Workshop on Probing Exotic Superconductors & Superfluids - 30th Oct. 2014

1. Workshop on Probing and Understanding Exotic Superconductors and Superfluids – 30 October 2014 Mobility edge of atoms in laser speckle potentials: exact calculations Vs self-consistent approaches Giuliano Orso Laboratoire Matériaux et Phénomènes Quantiques Université Paris Diderot (Paris) Work in collaboration with: Dominique Delande LKB, ENS and Université Pierre et Marie Curie (Paris)

2. Outline • Anderson localization with cold atoms in laser speckles • Numerical computation of the mobility edge • Self-consistent theory of localization • On site-distribution and blue-red asymmetry • Role of the spatial correlation function • Comparison with experiments

3. Anderson localization • Particle in a disordered (random) potential: • When , the particle is classically trapped in the potential wells. • When , the classical motion is ballistic in 1d, typically diffusive in dimension 2 and higher. • Quantum interference of scattering paths may inhibit diffusion at long times => • Anderson localization One-dimensional system Two-dimensional system Particle with energy E Disordered potential V(z) (typical value V0)

4. Anderson localization • Halt of diffusion due to exponentiallocalization of single-particle states Low disorder “deformed” Bloch state strong disorder localization length • difficult to see with electrons in disordered solids (due to e-e • interactions and phonon scattering) ⇒ ULTRA-COLD ATOMS? • easier to observe with classical waves: microwaves, light, acoustic • waves, seismic waves, etc.

5. Localization effects in water waves

6. Anderson localization with cold atoms • Orders of magnitude: • Velocity: cm/s • De Broglie wavelength: mm • Time: ms-ms • ⇒can directly observe the expansion of wave-packet • Interaction effects can be reduced or cancelled (use Feshbach resonances or very dilute samples), no phonon scattering • ⇒density profile = (atomic wave-function)2 • A tunable disorder is added using light-atom interaction: • - quasi-periodic potentials (1 dimension, Aubry-André model) • G. Modugno (Florence) • - speckle patterns (any dimension) • A. Aspect (Palaiseau), B. DeMArco (Urbana-C), G. Modugno Very favorable!

7. Speckle optical potential (2D version) • Speckle created by shining a laser on a diffusive glass plate: • Atoms feel optical dipole potential Amplitude of electric field follows Huygens principle Blue speckles: Red speckles:

8. 1) On-site potential distribution P(V) for speckles From central limit theorem, real and imaginary parts of field amplitude are independent gaussian random variables Disorder potential is modulo square of field amplitude ⇒P(V) is not gaussian but exponential: : Step function

9. 1) On-site potential distribution P(V) for speckles • We shift potential by its average value: • Very asymmetricdistributions (crucial for this talk!) • Bluespeckle has a strict lower energy bound, red does not • Even order (in V0) contributions are identical for blue and red • Odd order contributions have opposite signs P(V) V 0 e.g.

10. A typical realization of a 2D blue-detuned speckle potential Dark region (low potential, ocean floor, zero energy) Bright spot (high potential) Rigorous low energy bound, no high energy bound

11. 2) Spatial correlation of speckles • Correlation function for 2D circular aperture: • a : numerical aperture of the device imaging the speckle • 3D isotropic spherical speckle: • Define correlation energy: correlation length σ ∼ 1μm

12. Two (coherent) crossed speckles 3D speckles can be realized experimentally by crossing two orthogonal 2D speckles. They are anisotropic in general. courtesy V. Josse

13. Theory vs cold atoms experiments Interference effects depend crucially on the geometry of multiple scattering paths, i.e. on the dimension. Abrahams, Anderson, Licciardello,Ramakrishnan, PRL 42, 673 (1979) Dimension 1:(almost) always localized. : mean free path. Initial atomic density Final atomic density(after 1 second) Disordered potential (optical “speckle” potential) J. Billy et al, Institut d'Optique (Palaiseau, France), Nature, 453, 891 (2008) G. Roati et al., Nature, 453, 895 (2008) (Aubry-André model)

14. Theory vs cold atoms experiments Dimension 2: marginally localized. : particle wave-vector Weaklocalizationeffects (coherentbackscattering) observed. Hard to observe strong (Anderson) localization: • speckles have highthreshold value for classical percolation • for large k, localizationlengthcanexceed the system size!! Jendrzejewski et al, PRL 109,195302 (2012)

15. Theory vs cold atoms experiments localized Dimension 3: metal-insulator transition • Localized states near band edge • Extended states in interior • MobilityedgesEcseparatingtwo phases • Second order phase transition at E=Ec: extended Universal critical exponent Expect (orthogonal universality class) Paris-Lille collaboration: kicked-rotor model with cold atoms 3D Anderson transition in momentum space Test of universality exponent J. Chabé et al, PRL 101, 25 (2008) 255702

16. Experiments with 3D speckles Palaiseau Florence Jendrzejewski et al, Nat. Phys. 8, 398 (2012) Semeghini et al., arXiv:/1404.3528

17. Hard to locate the mobility edge experimentally: broad energy distribution of atoms; initial wave-packet contains both localized and extended states. So only a fraction of atoms actually localizes! 2) Comparison with theory is unclear ; there exist different estimates of Ec based on different implementations of Self-Consistent Theory of Localization (SCTL) Kuhn et al., NJP 9, 161 (2007) Yedjour and Van Tiggelen, Eur. Phys. J. D 59, 249 (2010) Piraud, Pezzé, and Sanchez-Palencia, NJP 15, 075007 (2013) These theories contain several approximations and seem to contradict each other: which one should we trust?

18. Can we make numerically exact predictions for the mobility edge of atomsin 3D speckles? • Non-Gaussian on-site distribution P(V) for blue/redspeckles • To compare with SCTL theoriesassume isotropic spatial correlation: Delande and Orso, PRL 113, 060601 (2014)

19. Outline • Anderson localization with cold atoms in laser speckles • Numerical computation of the mobility edge • Self-consistent theory of localization • On site-distribution and blue-red asymmetry • Role of the spatial correlation function • Comparison with experiments

20. First step: Mapping problem to Anderson model • Spatial discretization of the Schrödinger equation on a cubic grid of step D. • : 5-10 points per correlation length (error <1%) • Speckle generated numerically on the grid • Proper correlation function imprinted by an appropriate filter in Fourier space ⇒ Only states at bottom of the band are important

21. Second Step: Transfer Matrix Method • Take a long bar and compute recursively its total transmission using the transfer matrix method at fixed energy E. • Transverse periodic boundary conditions. • Longitudinal boundary condition does not matter • Total number of arithmetic operations scales like • Quasi-1D system always exponentially localized: MacKinnon and Kramer, Z. Phys. B 53, 1, (1983)

22. Second Step: Transfer Matrix Method • For a given disorder strength V0, compute lM(E) for various values of energy E and increasingvalues of M. • Requires relative accuracy in the 0.1-1% range ⇒35<M<90 • Speed up calculations by cutting the long bar in smaller pieces and averaging out results (eq. to disorder average) • In the localized regime: • In the diffusive regime, lM(E) diverges (like M2) for large M. • At the mobility edge, lM(E) is proportional to M. x(E): true 3D localization length => Study vs. M and E. Fixed point is the mobility edge!

23. Results of Transfer Matrix Calculation Spherical 3D speckle Mobility Edge? Too nice to be true …

24. Results of Transfer Matrix Calculation Spherical 3D speckle Apparent mobilityedge at small M True mobility edge Determination at +/- 0.01 is easy+/- 0.001 is difficult (200 000 hours computer time) D. Delande and G. Orso, PRL 113, 060601 (2014)

25. One-parameter scaling law • Finite-size scaling predicts: • Numerical results gathered for various values of M and E: Diffusive branch Critical point (mobility edge) Spherical 3D speckle Localized branch

26. One-parameter scaling law • Numerically determined localization/correlation length: Numerics (with error bars) Spherical 3D speckle Mobility edge Fit with Λc and ν same as for uncorrelated Anderson model (within error bar)

27. Numerical results for the mobility edge Average potential Mobility edge significantly below the average potential Forbiddenregion (below potentialminimum) D. Delande and G. Orso, PRL 113, 060601 (2014)

28. Outline • Anderson localization with cold atoms in laser speckles • Numerical computation of the mobility edge • Self-consistent theory of localization • On site-distribution and blue-red asymmetry • Role of the spatial correlation function • Comparison with experiments

29. Comparison with previous self-consistent results Naive self-consistent theory (Kuhn et al) Yedjour et al Improved self-consistenttheories Piraud et al Forbiddenregion (below potentialminimum) D. Delande and G. Orso, PRL 113, 060601 (2014)

30. Self-consistent theory of localization • Following Vollhardt and Wölfle (80's and 90's). • The starting point is the weak localization correction to the diffusion constant due to closed loops: • Self-consistent theory: • The onset of localization is characterized by: : density of states : Boltzmann diffusion constant : Diffusion constant (including interference) at small w

31. Intrinsic limits of the self-consistent theory of localization: • It is by itself an approximate theory (e.g. it predits the wrong critical exponent ν=1) • it requires the knowledge of disorder-averaged Green’s function: need further approximations (e.g. Born or Self-consistent Born approximation, CPA, etc.) • It is based on a hydrodynamic approach; Ec value depends on UV cut-off in momentum space: Vollhardt & Wölfle, PRL 48, 699 (1982); Economou & Soukoulis, PRB 28,1093 (1983)

32. Self-consistent theory of localization R. Kuhn et al, NJP 9, 161 (2007) • Very crude approximation: evaluate all quantities in the perturbative limit, at lowest order in V0. • on-shell approximation: • Always predicts the mobility edge above the average potential => badly wrong! There are states below E=0!

33. Self-consistent theory of localization A. Yedjour and B. v. Tiggelen, EPJD 59, 249 (2010) M. Piraud, L. Pezzé and L. Sanchez-Palencia, NJP 15, 075007 (2013) Improvement: take into account the shift of the lower bound of the energy spectrum (real part of self-energy) via self- consistent Born approximation Correctly predict that Ec is negative for blue speckle but … 1) the value of Ec is not very accurate; contributions from all order in V0 are important! 2) They predict same value of Ec for blue and red speckles, because the calculated self-energy is the same. This is wrong.

34. Numerical results for the mobility edge Singularity due to peculiarities of the3D speckle potentialcorrelation function Singularity is smoothed out in exact numerics Blue-detuned 3D spherical speckle

35. Self-consistent theory of localization A. Yedjour and B. v. Tiggelen, EPJD 59, 249 (2010) M. Piraud, L. Pezzé and L. Sanchez-Palencia, NJP 15, 075007 (2013) Improvement: take into account the shift of the lower bound of the energy spectrum (real part of self-energy) via self- consistent Born approximation Correctly predict that Ec is negative for blue speckles but… 1) the value of Ec is not very accurate; contributions from all order in V0 are important! 2) They predict same value of Ec for blue and red speckles, because the approximate self-energy is the same!

36. Outline • Anderson localization with cold atoms in laser speckles • Numerical computation of the mobility edge • Self-consistent theory of localization • On site-distribution and blue-red asymmetry • Role of the spatial correlation function • Comparison with experiments

37. Huge blue-redasymmetry red speckle ? towardsclassicallocalization(percolationthreshold) blue speckle • Naive and improved self-consistent theories predict the same mobility edge!!

38. Classical percolation argument • Classical allowed region at an energy half-way between the red and blue mobility edges (pictures in 2D!). Connected Not connected

39. Outline • Anderson localization with cold atoms in laser speckles • Numerical computation of the mobility edge • Self-consistent theory of localization • On site-distribution and blue-red asymmetry • Role of the spatial correlation function • Comparison with experiments

40. How important are details of spatial correlation function for speckle? • We compute Ec for different correlation functions having the same “width″ σ • Almost no effect!At the mobility edge, disorder is so strong that details of the spatial correlation function are completely smoothed out => only the correlation length s matters

41. Outline • Anderson localization with cold atoms in laser speckles • Numerical computation of the mobility edge • Self-consistent theory of localization • On site-distribution and blue-red asymmetry • Role of the spatial correlation function • Comparison with experiments

42. Comparison with experimental results • Three experimental measurements of the mobility edge. Mobility edge higher than our numerical predictions. • B. De Marco (Urbana Champaign). Much too high mobility edge, strange properties of the atomic momentum/energy distributions. • V. Josse (Palaiseau). Anisotropic disordered potential, relatively low fraction of atoms below the mobility edge. Measured mobility edge below zero, but still too high. • G. Modugno (Florence). Anisotropic disordered potential. Large localized fraction. Qualitative behavior of the mobility edge with V0 in fair agreement. Mobility edge seems a bit too high. • Anderson transition is second order transition => atoms with energy close to the mobility edge diffuse very slowly. Maybe responsible for overestimation of the mobility edge? • Numerical calculation with anisotropic disorder are needed, but difficult. Work in progress.

43. Comparison with experimental results Experiment (Josse et al., Palaiseau) WARNING: spatial correlationfunctions are different for numerics and experiment! Forbiddenregion (below potentialminimum) “Exact”numerical result

44. Summary • It is possible to compute numerically the mobility edge for non-interacting cold atoms in a 3D spatially correlated potential. • Can be computed for any type of on-site potential distribution and any not-too-anisotropic spatial correlation function. • Work in progress for anisotropic potentials. • Large blue-red asymmetry. • Partial failure of the self-consistent theory of localization, mainly because some quantities are computed at the Born approximation. • Main features can be understood from P(V) distribution