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arXiv:1510.05121 · Physical Review A

Anderson localization in optical lattices with correlated disorder

Elisa Fratini, Sebastiano Pilati

Published Dec 14, 2015• 23 citations
arXivHTMLPDF

Abstract

We study the Anderson localization of atomic gases exposed to simple-cubic optical lattices with a superimposed disordered speckle pattern. The two mobility edges in the first band and the corresponding critical filling factors are determined as a function of the disorder strength, ranging from vanishing disorder up to the critical disorder intensity where the two mobility edges merge and the whole band becomes localized. Our theoretical analysis is based both on continuous-space models that take into account the details of the spatial correlation of the speckle pattern, and also on a simplified tight-binding model with an uncorrelated distribution of the on-site energies. The mobility edges are computed via the analysis of the energy-level statistics, and we determine the universal value of the ratio between consecutive level spacings at the mobility edge. We analyze the role of the spatial correlation of the disorder, and we also discuss a qualitative comparison with available experimental data for interacting atomic Fermi gases obtained in the moderate interaction regime.

DOI
10.1103/physreva.92.063621

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We study the Anderson localization of atomic gases exposed to simple-cubic optical lattices with a superimposed disordered speckle pattern. The two mobility edges in the first band and the corresponding critical filling factors are determined as a function of the disorder strength, ranging from vanishing disorder up to the critical disorder intensity where the two mobility edges merge and the whole band becomes localized. Our theoretical analysis is based both on continuous-space models that take into account the details of the spatial correlation of the speckle pattern, and also on a simplified tight-binding model with an uncorrelated distribution of the on-site energies. The mobility edges are computed via the analysis of the energy-level statistics, and we determine the universal value of the ratio between consecutive level spacings at the mobility edge. We analyze the role of the spatial correlation of the disorder, and we also discuss a qualitative comparison with available experimental data for interacting atomic Fermi gases obtained in the moderate interaction regime.
We experimentally study localization and dynamics of ultracold fermions in speckle and optical lattice potentials to explore Anderson localization, many-body localization, and relaxation dynamics in strongly correlated systems. Anderson localization is probed by releasing non-interacting, spin-polarized gases into three dimensional, anisotropic disordered potentials produced from optical speckle. A fraction of the atoms are localized by the disorder, and a mobility edge is found separating localized from extended states. The length scale of the speckle is varied, and the localized state is found to scale linearly with the geometric mean of the speckle autocorrelation length. We realize the Fermi Hubbard model by loading atoms in a cubic optical lattice. Non-equilibrium momentum distributions are created via Raman transitions, and the excitation relaxation rate is measured in the lattice. Transport experiments were performed in a disordered optical lattice to explore the disordered Hubbard model. These experiments reveal localization in the presence of strong interactions and an interaction driven metal-to-insulator transition. The localized state is found to be insensitive to a doubling in the temperature of the gas and is consistent with predictions of many-body localization.
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.
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.
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 existence of Anderson localization, characterized by vanishing diffusion due to strong disorder, has been demonstrated in numerous ways. A systematic approach based on the Anderson quantum model of the Fermi gas in random lattices that can describe both diffusive and localized regimes has not yet been fully established. We build on a recent publication [V. Janiš, ] and present a microscopic theory of disordered electrons that covers both the metallic phase with extended Bloch waves and the localized phase, where a propagating particle forms a quantum bound state with the hole left behind at the origin. The general theory provides a framework for constructing controlled approximations to one- and two-particle Green functions that satisfy the necessary conservation laws and causality requirements across the full range of disorder strength. It is used explicitly to derive a local, mean-field-like approximation for the two-particle irreducible vertices, enabling quantitative analysis of the solution's dynamic properties in both metallic and localized phases, including critical behavior at the mobility edge. A new instability line for the dynamical electron-hole correlation function of the metallic phase is introduced.