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arXiv:1510.06857 · Physical Review A

Breakdown of Anderson localization in the transport of Bose-Einstein condensates through one-dimensional disordered potentials

Julien Dujardin, Thomas Engl, Peter Schlagheck

Published Jan 13, 2016• 14 citations
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Abstract

We study the transport of an interacting Bose--Einstein condensate through a 1D correlated disorder potential. We use for this purpose the truncated Wigner method, which is, as we show, corresponding to the diagonal approximation of a semiclassical van Vleck--Gutzwiller representation of this many-body transport process. We also argue that semiclassical corrections beyond this diagonal approximation are vanishing under disorder average, thus confirming the validity of the truncated Wigner method in this context. Numerical calculations show that, while for weak atom-atom interaction strengths Anderson localization is preserved with a slight modification of the localization length, for larger interaction strengths a crossover to a delocalized regime exists due to inelastic scattering. In this case, the transport is fully incoherent.

DOI
10.1103/physreva.93.013612

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We study the transport of an interacting Bose--Einstein condensate through a 1D correlated disorder potential. We use for this purpose the truncated Wigner method, which is, as we show, corresponding to the diagonal approximation of a semiclassical van Vleck--Gutzwiller representation of this many-body transport process. We also argue that semiclassical corrections beyond this diagonal approximation are vanishing under disorder average, thus confirming the validity of the truncated Wigner method in this context. Numerical calculations show that, while for weak atom-atom interaction strengths Anderson localization is preserved with a slight modification of the localization length, for larger interaction strengths a crossover to a delocalized regime exists due to inelastic scattering. In this case, the transport is fully incoherent.
This thesis presents investigations on the interplay of coherence and many-body effects in the quasi one-dimensional transport of Bose-Einstein condensates (BEC) through scattering potentials. Such configurations can be realized with guided atom lasers that provide a coherent atomic beam. An exact theoretical description of the dynamics is out of reach due to the presence atom-atom interactions. Different levels of approximations are nevertheless possible with their strengths and weaknesses. The mean-field approximation, where the dynamics of the BEC is governed by the Gross-Pitaevskii equation, is most commonly used in the field of ultracold atoms. In this thesis the truncated Wigner method is used to go beyond the standard Gross-Pitaevskii description. This method is adapted in order to study the scattering of Bose-Einstein condensates in one-dimensional waveguides where atom-atom interactions and external potentials are nonvanishing only in a finite region of space. In this case, the truncated Wigner method is combined with the smooth exterior complex scaling method and incorporates quantum noise that originate from the vacuum fluctuations in the waveguide. Inelastic scattering is shown to play a major role in the resonant transport of BEC through a symmetric double potential barrier effectively forming an atomic quantum dot. Indeed, fully resonant transmission is prohibited and incoherent atoms as well as collective oscillations are detected in the transmitted beam. It is also shown that inelastic scattering destroys Anderson localization in the case of transport through disordered potentials. The classical (incoherent) ohmic transmission is recovered for finite atom-atom interactions. The validity of the truncated Wigner method is then assessed using the semiclassical van Vleck-Gutzwiller propagator in the Fock space of the many-body system. It is shown that the truncated Wigner method corresponds to the so-called diagonal approximation, and it is possible to identify the leading correction to the truncated Wigner results, which is provided by the so-called coherent backscattering (CBS) contribution. Coherent basckattering in Fock space is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. For the case of closed Bose-Hubbard models, the relevance of CBS is confirmed by numerically comparing the (classical) truncated Wigner evolution probabilities to the exact quantum probabilities in Bose-Hubbard models: While a CBS-induced enhancement of the return probability to the initial state is clearly seen in the exact quantum simulations of the bosonic many-body system, this enhancement is absent in the classical calculations. The magnitude and dependence of the CBS contribution on gauge fields, which break time-reversal invariance, is numerically confirmed. For the case of disordered open systems, it can be shown that this contribution as well as next-to leading order contributions vanish thereby confirming the validity of the truncated Wigner method.
We theoretically investigate the localization of an expanding Bose–Einstein condensate (BEC) 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 the considered 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 as in the experiments reported by Clément et al 2005 Phys. Rev. Lett. 95 170409 and Fort et al 2005 Phys. Rev. Lett. 95 170410, 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.
We theoretically study the propagation of an interacting Bose-Einstein condensate in a two-dimensional disorder potential, following the principle of an atom laser. The constructive interference between time-reversed scattering paths gives rise to coherent backscattering, which may be observed under the form of a sharp cone in the disorder-averaged angular backscattered current. As is found by the numerical integration of the Gross-Pitaevskii equation, this coherent backscattering cone is inversed when a nonvanishing interaction strength is present, indicating a crossover from constructive to destructive interferences. Numerical simulations based on the truncated Wigner method allow one to go beyond the mean-field approach and show that dephasing renders this signature of antilocalization hidden behind a structureless and dominant incoherent contribution as the interaction strength is increased and the injected density decreased, in a regime of parameters far away from the mean-field limit. However, despite a partial dephasing, we observe that this weak antilocalization scenario prevails for finite interaction strengths, opening the way for an experimental observation with 87Rb^{87}\mathrm{Rb}87Rb atoms.
Anderson localization of waves in disordered media was originally predicted[1] fifty years ago, in the context of transport of electrons in crystals[2]. The phenomenon is much more general[3] and has been observed in a variety of systems, including light waves[4, 5]. However, Anderson localization has not been observed directly for matter waves. Owing to the high degree of control over most of the system parameters (in particular the interaction strength), ultracold atoms offer opportunities for the study of disorder-induced localization[6]. Here we use a non-interacting Bose-Einstein condensate to study Anderson localization. The experiment is performed with a one-dimensional quasi-periodic lattice - a system that features a crossover between extended and exponentially localized states, as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated through investigations of the transport properties and spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. This controllable system may be used to investigate the interplay of disorder and interaction (ref. 7 and references therein), and to explore exotic quantum phases[8, 9].
We consider the phase coherent transport of a quasi-one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length LLL. Among the possible different types of flow we identified [T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale ${L}^{\ensuremath{\ast}}$ for the disordered region, above which most of the realizations of the random potential lead to time-dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.
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 ${\ensuremath{\sigma}}_{R}$. For speckle potentials the Fourier transform of the correlation function vanishes for momenta $k>2/{\ensuremath{\sigma}}_{R}$ so that the Lyapunov exponent vanishes in the Born approximation for $k>1/{\ensuremath{\sigma}}_{R}$. Then, for the initial healing length of the condensate ${\ensuremath{\xi}}_{\mathrm{in}}>{\ensuremath{\sigma}}_{R}$ the localization is exponential, and for ${\ensuremath{\xi}}_{\mathrm{in}}<{\ensuremath{\sigma}}_{R}$ it changes to algebraic.