Hofstadter butterflies in nonlinear Harper lattices, and their optical realizations

Ofer Manela,Demetrios N Christodoulides,Detlef Kip,Mordechai Segev

doi:10.1088/1367-2630/12/5/053017

Abstract

The ubiquitous Hofstadter butterfly describes a variety of systems characterized by incommensurable periodicities, ranging from Bloch electrons in magnetic fields and the quantum Hall effect to cold atoms in optical lattices and more. Here, we introduce nonlinearity into the underlying (Harper) model and study the nonlinear spectra and the corresponding extended eigenmodes of nonlinear quasiperiodic systems. We show that the spectra of the nonlinear eigenmodes form deformed versions of the Hofstadter butterfly and demonstrate that the modes can be classified into two families: nonlinear modes that are a ‘continuation’ of the linear modes of the system and new nonlinear modes that have no counterparts in the linear spectrum. Finally, we propose an optical realization of the linear and nonlinear Harper models in transversely modulated waveguide arrays, where these Hofstadter butterflies can be observed. This work is relevant to a variety of other branches of physics beyond optics, such as disorder-induced localization in ultracold bosonic gases, localization transition processes in disordered lattices, and more.

Highlights

The study of wave propagation in nonlinear periodic or quasiperiodic structures, both discrete and continuous, deals with many fascinating issues [1]: how fast does a localized wavepacket spread [2] and under what circumstances will it self-localize to form a soliton [3]? How does
New Journal of Physics 12 (2010) 053017 1367-2630/10/053017+09$30.00 the energy carried by the wavepacket spread among the linear modes of the system [4, 5, 37]? What are the nonlinear modes of the system—both localized and extended, and what are their stability properties [3, 6, 7, 38]
The ubiquitous Hofstadter butterfly is known to describe a variety of incommensurate systems ranging from Bloch electrons in uniform magnetic fields [8]–[13] and cold atoms in optical lattices [14], to the Quantum Hall effect [15]–[17], and microwaves [18] and acoustic wave systems [19]

Summary

Introduction

Perhaps the best known case of a linear wave system with two incommensurable periodicities is the one described by the Harper model, whose set of eigenvalues comprise the Hofstadter butterfly. How does the nonlinearity affect the linear spectrum of nonlinear eigenmodes—the Hofstadter butterfly? We show that the sets of eigenvalues of the nonlinear Harper model give rise to deformed versions of the Hofstadter butterfly, for either sign of the nonlinearity (self-focusing and self-defocusing).

Objectives

Results

Conclusion