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Two- and three-dimensional strong localization of matter waves by atomic scatterers

Mauro Antezza

Published Dec 15, 2010• 0 citations
OpenAlex

Abstract

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.

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The possibility of using ultracold atoms to observe localization of matter waves is now a subject of significant interest. These systems offer unprecedented control over inter-particle interactions, imposed potentials and the level of disorder as compared to their condensed matter analogues. The two-dimensional (2D) case is of particular interest. The prevailing view has been that in the 2D electron gas there is no metallic state, but recent experiments are suggestive of a metal-insulator transition in very dilute systems. We investigate theoretically the possibility of observing strongly localized states, corresponding to the insulating phase, in the dilute 2D gas.

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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.
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.
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 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.
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.
In 1958, P.W. Anderson predicted the exponential localization1 of electronic wave functions in disordered crystals and the resulting absence of diffusion. It has been realized later that Anderson localization (AL) is ubiquitous in wave physics2 as it originates from the interference between multiple scattering paths, and this has prompted an intense activity. Experimentally, localization has been reported in light waves 3 microwaves4, sound waves5, and electron gases6 but to our knowledge there is no direct observation of exponential spatial localization of matter-waves (electrons or others). We present here the observation of Anderson localization7 of a Bose-Einstein condensate (BEC) released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle. We also show that, in our one-dimensional speckle potentials whose noise spectrum has a high spatial frequency cut-off, exponential localization occurs only when the de Broglie wavelengths of the atoms in the expanding BEC are larger than an effective mobility edge corresponding to that cut-off. In the opposite case, we find that the density profiles decay algebraically8.
The possibility of using ultracold atoms to observe localization of matter waves is now a subject of significant interest. These systems offer unprecedented control over inter-particle interactions, imposed potentials and the level of disorder as compared to their condensed matter analogues. The two-dimensional (2D) case is of particular interest. The prevailing view has been that in the 2D electron gas there is no metallic state, but recent experiments are suggestive of a metal-insulator transition in very dilute systems. We investigate theoretically the possibility of observing strongly localized states, corresponding to the insulating phase, in the dilute 2D gas.