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arXiv:1112.2859 · Europhysics Letters (EPL)

Matter wave transport and Anderson localization in anisotropic three-dimensional disorder

Marie Piraud, Luca Pezzè, Laurent Sanchez-Palencia

Published Sep 1, 2012• 50 citations
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Abstract

We study quantum transport of matter waves in anisotropic three-dimensional disorder.First, we show that structured correlations can induce rich effects, such as anisotropic suppression of the white-noise limit and inversion of the transport anisotropy.Second, we show that the localization threshold (mobility edge) is strongly affected by a disorder-induced shift of the energy states, which we calculate.Our work is directly relevant to ultracold-matter waves in optical disorder, and implications on recent experiments are discussed.It also offers scope for further studies of anisotropy effects in other systems with controlled disorder, where counterparts of the discussed effects can be expected.

DOI
10.1209/0295-5075/99/50003

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We study quantum transport of matter waves in anisotropic three-dimensional disorder.First, we show that structured correlations can induce rich effects, such as anisotropic suppression of the white-noise limit and inversion of the transport anisotropy.Second, we show that the localization threshold (mobility edge) is strongly affected by a disorder-induced shift of the energy states, which we calculate.Our work is directly relevant to ultracold-matter waves in optical disorder, and implications on recent experiments are discussed.It also offers scope for further studies of anisotropy effects in other systems with controlled disorder, where counterparts of the discussed effects can be expected.
The macroscopic transport properties in a disordered potential, namely diffusion and weak/strong localization, closely depend on the microscopic and statistical properties of the disorder itself.This dependence is rich in counter-intuitive consequences.It can be particularly exploited in matter wave experiments, where the disordered potential can be tailored and controlled, and anisotropies are naturally present.In this work, we apply a perturbative microscopic transport theory and the self-consistent theory of Anderson localization to study the transport properties of ultracold atoms in anisotropic two-dimensional (2D) and three-dimensional (3D) speckle potentials.In particular, we discuss the anisotropy of single-scattering, diffusion and localization.We also calculate disorder-induced shift of the energy states and propose a method to include it, which amounts to renormalizing energies in the standard on-shell approximation.We show that the renormalization of energies strongly affects the prediction for the 3D localization threshold (mobility edge).We illustrate the theoretical findings with examples which are relevant for current matter wave experiments, where the disorder is created with laser speckle.This paper provides a guideline for future experiments aiming at the precise location of the 3D mobility edge and study of anisotropic diffusion and localization effects in 2D and 3D.
Abstract We discuss effects of anisotropic scattering in transport properties of ultracold atoms in three-dimensional optical potentials. Within the realm of the first Born approximation, we calculate the self energy, the scattering mean free time, the scattering mean free path, and the anisotropy factor. The behavior of the diffusion constant as a function of the wavenumber is also examined in diffusive and weak localization regimes. We show that these quantities are affected by quantum corrections due to the interference caused by disorder. The dimensionless conductance is evaluated using the scaling theory of localization. Our results are compared with previous theoretical and the experimental results.
Recent experiments in noninteracting ultracold atoms in three-dimensional speckle potentials have yielded conflicting results regarding the so-called mobility edge, i.e., the energy threshold separating Anderson localized from diffusive states. At the same time, there are theoretical indications that most experimental data overestimate this critical energy, sometimes by a large amount. Using extensive numerical simulations, we show that the effect of anisotropy in the spatial correlations of realistic disorder configurations alone is not sufficient to explain the experimental data. In particular, we find that the mobility edge obeys a universal scaling behavior, independently of the speckle geometry.
Anderson localization of particles -- the complete halt of wave transport through multiple scattering and phase coherence -- is a paradigmatic manifestation of quantum interference in disordered media. In three dimensions, the scaling theory predicts a quantum phase transition at a critical energy, the mobility edge, separating localized from diffusive states and underpinning metal-insulator transitions in electronic systems. Despite decades of experimental efforts, a direct observation of this emblematic transition for matter waves has remained elusive. Previous attempts with ultracold atoms were hindered by strong and uncontrolled energy broadening, resulting in indirect, sometimes inaccurate, and model-dependent estimates of the mobility edge. Here we implement a novel energy-resolved scheme to prepare atomic matter waves with a narrow energy distribution and track their expansion dynamics over long timescales. This allows for a direct observation of the three-dimensional Anderson transition in a laser-speckle disordered potential, and for a precise measurement of the mobility edge that is independent of any underlying theoretical modeling. Our measurements show excellent agreement with state-of-the-art numerical predictions over a wide range of disorder strengths, resolving long-standing discrepancies between prior experiments and theory. Beyond the three-dimensional Anderson transition, our approach opens new avenues for quantitative investigations of quantum critical phenomena in spatially disordered systems, including the roles of dimensionality, symmetry class, and interactions.
The development of atom cooling and trapping techniques and the ability to load optical lattices or disordered potentials with Bose-Einstein condensates or degenerate Fermi gases has opened a whole field of research around the strong localization of matter waves. In this theoretical work we study the coherent transport of matter waves evolving in disordered optical potentials (speckle). The influence of correlated disorder is first studied numerically in the framework of the Anderson model. Subsequently, a self-consistent diagrammatic calculation allows us to determine analytically the basic transport parameters in the regime of weak disorder: mean free path, transport mean free path, diffusion constant, localization length. A crucial quantity for these analytical calculations is the spatial correlation function of speckle fluctuations. It determines the degree of anisotropy of a scattering event. We consider in particular the transition from weak localization to strong localization. In this case the diffusion constant of matter waves decreases and approaches zero at the threshold of strong localization. We calculate the renormalization of the diffusion constant due to the interference of matter waves by explicitly considering the correlation of fluctuations of the disordered potential.