1. HKUST april 20101 3D Anderson Localization of Noninteracting Cold Atoms Bart van Tiggelen Université Joseph Fourier – Grenoble 1 / CNRS Warsaw may 2011

2. HKUST april 20102 My precious collaborators Sergey Skipetrov, Anna Minguzzi (Grenoble) Afifa Yedjour (PhD) (Grenoble and Oran-Algeria)

3. HKUST april 20103 Localization [..] very few believed it at the time, and even fewer saw its importance, among those who failed was certainly its author. It has yet to receive adequate mathematical treatment, and one has to resort to the indignity of numerical simulations to settle even the simplest questions about it. P.W. Anderson, Nobel lecture, 1977 50 years of Anderson localization …..and now we have (numerical) experiments !

4. HKUST april 20104 c Physics Today, August 2009 50 years of Anderson Localization http://www.andersonlocalization.com/

5. HKUST april 20105 Diffusion of Waves diffusion constant Diffusion = random walk of waves

6. HKUST april 20106 Small diffusion constant ≠ localization photon Trapped Rb 85 Temperature T = 0,0001 K (v=15 cm/s) 10 10 atomes 5 mm Random walk of photons ℓ ℓ Labeyrie, Miniatura, Kaiser (2006, Nice)

7. HKUST april 20107 V(r) r Dimension < 3 « Trivial »Localization (most mathematical proofs)

8. HKUST april 20108 V(r) r Dimension < 3 « Trivial »Localization « Tunnel /percolation assisted »localization (Anderson model) (most mathematical proofs)

9. HKUST april 20109 V(r) r Dimension < 3 « Trivial »Localization « genuine »Localization E > V max (Classical waves, cold atoms ??) (Anderson model) (most mathematical proofs) « Tunnel /percolation assisted »localization

10. HKUST april 201010 V(r) r Dimension = 3 Mobilityedge « metal » «insulator»

11. HKUST april 201011 Mott minimum conductivity Thouless criterion and scaling theoryThouless criterion and scaling theory Quantum Hall effectQuantum Hall effect MIT and role of interactions MIT and role of interactions dense point spectrum dense point spectrum Chaos theory (DMPK equation) Multifractal eigenfunctions Multifractal eigenfunctions Full statistics of conductance and transmission Random laser Transverse localizationTransverse localization Anderson tight binding model & Kicked rotor Anderson tight binding model & Kicked rotor

12. HKUST april 201012 Mesoscopic Wave Transport One particle Green function Dyson Green function Self energy Mean free path Spectral function Average LDOS k=1/2 ℓ : strong scattering reciprocity

13. HKUST april 201013 Mesoscopic Wave Transport Dyson Green function Mean free path: is strongly scattering (localized) OK for white noise fluctuations:

14. HKUST april 201014 Mesoscopic Wave Transport Two particle Green function Momentum conservation Wigner function (looks like phase space distribution) Proba density Proba current density

15. HKUST april 201015 Diffusion approximation Proba of quantum diffusion reciprocity normalization Kubo Greenwood formula

16. HKUST april 201016 Diffusion approximation near mobility edge k ℓ =1 k+q/2 k’+q/2 k-q/2-k’-q/2 E+h Ω /2 E-h Ω /2 x x x x xx Boltzmann approximation

17. HKUST april 201017 Diffusion approximation k+q/2 k’+q/2 k-q/2k’-q/2 E+h Ω /2 E-h Ω /2 x x x x xx + k+q/2 k’+q/2 k-q/2k’-q/2 E+h Ω /2 E-h Ω /2 x x x x xx « ladder » « most-crossed »

18. HKUST april 201018 Diffusion approximation Diffuse return Green function Diverges in 3D: q < 1/ ℓ or 1/ ℓ *? Infinite medium with white noise Critical exponent =1

19. HKUST april 201019 Inhibition of transport of Q1D BEC in random potential Palaiseau group, Firenze group PRL oct 2005 Time after trap extinction expansion n(x,t) V(x)

20. HKUST april 201020 mobility edge kℓ ~1 Diffusive regime Localization with  > ℓ Localization with  < ℓ Localization of noninteracting cold atoms in 3D white noise band edge Trap stage µ chemical potential expansion stage (t=0) Random potential Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008 Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008

21. HKUST april 201021 Density profile of atoms at large times localized anomalous diffusion n loc (r) Selfconsistent theory with white noise ( ν =s=1) Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008 Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008 45 % localized 3 % localized

22. HKUST april 201022 Cold atoms in a 3D speckle potential Mott minimum Kuhn, Miniatura, Delande etal NJP 2007 Yedjour, BavT, EPJD 2010 nonGaussian!

23. HKUST april 201023 Self-consistent Born Approximation Mean free path?

24. HKUST april 201024 Selfconsistent theory of localization

25. HKUST april 201025 0 D/D B D/D B {1-K}

26. HKUST april 201026 Kuhn, Miniatura, etal FBA (2007): k ℓ =0.95 (1- ) Is 3D cold atom localization « trivial »? k ℓ =1.12

27. HKUST april 201027 V(r) r Cold atoms in 3D speckle Mobilityedge « metal » «insulator»

28. HKUST april 201028 Energy distribution Fraction of localized atoms * * 45 % in white noise (Skipetrov etal 2008) U=E ξ 2

29. HKUST april 201029 Anderson Localization is still a major theme in condensed matter physics, full of surprises New experiments (in high dimensions and with « new » matter waves) exist and are underway. Need of accurate description of self-energy Thank you for your attention