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Direct observation of Anderson localization of matter waves in a controlled disorder

Abstract

In 1958, Anderson predicted the localization1 of electronic wavefunctions in disordered crystals and the resulting absence of diffusion. It is now recognized that Anderson localization is ubiquitous in wave physics2 because it originates from the interference between multiple scattering paths. Experimentally, localization has been reported for light waves3,4,5,6,7, microwaves8,9, sound waves10 and electron gases11. However, there has been no direct observation of exponential spatial localization of matter waves of any type. Here we observe exponential localization of a Bose–Einstein condensate released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle12. We operate in a regime of pure Anderson localization, that is, with weak disorder—such that localization results from many quantum reflections of low amplitude—and an atomic density low enough to render interactions negligible. We directly image the atomic density profiles as a function of time, and find that weak disorder can stop the expansion and lead to the formation of a stationary, exponentially localized wavefunction—a direct signature of Anderson localization. We extract the localization length by fitting the exponential wings of the profiles, and compare it to theoretical calculations. The power spectrum of the one-dimensional speckle potentials has a high spatial frequency cutoff, causing exponential localization to occur only when the de Broglie wavelengths of the atoms in the expanding condensate are greater than an effective mobility edge corresponding to that cutoff. 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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Figure 1: Observation of exponential localization.
Figure 2: Stationarity of the localized profile.
Figure 3: Localization length versus amplitude of the disordered potential.
Figure 4: Algebraic and exponential regimes in a one-dimensional speckle potential.

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Acknowledgements

The authors are indebted to P. Chavel, T. Giamarchi, M. Lewenstein and G. Shlyapnikov for many discussions, to P. Georges and G. Roger for assistance with the laser, and to F. Moron, A. Villing and G. Colas for technical assistance on the experimental apparatus. This research was supported by the Centre National de la Recherche Scientifique (CNRS), the Délégation Générale de l’Armement (DGA), the Ministère de l’Education Nationale, de la Recherche et de la Technologie (MENRT), the Agence Nationale de la Recherche (ANR), the Institut Francilien de Recherche sur les Atomes Froids (IFRAF) and IXSEA; by the STREP programme FINAQS of the European Union; and by the programme QUDEDIS of the European Science Foundation (ESF).

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Billy, J., Josse, V., Zuo, Z. et al. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453, 891–894 (2008). https://doi.org/10.1038/nature07000

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