Dynamics of localized waves in one-dimensional random potentials: Statistical theory of the coherent forward scattering peak

Abstract

As recently discovered [PRL 109 190601(2012)], Anderson localization in a bulk disordered system triggers the emergence of a coherent forward scattering (CFS) peak in momentum space, which twins the well-known coherent backscattering (CBS) peak observed in weak localization experiments. Going beyond the perturbative regime, we address here the long-time dynamics of the CFS peak in a 1D random system and we relate this novel interference effect to the statistical properties of the eigenfunctions and eigenspectrum of the corresponding random Hamiltonian. Our numerical results show that the dynamics of the CFS peak is governed by the logarithmic level repulsion between localized states, with a time scale that is, with good accuracy, twice the Heisenberg time. This is in perfect agreement with recent findings based on the nonlinear σ-model. In the stationary regime, the width of the CFS peak in momentum space is inversely proportional to the localization length, reflecting the exponential decay of the eigenfunctions in real space, while its height is exactly twice the background, reflecting the Poisson statistical properties of the eigenfunctions. Our results should be easily extended to higher dimensional systems and other symmetry classes.

Coherent forward scattering in two-dimensional disordered systems

“…IV). This shows in particular that the CFS peak is intrinsically connected with the spectral properties of the disordered system [16]. In real space, the form factor also governs the dynamics of the "mesoscopic echo effect," i.e., the enhancement of the probability for a spatially narrow wave packet to return to the origin in the presence of disorder [32].…”

“…In fact, the relation (9) between the CFS peak and the form factor defined by (14) turns out to hold very generally, for any ratio 063602-6 of L and ξ . This can be explicitly shown by a modal decomposition of the wave function written in momentum space, a task that was accomplished in [16]. With the help of this relation, the problem of calculating the CFS contrast as a function of time boils down to the analysis of the frequency dependence of the correlation function K E (ω).…”

“…This signature has been largely exploited in a number of experiments searching for Anderson localization of classical waves in disordered media [3][4][5] or of matter waves subjected to time-periodic [6,7] and random [8][9][10][11][12] optical potentials. At the same time, recent works [13][14][15][16] pointed out that in ultracold-atom setups, the momentum distribution of a matter wave in a random potential can exhibit a highly nontrivial dynamics due to localization if it is initially launched with a nonzero mean wave vector k 0 . The scenario is then the following.…”

“…From a theoretical point of view, while the physics of CBS is today well understood (it stems from wave amplitudes traveling along the same multiple scattering sequence but in opposite directions [20]), the mechanism of CFS is much less obvious. In fact, the building of the CFS peak relies on interference sequences where particles are scattered back and forth several times between the directions −k 0 and +k 0 [14][15][16]. While this mechanism is inefficient in a purely diffusive system, it becomes strongly enhanced when the wave gets confined in a limited region of space: wave interference can then accumulate and make the CFS peak macroscopic.…”

“…We consider a random potential of speckle type, routinely used in current experiments with ultracold atoms [21], and focus on the two-dimensional (2D) geometry for which the CFS peak clearly distinguishes itself from the isotropic part of the momentum distribution [14] (we refer the reader to [15,16] for the one-dimensional and quasi-one-dimensional cases). The main concepts discussed in the paper as well as the theoretical framework are introduced in Sec.…”