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arXiv:cond-mat/0612670 · Physical Review Letters

Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials

Laurent Sanchez-Palencia, David Clément, Pierre Lugan, Philippe Bouyer, Georgy V. Shlyapnikov, Alain Aspect

Published May 23, 2007• 313 citations
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Abstract

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.

DOI
10.1103/physrevlett.98.210401

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We study the Anderson localization of Bogolyubov quasiparticles in an interacting Bose-Einstein condensate (with a healing length $\ensuremath{\xi}$) subjected to a random potential (with a finite correlation length ${\ensuremath{\sigma}}_{R}$). We derive analytically the Lyapunov exponent as a function of the quasiparticle momentum kkk, and we study the localization maximum kmax{k}_{\mathrm{max}}kmax​. For 1D speckle potentials, we find that ${k}_{\mathrm{max}}\ensuremath{\propto}1/\ensuremath{\xi}$ when $\ensuremath{\xi}\ensuremath{\gg}{\ensuremath{\sigma}}_{R}$ while ${k}_{\mathrm{max}}\ensuremath{\propto}1/{\ensuremath{\sigma}}_{R}$ when $\ensuremath{\xi}\ensuremath{\ll}{\ensuremath{\sigma}}_{R}$, and that the localization is strongest when $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\sigma}}_{R}$. Numerical calculations support our analysis, and our estimates indicate that the localization of the Bogolyubov quasiparticles is accessible in experiments with ultracold atoms.
The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice Boltzmann (QLB) method is used, and compared with a standard Crank-Nicolson scheme. The QLB simulations show evidence of Anderson localization even for relatively low-energy condensates, with a healing length as large as one-tenth of the Thomas-Fermi length. Moreover, very long-time simulations, lasting up to 15 000 optical confinement periods, indicate that the Anderson localization degrades in time, although at a very slow pace. In particular, the inverse localization length is found to decay according to a t;{-1/3} law. This lends support to the idea that localized wave functions, although not strictly ground states, represent extremely long-lived metastable states of the expanding condensate.
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.
By numerical and variational solutions of the Gross-Pitaevskii equation, we studied the localization of a noninteracting and weakly interacting Bose-Einstein condensate (BEC) in a disordered cold atom lattice and a speckle potential. In the case of a single BEC fragment, the variational analysis produced good results. For a weakly disordered potential, the localized BEC's are found to have an exponential tail as in the weak Anderson localization. We also investigated the expansion of a noninteracting BEC in these potentials. We find that the BEC will be locked in an appropriate localized state after an initial expansion and will execute breathing oscillation around a mean shape when a BEC at equilibrium in a harmonic trap is suddenly released into a disorder potential.
We numerically study the Anderson localization in an oscillating one-dimensional Rydberg-dressed Bose-Einstein condensate with weak random disorder in which the range of the interaction, the blockade radius Rc{R}_{\mathrm{c}}Rc​, is variable. Without disorder, the uniform system can undergo a superfluid-supersolid transition at ${R}_{\text{c}}={l}_{\mathrm{c}}\ensuremath{\simeq}1.7{\ensuremath{\xi}}_{0}$ with ${\ensuremath{\xi}}_{\text{0}}$ the zero-range healing length. When ${\ensuremath{\xi}}_{0}$ exceeds the disorder correlation length ${\ensuremath{\sigma}}_{\mathrm{D}}$, we show that exponential localization occurs in the equilibrium condensate when ${R}_{\mathrm{c}}\ensuremath{\le}{l}_{\mathrm{c}}$, while Gaussian localization occurs when Rc>lc{R}_{\mathrm{c}}>{l}_{\mathrm{c}}Rc​>lc​. The latter suggests that the kkk wave for a long-ranged interacting system could decay Gaussianly with weak random disorder.
We study the interplay of disorder localization and strong local interactions within the Anderson-Hubbard model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. For fermions, we find rich nonmonotonic behavior of the localization length ξ, particularly in two-dimensional systems, including an interaction-induced exponential enhancement of ξ for small and intermediate disorders and a strong reduction of ξ due to hopping suppression by strong interactions. In three dimensions, we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbardgapped phases. For small U there is re-entrant behavior from the Anderson localized phase to the metallic phase. For bosons, the unrestricted particle occupation number per lattice site yields a monotonic enhancement of ξ as a function of decreasing interaction, which we assume to persist until the superfluid Bose- Einstein condensate phase is entered. Besides, we study cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.