Theory of Nonlinear Matter Waves in Optical Lattices

TL;DR

This paper develops theoretical models for nonlinear matter wave dynamics in Bose-Einstein condensates within optical lattices, analyzing stability, solitons, and effects of external influences, providing insights into complex wave behaviors in these systems.

Contribution

It introduces approximate evolution equations and lattice models for matter waves, including the tight-binding approximation, and explores various nonlinear wave phenomena and their responses to external factors.

Findings

Derivation of approximate evolution equations for low-density condensates

Analysis of modulational instability and soliton solutions

Effects of Feshbach resonance, linear forces, and lattice defects on wave dynamics

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.