Theoretical studies of Bose-Hubbard and discrete nonlinear Schrödinger models : Localization, vortices, and quantum-classical correspondence

Abstract

This thesis is mainly concerned with theoretical studies of two types of models: quantum mechanical Bose-Hubbard models and (semi-)classical discrete nonlinear Schrödinger (DNLS) models.Bose-Hubbard models have in the last few decades been widely used to describe Bose-Einstein condensates placed in periodic optical potentials, a hot research topic with promising future applications within quantum computations and quantum simulations. The Bose-Hubbard model, in its simplest form, describes the competition between tunneling of particles between neighboring potential wells ('sites') and their on-site interactions (can be either repulsive or attractive). We will also consider extensions of the basic models, with additional interactions and tunneling processes.While Bose-Hubbard models describe the behavior of a collection of particles in a lattice, the DNLS description is in terms of a classical field on each site. DNLS models can also be applicable for Bose-Einstein condensates in periodic potentials, but in the limit of many bosons per site, where quantum fluctuations are negligible and a description in terms of average values is valid. The particle interactions of the Bose-Hubbard models become nonlinearities in the DNLS models, so that the DNLS model, in its simplest form, describes a competition between on-site nonlinearity and tunneling to neighboring sites. DNLS models are however also applicable for several other physical systems, most notably for nonlinear waveguide arrays, another rapidly evolving research field.The research presented in this thesis can be roughly divided into two parts: 1) We have studied certain families of solutions to the DNLS model. First, we have considered charge flipping vortices in DNLS trimers and hexamers. Vortices represent a rotational flow of energy, and a charge flipping vortex is one where the rotational direction (repeatedly) changes. We have found that charge flipping vortices indeed exist in these systems, and that they belong to continuous families of solutions located between two stationary solutions.Second, we have studied discrete breathers, which are spatially localized and time-periodic solutions, in a DNLS models with the geometry of a ring coupled to an additional, central site. We found under which parameter values these solutions exist, and also studied the properties of their continuous solution families. We found that these families undergo different bifurcations, and that, for example, the discrete breathers which have a peak on one and two (neighboring) sites, respectively, belong to the same family below a critical value of the ring-to-centralv vi site coupling, but to separate families for values above it.2) Since Bose-Hubbard models can be approximated with DNLS models in the limit of a large number of bosons per site, we studied signatures of certain classical solutions and structures of DNLS models in the corresponding Bose-Hubbard models.These studies have partly focused on quantum lattice compactons. The corresponding classical lattice compac...

Flat bands and compactons in mechanical lattices

“…Moreover, linear stability analysis is performed numerically, in an approximate fashion. A survey of different results related to local-support solutions in the aforementioned physical systems is given in [39].…”