Symmetry-breaking instabilities in perturbed optical lattices: Nonlinear nonreciprocity and macroscopic self-trapping Andrea Fratalocchi * and Gaetano Assanto † Nonlinear Optics and OptoElectronics Lab (NooEL), INFN & CNISM, University ROMA TRE, Via della Vasca Navale 84, 00146, Rome, Italy Received 9 June 2006; published 27 June 2007 We develop an asymptotic analysis of nonlinear energy propagation in lattices subject to slowly varying perturbations, investigating symmetry breaking and its effects. We derive a general set of evolution equations and study them by using catastrophe theory, revealing a wealth of system dynamics. Below a power threshold, symmetry breaking drives nonreciprocal oscillations; beyond that, symmetry breaking yields an effect of “macroscopic” self-trapping, which supports a self-maintained energy imbalance between Bloch bands. We numerically verify the theoretical results and discuss their possible implementation in waveguide arrays. DOI: 10.1103/PhysRevA.75.063828 PACS numbers: 42.65.Sf, 03.75.Lm I. INTRODUCTION Symmetry breaking is ubiquitous in modern nonlinear physics. It occurs when the symmetry governing the dynam- ics of a nonlinear system is broken, with the establishment of one or more asymmetric states which no longer conserve the properties of the original solution and usually link to radi- cally different physics. Examples of symmetry breaking can be found in several areas, including nonlinear optics 1–3, molecular materials 4, quantum systems 5,6, pattern dy- namics 7, catastrophes 8, Bose-Einstein condensates 9, and statistical and gravitational field theory 10,11. In recent years energy propagation and localization in op- tical lattices has attracted a growing interest 12–15. Such systems find one-dimensional implementations in several media, including photorefractives 13,14, semiconductors 16, liquid crystals 17, and ultracold atoms 15. Both fun- damental dynamics and applications of optical lattices have been addressed 1,12,15,18–24, with great attention being recently paid to understanding the role of localized and/or extended transverse defects in the periodic structure 18–20,22–25. To date, however, nonlinear lattice dynamics in the latter cases remains undisclosed. In this paper, we study the dynamics of a nonlinear optical lattice in the presence of a slowly varying perturbation, ex- ploring symmetry breaking and its effects. Considering the adiabatic regime 26, when no linear interactions between Floquet-Bloch FB bands take place, we carry out an asymptotic analysis and employ catastrophe theory in order to assess the results 8. We demonstrate that such a system is described by a general set of equations: depending on the control parameters they can model diverse physical systems, from spin particle ensembles to superconducting Josephson junctions. In the general case, the system breaks specific symmetries and acquires a nonreciprocal oscillatory charac- ter for excitations below a threshold. In this regime, an in- terchange of input conditions between interacting FB bands results in an entirely different evolution of the system, in formal analogy to mismatched nonlinear couplers NCs 27. For powers above threshold, conversely, symmetry breaking sustains a “macroscopic” self-trapping, resulting in a self-maintained energy imbalance between FB bands. Un- less previously investigated dynamics, this trapping does not resemble the macroscopic quantum trapping effect of Boson- Josephson junctions 28,29. II. MODEL AND ANALYTICAL APPROACH We consider a dimensionless equation of the type  = N, with  = i   z + H + V S , N =- || 2 , 1 with H = 1 2  2 y 2 + V F y being a Hamiltonian operator, V F y = V F y +  a periodic potential that makes up the lattice, V S a generic defect or perturbation along y, and  denoting an optical wave envelope or a wave function in BEC. For V S =0, Eq. 1 is the nonlinear Schrödinger equation model- ing lattice dynamics in several materials 12–15. By assum- ing that V S is analytic and varies on a slower spatial scale  than the period   ≪ , we can expand it in Taylor series V S y =  0 + y + O 2 we take the defect length to be d  1/  and define the system metric with respect to this characteristic scale by applying periodic boundary conditions of period d. Any constant term  0 can be gauged away by the substitution  →  expi 0 z and therefore does not af- fect the system evolution. Here we consider the dynamics at order O, the resulting potential V S of which encompasses several possible experimental realizations see, e.g., 21,23,24, and references therein. We perform an asymptotic expansion 30 by introducing the scales z n =  n/2 z, y n =  n/2 y and setting  =  n  n+1/2  n with  0 y, z = expi 0 z 0   p=1,2 A p 0 z 1 , y 1  p,k 0 y 0  ,  n0 y, z =  q1,2 A q n z 1 , y 1  q,k 0 y 0 expi 0 z 0  , 2 with k 0 an integer multiple of  / . At leading order we take the interaction of two slowly modulated Bloch modes  p,k 0 *Electronic address: frataloc@uniroma3.it † Electronic address: assanto@uniroma3.it PHYSICAL REVIEW A 75, 063828 2007 1050-2947/2007/756/0638285 ©2007 The American Physical Society 063828-1