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Transmission thresholds in time-periodically driven nonlinear disordered systems

M. Johansson, G. Kopidakis, S. Lepri, S. Aubry

EPL, 86 (2009) 10009

Since the pioneering work of Anderson in 1958 (Nobel prize 1977), it is now well understood that for any kind of wave that can be described by a linear wave equation, a sufficiently strong randomness in the propagation medium causes it to localize in space. In particular, transport of energy through the system is prohibited ("absence of diffusion"), since any initially localized wave packet remains localized forever in the same spatial region. This "Anderson localization" was originally proposed for electronic motion in disordered solids, but it appears also for classical waves such as light propagating in media with random fluctuations of refractive index, and for quantum mechanical matter waves such as Bose-Einstein condensates of ultracold atoms in disordered optical potentials.

However, many situations require that nonlinearities are included to properly describe interactions. For a linear system, superposition implies an equivalence between absence of diffusion and localization of time-periodic eigenmodes (stationary solutions for a quantum wave). But nonlinearities break the superposition principle, and it is still largely an open question to what extent Anderson localization survives.

We have studied a general class of disordered anharmonic chains, where a time-periodic driving force is applied to one end. For small-amplitude driving, the system is almost linear, and Anderson localization implies that the input energy spreads only to a few lattice sites close to the driver. When slowly increasing the driving strength, we find that for any given frequency there is a well-defined amplitude threshold, such that above this threshold, localization is destroyed and energy starts to propagate throughout the chain. The dependence of the threshold amplitude with frequency for a typical disordered chain is very complicated (see figure): we conjecture it to be an upper semicontinuous function, which generally is nonzero, but becomes zero at an infinite but discrete set of points, associated with resonances with linear Anderson eigenmodes.

Threshold amplitude versus driving frequency for a typical disordered nonlinear chain. Inset: Distribution of energy density in the lattice versus time, when driving becomes larger than threshold.

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Last updated: 06/01/09