coherent-state picture which allowed us to find the semi-classical equations determining the distribution of boson components in the ring lattice. Their fully analytic solutions, in the limit of large boson numbers, provide the boson populations at each well as a function of the interspecies interaction and of other significant model parameters, while allowing to reconstruct the non-trivial architecture of the ground-state four-well phase diagram. The comparison with the L-well (L=2,3) phase diagrams highlights how increasing the number of wells considerably modifies the phase diagram structure and the transition mechanism from the full-mixing to the full-demixing phase controlled by the interspecies interaction. Despite the fact that the phase diagrams for L=2,3,4 share various general properties, we show that, unlike attractive binary mixtures, repulsive mixtures do not feature a transition mechanism which can be extended to an arbitrary lattice of size L.
We study a quasi-one-dimensional attractive Bose gas confined in an optical lattice with a superimposed harmonic potential by analyzing the one-dimensional Bose-Hubbard Hamiltonian of the system. Starting from the three-dimensional many-body quantum Hamiltonian, we derive strong inequalities involving the transverse degrees of freedom under which the one-dimensional Bose-Hubbard Hamiltonian can be safely used. To have a reliable description of the one-dimensional ground state, which we call a quantum bright soliton, we use the density-matrix-renormalization-group (DMRG) technique. By comparing DMRG results with mean-field (MF) ones, we find that beyond-mean-field effects become relevant by increasing the attraction between bosons or by decreasing the frequency of the harmonic confinement. In particular, we find that, contrary to the MF predictions based on the discrete nonlinear Schr\"odinger equation, average density profiles of quantum bright solitons are not shape-invariant. We also use the time-evolving-block-decimation method to investigate the dynamical properties of bright solitons when the frequency of the harmonic potential is suddenly increased. This quantum quench induces a breathing mode whose period crucially depends on the final strength of the superimposed harmonic confinement.
The ground-state properties of attractive bosons trapped in a ring lattice including a single attractive potential well with an adjustable depth are investigated. The energy spectrum is reconstructed both in the strong-interaction limit and in the superfluid regime within the Bogoliubov picture. The analytical results thus obtained are compared with those found numerically from the exact Hamiltonian, in order to identify the regions in the parameter space where this picture is effective. The single potential introduced is the simplest way to break the translational symmetry and to observe, through a completely analytical approach, how the absence of symmetry affects the properties of the low-excited eigenstates of the system. This model gives a first insight into the properties of systems including more complex potentials.
Abstract The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.
Motivated by recent experiments on trapped ultracold bosonic atoms in an optical lattice potential, we consider the nonequilibrium dynamic properties of such bosonic systems for a number of experimentally relevant situations. When the number of bosons per lattice site is large, there is a wide parameter regime where the effective boson interactions are strong, but the ground state remains a superfluid (and not a Mott insulator): we describe the conditions under which the dynamics in this regime can be described by a discrete Gross-Pitaevskii equation. We describe the evolution of the phase coherence after the system is initially prepared in a Mott insulating state, and then allowed to evolve after a sudden change in parameters places it in a regime with a superfluid ground state. We also consider initial conditions with a ``\ensuremath{\pi} phase'' imprint on a superfluid ground state (i.e., the initial phases of neighboring wells differ by \ensuremath{\pi}), and discuss the subsequent appearance of the density wave order and ``Schr\"odinger cat,'' i.e., macroscopic quantum interference, states.