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 report on the measurement of the time required for a wave packet to tunnel through the potential barriers of an optical lattice. The experiment is carried out by loading adiabatically a Bose-Einstein condensate into a 1D optical lattice. A sudden displacement of the lattice by a few tens of nanometers excites the micromotion of the dipole mode. We then directly observe in momentum space the splitting of the wave packet at the turning points and measure the delay between the reflected and the tunneled packets for various initial displacements. Using this atomic beam splitter twice, we realize a chain of coherent micron-size Mach-Zehnder interferometers at the exit of which we get essentially a wave packet with a negative momentum, a result opposite to the prediction of classical physics.

We report on the efficient design of quantum optimal control protocols to manipulate the motional states of an atomic Bose-Einstein condensate (BEC) in a one-dimensional optical lattice. Our protocols operate on the momentum comb associated with the lattice. In contrast to previous works also dealing with control in discrete and large Hilbert spaces, our control schemes allow us to reach a wide variety of targets by varying a single parameter, the lattice position. With this technique, we experimentally demonstrate a precise, robust and versatile control: we optimize the transfer of the BEC to a single or multiple quantized momentum states with full control on the relative phase between the different momentum components. This also allows us to prepare the BEC in a given eigenstate of the lattice band structure, or superposition thereof.

We present a guided atom laser. A Bose-Einstein condensate (BEC) is created in a crossed hybrid magnetic and an elongated optical trap, which acts as a matterwave guide. Atoms are extracted from the BEC by radio frequency (rf) outcoupling and then guided in the horizontal optical matterwave guide. This method allows to control the acceleration of the beam and to achieve large de Broglie wavelength. We also measure the longitudinal energy of the guided atom laser using atom optical elements based on a blue light barrier. * Electronic address: juliette.billy@institutoptique.fr; URL: http://www.atomoptic.fr

We report on a new method to calibrate the depth of an optical lattice. It consists in triggering the intrasite dipole mode of the cloud by a sudden phase shift. The corresponding oscillatory motion is directly related to the intraband frequencies on a large range of lattice depths. Remarkably, for a moderate displacement, a single frequency dominates this oscillation for the zeroth and first order interference pattern observed after a sufficiently long time-of-flight. The method is robust against atom-atom interactions and the exact value of the extra external confinement of the initial trapping potential.

The dynamical transition of an atomic Bose-Einstein condensate from a spatially periodic state to a staggered state with alternating sign in its wavefunction is experimentally studied using a onedimensional phase modulated optical lattice. We observe the crossover from quantum to thermal fluctuations as the triggering mechanism for the nucleation of staggered states. In good quantitative agreement with numerical simulations based on the truncated Wigner method, we experimentally investigate how the nucleation time varies with the renormalized tunneling rate, the atomic density, and the driving frequency. The effective inverted energy band in the driven lattice is identified as the key ingredient which explains the emergence of gap solitons as observed in numerics and the possibility to nucleate staggered states from interband excitations as reported experimentally.

In the presence of a complex classical dynamics associated with a mixed phase space, a quantum wave function can tunnel between two stable islands through the chaotic sea, an effect that has no classical counterpart. This phenomenon, referred to as chaos assisted tunneling, is characterized by large fluctuations of the tunneling rate when a parameter is varied. To date the full extent of this effect as well as the associated statistical distribution have never been observed in a quantum system. Here we analyze the possibility of characterizing these effects accurately in a cold atom experiment.Using realistic values of the parameters of an experimental setup, we examine through analytical estimates and extensive numerical simulations a specific system that can be implemented with cold atoms, the atomic modulated pendulum. We assess the efficiency of three possible routes to observe in detail chaos assisted tunneling properties. Our main conclusion is that due to the fragility of the symmetry between positive and negative momenta as a function of quasimomentum, it is very challenging to use tunneling between classical islands centered on fixed points with opposite momentum. We show that it is more promising to use islands symmetric in position space, and characterize the regime where it could be done. The proposed experiment could be realized with current state-of-the-art technology.