Formation of nonlinear modes in one-dimensional quasiperiodic lattices with a mobility edge

TL;DR

This paper explores how nonlinear localized modes form in one-dimensional quasiperiodic lattices with a mobility edge, revealing bifurcation behaviors and differences across the mobility edge, with implications for Bose-Einstein condensates and photonic lattices.

Contribution

It provides a detailed analysis of the bifurcation mechanisms of nonlinear modes in quasiperiodic potentials, highlighting the role of the mobility edge and phase shifts in mode formation.

Findings

Nonlinear modes undergo symmetry-breaking and saddle-node bifurcations.

Mode properties differ depending on their position relative to the mobility edge.

Localized modes form through cascades of saddle-node bifurcations in generic quasiperiodic potentials.

Abstract

We investigate the formation of steady states in one-dimensional Bose-Einstein condensates of repulsively interacting ultracold atoms loaded into a quasiperiodic potential created by two incommensurate periodic lattices. We study the transformations between linear and nonlinear modes and describe the general patterns that govern the birth of nonlinear modes emerging in spectral gaps near band edges. We show that nonlinear modes in a symmetric potential undergo both symmetry-breaking pitchfork bifurcations and saddle-node bifurcations, mimicking the prototypical behaviors of symmetric and asymmetric double-well potentials. The properties of the nonlinear modes differ for bifurcations occurring below and above the mobility edge. In the generic case, when the quasiperiodic potential consists of two incommensurate lattices with a nonzero phase shift between them, the formation of localized modes in the spectral gaps occurs through a cascade of saddle-node bifurcations. Because of the analogy between the Gross-Pitaevskii equation and the nonlinear Schr\"odinger equation, our results can also be applied to optical modes guided by quasiperiodic photonic lattices.