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The ability to load ultracold atoms at a well-defined energy in a disordered potential is a crucial tool to study quantum transport, and in particular Anderson localization. In this paper, we present a new method for achieving that goal by rf transfer of atoms of an atomic Bose-Einstein condensate from a disorder insensitive state to a disorder sensitive state. It is based on a bichromatic laser speckle pattern, produced by two lasers whose frequencies are chosen so that their light-shifts cancel each other in the first state and add-up in the second state. Moreover, the spontaneous scattering rate in the disorder-sensitive state is low enough to allow for long observation times of quantum transport in that state. We theoretically and experimentally study the characteristics of the resulting potential., Comment: 14 pages, 7 figures

We study the full counting statistics (FCS) of quantum gases in samples of thousands of interacting bosons, detected atom-by-atom after a long free-fall expansion. In this far-field configuration, the FCS reveals the many-body coherence from which we characterize iconic states of interacting lattice bosons, by deducing the normalized correlations $g^{(n)}(0)$ up to the order $n=6$. In Mott insulators, we find a thermal FCS characterized by perfectly-contrasted correlations $g^{(n)}(0)= n!$. In interacting Bose superfluids, we observe small deviations to the Poisson FCS and to the ideal values $g^{(n)}(0)=1$ expected for a pure condensate. To describe these deviations, we introduce a heuristic model that includes an incoherent contribution attributed to the depletion of the condensate. The predictions of the model agree quantitatively with our measurements over a large range of interaction strengths, that includes the regime where the condensate is strongly depleted by interactions. These results suggest that the condensate component exhibits a full coherence $g^{(n)}(0) =1$ at any order $n$ up to $n=6$ and at arbitrary interaction strengths. The approach demonstrated here is readily extendable to characterize a large variety of interacting quantum states and phase transitions., Comment: 12 pages, 7 figures

We measure the momentum density in a Bose-Einstein condensate (BEC) with dilute spin impurities after an expansion in the presence of interactions. We observe tails decaying as $1/k^4$ at large momentum $k$ in the condensate and in the impurity cloud. These algebraic tails originate from the impurity-BEC interaction, but their amplitudes greatly exceed those expected from two-body contact interactions at equilibrium in the trap. Furthermore, in the absence of impurities, such algebraic tails are not found in the BEC density measured after the interaction-driven expansion. These results highlight the key role played by impurities when present, a possibility that had not been considered in our previous work [Phys. Rev. Lett. 117, 235303 (2016)]. Our measurements suggest that these unexpected algebraic tails originate from the non-trivial dynamics of the expansion in the presence of impurity-bath interactions., Comment: 6 pages, 6 figures

Spectral function is a key tool for understanding the behavior of Bose-Einstein condensates of cold atoms in random potentials generated by a laser speckle. In this paper we introduce a new method for computing the spectral functions in disordered potentials. Using a combination of the Wigner-Weyl approach with the landscape theory, we build an approximation for the Wigner distributions of the eigenstates in the phase space and show its accuracy in all regimes, from the deep quantum regime to the intermediate and semiclassical. Based on this approximation, we devise a method to compute the spectral functions using only the landscape-based effective potential. The paper demonstrates the efficiency of the proposed approach for disordered potentials with various statistical properties without requiring any adjustable parameters., Comment: 10 pages, 10 figures

We report the realisation of a Hanbury-Brown and Twiss (HBT)-like experiment with a gas of strongly interacting bosons at low temperatures. The regime of large interactions and low temperatures is reached in a three-dimensional optical lattice and atom-atom correlations are extracted from the detection of individual metastable Helium atoms after a long free-fall. We observe a HBT bunching in the non-condensed fraction of the gas whose properties strongly deviate from the HBT signals expected for non-interacting bosons. In addition, we show that the measured correlations reflect the peculiar quantum statistics of atoms belonging to the quantum depletion and of the Bogoliubov phonons, i.e., of collective excitations of the many-body quantum state. Our results demonstrate that atom-atom correlations provide information about the quantum state of strongly-interacting particles, extending the interest of HBT-like experiments beyond the case of non-interacting particles., Comment: 10 pages, 7 figures

In this lecture, I first present my views on the second vs the first quantum revolution, then describe the Hanbury Brown and Twiss effect with photons, and indicate why it was so important in the development of modern quantum optics. The presentation of our experiments on the HBT effect with atoms will allow me to emphasize the analogies but also the increased richness of the effect when going from photons to atoms. I will similarly describe the HOM effect for photons and its significance, and then present the analogous experiment with atoms. In conclusion, I will put these two effects in the long list of landmarks in the development of quantum optics, and indicate what has been done and what remains to be done with atoms in lieu of photons., Comment: 22 pages, 13 figures. Published as Chapter 12 in the proceedings of the Les Houches Summer School 1996 : {\it Current Trends in Atomic Physics.} Edited by Antoine Browaeys, Trey Porto, Charles S. Adams, Matthias Weidemuller, and Leticia F. Cugliandolo. Oxford University Press (2019). DOI: 10.1093/oso/9780198837190.003.0012

We study the elastic scattering time $\tau_\mathrm{s}$ of ultracold atoms propagating in optical disordered potentials in the strong scattering regime, going beyond the recent work of J. Richard \emph{et al.} \textit{Phys. Rev. Lett.} \textbf{122} 100403 (2019). There, we identified the crossover between the weak and the strong scattering regimes by comparing direct measurements and numerical simulations to the first order Born approximation. Here we focus specifically on the strong scattering regime, where the first order Born approximation is not valid anymore and the scattering time is strongly influenced by the nature of the disorder. To interpret our observations, we connect the scattering time $\tau_\mathrm{s}$ to the profiles of the spectral functions that we estimate using higher order Born perturbation theory or self-consistent Born approximation. The comparison reveals that self-consistent methods are well suited to describe $\tau_\mathrm{s}$ for Gaussian-distributed disorder, but fails for laser speckle disorder. For the latter, we show that the peculiar profiles of the spectral functions, as measured independently in V. Volchkov \emph{et al.} \textit{Phys. Rev. Lett.} \textbf{120}, 060404 (2018), must be taken into account. Altogether our study characterizes the validity range of usual theoretical methods to predict the elastic scattering time of matter waves, which is essential for future close comparison between theory and experiments, for instance regarding the ongoing studies on Anderson localization., Comment: 21 pages, 6 figures

We report on the investigation of the three-dimensional single-atom-resolved distributions of bosonic Mott insulators in momentum-space. Firstly, we measure the two-body and three-body correlations deep in the Mott regime, finding a perfectly contrasted bunching whose periodicity reproduces the reciprocal lattice. In addition, we show that the two-body correlation length is inversely proportional to the in-trap size of the Mott state with a pre-factor in agreement with the prediction for an incoherent state occupying a uniformly filled lattice. Our findings indicate that the momentum-space correlations of a Mott insulator at small tunnelling is that of a many-body ground-state with Gaussian statistics. Secondly, in the Mott insulating regime with increasing tunnelling, we extract the spectral weight of the quasi-particles from the momentum density profiles. On approaching the transition towards a superfluid, the momentum spread of the spectral weight is found to decrease as a result of the increased mobility of the quasi-particles in the lattice. While the shapes of the observed spectral weight agree with the ones predicted by perturbative many-body calculations, the fitted mobilities are larger than the theoretical ones. This discrepancy is similar to that previously reported on the time-of-flight visibility., Comment: 13 pages, 10 figures

We report on an extensive study of the elastic scattering time $\tauS$ of matter-waves in optical disordered potentials. Using direct experimental measurements, numerical simulations and comparison with first-order Born approximation based on the knowledge of the disorder properties, we explore the behavior of $\tauS$ over more than three orders of magnitude, spanning 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 $k\lS\sim 1$. 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., Comment: 6+4 pages, 3+4 figures

We propose to use the method introduced by Volchkov et al., based on state dependent disordered ultracold bosons, to address the critical state at the mobility edge of the Anderson localization transition, and to observe its intriguing multifractal structure. An optimally designed external radio frequency pulse can be applied to generate transitions to eigenstates in a narrow energy window close to the mobility edge, where critical scaling and multifractality emerge. Two-photon laser scanning microscopy will be used to address individual localized states even close to the transition. The projected image of the cloud is shown to inherit multifractality and to display universal density correlations. Time of flight images of the excited states are predicted to show interference fringes in the localized phase, while they allow one to map equal energy surfaces deep in the metallic phase.

We report on the measurement of the spectral functions of noninteracting ultracold atoms in a three-dimensional disordered potential resulting from an optical speckle field. Varying the disorder strength by 2 orders of magnitude, we observe the crossover from the "quantum" perturbative regime of low disorder to the "classical" regime at higher disorder strength, and find an excellent agreement with numerical simulations. The method relies on the use of state-dependent disorder and the controlled transfer of atoms to create well-defined energy states. This opens new avenues for experimental investigations of three-dimensional Anderson localization.

We present a free-space interferometer to observe two-particle interference of a pair of atoms with entangled momenta. The source of atom pairs is a Bose--Einstein condensate subject to a dynamical instability, and the interferometer is realized using Bragg diffraction on optical lattices, in the spirit of our recent Hong--Ou--Mandel experiment. We report an observation consistent with an entangled state at the input of the interferometer. We explain how our current setup can be extended to enable a test of a Bell inequality on momentum observables.

We study the transmission of a disordered waveguide subjected to a finite bias field. The statisticaldistribution of transmission is analytically shown to take a universal form. It depends on a singleparameter, the system length expressed in a rescaled metrics, which encapsulates all the microscopicfeatures of the medium and the bias field. Excellent agreement with numerics is found for variousmodels of disorder and bias field. For white-noise disorder and a linear bias field, we demonstratethe algebraic nature of the decay of the transmission with distance, irrespective of the value ofthe bias field. It contrasts with the expansion of a wave packet, which features a delocalizationtransition for large bias field. The difference is attributed to the different boundary conditionsfor the transmission and expansion schemes. The observability of these effects in conductancemeasurements for electrons or ultracold atoms is discussed, taking into account key features, suchas finite-range disorder correlations, nonlinear bias fields, and finite temperatures.

The celebrated Hong, Ou and Mandel (HOM) effect is one of the simplest illustrations of two-particle interference, and is unique to the quantum realm. In the original experiment, two photons arriving simultaneously in the input channels of a beam-splitter were observed to always emerge together in one of the output channels. Here, we report on the realisation of a closely analogous experiment with atoms instead of photons. This opens the prospect of testing Bell's inequalities involving mechanical observables of massive particles, such as momentum, using methods inspired by quantum optics, with an eye on theories of the quantum-to-classical transition. Our work also demonstrates a new way to produce and benchmark twin-atom pairs that may be of interest for quantum information processing and quantum simulation.

We report on the observation of suppression and revival of coherent backscattering of ultra-cold atoms launched in an optical disorder and submitted to a short dephasing pulse, as proposed in a recent paper of T. Micklitz \textit{et al.} [arXiv:1406.6915]. This observation, in a quasi-2D geometry, demonstrates a novel and general method to study weak localization by manipulating time reversal symmetry in disordered systems. In future experiments, this scheme could be extended to investigate higher order localization processes at the heart of Anderson (strong) localization., Comment: 5 pages, 5 figures

Many predictions of Doppler cooling theory of two-level atoms have never been verified in a three-dimensional geometry, including the celebrated minimum achievable temperature $\hbar \Gamma/2 k_B$, where $\Gamma$ is the transition linewidth. Here, we show that, despite their degenerate level structure, we can use Helium-4 atoms to achieve a situation in which these predictions can be verified. We make measurements of atomic temperatures, magneto-optical trap sizes, and the sensitivity of optical molasses to a power imbalance in the laser beams, finding excellent agreement with the Doppler theory. We show that the special properties of Helium, particularly its small mass and narrow transition linewidth, prevent effective sub-Doppler cooling with red-detuned optical molasses., Comment: 8 pages, 5 figures

We report the all-optical production of Bose Einstein condensates (BEC) of $^{39}$K atoms. We directly load $3 \times 10^{7}$ atoms in a large volume optical dipole trap from gray molasses on the D1 transition. We then apply a small magnetic quadrupole field to polarize the sample before transferring the atoms in a tightly confining optical trap. Evaporative cooling is finally performed close to a Feshbach resonance to enhance the scattering length. Our setup allows to cross the BEC threshold with $3 \times 10^5$ atoms every 7s. As an illustration of the interest of the tunability of the interactions we study the expansion of Bose-Einstein condensates in the 1D to 3D crossover.

We use a one-dimensional optical lattice to modify the dispersion relation of atomic matter waves. Four-wave mixing in this situation produces atom pairs in two well defined beams. We show that these beams present a narrow momentum correlation, that their momenta are precisely tunable, and that this pair source can be operated both in the regime of low mode occupancy and of high mode occupancy.

We present a study of two di fferent sets of Micro-Channel Plates used for time and space resolved single particle detection. We investigate the eff ects of the gold coating and that of introducing an interplate voltage between the spatially separated plates. We fi nd that the gold coating increases the count rate of the detector and the pulse amplitude as previously reported for non-spatially resolved setups. The interplate voltage also increases count rates. In addition, we nd that a non-zero interplate voltage improves the spatial accuracy in determining the arrival position of incoming single particles (by 20%) while the gold coating has a negative e ect (by 30%).

We report on the direct observation of coherent backscattering (CBS) of ultracold atoms, in a quasi-two-dimensional configuration. Launching atoms with a well-defined momentum in a laser speckle disordered potential, we follow the progressive build up of the momentum scattering pattern, consisting of a ring associated with multiple elastic scattering, and the CBS peak in the backward direction. Monitoring the depletion of the initial momentum component and the formation of the angular ring profile allows us to determine microscopic transport quantities. The time resolved evolution of the CBS peak is studied and is found a fair agreement with predictions, at long times as well as at short times. The observation of CBS can be considered a direct signature of coherence in quantum transport of particles in disordered media. It is responsible for the so called weak localization phenomenon, which is the precursor of Anderson localization., Comment: 5 pages, 4 figures

We present an experimental study of a four beam optical lattice using the light scattered by the atoms in the lattice. We use both intensity correlations and observations of the transient behavior of the scattering when the lattice is suddenly switched on. We compare results for 3 different configurations of the optical lattice. We create situations in which the Lamb-Dicke effect is negligible and show that, in contrast to what has been stated in some of the literature, the damping rate of the 'coherent' atomic oscillations can be much smaller than the inelastic photon scattering rate., Comment: An old paper

We experimentally study the effect of disorder on trapped quasi two-dimensional (2D) 87Rb clouds in the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition. The disorder correlation length is of the order of the Bose gas characteristic length scales (thermal de Broglie wavelength, healing length) and disorder thus modifies the physics at a microscopic level. We analyze the coherence properties of the cloud through measurements of the momentum distributions, for two disorder strengths, as a function of its degeneracy. For moderate disorder, the emergence of coherence remains steep but is shifted to a lower entropy. In contrast, for strong disorder, the growth of coherence is hindered. Our study is an experimental realization of the dirty boson problem in a well controlled atomic system suitable for quantitative analysis.

We measure the momentum distribution of a 2D trapped Bose gas and observe the increase of the range of coherence around the Berezinskii-Kosterlitz-Thouless (BKT) transition. We quantitatively compare our observed profiles to both a Hartee-Fock mean-field theory and to quantum Monte-Carlo simulations. In the normal phase, we already observe a sharpening of the momentum distribution. This behavior is partially captured in a mean-field approach, in contrast to the physics of the BKT transition.

We report a study of three-dimensional (3D) localization of ultracold atoms suspended against gravity, and released in a 3D optical disordered potential with short correlation lengths in all directions. We observe density profiles composed of a steady localized part and a diffusive part. Our observations are compatible with the self-consistent theory of Anderson localization, taking into account the specific features of the experiment, and in particular the broad energy distribution of the atoms placed in the disordered potential. The localization we observe cannot be interpreted as trapping of particles with energy below the classical percolation threshold., Comment: published in Nature Physics; The present version is the initial manuscript (unchanged compared to version 1); The published version is available online at http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys2256.html

We show that, in contrast to immediate intuition, Anderson localization of noninteracting particles induced by a disordered potential in free space can increase (i.e., the localization length can decrease) when the particle energy increases, for appropriately tailored disorder correlations. We predict the effect in one, two, and three dimensions, and propose a simple method to observe it using ultracold atoms placed in optical disorder. The increase of localization with the particle energy can serve to discriminate quantum versus classical localization.

We theoretically study the Anderson localization of a matter wave packet in a one-dimensional disordered potential. We develop an analytical model which includes the initial phase-space density of the matter wave and the spectral broadening induced by the disorder. Our approach predicts a behavior of the localized density profile significantly more complex than a simple exponential decay. These results are confirmed by large-scale and long-time numerical calculations. They shed new light on recent experiments with ultracold atoms and may impact their analysis.

We study in detail the flux properties of a radiofrequency outcoupled horizontally guided atom laser, following the scheme demonstrated in [Guerin W et al. 2006 Phys. Rev. Lett. 97 200402]. Both the outcoupling spectrum (flux of the atom laser versus rf frequency of the outcoupler) and the flux limitations imposed to operate in the quasicontinuous regime are investigated. These aspects are studied using a quasi-1D model, whose predictions are shown to be in fair agreement with the experimental observations. This work allows us to identify the operating range of the guided atom laser and to confirm its good promises in view of studying quantum transport phenomena.

We demonstrate sub-Poissonian number differences in four-wave mixing of Bose-Einstein condensates of metastable helium. The collision between two Bose-Einstein condensates produces a scattering halo populated by pairs of atoms of opposing velocities, which we divide into several symmetric zones. We show that the atom number difference for opposing zones has sub-Poissonian noise fluctuations whereas that of nonopposing zones is well described by shot noise. The atom pairs produced in a dual number state are well adapted to sub shot-noise interferometry and studies of Einstein-Podolsky-Rosen-type nonlocality tests., Comment: 4 pages, 3 figures

Nontrivial symmetry of order parameters is crucial in some of the most interesting quantum many-body states of ultracold atoms and condensed matter systems. Examples in cold atoms include p-wave Feshbach molecules and d-wave paired states of fermions that could be realized in optical lattices in the Hubbard regime. Identifying these states in experiments requires measurements of the relative phase of different components of the entangled pair wavefunction. We propose and discuss two schemes for such phase sensitive measurements, based on two-particle interference revealed in atom-atom or atomic density correlations. Our schemes can also be used for relative phase measurements for non-trivial particle-hole order parameters, such as d-density wave order., Comment: 4 pages, 4 figures

We have observed the interferometric suspension of a free-falling Bose-Einstein condensate periodically submitted to multiple-order diffraction by a vertical 1D standing wave. The various diffracted matter waves recombine coherently, resulting in high contrast interference in the number of atoms detected at constant height. For long suspension times, multiple-wave interference is revealed through a sharpening of the fringes. We use this scheme to measure the acceleration of gravity.

We study Anderson localization of ultracold atoms in weak, one-dimensional speckle potentials, using perturbation theory beyond Born approximation. We show the existence of a series of sharp crossovers (effective mobility edges) between energy regions where localization lengths differ by orders of magnitude. We also point out that the correction to the Born term explicitly depends on the sign of the potential. Our results are in agreement with numerical calculations in a regime relevant for experiments. Finally, we analyze our findings in the light of a diagrammatic approach.

We study a Fabry-Perot cavity formed from a ridge waveguide on a AlGaAs substrate. We experimentally determined the propagation losses in the waveguide at 780 nm, the wavelength of Rb atoms. We have also made a numerical and analytical estimate of the losses induced by the presence of the gap which would allow the interaction of cold atoms with the cavity field. We found that the intrinsic finesse of the gapped cavity can be on the order of F ~ 30, which, when one takes into account the losses due to mirror transmission, corresponds to a cooperativity parameter for our system C ~ 1.

We report on light-shift tomography of a cloud of 87 Rb atoms in a far-detuned optical-dipole trap at 1565 nm. Our method is based on standard absorption imaging, but takes advantage of the strong light-shift of the excited state of the imaging transition, which is due to a quasi-resonance of the trapping laser with a higher excited level. We use this method to (i) map the equipotentials of a crossed optical-dipole trap, and (ii) study the thermalisation of an atomic cloud by following the evolution of the potential-energy of atoms during the free-evaporation process.

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 . 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 [Phys. Rev. Lett. 98, 210401 (2007)]. The method presented here can be extended to localization of atomic quantum gases in higher dimensions, and with controlled interactions.

Atom optics, a field which takes much inspiration from traditional optics, has advanced to the point that some of the fundamental experiments of quantum optics, involving photon correlations, have found atomic analogs. We discuss some recent experiments on atom bunching and anti-bunching as well as some prospects for extending them to the field of many body physics., Comment: English version of "Optique atomique quantique : apr\`es les bosons, les fermions"

We theoretically investigate the localization of an expanding Bose-Einstein condensate with repulsive atom-atom interactions in a disordered potential. We focus on the regime where the initial inter-atomic interactions dominate over the kinetic energy and the disorder. At equilibrium in a trapping potential and for small disorder, the condensate shows a Thomas-Fermi shape modified by the disorder. When the condensate is released from the trap, a strong suppression of the expansion is obtained in contrast to the situation in a periodic potential with similar characteristics. This effect crucially depends on both the momentum distribution of the expanding BEC and the strength of the disorder. For strong disorder, the suppression of the expansion results from the fragmentation of the core of the condensate and from classical reflections from large modulations of the disordered potential in the tails of the condensate. We identify the corresponding disorder-induced trapping scenario for which large atom-atom interactions and strong reflections from single modulations of the disordered potential play central roles. For weak disorder, the suppression of the expansion signals the onset of Anderson localization, which is due to multiple scattering from the modulations of the disordered potential. We compute analytically the localized density profile of the condensate and show that the localization crucially depends on the correlation function of the disorder. In particular, for speckle potentials the long-range correlations induce an effective mobility edge in 1D finite systems. Numerical calculations performed in the mean-field approximation support our analysis for both strong and weak disorder., Comment: New Journal of Physics; focus issue "Quantum Correlations in Tailored Matter - Common perspectives of mesoscopic systems and quantum gases"; 30 pages, 10 figures

We report an experimental test of complementarity using clock-triggered single-photon pulses emitted by an individual N-V color center in a diamond nanocrystal. The single photons are sent into a Mach-Zehnder interferometer with an output beamsplitter of adjustable reflection coefficient R. In addition, the choice of introducing or removing this beamsplitter is random and relativistically space-like separated from the entering of the photon inside the interferometer, as required for the Wheeler's delayed-choice regime. Each set value of R allows us to observe interference with visibility V and to obtain incomplete which-path information characterized by the distinguishability D. The measured values of V and D are found to obey the complementarity relation V^2 + D^2 =< 1.

Wheeler has strikingly illustrated the wave-particle duality by the delayed-choice thought experiment, in which the configuration of a 2-path interferometer is chosen after a single-photon light-pulsed has entered it. We present a quantitative theoretical analysis of an experimental realization of Wheeler's proposal.

We observe large density modulations in time-of-flight images of elongated Bose-Einstein condensates, initially confined in a harmonic trap and in the presence of weak disorder. The development of these modulations during the time-of-flight and their dependence with the disorder are investigated. We render an account of this effect using numerical and analytical calculations. We conclude that the observed large density modulations originate from the weak initial density modulations induced by the disorder, and not from initial phase fluctuations (thermal or quantum)., Comment: Published version; 4+ pages; 4 figures

We study the Anderson localization of Bogolyubov quasiparticles in an interacting Bose-Einstein condensate (with healing length \xi) subjected to a random potential (with 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} is proportional to 1/\xi when \xi is much larger than \sigma_R while k_{max} is proportional to 1/\sigma_R when \xi is much smaller than \sigma_R, and that the localization is strongest when \xi is of the order of \sigma_R. Numerical calculations support our analysis and our estimates indicate that the localization of the Bogolyubov quasiparticles is accessible in current experiments with ultracold atoms., Comment: published version (no significant changes compared to last version)

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 > \sigma_R the localization is exponential, and for \xi < \sigma_R it changes to algebraic., Comment: published versioon (no significant change compared to last version)

Fifty years ago, Hanbury Brown and Twiss (HBT) discovered photon bunching in light emitted by a chaotic source, highlighting the importance of two-photon correlations and stimulating the development of modern quantum optics . The quantum interpretation of bunching relies upon the constructive interference between amplitudes involving two indistinguishable photons, and its additive character is intimately linked to the Bose nature of photons. Advances in atom cooling and detection have led to the observation and full characterisation of the atomic analogue of the HBT effect with bosonic atoms. By contrast, fermions should reveal an antibunching effect, i.e., a tendency to avoid each other. Antibunching of fermions is associated with destructive two-particle interference and is related to the Pauli principle forbidding more than one identical fermion to occupy the same quantum state. Here we report an experimental comparison of the fermion and the boson HBT effects realised in the same apparatus with two different isotopes of helium, 3He (a fermion) and 4He (a boson). Ordinary attractive or repulsive interactions between atoms are negligible, and the contrasting bunching and antibunching behaviours can be fully attributed to the different quantum statistics. Our result shows how atom-atom correlation measurements can be used not only for revealing details in the spatial density, or momentum correlations in an atomic ensemble, but also to directly observe phase effects linked to the quantum statistics in a many body system. It may thus find applications to study more exotic situations >., Comment: Nature 445, 402 (2007). V2 includes the supplementary information

The quantum "mystery which cannot go away" (in Feynman's words) of wave-particle duality is illustrated in a striking way by Wheeler's delayed-choice GedankenExperiment. In this experiment, the configuration of a two-path interferometer is chosen after a single-photon pulse has entered it : either the interferometer is \textit{closed} (\textit{i.e.} the two paths are recombined) and the interference is observed, or the interferometer remains \textit{open} and the path followed by the photon is measured. We report an almost ideal realization of that GedankenExperiment, where the light pulses are true single photons, allowing unambiguous which-way measurements, and the interferometer, which has two spatially separated paths, produces high visibility interference. The choice between measuring either the 'open' or 'closed' configuration is made by a quantum random number generator, and is space-like separated -- in the relativistic sense -- from the entering of the photon into the interferometer. Measurements in the closed configuration show interference with a visibility of 94%, while measurements in the open configuration allow us to determine the followed path with an error probability lower than 1%.

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., Comment: published version

We discuss experiments to produce and detect atom correlations in a degenerate or nearly degenerate gas of neutral atoms. First we treat the atomic analog of the celebrated Hanbury Brown Twiss experiment, in which atom correlations result simply from interference effects without any atom interactions.We have performed this experiment for both bosons and fermions. Next we show how atom interactions produce correlated atoms using the atomic analog of spontaneous four-wavemixing. Finally, we briefly mention experiments on a one dimensional gas on an atom chip in which correlation effects due to both interference and interactions have been observed., Comment: to appear in conference proceedings "Atomic Physics 20"