1. 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) Mobility edge of atoms in laser speckle potentials: exact calculations Vs self-consistent approaches Workshop on Probing and Understanding Exotic Superconductors and Superfluids – 30 October 2014

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 Particle with energy E Disordered potential V(z) (typical value V 0 ) One-dimensional system Two-dimensional system

4. Halt of diffusion due to exponential localization of single-particle states Low disorder strong disorder localization length “deformed” Bloch state 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. Anderson localization

5. Localization effects in water waves

6. Orders of magnitude: Velocity: cm/s De Broglie wavelength:  m Time:  s-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! Anderson localization with cold atoms

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. 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: 1) On-site potential distribution P(V) for speckles : Step function

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

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:  : numerical aperture of the device imaging the speckle 3D isotropic spherical speckle: Define correlation energy: correlation length σ ∼ 1μm

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

13. 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. Theory vs cold atoms experiments J. Billy et al, Institut d'Optique (Palaiseau, France), Nature, 453, 891 (2008) G. Roati et al., Nature, 453, 895 (2008) (Aubry-André model) Disordered potential (optical “speckle” potential) Final atomic density (after 1 second) Initial atomic density

14. Dimension 2: marginally localized. : particle wave-vector Theory vs cold atoms experiments Weak localization effects (coherent backscattering) observed. Hard to observe strong (Anderson) localization: speckles have high threshold value for classical percolation for large k, localization length can exceed the system size!! Jendrzejewski et al, PRL 109,195302 (2012)

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

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

17. Hard to locate the mobility edge experimentally: 1)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 atoms in 3D speckles? o Non-Gaussian on-site distribution P(V) for blue/red speckles o To compare with SCTL theories assume 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 . : 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. For a given disorder strength V 0, compute M (E) for various values of energy E and increasing values 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, M ( E ) diverges (like M 2 ) for large M. At the mobility edge, M ( E ) is proportional to M.  (E): true 3D localization length => Study vs. M and E. Fixed point is the mobility edge! Second Step: Transfer Matrix Method

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 True mobility edge Apparent mobility edge at small M 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 : Localized branch Diffusive branch Critical point (mobility edge) Spherical 3D speckle

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 Forbidden region (below potential minimum) Average potential Mobility edge significantly below the average potential 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 Forbidden region (below potential minimum) Naive self-consistent theory (Kuhn et al) Improved self-consistent theories Yedjour et al Piraud et al 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: : Boltzmann diffusion constant : Diffusion constant (including interference) : density of states at small 

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 V 0. on-shell approximation: Always predicts the mobility edge above the average potential => badly wrong! There are states below E=0!

33. 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 V 0 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. Self-consistent theory of localization

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

35. 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 V 0 are important! 2) They predict same value of Ec for blue and red speckles, because the approximate self-energy is the same! Self-consistent theory of localization

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-red asymmetry towards classical localization (percolation threshold) ? red speckle blue speckle o 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  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 V 0 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. Forbidden region (below potential minimum) “Exact” numerical result Experiment (Josse et al., Palaiseau) WARNING: spatial correlation functions are different for numerics and experiment! Comparison with experimental results

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