Anderson localization of matter waves in tailored disordered potentials

Abstract

We show that, in contrast to immediate intuition, Anderson localization of noninteracting particles induced by a disordered potential in free space can increase (i.e. the localization length can decrease) when the particle energy increases, for appropriately tailored disorder correlations. We predict the effect in one, two and three dimensions, and propose a simple method to observe it using ultracold atoms placed in optical disorder. The increase of localization with the particle energy can serve to discriminate quantum versus classical localization.PACS numbers: 03.75. Kk, 05.60.Gg, 03.75.Nt, 05.30.Jp The transport properties of a coherent wave in a disordered medium are inherently determined by interference of multiple scattering paths, which can lead to spatial localization and absence of diffusion [1]. This effect, known as Anderson localization (AL), was first predicted for electrons in disordered crystals [2] and then extended to classical waves [3], which permitted observation of AL in a variety of systems (see Ref.[4] and references therein). The most fundamental features of AL are ubiquity and universality [5]. For instance, in conventional cases, all states are known to be localized in one (1D) and two (2D) dimensions, while in three dimensions (3D) the spectrum splits into regions of localized states and regions of extended states, separated by so-called mobility edges [6]. Nevertheless, the observable features of AL strongly depend on the system details.Consider a wave propagating among randomly distributed point scatterers (point-impurity disorder). In the absence of interference, the propagation is dominated by normal diffusion. It devises a diffusive medium characterized by the length scale (transport Boltzmann mean free path) l B = vτ , with v = |∂ω(k)/∂k| the wave velocity [ω(k) is the dispersion relation] and τ the scattering time. Then, localization arises from the interference of the diffusive paths. The more the wavelength exceeds the mean free path, the stronger interference affects the transport. It can thus be inferred that the Lyapunov exponent (inverse localization length), which characterizes the localization strength, readswhere the function F d strongly depends on the spatial dimension d and is a decreasing function of the interference parameter kl B . For a particle (scalar matter wave) in free space and a weak point-impurity disorder, v ∝ k, τ is proportional to the inverse of the density of states (ρ ∝ k d−2 ), as given by the Fermi golden rule, and finally l B ∝ 1/k d−3 . Then, for any d ≤ 3, γ is a decreasing function of k. In other words, the localization gets weaker when the particle energy E = ω(k) = 2 k 2 /2m increases, which conforms to natural intuition. This decrease of γ(E) with E however relies on the microscopic details of the system, namely on the dispersion relation and on the properties of the scattering time assumed above, and can be altered in different ways. For instance, it does not hold for lattice systems, such as electrons in disordered crystals, because ...

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