We investigate, both experimentally and theoretically, possible routes towards Anderson-like localization of Bose-Einstein condensates in disordered potentials. The dependence of this quantum interference effect on the nonlinear interactions and the shape of the disorder potential is investigated. Experiments with an optical lattice and a superimposed disordered potential reveal the lack of Anderson localization. A theoretical analysis shows that this absence is due to the large length scale of the disorder potential as well as its screening by the nonlinear interactions. Further analysis shows that incommensurable superlattices should allow for the observation of the cross-over from the nonlinear screening regime to the Anderson localized case within realistic experimental parameters.Disordered systems have played a central role in condensed matter physics in the last 50 years. Recently, it was proposed that ultracold atomic gases may serve as a laboratory for disordered quantum systems [1,2] and allow for the experimental investigation of various open problems in that field [3]. Some of these problems concern strongly correlated systems [4], the realization of Bose [5,6] or Fermi glasses [7], quantum spin glasses [8] and quantum percolation [9]. This letter addresses one of the most important issues, namely the interplay of Anderson localization (AL) [10] and repulsive interactions [11]. This interplay may lead to the creation of delocalized phases both for fermions [12] and bosons [6]. The possible occurrence of AL has also been investigated theoretically for weakly interacting Bose-Einstein condensates (BEC) [13], and in this case it was shown that even moderate nonlinear interaction counteracts the localization. As a main result of this letter we show that despite this difficulty there exists an experimentally accessible regime where Anderson-like localization can be realized with present day techniques.Several methods have been proposed to produce a disordered, or quasi-disordered potential for trapped atomic gases. They include the use of speckle radiation [14], incommensurable optical lattices [15], impurity atoms in the sample [16] and the disorder that appears close to the surface of atom chips [17]. Recently, first experiments searching for effects of disorder in the dynamics of weakly interacting BECs were realized [18].In this letter we shed new light on the interplay between disorder and interactions by studying trapped BECs under the influence of a disordered potential and a one dimensional (1D) optical lattice (OL). The OL creates a periodic potential and the randomness of the disordered potential leads to AL for noninteracting particles [1]. We study how the presence of interactions affects nontrivial localization in our necessarily finite system.Our experiments were performed with 87 Rb BoseEinstein condensates in an elongated magnetic trap (MT) with axial and radial frequencies of ω z = 2π × 14 Hz and ω ⊥ = 2π × 200 Hz, respectively. Further details of our experimental apparatus were described...

We measure the intensity correlation function of two interfering spatially displaced copies of phase fluctuating Bose-Einstein condensates. It is shown that this corresponds to a measurement of the phase correlation properties of the initial condensate. Analogous to the method used in the stellar interferometer experiment of Hanbury Brown and Twiss, we use spatial intensity correlations to determine the phase coherence lengths of elongated condensates. We find good agreement with our prediction of the correlation function and confirm the expected coherence length.

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