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Physical Review B

Boundary-induced energy localization in a nonlinear quantum lattice

Vincent Pouthier

Published Dec 19, 2007• 6 citations
Publisher

Abstract

The dynamics of $v$ 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 $v$ 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 $v$ 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.

DOI
10.1103/physrevb.76.224302

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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.
We investigate the dynamics of two bosons trapped in an infinite one-dimensional optical lattice potential within the framework of the Bose-Hubbard model and derive an exact expression for the wave function at finite time. As initial condition we chose localized atoms that are separated by a distance of ddd lattice sites and carry a center-of-mass quasimomentum. An initially localized pair (d=0d=0d=0) is found to be more stable as quantified by the pair probability (probability to find two atoms at the same lattice site) when the interaction and/or the center-of-mass quasimomentum is increased. For initially separated atoms ($d\ensuremath{\ne}0$) there exists an optimal interaction strength for pair formation. Simple expressions for the wave function, the pair probability, and the optimal interaction strength for pair formation are computed in the limit of infinite time. Whereas the time-dependent wave function differs for values of the interaction strength that differ only by the sign, important observables such as the density and the pair probability do not. With a symmetry analysis this behavior is shown to extend to the NNN-particle level and to fermionic systems. Our results provide a complementary understanding of the recently observed [Winkler et al., Nature (London) 441, 853 (2006)] dynamical stability of atom pairs in a repulsively interacting lattice gas.
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We investigate the existence of quantum quasi-phase-transitions for an ensemble of ultracold bosons in a one-dimensional optical lattice performing exact diagonalizations of the Bose-Hubbard Hamiltonian. When an external parabolic potential is added to the system quasi-phase-transitions are induced by the competition of on-site mean-field energy, hopping energy, and energy offset among lattice sites due to the external potential and lead to the coexistence of regions of particle localization and delocalization in the lattice. We clarify the microscopic mechanisms responsible for these quasi-phase-transitions as a function of the depth of the external potential when the on-site mean-field energy is large compared to the hopping energy. In particular, we show that a model Hamiltonian involving a few Fock states can describe the behavior of energy gap, mean particle numbers per site, and number fluctuations per site almost quantitatively. The role of symmetry on the gap as a function of the depth of the external trapping potential is elucidated. We discuss possible experimental signatures of quasi-phase-transitions studying the single-particle density matrix and explain microscopically the occurrence of local maxima in the momentum distribution. The role of a thermal population of the excited states on the momentum distribution is discussed.
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.
We discuss the possibility of exponential quantum localization in systems of ultracold bosonic atoms with repulsive interactions in open optical lattices without disorder. We show that exponential localization occurs in the maximally excited state of the lowest energy band. We establish the conditions under which the presence of the upper energy bands can be neglected, determine the successive stages and the quantum phase boundaries at which localization occurs, and discuss schemes to detect it experimentally by visibility measurements. The discussed mechanism is a particular type of quantum localization that is intuitively understood in terms of the interplay between nonlinearity and a bounded energy spectrum.
We extend the self-energy functional theory to the case of interacting lattice bosons in the presence of symmetry breaking and quenched disorder. The self-energy functional we derive depends only on the self-energies of the disorder-averaged propagators, allowing for the construction of general non-perturbative approximations. Using a simple single-site reference system with only three variational parameters, we are able to reproduce numerically exact quantum Monte Carlo (QMC) results on local observables of the Bose–Hubbard model with box disorder with high accuracy. At strong interactions, the phase boundaries are reproduced qualitatively but shifted with respect to the ones observed with QMC due to the extremely low condensate fraction in the superfluid phase. Deep in the strongly-disordered weakly-interacting regime, the simple reference system employed is insufficient and no stationary solutions can be found within its restricted variational subspace. By systematically analyzing thermodynamical observables and the spectral function, we find that the strongly interacting Bose glass is characterized by different regimes, depending on which local occupations are activated as a function of the disorder strength. We find that the particles delocalize into isolated superfluid lakes over a strongly localized background around maximally-occupied sites whenever these sites are particularly rare. Our results indicate that the transition from the Bose glass to the superfluid phase around unit filling at strong interactions is driven by the percolation of superfluid lakes which form around doubly occupied sites.