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arXiv:1006.4429 · Physical Review A

Quantitative study of two- and three-dimensional strong localization of matter waves by atomic scatterers

Mauro Antezza, Yvan Castin, David A. W. Hutchinson

Published Oct 7, 2010• 19 citations
arXivHTMLPDF

Abstract

We study the strong localization of atomic matter waves in a disordered potential created by atoms pinned at the nodes of a lattice, for both three-dimensional (3D) and two-dimensional (2D) systems. The localization length of the matter wave, the density of localized states, and the occurrence of energy mobility edges (for the 3D system), are numerically investigated as a function of the effective scattering length between the atomic matter wave and the pinned atoms. Both positive and negative matter wave energies are explored. Interesting features of the density of states are discovered at negative energies, where maxima in the density of bound states for the system can be interpreted in terms of bound states of a matter wave atom with a few pinned atomic scatterers. In 3D we found evidence of up to three mobility edges, one at positive energies, and two at negative energies, the latter corresponding to transitions between extended and localized bound states. In 2D, no mobility edge is found, and a rapid exponential-like increase of the localization length is observed at high energy.

DOI
10.1103/physreva.82.043602

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We study the strong localization of atomic matter waves in a disordered potential created by atoms pinned at the nodes of a lattice, for both three-dimensional (3D) and two-dimensional (2D) systems. The localization length of the matter wave, the density of localized states, and the occurrence of energy mobility edges (for the 3D system), are numerically investigated as a function of the effective scattering length between the atomic matter wave and the pinned atoms. Both positive and negative matter wave energies are explored. Interesting features of the density of states are discovered at negative energies, where maxima in the density of bound states for the system can be interpreted in terms of bound states of a matter wave atom with a few pinned atomic scatterers. In 3D we found evidence of up to three mobility edges, one at positive energies, and two at negative energies, the latter corresponding to transitions between extended and localized bound states. In 2D, no mobility edge is found, and a rapid exponential-like increase of the localization length is observed at high energy.
We study the strong localization of atomic matter waves in a disordered potential created by atoms pinnedat the nodes of a lattice, for both three-dimensional (3D) and two-dimensional (2D) systems. The localizationlength of the matter wave, the density of localized states, and the occurrence of energy mobility edges (for the 3Dsystem), are numerically investigated as a function of the effective scattering length between the atomic matterwave and the pinned atoms. Both positive and negative matter wave energies are explored. Interesting featuresof the density of states are discovered at negative energies, where maxima in the density of bound states for thesystem can be interpreted in terms of bound states of a matter wave atom with a few pinned atomic scatterers.In 3D we found evidence of up to three mobility edges, one at positive energies, and two at negative energies,the latter corresponding to transitions between extended and localized bound states. In 2D, no mobility edge isfound, and a rapid exponential-like increase of the localization length is observed at high energy.
The possibility of using ultracold atoms to observe strong localization of matter waves is now a subject of great interest, as undesirable decoherence and interactions can be made negligible in these systems. It was proposed that a static-disordered potential can be realized by trapping atoms of a given species in randomly chosen sites of a deep three-dimensional (3D) optical lattice with no multiple occupation. We analyze in detail the prospects of this scheme for observing localized states in 3D for a matter wave of a different atomic species that interacts with the trapped particles and that is sufficiently far detuned from the optical lattice to be insensitive to it. We demonstrate that at low energy a large number of 3D strongly localized states can be produced for the matter wave, if the effective scattering length describing the interaction of the matter wave with a trapped atom is of the order of the mean distance between the trapped particles. Such high values of the effective scattering length can be obtained by using a Feshbach resonance to adjust the free-space interspecies scattering length and by taking advantage of confinement-induced resonances induced by the trapping of the scatterers in the lattice.
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.
The possibility of using ultracold atoms to observe strong localization of matter waves is now the subject of a great interest, as undesirable decoherence and interactions can be made negligible in these systems. It was proposed that a static disordered potential can be realized by trapping atoms of a given species in randomly chosen sites of a deep 3D optical lattice with no multiple occupation. We analyze in detail the prospects of this scheme for observing localized states in 3D for a matter wave of a different atomic species that interacts with the trapped particles and that is sufficiently far detuned from the optical lattice to be insensitive to it. We demonstrate that at low energy a large number of 3D strongly localized states can be produced for the matter wave, if the effective scattering length describing the interaction of the matter wave with a trapped atom is of the order of the mean distance between the trapped particles. Such high values of the effective scattering length can be obtained by using a Feshbach resonance to adjust the free space inter-species scattering length and by taking advantage of confinement-induced resonances induced by the trapping of the scatterers in the lattice.
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.
We study the critical dynamics of matter waves at the three-dimensional Anderson mobility edge in cold-atom disorder quench experiments. General scaling arguments are supported by precision numerics for the spectral function, diffusion coefficient, and localization length in isotropic blue-detuned speckle potentials. We discuss signatures of critical slowdown in the time-dependent central column density of a spreading wave packet, and evaluate the prospects of observing anomalous diffusion right at criticality.