Theory of Nonlinear Matter Waves in Optical Lattices

Abstract

We consider several effects of the matter wave dynamics which can be observed in Bose-Einstein condensates embedded into optical lattices. For low-density condensates we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross-Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force, and lattice defects affect the nonlinear matter waves. *

Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices

“…For a discussion on the limits of applicability of this 1D reduction see [25]. Introducing the dimensionless variables…”

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

“…In BEC, it describes an external potential applied to the condensate. The potential can be localized (e.g., a single waveguide in nonlinear optics [39,32]), parabolic (e.g., a magnetic trap in BEC [1,31]) or periodic (e.g., a waveguide array or photonic crystal lattice in nonlinear optics [42]). …”

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

“…If this happens, the mode represents a small amplitude gap soliton governed by the nonlinear Schrödinger equation (see e.g. [3,9])…”

“…This is, in particular, the case of 2D and 3D lattices [6,8,11]. In order to explore such a possibility in the 1D model (1) we recall that the condition for the modulational [3,9] (also referred to as dynamical [20]) instability now reads M (σ) n χ (σ) n < 0. For the existence of small amplitude solitons this condition must be satisfied near at least one of the gap edges.…”

“…The phenomenon was explained [8] by the fact, that in 1D lattices there exists no threshold for exciting localized states, which at small amplitudes become matter gap-solitons (see e.g. [3,4,9]). Such a threshold appears in 2D and 3D systems.…”