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

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

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

HKUST april c Physics Today, August years of Anderson Localization

HKUST april Diffusion of Waves diffusion constant Diffusion = random walk of waves

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

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

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

HKUST april 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

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

HKUST april 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

HKUST april 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

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

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

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

HKUST april 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

HKUST april 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 »

HKUST april 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 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)

HKUST april 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

HKUST april 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, % localized 3 % localized

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

HKUST april Self-consistent Born Approximation Mean free path?

HKUST april Selfconsistent theory of localization

HKUST april D/D B D/D B {1-K}

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

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

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

HKUST april 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