Anderson Localization or Nonlinear Waves: A Matter of Probability

Abstract

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems is under intense theoretical debate and experimental study. We resolve this dispute showing that at any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results are generalized to higher dimensions as well. [6,7].The pioneering experiments [8] demonstrate both the principal possibility to study this effect, but also the severe length and time scale limitations which restrict a quantitative analysis. Theoretical studies become therefore central here. Two faces of Anderson localization are usually probed: the absence of the wavepacket spreading [11][12][13][14][15][16][17][18][19][20] (the focus of this paper) and vanishing conductivity [9,10]. Wave packets, for not too weak nonlinearities, have been found to spread subdiffusively, disobeying AL at least over many time scales [11][12][13], and remarkable progress in studying the regimes and characteristics of wave packet spreading has been achieved [16][17][18][19][20].Still, the original questions are open. It remains debated, whether the observed spreading will continue infinitely or slow down and even stop to restore AL, once the wave packet densities become substantially small, as conjectured in Ref. [14,15,24]. It is unknown, whether there exists a lower bound on the nonlinearity strength, beyond which wave packets do obey AL, and therefore do not spread at all; or are some time scales diverging in this limit? Unavoidable finite size, time, energy, and precision limitations will make the most advanced numerical results not entirely convincing. To achieve a breakthrough, a novel approach is in need.The probabilistic description of AL in terms of the measure of localized regular trajectories in phase space (periodic orbits and tori) appears a promising solution for nonlinear disordered systems. Indeed, the mere assumption that a wave packet is launched on a chaotic trajectory leads to the conclusion that chaos remains forever: Arnol'd conjecture, unproved but widely accepted, states the uniqueness of the chaotic region in phase space [21]. If initial conditions are chaotic, then the trajectory will be unbounded in phase space, corresponding to unlimited spreading of a wavepacket. If they belong to a localized periodic orbit or torus, then the spreading is absent.The progress has been quite limited up to now due to the difficulty of the task. Persistence of tori has been proved for a special class of infinite systems [22] and for finite tori dimensionality [23] only. The non-zero probability of existence of infinite-dimensional tori in generic case (for small enough ener...

Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves

“…It is straightforward to show that the exponent α satisfies the relations (24) and (25) respectively [25,28,29,23]. We can also conclude that a large system at equilibrium will show a conductivity which is proportional to D.…”

Nonlinear Lattice Waves in Random Potentials

“…Another type of initial states are the ones mostly considered in this chapter -compact localized wave packets in a zero density surrounding. Then, if nonlinearity is lowered, several papers study the fate of these states [14], [15], [16]. The main outcome appears to be, that for a given and fixed initial state, at small enough nonlinearity, the dynamics will be in a KAM regime, i.e.…”

“…19 Schematic dependence of the probability P V for wave packets to stay localized (dark area) together with the complementary light area of spreading wave packets versus the wave packet volume V (either initial or attained at some time t) for three different orders of nonlinearity γ < 4, γ = 4 and γ > 4. Adapted from [16] V, t g > 4 g = 4 g < 4 V P will occur. But then, there is the complementary probability P Ch = 1 − P R to miss the torus, and instead to be launched on a chaotic trajectory, where dynamics is irregular, phase coherence is lost, and spreading may occur.…”

Inelastic Multiple Scattering of Interacting Bosons in Weak Random Potentials

“…All the real space experiments were explicitly done for non-interacting particles and, for 3D, in the regime of a very strong disorder potential and/or for very low single-particle energies (in agreement with k dis 1). The inclusion of inter-particle interaction rather leads to a suppression of Anderson localization [28][29][30] and, as demonstrated recently, to sub-diffusive behavior in 1D [31]. In 2D and 3D, however, the understanding is still far from complete, see [32][33][34] and references therein.…”

“…The separation (29) can be also obtained for the two-body T -matrixT (2) U with total energy E 1 + E 2 and the symmetrized single-particle states (30), upon the introduction of the total and reduced mass, M = 2m and µ = m/2, respectively, see Appendix B.1:…”