Measurement of the mobility edge for 3D Anderson localization


Anderson localization is a universal phenomenon affecting non-interacting quantum particles in a disordered environment. In three spatial dimensions, theory predicts a quantum phase transition from localization to diffusion at a critical energy, the mobility edge, which depends on the disorder strength. Although it has been recognized already long ago as a prominent feature of disordered systems, a complete experimental characterization of the mobility edge is still missing. Here we report the measurement of the mobility edge for ultracold atoms in a disordered potential created by laser speckles. We are able to control both the disorder strength and the energy of the system, so as to probe the position of the localization threshold in the disorder–energy plane. Our results might allow a direct experiment–theory comparison, which is a prerequisite to study the even more challenging problem of disorder and interactions.

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Figure 1: 3D speckle disorder.
Figure 2: Expansion and localization dynamics.
Figure 3: Momentum and energy distribution.
Figure 4: Excitation spectrum.
Figure 5: Measured mobility edge versus the disorder strength.


  1. 1

    Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

  2. 2

    Mott, N. F. Metal–insulator transitions. Phys. Today 31(11), 42–47 (1978).

  3. 3

    Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008).

  4. 4

    Abrahams, E. (ed.) 50 Years of Anderson Localization (World Scientific, 2012).

  5. 5

    Katsumoto, S., Komori, F., Sano, N. & Kobayashi, S. Fine tuning of metal-insulator transition in Al0.3Ga0.7As using persistent photoconductivity. J. Phys. Soc. Jpn 56, 2259–2262 (1987).

  6. 6

    Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).

  7. 7

    Vollhardt, D. & Wölfle, P. in Electronic Phase Transitions (eds Hanke, W. & Kopaev, Yu. V.) 1–78 (Elsevier, 1992).

  8. 8

    Kramer, B. & MacKinnon, A. Localization: Theory and experiment. Rep. Prog. Phys. 56, 1469–1564 (1993).

  9. 9

    Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).

  10. 10

    Hu, H. et al. Localization of ultrasound in a three-dimensional elastic network. Nature Phys. 4, 845–848 (2008).

  11. 11

    Sperling, T. et al. Direct determination of the transition to localization of light in three dimensions. Nature Photon. 7, 48–52 (2013).

  12. 12

    Chabé, J. et al. Experimental observation of the Anderson metal–insulator transition with atomic matter waves. Phys. Rev. Lett. 101, 255702 (2008).

  13. 13

    Lemarié, G., Lignier, H., Delande, D., Szriftgiser, P. & Garreau, J. C. Critical state of the Anderson transition: Between a metal and an insulator. Phys. Rev. Lett. 105, 090601 (2010).

  14. 14

    Lopez, M., Clément, J-F., Szriftgiser, P., Garreau, J. C. & Delande, D. Experimental test of universality of the Anderson transition. Phys. Rev. Lett. 108, 095701 (2012).

  15. 15

    Lopez, M. et al. Phase diagram of the anisotropic Anderson transition with the atomic kicked rotor: Theory and experiment. New J. Phys. 15, 065013 (2013).

  16. 16

    Billy, J. et al. Direct observation of Anderson localization of matter-waves in a controlled disorder. Nature 453, 891–894 (2008).

  17. 17

    Roati, G. et al. Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453, 895–898 (2008).

  18. 18

    Kondov, S. S. et al. Three-dimensional Anderson localization of ultracold matter. Science 334, 66–68 (2011).

  19. 19

    Jendrzejewski, F. et al. Three-dimensional localization of ultracold atoms in an optical disordered potential. Nature Phys. 8, 398–403 (2012).

  20. 20

    Roati, G. et al. 39K Bose–Einstein condensate with tunable interactions. Phys. Rev. Lett. 99, 010403 (2007).

  21. 21

    Shapiro, B. Cold atoms in the presence of disorder. J. Phys. A 45, 143001 (2012).

  22. 22

    Yedjour, A. & Van Tiggelen, B. A. Diffusion and localization of cold atoms in 3D optical speckle. Eur. Phys. J. D 59, 249–255 (2010).

  23. 23

    Piraud, M., Pezzé, L. & Sanchez-Palencia, L. Matter wave transport and Anderson localization in anisotropic three-dimensional disorder. Eur. Phys. Lett. 99, 50003 (2012).

  24. 24

    Piraud, M., Pezzé, L. & Sanchez-Palencia, L. Quantum transport of atomic matter waves in anisotropic two-dimensional and three-dimensional disorder. New J. Phys. 15, 075007 (2013).

  25. 25

    Delande, D. & Orso, G. Mobility edge for cold atoms in laser speckle potentials. Phys. Rev. Lett. 113, 060601 (2014).

  26. 26

    Landini, M. et al. Direct evaporative cooling of 39K atoms to Bose–Einstein condensation. Phys. Rev. A 86, 033421 (2012).

  27. 27

    Kuhn, R. C., Sigwarth, O., Miniatura, C., Delande, D. & Müller, C. A. Coherent matter wave transport in speckle potentials. New J. Phys. 9, 161 (2007).

  28. 28

    Mahan, G. D. Many Particle Physics (Springer, 1990).

  29. 29

    Lifshits, I. M., Gredeskui, S. A. & Pastur, L. A. Introduction to the Theory of Disordered Systems (Wiley, 1988).

  30. 30

    Piraud, M., Sanchez-Palencia, L. & Van Tiggelen, B. Anderson localization of matter waves in 3D anisotropic disordered potentials. Phys. Rev. A 90, 063639 (2014).

  31. 31

    Fratini, E. & Pilati, S. Anderson localization of matter waves in quantum-chaos theory. Preprint at (2015).

  32. 32

    McGehee, W. R., Kondov, S. S., Xu, W., Zirbel, J. J. & DeMarco, B. Three-dimensional Anderson localization in variable scale disorder. Phys. Rev. Lett. 111, 145303 (2013).

  33. 33

    Müller, C. A. & Shapiro, B. Comment on “Three-Dimensional Anderson Localization in Variable Scale Disorder”. Phys. Rev. Lett. 113, 099601 (2014).

  34. 34

    McGehee, W. R., Kondov, S. S., Xu, W., Zirbel, J. J. & DeMarco, B. McGehee et al. Reply. Phys. Rev. Lett. 113, 099602 (2014).

  35. 35

    Huang, K. & Meng, H-F. Hard-sphere Bose gas in random external potentials. Phys. Rev. Lett. 69, 644–647 (1992).

  36. 36

    Nattermann, T. & Pokrovsky, V. L. Bose–Einstein condensates in strongly disordered traps. Phys. Rev. Lett. 100, 060402 (2008).

  37. 37

    Pilati, S., Giorgini, S. & Prokof’ev, N. Superfluid transition in a Bose gas with correlated disorder. Phys. Rev. Lett. 102, 150402 (2009).

  38. 38

    Crowell, P. A., Van Keulz, F. W. & Reppy, J. D. Onset of superfluidity in 4He films adsorbed on disordered substrates. Phys. Rev. B 55, 12620–12634 (1997).

  39. 39

    Lye, J. E. et al. Bose–Einstein condensate in a random potential. Phys. Rev. Lett. 95, 070401 (2005).

  40. 40

    Zapf, V., Jaime, M. & Batista, C. D. Bose–Einstein condensation in quantum magnets. Rev. Mod. Phys. 86, 563–614 (2014).

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We acknowledge discussions with V. Josse and L. Pezzé. This work was supported by ERC (grants 247371 and 258325), and partially by EU - H2020 research and innovation programme (grant 641122), INFN (MICRA collaboration) and MIUR (grant RBFR08H058).

Author information

G.Semeghini and M.L. designed the experiment; G.Semeghini, M.L. and G.M. analysed the data and performed the numerical simulations; all the other authors participated to the experiment, data analysis, discussion of the results and writing of the manuscript.

Correspondence to G. Modugno.

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The authors declare no competing financial interests.

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Semeghini, G., Landini, M., Castilho, P. et al. Measurement of the mobility edge for 3D Anderson localization. Nature Phys 11, 554–559 (2015).

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