Anderson Anderson LocalizationLocalization19581958--20082008

FromFrom electronselectrons, , towardstowards ultrasoundultrasoundand and …… cold cold atomsatoms

Bart van TiggelenBart van TiggelenLaboratoire de Physique et ModLaboratoire de Physique et Modéélisation des Milieux lisation des Milieux

CondensCondensééssUniversitUniversitéé Joseph Fourier /CNRSJoseph Fourier /CNRS

ScatteringScattering Systems Systems withwith complexcomplex Dynamics FOR760 2008: Dynamics FOR760 2008: OctoberOctober 6 20086 2008

Localization [..] very few believed it at the time, and even fewer saw itsimportance, 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

1958 Anderson « vanishing of diffusion »

1960 Mott/Ioffe-Regel

1965 Mott Minimum conductivityVariable range hopping

1972 Thouless Sensitivity to BC: g < 1(Thouless criterion)

>1982 Sharvin, Lagendijk,Maret, Maynard,…

Weak localizationMesoscopic physics!

1977 Anderson/Mott Nobel Prize

1980 « gang of four » Scaling theory

1980 Götze, Vollhardt, Wölfle Self-consistent theory

1982 Halperin, Pruisken Scaling theory of Quantum Hall effect

πλ2≤l

)log()(log

Lg

∂∂

Half a century…….

1983 Fröhlich & Spencer Mathematical proof for 3D Anderson model

1984 Anderson 25 years localization« unrecognizable monster »

1986 Anderson « Theory of white paint »

1987 Papanicolaou, Sheng Prediction of Localization of Seismic Waves in layered Earth

Crust1987 Souillard Localization of Gravitational Waves

in Universe?

1986 Kramer, Mackinnon, Economou,

Soukoulis, Schreiber

Tight binding model

1990 Dorokhorov, Melloetal, Beenakker

Altschuler

DMPK equation for wire

interactions

1988 John Prediction of Localization of light in Photonic crystals

>1991

Bell-labsWeaverGenack

Exxon (Sheng etal)

2D localization microwaves2D localization of ultrasoundQ1D localization microwaves

2D localization of bending waves>1995 BEC community Localization of light in BEC cold atomes

gazes or the contrary?

2000 GenackBeenakker, …Van Tiggelen/

Lagendijk/Wiersma

Statistics in localized regime (exp)Idem (theo)

D(r) in localized regime

2006 Maret Anomalous dynamic transmission of lightnear mobility edge

2007 Fishman/Segev Transverse Light Localization in 2D lattices

1997 Wiersma, Lagendijk 3D localizationof infrared light

>1998

Cao, Wiersma etal, Soukoulis, Sebbah, …

Random lasering from(pre) localized states

2008 Page/Skipetrov/Van Tiggelen 3D localization of ultrasound

)()(),(),( 2St tStDt rrrr −=∇−∂ δδρρ

tDt

tt 6

),(),(

)(2

2 ==r

rrr

ρ

ρ

Diffusion Diffusion ofof WavesWaves

diffusion constantdiffusion constant

*v31 l=D

Diffusion = Diffusion = randomrandom walkwalk ofof waveswaves

nmmL

73920

==

λμ

Diffusion, works even better than expected!

GaP poreux

transmission

réflexion )1.2(250

/23**

2

==

=

ll knm

smD

1/t3/2

Diffusion equation

Lagendijk etal, PRE 2003

1 µm

Anderson localisation?Anderson localisation?

••suppressedsuppressed diffusion : rdiffusion : r22(t)(t) constant constant «« D(t) ~ 1/tD(t) ~ 1/t »»

••Dense Dense «« pure pointpure point »» spectrumspectrum / / ((nono levellevel repulsionrepulsion: : infiniteinfinite media)media)

••GiantGiant nongaussiannongaussian fluctuations (fluctuations (lognormallognormal))

•• complexcomplex dynamicsdynamics (open media): T(t) (open media): T(t) ~ exp(~ exp(--D(tD(t) t /L) t /L22))((anomalousanomalous leakingleaking))

•• T(L), G(L) T(L), G(L) ~ exp(~ exp(--L/L/ξξ))

Mesurable?

Mesurable?

••NeedNeed goodgood observable observable withwith continuouscontinuous control control parameterparameter: L,t, f, E ..: L,t, f, E ..••NeedNeed a a goodgood theorytheory thatthat allowsallows engineeringengineering

R(t) ~1/tR(t) ~1/t22

«« UnrecognizableUnrecognizable monstermonster »»

Trivial Localization: tunnelling is inefficient

r

1D 1D mattermatter wavewave

m

Localisation is «not trivial yet reasonable »

r

1D 1D mattermatter wavewave

m

V(r)

Localisation is « non trivial »: phase!

r

0),(),()( 22 =∇−∂ ttt rrr ψψε

⎪⎩

⎪⎨⎧

=

−=⇒−=

2

2)](1[)()exp()(

ω

ωεωψψ

EV

tirr

r VE >

ClassicalClassicalwaveswaves

m

r

MobilityMobilityedgeedge

k(E)k(E)ℓℓ(E)=1(E)=1

3D: 3D: strongstrong disorderdisorderExtendedExtended statesstates

LocalizedLocalized statesstates

m

Distribution of leakage ratesN dipoles in a 3D sphere

Kottos & Weiss 2003; Pinheiro/van Tiggelen PRE 2004

Γ-1

13 =ρλ

603=ρλ

P(Γ)

Leakage rate/ Thouless frequency

Γ∝Γ

1)(P

LocalizationLocalization in open mediain open media

1>>lk *ΨΨ

*31 lEB vD =

1≈lk

( )ωπ NvC

DD EB

),(11 rr+=

Ψ

*Ψ

VollhardtVollhardt & & WWöölflelfle, 1980, 1980

SchwarzshildSchwarzshild//MilneMilne , 1900, 1900Chandrasekhar, 1950Chandrasekhar, 1950

Van de Van de HulstHulst, 1950, 1950

TheoreticalTheoretical DescriptionDescription

Boltzmann equation

Self consistent theory

( ) ( )

( ) ( )ωπ

δπ

NvG

DD

GD

EB

),(11

'4'),(

rrr

rrrrr

+=

−=∇⋅∇−l

reciprocityreciprocity ),(),( rrrr GC =⇒

==

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⇒

==−= ∫ <

2

2/13

11

14)0(),(');('),(

l

l l

kDD

DqdGGGG

B

qqrrrrrr π

3D 3D unboundedunbounded mediummedium

OK OK MottMott

QuasiQuasi--1D 1D halfhalf--wirewirezz

( )

)(

2exp)(211:

4),('4'),()(2

l

ll

N

zDzDDDdz

d

GGzDdzd

BB

∝

⎟⎠⎞

⎜⎝⎛−=⇒+==⇒

=⇒−=∂−⇒≡

ξ

ξτξτ

τπττττδπτττ τ

OK DMPKOK DMPK

NN

ComplexComplex DynamicsDynamics in in finitefinite open mediaopen mediaSkipetrovSkipetrov & Van Tiggelen, PRL 2004,2005& Van Tiggelen, PRL 2004,2005

),,,(21),(

12 qzzG

kDzD B l+=

ssq2

3/

dq∫< l

)'(),,',(),,',(),(),,',( 2 zzqzzGqqzzGzDqzzG z −=+∂+− δss ss ss ss ss

ss

Diffuse Diffuse regimeregime: simple : simple polespoles LocalizedLocalized regimeregime: Riemann : Riemann sheetssheets

ΩΩ

ss

ΩΩ

ss

I(I(ΩΩ))

P(s)P(s)

ComplexComplex FrequencyFrequency ΩΩ++isis

( )stszzGdstT s −∝ ∫∞

exp),,()(0

tt--3/23/2

tt--22

time/(time/(ζζ2 /2 /DDBB))

R(t)R(t)

3D, 3D, localizedlocalized HalfHalf spacespace : k: kℓℓ=0.7=0.7

zz

1D sismologie : 1D sismologie : ShengSheng PapanicolaouPapanicolaou, 1987, 1987Q1D (DMKP) Q1D (DMKP) TitovTitov, , BeenakkerBeenakker, 2000, 2000

2

1)(t

tR ∝

3D Transverse localisation 3D Transverse localisation ofof ultrasoundultrasoundPage (Winnipeg), Page (Winnipeg), SkipetrovSkipetrov, Van Tiggelen, , Van Tiggelen, CherroretCherroret

diffuse localized

10 mm

8 – 23 mm

Diffuse: <ρ2> = 4Dttransition: <ρ2>~ L2, not t2/3

Localized: <ρ2>~ Lξ

3D Transverse localisation 3D Transverse localisation ofof ultrasoundultrasoundPage (Winnipeg), Page (Winnipeg), SkipetrovSkipetrov, Van Tiggelen, , Van Tiggelen, CherroretCherroret

ρ

L=15 mm

3D Transverse localisation 3D Transverse localisation ofof ultrasoundultrasoundPage , Page , SkipetrovSkipetrov, Van Tiggelen, , Van Tiggelen,

Nature Nature PhysicsPhysics OctoberOctober 20082008

Time (ms)

ρ = 15 mmρ = 20 mmρ = 25 mmρ = 30 mm4 D0t

kℓ ≈ 1,82vE=17,4 km/s=3.5 vp

ξ= 16,3 mm

Transversesize

Transverse position

Near fieldUltrasound

energy

TimeTime--dependentdependent transmissiontransmission

T(t))(tδ

SpeckleSpeckle distribution of transmissiondistribution of transmission

Rayleigh Random matrixg=0.8 +/- 0.02

Inhibition of transport of Q1D BEC in Inhibition of transport of Q1D BEC in randomrandom potentialpotential

Orsay group, Firenze groupOrsay group, Firenze groupPRL PRL octoct 20052005

Time after trap extinction

expansion

1<μδV

n(x,t)V(x)

mobility edge

Diffusive regime Localizationwith ξ > ℓ

Localizationwith ξ < ℓ

LocalizationLocalization of of noninteractingnoninteracting cold cold atomsatoms in 3D white noisein 3D white noise

⎟⎟⎠

⎞⎜⎜⎝

⎛−∝=

mkt

2)0,(

22hμθψ k

band edge

TrapTrap stagestage

µchemical potential

2)(rψ

expansion stage (t=0)expansion stage (t=0)

gVet ti )(),( rr −

= − μψ μ

∞→

==

tttn ??),(),( 2rr ψ

RandomRandom potentialpotential

εε ∝)(D

with Sergey Skipetrov, Anna Minguzzi and Boris Shapiro

DensityDensity profile of profile of atomsatoms atat large timeslarge times

),(),()()2(

),( 23

3

tPEAdEdtn E rkkkr ∫∫∞

∞−= φ

π

Distribution of velocitiesmv = ћk

Spectral function

Probability of quantum diffusion

EiUUkE +−−−

4/1Im122π

)'(4)'()( rrrr −= δπUVV

Mean free path: U1

=l 2UE < localized

DensityDensity profile of profile of atomsatoms atat large timeslarge times

Potential fluctuation/chemical potential

FractionOf

Localized atoms

DensityDensity profile of profile of atomsatoms atat large timeslarge times

-2 -1 0 1 2 3 4-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

),()(),( loc tnntn AD rrr Δ+=

ssADs

trtnD c /1/23

1),()()( −∝Δ⇒−∝ rεεεννεε

εξ /13loc1)(

)(1)( +∝⇒−

∝r

nc

r

localized anomalous diffusion

nloc(r)

Selfconsistent theory (ν=s=1)

50 50 yearsyears of Anderson of Anderson LocalizationLocalizationParis, Institut Henry PoincarParis, Institut Henry Poincaréé

4/5 4/5 decemberdecember 20082008

http://www.andersonlocalization.com

SpeakersSpeakers: P.: P.W.AndersonW.Anderson, D.J. , D.J. ThoulessThoulessA. Aspect, S. John, P. A. Aspect, S. John, P. WWöölflelfle, , G. Maret, R. Weaver, A.Z. G. Maret, R. Weaver, A.Z. GenackGenack, , A. A. LagendijkLagendijk, A. , A. MirlinMirlin, M. Schreiber, M. Schreiber

TheoryTheory and and experimentsexperiments, , oldold and new and new thingsthings

Localized? : kℓ < 4,2 D(t) ~1/t1/3Mobility edge: kℓ ≈ 4,2D(t) ~1/t

G. Maret G. Maret etaletal, EPL 2006, EPL 2006tr

ansm

issi

on

time

TiO2 powder ; L/ℓ =10 000

Sample and CBS

T(t))(tδ

ClichClichéé, clich, clichéé !!LL’’article darticle d’’AndersonAnderson’’s en 1958 a s en 1958 a ééttéé «« oftenoften quotedquoted but but nevernever readread »»Il y a autant de dIl y a autant de dééfinitions et dfinitions et d’’approches quapproches qu’’il y a de publications.il y a de publications.La moitiLa moitiéé des publications est des publications est ……....

•• …….. soit fausse.. soit fausse•• …….. soit triviale .. soit triviale •• …….. soit d.. soit dééjjàà contenue dans le papier dcontenue dans le papier d’’Anderson en 1958 Anderson en 1958 •• …….. soit trop compliqu.. soit trop compliquéée ou trop phe ou trop phéénomnoméénologiquenologique

pour comprendre les exppour comprendre les expéériencesriences

ProtectProtect me me fromfrom knowingknowing whatwhat I I dondon’’tt needneed to knowto knowProtectProtect me me eveneven fromfrom knowingknowing thatthat therethere are are thingsthings to know to know thatthat I I dondon’’tt knowknowProtectProtect fromfrom knowingknowing thatthat I I decideddecided not to know not to know

about the about the thingsthings thatthat I I decideddecided not to know aboutnot to know about

Douglas Adams, Douglas Adams, MostlyMostly HarmlessHarmless, 1992, 1992

Distance in units of mean free pathdistance

density of atoms