We study the transport dynamics of matter-waves in the presence of disorder and nonlinearity. An atomic Bose-Einstein condensate that is localized in a quasiperiodic lattice in the absence of atom-atom interaction shows instead a slow expansion with a subdiffusive behavior when a controlled repulsive interaction is added. The measured features of the subdiffusion are compared to numerical simulations and a heuristic model. The observations confirm the nature of subdiffusion as interaction-assisted hopping between localized states and highlight a role of the spatial correlation of the disorder. [1-3, 7, 11, 12, 14-16, 18]. Most authors agree that the nonlinearity should prevent localization and that the wavepacket should expand in a way which is slower than normal diffusion. However, experimental evidence of such subdiffusive expansion is to date still missing.Here we study the dynamics of Bose-Einstein condensates with controllable nonlinearity expanding along a one-dimensional quasiperiodic lattice. Despite its large spatial correlation, this kind of potential is known [20,21] to feature exponentially localized states that are equivalent to those appearing in lattices with uncorrelated disorder described by the Anderson model [22]. Such a system has been successfully exploited for the investigation of the delocalizing effect of repulsive interactions on a trapped system in equilibrium [23,24]. If a noninteracting gas is let free to expand along the quasiperiodic lattice, no transport is observed [25] because all single-particle eigenstates are localized. By adding a controlled interatomic repulsion, we now observe a slow increase of the width σ of the sample that asymptotically follows a subdiffusive law: σ(t) ∝ t α , with α = 0.2 − 0.4. We find that the exponent increases with the interaction energy, in qualitative agreement with both numerical simulations based on a 1D discrete nonlinear Schrödinger equation (DNLSE) of the quasiperiodic lattice [7] and the predictions of a heuristic model. Our observation confirms the nature of subdiffusion as interaction-assisted hopping between localized states. The observed exponents are however larger than the one calculated for uncorrelated disordered potentials [1-3, 11, 12, 14-16, 18], suggesting a role of the spatial correlation of the disorder.The one-dimensional quasiperiodic potential is created by perturbing a primary optical lattice with a weaker incommensurate lattice [26]:V (x)=V 1 cos 2 (k 1 x) + V 2 cos 2 (k 2 x). Here k i =2π/λ i are the wavevectors of the lattices (λ 1 =1064.4 nm and λ 2 =859.6 nm). This potential is characterized by the spacing d=λ 1 /2 and the tunneling energy J of the primary lattice, and by the disorder strength ∆, which scales linearly with V 2 [27]. In the case of non-interacting particles this system constitutes an experimental realization of the Harper or AubryAndré model [20,21] which shows a transition between extended and localized states for a finite value of the disorder ∆/J = 2 [23,27]. Above this threshold all single-partic...

We study the behaviour of an ultracold atomic gas of bosons in a bichromatic lattice, where the weaker lattice is used as a source of disorder. We numerically solve a discretized mean-field equation, which generalizes the one-dimensional Aubry-Andrè model for particles in a quasi-periodic potential by including the interaction between atoms. We compare the results for commensurate and incommensurate lattices. We investigate the role of the initial shape of the wavepacket as well as the interplay between two competing effects of the interaction, namely self-trapping and delocalization. Our calculations show that, if the condensate initially occupies a single lattice site, the dynamics of the interacting gas is dominated by self-trapping in a wide range of parameters, even for weak interaction. Conversely, if the diffusion starts from a Gaussian wavepacket, self-trapping is significantly suppressed and the destruction of localization by interaction is more easily observable.

We study the time evolution of wave packets in one-dimensional quasiperiodic lattices which localize linear waves. Nonlinearity (related to twobody interactions) has a destructive effect on localization, as observed recently for interacting atomic condensates (Lucioni et al 2011 Phys. Rev. Lett. 106 230403). We extend the analysis of the characteristics of the subdiffusive dynamics to large temporal and spatial scales. Our results for the second moment m 2 consistently reveal an asymptotic m 2 ∼ t 1/3 and an intermediate m 2 ∼ t 1/2 law. At variance with purely random systems (Laptyeva et al 2010 Europhys. Lett. 91 30001), the fractal gap structure of the linear wave spectrum strongly favours intermediate self-trapping events. Our findings give a new dimension to the theory of wave packet spreading in localizing environments.

We characterize the disorder-induced localization in momentum space for ultracold atoms in one-dimensional incommensurate lattices, according to the dual Aubry-André model. For low disorder the system is localized in momentum space, and the momentum distribution exhibits time-periodic oscillations of the relative intensity of its components. The behavior of these oscillations is explained by means of a simple three-mode approximation. We predict their frequency and visibility by using typical parameters of feasible experiments. Above the transition the system diffuses in momentum space, and the oscillations vanish when averaged over different realizations, offering a clear signature of the transition.

Cataloged from PDF version of article.We study the localization properties of a test dipole feeling the disordered potential induced by dipolar impurities trapped at random positions in an optical lattice. This random potential is marked by correlations which are a convolution of short-range and long-range ones. We show that when short-range correlations are dominant, extended states can appear in the spectrum. Introducing long-range correlations, the extended states, if any, are wiped out and localization is restored over the whole spectrum. Moreover, long-range correlations can either increase or decrease the localization length at the center of the band, which indicates a richer behavior than previously predicted