In 1958, P.W. Anderson predicted the exponential localization 1 of electronic wave functions in disordered crystals and the resulting absence of diffusion. It has been realized later that Anderson localization (AL) is ubiquitous in wave physics 2 as it originates from the interference between multiple scattering paths, and this has prompted an intense activity. Experimentally, localization has been reported in light waves 3,4,5,6,7 , microwaves 8,9 , sound waves 10 , and electron 11 gases but to our knowledge there is no direct observation of exponential spatial localization of matter-waves (electrons or others). Here, we report the observation of exponential localization of a Bose-Einstein condensate (BEC) released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle 12 . We operate in a regime allowing AL: i) weak disorder such that localization results from many quantum reflections of small amplitude; ii) atomic density small enough that interactions are negligible. We image directly the atomic density profiles vs time, and find that weak disorder can lead to the stopping of the expansion and to the formation of a stationary exponentially localized wave function, a direct signature of AL. Fitting the exponential wings, we extract the localization length, and compare it to theoretical calculations. Moreover we show that, in our one-dimensional speckle potentials whose noise spectrum has a high spatial frequency cut-off, exponential localization occurs only when the de Broglie wavelengths of the atoms in the expanding BEC are larger than an effective mobility edge corresponding to that cut-off. In the opposite case, we find that the density profiles decay algebraically, as predicted in ref 13. The method presented here can be extended to localization of atomic quantum gases in higher dimensions, and with controlled interactions.

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We present a detailed analysis of the 1D expansion of a coherent interacting matterwave (a Bose-Einstein condensate) in the presence of disorder. A 1D random potential is created via laser speckle patterns. It is carefully calibrated and the self-averaging properties of our experimental system are discussed. We observe the suppression of the transport of the BEC in the random potential. We discuss the scenario of disorder-induced trapping taking into account the radial extension in our experimental 3D BEC and we compare our experimental results with the theoretical predictions.

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We report the Bragg spectroscopy of interacting one-dimensional Bose gases loaded in optical lattices across the superfluid to Mott-insulator phase transition. Elementary excitations are created with a non-zero momentum and the response of the correlated 1D gases is in the linear regime. The complexity of the strongly correlated quantum phases is directly displayed in the spectra which exhibit novel features. This work paves the way for a precise characterization of the state of correlated atomic phases in optical lattices.PACS numbers: 67.85. Hj, 67.85.De Cold atomic gases loaded in optical lattices have been routinely used during the past few years as a versatile and powerful experimental system to study many physical problems [1]. In particular, they allow to realize and manipulate strongly correlated quantum phases [2,3,4] and constitute a promising candidate for implementing quantum information processing and quantum simulation schemes [5]. To achieve those goals, a corner stone consists in a precise characterization of the correlated gaseous phases.As in similar problems of condensed-matter, the presence of strong correlations makes it hard to draw a complete picture, both from the experimental and theoretical point of view. The implementation of spectroscopic probes, such as angle-resolved photo-emission spectroscopy for high-Tc superconductors where electronelectron correlations play a major role [6], are crucial. For the Mott-insulator phase created with trapped atomic gases, experiments have demonstrated the existence of a gap in the spectrum [2,7] and have investigated the shell structure [8], the spatial order via noise correlation techniques [9] and the suppression of compressibility [4]. Yet an experimental probe giving direct access to important information, such as the temperature or the elementary excitations on which the dynamical properties of the many-body system depend, is still missing. It has been proposed that inelastic light scattering (Bragg spectroscopy) performed at non-zero momentum in the linear response regime could provide such a tool [10,11,12,13,14].In this letter we report the measurement of the linear response of interacting one-dimensional (1D) gases across the superfluid (SF) to Mott-insulator (MI) transition. The 1D gases are loaded in an optical lattice whose amplitude drives the atomic sample from a SF to an inhomogeneous MI state. The elementary excitations are created at a non-zero momentum using twophoton Bragg transition. The presence of an additional mode to the phonon mode in the SF state is suggested. In the inhomogeneous MI state, multiple resonances are observed which give information about the particle-hole excitation energy, the inhomogeneity of the trapped system and which exhibit novel features at low energies that could be related to the temperature of the atomic sample. From the continuous modification of the spectra we also get quantitative information on the critical lattice amplitude for entering the MI phase.Amount of excitation (a.u.)Amount of excit...

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)

Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In this work, we trace these elusive excitations via their effects on the dynamical structure factor of 1D strongly interacting Bose gases at low temperature. An array of 1D Bose gases is obtained by loading a 87 Rb condensate in a two-dimensional lattice potential. The dynamical structure factor of the system is probed by energy deposition through low-momentum Bragg excitations. The experimental signals are compared to recent theoretical predictions for the dynamical structure factor of the Lieb-Liniger model at T > 0. Our results demonstrate that the main contribution to the spectral widths stems from the dynamics of the interaction-induced excitations in the gas, which cannot be described by the Luttinger liquid theory.

We study an ultracold Bose gas in the presence of 1D disorder for repulsive inter-atomic interactions varying from zero to the Thomas-Fermi regime. We show that for weak interactions the Bose gas populates a finite number of localized single-particle Lifshits states, while for strong interactions a delocalized disordered Bose-Einstein condensate is formed. We discuss the schematic quantum-state diagram and derive the equations of state for various regimes.