Anderson localization of matter waves in quantum-chaos theory

We study the Anderson localization of atomic gases exposed to three-dimensional optical speckles by analyzing the statistics of the energy-level spacings. This method allows us to consider realistic models of the speckle patterns, taking into account the strongly anisotropic correlations which are realized in concrete experimental configurations. We first compute the mobility edge $E_c$ of a speckle pattern created using a single laser beam. We find that $E_c$ drifts when we vary the anisotropy of the speckle grains, going from higher values when the speckles are squeezed along the beam propagation axis, to lower values when they are elongated. We also consider the case where two speckle patterns are superimposed forming interference fringes, and we find that $E_c$ is increased compared to the case of idealized isotropic disorder. We discuss the important implications of our findings for cold-atoms experiments.