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

Outline

Anderson localization with cold atoms in laser specklesNumerical computation of the mobility edgeSelf-consistent theory of localizationOn site-distribution and blue-red asymmetryRole of the spatial correlation functionComparison with experiments

Anderson localizationParticle 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 V0)

One-dimensional system Two-dimensional system

•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

Localization effects in water waves

Orders of magnitude:Velocity: cm/sDe Broglie wavelength: mTime: 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

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:

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

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 V0) 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

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

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∼

Two (coherent) crossed speckles

courtesy V. Josse

3D speckles can be realized experimentally by crossing two orthogonal 2D speckles. They are anisotropic in general.

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

Dimension 2: marginally localized.

: particle wave-vector

`

k

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)

Dimension 3: metal-insulator transition

• Localized states near band edge • Extended states in interior

• Mobility edges Ec separating two phases

• Second order phase transition at E=Ec:

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

Experiments with 3D speckles

Semeghini et al., arXiv:/1404.3528

Jendrzejewski et al, Nat. Phys. 8, 398 (2012)

Palaiseau

Florence

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?

Can we make numerically exact predictionsfor 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)

Outline

Anderson localization with cold atoms in laser specklesNumerical computation of the mobility edgeSelf-consistent theory of localizationOn site-distribution and blue-red asymmetryRole of the spatial correlation functionComparison with experiments

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

⇒

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)

For a given disorder strength V0, 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 M2) 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

Results of Transfer Matrix Calculation

Spherical 3D speckle

Mobility Edge?

Too nice to be true …

Results of Transfer Matrix Calculation

Spherical 3D speckle

True mobility edge

Apparent mobilityedge 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)

One-parameter scaling law

Finite-size scaling predicts:

Numerical results gathered for various values of M and E:

Localized branch

Diffusive branchCritical point(mobility edge)

Spherical 3D speckle

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)

Numerical results for the mobility edge

Forbiddenregion (below potentialminimum)

Average potential

Mobility edge significantly below the average potential

D. Delande and G. Orso, PRL 113, 060601 (2014)

Outline

Anderson localization with cold atoms in laser specklesNumerical computation of the mobility edgeSelf-consistent theory of localizationOn site-distribution and blue-red asymmetryRole of the spatial correlation functionComparison with experiments

Comparison with previous self-consistent results

Forbiddenregion (below potentialminimum)

Naive self-consistent theory (Kuhn et al)

Improved self-consistenttheories

Yedjour et al

Piraud et al

D. Delande and G. Orso, PRL 113, 060601 (2014)

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

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)

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!

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.

Self-consistent theory of localization

Numerical results for the mobility edge

Blue-detuned 3D spherical speckle

Singularity due topeculiarities of the3D speckle potentialcorrelation function

Singularity is smoothed out in exact numerics

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!

Self-consistent theory of localization

Outline

Anderson localization with cold atoms in laser specklesNumerical computation of the mobility edgeSelf-consistent theory of localizationOn site-distribution and blue-red asymmetryRole of the spatial correlation functionComparison with experiments

Huge blue-red asymmetry

towardsclassicallocalization(percolationthreshold)

?

red speckle

blue speckle

o Naive and improved self-consistent theories predict the same mobility edge!!

Classical percolation argument

Classical allowed region at an energy half-way between the red and blue mobility edges (pictures in 2D!).

Connected

Not connected

Outline

Anderson localization with cold atoms in laser specklesNumerical computation of the mobility edgeSelf-consistent theory of localizationOn site-distribution and blue-red asymmetryRole of the spatial correlation functionComparison with experiments

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

Outline

Anderson localization with cold atoms in laser specklesNumerical computation of the mobility edgeSelf-consistent theory of localizationOn site-distribution and blue-red asymmetryRole of the spatial correlation functionComparison with experiments

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.

Forbiddenregion (below potentialminimum) “Exact”

numericalresult

Experiment (Josse et al., Palaiseau)

WARNING: spatial correlationfunctions are different for numericsand experiment!

Comparison with experimental results

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