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3D Anderson Localization of Noninteracting Cold Atoms Bart van Tiggelen

3D Anderson Localization of Noninteracting Cold Atoms Bart van Tiggelen Université Joseph Fourier – Grenoble 1 / CNRS. Warsaw may 2011. My precious collaborators Sergey Skipetrov, Anna Minguzzi (Grenoble) Afifa Yedjour (PhD) (Grenoble and Oran-Algeria).

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3D Anderson Localization of Noninteracting Cold Atoms Bart van Tiggelen

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  1. 3D Anderson Localization of Noninteracting Cold Atoms Bart van Tiggelen Université Joseph Fourier – Grenoble 1 / CNRS HKUST april 2010 Warsaw may 2011

  2. My precious collaborators • Sergey Skipetrov, Anna Minguzzi (Grenoble) • Afifa Yedjour (PhD) (Grenoble and Oran-Algeria) HKUST april 2010

  3. 50 years of Anderson localization 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 …..and now we have (numerical) experiments ! HKUST april 2010

  4. Physics Today, August 2009 50 years of Anderson Localization http://www.andersonlocalization.com/ c HKUST april 2010

  5. Diffusion of Waves Diffusion = random walk of waves HKUST april 2010 diffusion constant

  6. Small diffusion constant ≠ localization Trapped Rb85 Temperature T = 0,0001 K (v=15 cm/s) 1010 atomes ℓ ℓ Labeyrie, Miniatura, Kaiser (2006, Nice) photon 5 mm Random walk of photons HKUST april 2010

  7. Dimension < 3 V(r) r « Trivial » Localization (most mathematical proofs) HKUST april 2010

  8. Dimension < 3 V(r) « Tunnel /percolation assisted » localization (Anderson model) r « Trivial » Localization (most mathematical proofs) HKUST april 2010

  9. Dimension < 3 « genuine » Localization E > Vmax (Classical waves, cold atoms ??) V(r) « Tunnel /percolation assisted » localization (Anderson model) r « Trivial » Localization (most mathematical proofs) HKUST april 2010

  10. Dimension = 3 V(r) « metal » Mobility edge r «insulator» HKUST april 2010

  11. Mott minimum conductivity • Thouless criterion and scaling theory • Quantum Hall effect • MIT and role of interactions • dense point spectrum • Chaos theory (DMPK equation) • Multifractal eigenfunctions • Full statistics of conductance and transmission • Random laser • Transverse localization • Anderson tight binding model & Kicked rotor HKUST april 2010

  12. Mesoscopic Wave Transport One particle Green function reciprocity Dyson Green function k=1/2ℓ : strong scattering Self energy Mean free path Spectral function HKUST april 2010 Average LDOS

  13. Mesoscopic Wave Transport Dyson Green function OK for white noise fluctuations: Mean free path: is strongly scattering (localized) HKUST april 2010

  14. Mesoscopic Wave Transport Two particle Green function Momentum conservation Wigner function (looks like phase space distribution) Proba current density Proba density HKUST april 2010

  15. Diffusion approximation reciprocity Proba of quantum diffusion normalization Kubo Greenwood formula HKUST april 2010

  16. Diffusion approximation k+q/2 E+hΩ/2 k’+q/2 x x x Boltzmann approximation k-q/2 -k’-q/2 x x x E-hΩ/2 near mobility edge kℓ=1 HKUST april 2010

  17. Diffusion approximation k+q/2 E+hΩ/2 k’+q/2 x x x k-q/2 k’-q/2 x x x « ladder » E-hΩ/2 k+q/2 E+hΩ/2 k’+q/2 + x x x k-q/2 k’-q/2 x x x « most-crossed » E-hΩ/2 HKUST april 2010

  18. Diffusion approximation Diffuse return Green function Diverges in 3D: q < 1/ℓ or 1/ℓ*? Infinite medium with white noise Critical exponent =1 HKUST april 2010

  19. Inhibition of transport of Q1D BEC in random potential n(x,t) V(x) expansion Time after trap extinction Palaiseau group, Firenze group PRL oct 2005 HKUST april 2010

  20. µ chemical potential Localization of noninteracting cold atoms in 3D white noise Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008 expansion stage (t=0) Random potential Trap stage Localization with x < ℓ Localization with x > ℓ Diffusive regime mobility edge kℓ ~1 band edge HKUST april 2010

  21. Density profile of atoms at large times Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008 anomalous diffusion localized Selfconsistent theory with white noise (ν=s=1) 3 % localized nloc(r) 45 % localized HKUST april 2010

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

  23. Self-consistent Born Approximation Mean free path? HKUST april 2010

  24. Selfconsistent theory of localization HKUST april 2010

  25. <V> D/DB D/DB {1-K} 0 HKUST april 2010

  26. Is 3D cold atom localization « trivial »? Kuhn, Miniatura, etal FBA (2007): kℓ=0.95 (1-<cos ϑ>) kℓ=1.12 HKUST april 2010

  27. Cold atoms in 3D speckle V(r) « metal » Mobility edge r «insulator» HKUST april 2010

  28. U=Eξ2 Energy distribution Fraction of localized atoms * * 45 % in white noise (Skipetrov etal 2008) HKUST april 2010

  29. 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 HKUST april 2010

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