Nonlinear Wave Localization and Propagation in Disordered Systems

Georgios Kopidakis Department of Materials Science and Technology

University of Crete

Localization – delocalization of excitations •  Localized - extended states in large (infinite) systems •  Discrete linear models

–  Periodic (homogeneous lattices), propagation –  Disordered (inhomogeneous), Anderson localization

•  Discrete nonlinear models –  Periodic, localization without disorder –  Disordered ?

•  Interplay of disorder and nonlinearity •  Finite amount of energy injected in nonlinear Hamiltonian models

–  Localized initial excitation in isolated systems –  Final state in disordered systems

Universal wave packet spreading? •  Externally driven systems

–  Transmission thresholds –  Transparency

Simple discrete (lattice) models of wide applicability •  Chains of classical interacting oscillators

Nonlinearity in on-site term of potential (Klein-Gordon), in coupling term (Fermi-Pasta-Ulam), or in both terms.

•  Discrete Nonlinear Schrödinger models

(also derived as small amplitude oscillations limit of KG) Nonlinear optics, photonics, Bose-Einstein condensates,

polarons, Davydov solitons, etc

•  Models with any complexity, within a variety of physical contexts (all length scales)

Linear systems (small amplitude approximation) Linear combinations of eigenmodes

•  Spatially periodic systems, eigenmodes are plane waves Initially localized wave packet will disperse (zero amplitude at infinite time)

•  Disordered systems with discrete spectrum Anderson localized eigenmodes Wave packet will remain localized

random

Nonlinear Periodic Systems no superposition of plane waves

Discrete breathers (DBs) or Intrinsic Localized Modes (ILM) Spatially localized and time-periodic solutions of discrete nonlinear systems of coupled oscillators.

Discreteness: bounded linear spectrum Nonlinearity: frequency is amplitude dependent

“Extraband” Discrete Breathers

Theory for DBs is mathematically well founded and provides conceptual, theoretical, and computational tools to understand localized modes in nonlinear discrete models.

DBs in periodic nonlinear lattices: Localization without disorder

Discrete breathers in simulations and experiments

•  Ab initio level MD, systems with carbon and hydrogen small amplitudes, normal modes large amplitudes, DBs (local modes “rediscovered”)

GK, Aubry PB 2001

•  Experiments in macroscopic and mesoscopic systems confirm theory and numerics:

optically induced lattices, arrays of nonlinear optical waveguides, micromechanical oscillators, nonlinear optical waveguides, Josephson junctions, …

Nonlinearity and Disorder

Effect of nonlinearity on Anderson modes

Initially localized wave packets Localized or extended final state?

Localization in disordered nonlinear models Early attempts to understand interplay of disorder and nonlinearity Initial interest in electronic properties Polarons in disordered systems Tight-binding electron interacting with vibrations

Economou et al, PRL1992, PRB1993 G.K., Soukoulis, Economou PRB 1994, 1995, EPL1996

Long time dynamics in disordered nonlinear models •  simulations for relatively small systems and short times

(less than 1000 sites and t=105) •  variety of electronic initial conditions, including localized wave packets

lattice vibrations initially zero •  1-D models mimicking 3-D

only localized states above •  in many cases, participation number (localization length) increases with

nonlinear parameter (el-ph coupling) but seems to saturate even when second moment seems to continue growing

•  studies with DNLS Johansson et al PRB95

•  limited by real time simulations

GK, Soukoulis, Economou EPL 1996

Exact solutions - Intraband Discrete Breathers

• The strict continuation of Anderson modes gives spatially extended states • Localized time-periodic solutions with frequencies inside the linear band not in continuation of Anderson modes

As frequency varies, solution crosses resonant linear mode and develops new peak

G.K., Aubry, PhysicaD 1999, 2000, PRL 2000

Disordered 1D waveguide lattices Lahini et al PRL 2008

randomly modulated widths in [ ] “weak” disorder For very low intensities is zero Excite localized linear eigenmodes

Direct experimental observations

• Disordered 1D waveguide lattices Lahini et all PRL08 Nonlinear perturbations on localized eigenmodes by increasing input power beam (keeping low intensity, weak nonlinearity)

-Trend for enhanced localization at one band edge Shifted out of linear spectrum and remain localized (“extraband” DBs) -Trend for delocalization at the other band edge Shifted within linear spectrum, resonances with other modes (“Intraband” multi-DBs)

• Similar results in fiber arrays

Pertsch et al PRL 2004

Direct experimental observations

Wave packet dynamics in nonlinear disordered models •  “Double game” of disorder with nonlinearity for finite systems •  Isolated infinite systems (no temperature, no external driving)

–  Initial excitations in the form of stable periodic solutions: localized discrete breathers (from “exact” numerics) remain localized Localization not destroyed by nonlinearity

–  Quasi-periodic solutions Johansson, GK, Aubry EPL 2010 –  Localized initial condition which is not exact solution

Converge to an exact localized solution? Spread, zero amplitudes everywhere? Part remains localized?

•  Intuition: after possible initial spreading, amplitudes become small, quasi-linear system, i.e., localized final state

•  But: some arguments and numerical evidence for destruction of Anderson localization by

nonlinearity, “unlimited subdiffusive spreading” second moment diverges

Pikovsky, Sepelyansky PRL 2008

DNLS, excitations initially localized on a single site

with in

linear spectrum

participation number

second moment

numerical integration for long time evolution

No spreading for strong nonlinearity (rigorous result)

Linear energy part is bounded

Assuming wave packet spreads to infinity,

Inequality should always be fulfilled

Example: initial single site excitation may spread if

GK, Komineas, Flach, Aubry PRL 2008

Pedagogical example: periodic DNLS

Χ=0 Χ=1

Χ=2 Χ=3

Χ=4 Χ=5

Pedagogical example: periodic DNLS

Rigorous prediction for no (complete) spreading for χ = 4 and 5 Ballistic escape, even when part of initial excitation stays localized

Disordered DNLS

Rigorous results for DNLS numerically confirmed for Klein-Gordon Divergence of m2 does not imply destruction of localization Indications for quasiperiodic spectrum GK, Komineas, Flach, Aubry, PRL 2008

Summary for wave packet diffusion •  Rigorous results for DNLS and DNLS-like models

Numerical work shows they are valid in general

•  Incomplete wave packet spreading for large initial amplitude (nonlinearity) Rigorous and numerical results

•  Quasi-periodic solutions (KAM tori?) for very small initial amplitude Rigorous and numerical results Johansson, GK, Aubry EPL 2010

•  In many cases, localization persists even when rigorous results are not applicable Numerical results

•  There exist subdiffusive regimes when rigorous result is not applicable Numerical results

•  Final state?

Transmission through a driven system

•  Energy source due to external driving force on a single edge site semi-infinite disordered DNLS, KG, FPU chains

•  Results for DNLS

•  Dynamics determined by driving strength, disorder, driving frequency

•  For frequencies inside the “band” of linear Anderson modes: –  Weak driving below driving amplitude threshold:

localization close to driving site, no spreading, similar to linear –  Intermediate driving close to threshold:

initially slow propagation followed by rapid spreading –  Strong driving above threshold:

immediately rapid (diffusive) spreading

Johansson, GK, Lepri, Aubry EPL09

Thresholds related to turning points of the Nonlinear Response Manifold Adiabatic increase of driving amplitude from zero: system follows branch up to first turning point

Johansson, GK, Lepri, Aubry EPL09

Conclusions •  Long (infinite) time evolution of initially localized states in DNLS and

DNLS-like models

•  Moments divergence does not imply delocalization, participation number may not diverge

•  Localization persists for “special” solutions (periodic, quasiperiodic) as initial condition

•  In general, incomplete spreading of initially localized wave packets is supported by rigorous results and numerics for “strong” and “weak” nonlinearity

•  Numerical observations of subdiffusive regimes for “intermediate” nonlinearity

•  Propagation of finite amount of energy locally injected in isolated disordered nonlinear systems different from transmission through externally driven systems

•  Transmission thresholds in time-periodically driven nonlinear disordered systems (self-induced transparency)

Acknowledgements

C.M. Soukoulis Ames - Crete E.N. Economou

S. Aubry Saclay - Dresden M. Johansson