We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions.

We present a numerical study of dispersion manipulation and formation of matter-wave gap solitons in a Bose-Einstein condensate trapped in an optical superlattice. We demonstrate a method for controlled generation of matter-wave gap solitons in a stationary lattice by using an interference pattern of two condensate wavepackets, which mimics the structure of the gap soliton near the edge of a spectral band. The efficiency of this method is compared with that of gap soliton generation in a moving lattice recently demonstrated experimentally by Eiermann et al. [Phys. Rev. Lett., 92, 230401 (2004)]. We show that, by changing the relative depths of the superlattice wells, one can fine-tune the effective dispersion of the matter waves at the edges of the mini-gaps of the superlattice Bloch-wave spectrum and therefore effectively control both the peak density and the spatial width of the emerging gap solitons.

We analyze the Floquet-Bloch matter-wave spectrum of Bose-Einstein condensates loaded into single-periodic optical lattices and double-periodic superlattices. In the framework of the Gross-Pitaevskii equation, we describe the structure and analyze the mobility properties of dark matter-wave solitons residing on the background of extended nonlinear Bloch-type states. We demonstrate that interaction between dark solitons can be effectively controlled in optical superlattices.

We investigate the feasibility of a particular scheme for generating macroscopic quantum superposition states in two-species dilute gas Bose-Einstein condensates. The scheme utilizes two-body interactions and Josephson coupling between the species. We report numerical studies that extend a previous two-mode model to include dissipation and extra modes.

In this work we systematically investigate the condensate properties, superfluid properties and quantum phase transitions in interacting Bose gases trapped in disordered optical potentials. We numerically solve the Bose-Hubbard Hamiltonian exactly for different: (a) types of disorder, (b) disorder strengths, and (c) interatomic interactions. The three types of disorder studied are: quasiperiodic disorder, uniform random disorder and random speckle-type disorder. We find that the Bose glass, as identified by Fisher et al. [Phys. Rev. B 40, 546 (1989) [2] were able to describe an insulating state in disordered lattices which exists due to the cooperative effect of repulsive interactions and disorder. This 'Bose glass' (BG) is gapless and compressible. Taking an alternative approach with a microscopic low-density model involving a dilute low-temperature hard-sphere gas with random scatterers in the limit of weak disorder, Huang and Meng [3] and Giorgini et al. [4] found that the superfluid is more depleted by the disorder than the condensate. That is, there exists BEC without superfluidity.It was shown previously using the disordered BoseHubbard model that it is possible to obtain a normal condensate fraction for certain parameters [5] and that there is a normal condensate fraction in the Bose glass phase for quasiperiodic disorder [6]. In this work we show that the parameter regime where this occurs can be identified with the BG region described by Fisher et al. [2] by showing that the BG phase and the normal condensate phase occur together as we change the interaction strength, disorder strength and the type of disorder. We also find that in the BG regime the spatial correlations go to zero with distance but remain significant over multiple lattice sites.Our second goal is to show how the properties of 'dirty' Bose gases change with: (a) the type of disorder, (b) the disorder strength, and (c) the interatomic interaction strength, especially with respect to the existence of condensate without superfluidity. We show that there are significant differences in the phase diagram as we change the type of disorder and the nature of the differences depend on the type of disorder.To date the majority of the work on 'dirty' bosons has focused on liquid 4 He in porous materials such as Vycor glass or aerogel [7]. In recent years however, weaklyinteracting dilute Bose gases, especially those trapped in optical dipole potentials, have become increasingly important in this field [8,9,10,11]. An important advantage of these systems is their flexibility and their ease of control over important experimental parameters. This can be seen in recent experiments in which a variety of different types of disordered optical potentials of controllable strength were studied e.g., quasiperiodic lattices [11], speckle potentials [8,9], and lattices with superimposed disorder [10]. These recent experimental developments make an exploration of how condensate and superfluid properties change with the properties of the disorder increasingly rele...