Localization of matter waves in two-dimensional disordered optical potentials

Observation of two-dimensional Anderson localisation of ultracold atoms

Anderson localisation -the inhibition of wave propagation in disordered media- is a surprising interference phenomenon which is particularly intriguing in two-dimensional (2D) systems. While an ideal, non-interacting 2D system of infinite size is always localised, the localisation length-scale may be too large to be unambiguously observed in an experiment. In this sense, 2D is a marginal dimension between one-dimension, where all states are strongly localised, and three-dimensions, where a well-defined phase transition between localisation and delocalisation exists as the energy is increased. Here, we report the results of an experiment measuring the 2D transport of ultracold atoms between two reservoirs, which are connected by a channel containing pointlike disorder. The design overcomes many of the technical challenges that have hampered observation of localisation in previous works. We experimentally observe exponential localisation in a 2D ultracold atom system.

Elastic Scattering Time of Matter Waves in Disordered Potentials

We report on an extensive study of the elastic scattering time τ_{s} of matter waves in optical disordered potentials. Using direct experimental measurements, numerical simulations, and comparison with the first-order Born approximation based on the knowledge of the disorder properties, we explore the behavior of τ_{s} over more than 3 orders of magnitude, ranging from the weak to the strong scattering regime. We study in detail the location of the crossover and, as a main result, we reveal the strong influence of the disorder statistics, especially on the relevance of the widely used Ioffe-Regel-like criterion kl_{s}∼1. While it is found to be relevant for Gaussian-distributed disordered potentials, we observe significant deviations for laser speckle disorders that are commonly used with ultracold atoms. Our results are crucial for connecting experimental investigation of complex transport phenomena, such as Anderson localization, to microscopic theories.

Anderson Localization of Ultracold Atoms: Where is the Mobility Edge?

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.

Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials

We investigate the metal-insulator transition occurring in two-dimensional (2D) systems of noninteracting atoms in the presence of artificial spin-orbit interactions and a spatially correlated disorder generated by laser speckles. Based on a high order discretization scheme, we calculate the precise position of the mobility edge and verify that the transition belongs to the symplectic universality class. We show that the mobility edge depends strongly on the mixing angle between Rashba and Dresselhaus spin-orbit couplings. For equal couplings a non-power-law divergence is found, signaling the crossing to the orthogonal class, where such a 2D transition is forbidden.

Effective Confining Potential of Quantum States in Disordered Media

The amplitude of localized quantum states in random or disordered media may exhibit long-range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long-range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.

Mobility edge for cold atoms in laser speckle potentials

Using the transfer-matrix method, we numerically compute the precise position of the mobility edge of atoms exposed to a laser speckle potential and study its dependence versus the disorder strength and correlation function. Our results deviate significantly from previous theoretical estimates using an approximate, self-consistent approach of localization. In particular, we find that the position of the mobility edge in blue-detuned speckles is much lower than in the red-detuned counterpart, pointing out the crucial role played by the asymmetric on-site distribution of speckle patterns.

Coherent forward scattering peak induced by Anderson localization

Numerical simulations show that, at the onset of Anderson localization, the momentum distribution of a coherent wave packet launched inside a random potential exhibits, in the forward direction, a novel interference peak that complements the coherent backscattering peak. An explanation of this phenomenon in terms of maximally crossed diagrams predicts that the signal emerges around the localization time and grows on the scale of the Heisenberg time associated with the localization volume. Together, coherent back and forward scattering provide evidence for the occurrence of Anderson localization.

Conduction of ultracold fermions through a mesoscopic channel

In a mesoscopic conductor, electric resistance is detected even if the device is defect-free. We engineered and studied a cold-atom analog of a mesoscopic conductor. It consists of a narrow channel connecting two macroscopic reservoirs of fermions that can be switched from ballistic to diffusive. We induced a current through the channel and found ohmic conduction, even when the channel is ballistic. We measured in situ the density variations resulting from the presence of a current and observed that density remains uniform and constant inside the ballistic channel. In contrast, for the diffusive case with disorder, we observed a density gradient extending through the channel. Our approach opens the way toward quantum simulation of mesoscopic devices with quantum gases.

Dynamics of wave packets in two-dimensional random systems with anisotropic disorder

A theoretical model is proposed to describe narrowband pulse dynamics in two-dimensional systems with arbitrary correlated disorder. In anisotropic systems with elongated cigarlike inhomogeneities, fast propagation is predicted in the direction across the structure where the wave is exponentially localized and tunneling of evanescent modes plays a dominant role in typical realizations. Along the structure, where the wave is channeled as in a waveguide, the motion of the wave energy is relatively slow. Numerical simulations performed for ultra-wide-band pulses show that even at the initial stage of wave evolution, the radiation diffuses predominantly in the direction along the major axis of the correlation ellipse. Spectral analysis of the results relates the long tail of the wave observed in the transverse direction to a number of frequency domain "lucky shots" associated with the long-living resonant modes localized inside the sample.

Coherent backscattering of Bose-Einstein condensates in two-dimensional disorder potentials

We study quantum transport of an interacting Bose-Einstein condensate in a two-dimensional disorder potential. In the limit of a vanishing atom-atom interaction, a sharp cone in the angle-resolved density of the scattered matter wave is observed, arising from constructive interference between amplitudes propagating along reversed scattering paths. Weak interaction transforms this coherent backscattering peak into a pronounced dip, indicating destructive instead of constructive interference. We reproduce this result, obtained from the numerical integration of the Gross-Pitaevskii equation, by a diagrammatic theory of weak localization in the presence of nonlinearity.

Anderson localization of Bogolyubov quasiparticles in interacting Bose-Einstein condensates

We study the Anderson localization of Bogolyubov quasiparticles in an interacting Bose-Einstein condensate (with a healing [corrected] length xi) subjected to a random potential (with a finite correlation length sigma(R)). We derive analytically the Lyapunov exponent as a function of the quasiparticle momentum k, and we study the localization maximum k(max). For 1D speckle potentials, we find that k(max) proportional variant 1/xi when xi>>sigma(R) while k(max) proportional variant 1/sigma(R) when xi<<sigma(R), and that the localization is strongest when xi approximately sigma(R). Numerical calculations support our analysis, and our estimates indicate that the localization of the Bogolyubov quasiparticles is accessible in experiments with ultracold atoms.

Expansion of a Bose-Einstein condensate in the presence of disorder

Expansion of a Bose-Einstein condensate (BEC) is studied in the presence of a random potential. The expansion is controlled by a single parameter, (microtau(eff)/variant Planck's over 2pi), where micro is the chemical potential, prior to the release of the BEC from the trap, and tau(eff) is a transport relaxation time which characterizes the strength of the disorder. Repulsive interactions (nonlinearity) facilitate transport and can lead to diffusive spreading of the condensate which, in the absence of interactions, would have remained localized in the vicinity of its initial location.

Quantum glass phases in the disordered Bose-Hubbard model

The phase diagram of the Bose-Hubbard model in the presence of off-diagonal disorder is determined using quantum Monte Carlo simulations. A sequence of quantum glass phases intervene at the interface between the Mott insulating and the superfluid phases of the clean system. In addition to the standard Bose glass phase, the coexistence of gapless and gapped regions close to the Mott insulating phase leads to a novel Mott glass regime which is incompressible yet gapless. Numerical evidence for the properties of these phases is given in terms of global (compressibility, superfluid stiffness) and local (compressibility, momentum distribution) observables.

Superfluidity versus Anderson localization in a dilute Bose gas

We consider the motion of a quasi-one-dimensional beam of Bose-Einstein condensed particles in a disordered region of finite extent. Interaction effects lead to the appearance of two distinct regions of stationary flow. One is subsonic and corresponds to superfluid motion. The other one is supersonic and dissipative and shows Anderson localization. We compute analytically the interaction-dependent localization length. We also explain the disappearance of the supersonic stationary flow for large disordered samples.

Anderson localization of expanding Bose-Einstein condensates in random potentials

We show that the expansion of an initially confined interacting 1D Bose-Einstein condensate can exhibit Anderson localization in a weak random potential with correlation length sigma(R). For speckle potentials the Fourier transform of the correlation function vanishes for momenta k>2/sigma(R) so that the Lyapunov exponent vanishes in the Born approximation for k>1/sigma(R). Then, for the initial healing length of the condensate xi(in)>sigma(R) the localization is exponential, and for xi(in)<sigma(R) it changes to algebraic.