Anderson Localization of Light and Ultrasound Bart...

HKUST april 2010 1

Anderson Anderson LocalizationLocalizationof of

Light and Light and UltrasoundUltrasound

Bart van Bart van TiggelenTiggelen

UniversitUniversit éé Joseph Fourier Joseph Fourier –– Grenoble 1 / CNRSGrenoble 1 / CNRS

HKUST april 2010 2

Contents of the course1. Introduction to localized waves

• Historical perspective• Diffusion of waves and mean free path• Localization in a nutshell

2. Theories of Localization• Random Matrix theory• Ab-initio methods• Supersymmetric theory• Selfconsistent theory of localization

3. Mesoscopic transport theory• Dyson Green function• Bethe-Salpeter equation• Diffusion approximation• Interference in diffusion

4. From matter towards classical waves• Analogies and differences• Mesoscopic regime for different waves• Energy velocity of classical waves

HKUST april 2010 3

Contents of the course5. Enhanced backscattering as a precursor of strong localization

• Reciprocity principle• Observation of enhanced backscattering of light and sound• Return probability in infinite and open media

6 Speckles and correlations in wave transport • Gaussian statistics• Short and long-range correlations

7 . Random laser– Historical perspective– Recent experiments and link with localization

8. Observation of Anderson localization in high dimensions• 3D light localization• 2D transverse localization• Quasi 1D localization of microwaves• 3D localization of ultrasound

- dynamics in transmission- transverse confinement- selfconsistent theory- speckle distribution- multifractal wave function

HKUST april 2010 4

Contents of the course5. Anderson localization of noninteracting atoms in 3D

• Self-consistent Born approximation in speckle potential• Phase diagram from selfconsistent theory• Energy distribution of atoms

HKUST april 2010 5

I.I. Introduction to Introduction to Anderson Anderson LocalizationLocalization

HKUST april 2010 6

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

50 50 yearsyears of Anderson of Anderson localizationlocalization

…..and now we have (numerical) experiments !

HKUST april 2010 7

cc

PhysicsPhysics TodayToday , August 2009, August 200950 50 yearsyears of Anderson of Anderson LocalizationLocalization

HKUST april 2010 8

AndersonAnderson ’’s 1958 paper has s 1958 paper has «« often been quoted but hardly ever readoften been quoted but hardly ever read »»and even less understoodand even less understood

Several of his results have become mathematical the oremsSeveral of his results have become mathematical the oremsAnderson localization has become an Anderson localization has become an «« unrecognizable monsterunrecognizable monster »»Weak localization is (just) a precursor of strong l ocalizationWeak localization is (just) a precursor of strong l ocalizationP.W. Anderson created condensed matter physicsP.W. Anderson created condensed matter physics

Yes

AndersonAnderson ’’ss definitiondefinition isis zerozero diffusion constantdiffusion constantAnderson shows how Anderson shows how wavewave loopsloops induceinduce localizationlocalizationA local A local defectdefect in a in a metalmetal inducesinduces Anderson Anderson localizationlocalizationAnderson transition Anderson transition isis discontinuousdiscontinuous

90 % of the publications 90 % of the publications isis eithereither wrongwrong , trivial, , trivial, alreadyalready shownshown by Anderson, or not relevantby Anderson, or not relevant

Anderson Anderson localizationlocalization wouldwould nevernever have been have been foundfound «« accidentallyaccidentally »»in a computer simulation in a computer simulation

AfterAfter thisthis lecture lecture youyou willwill finallyfinally understandunderstand Anderson Anderson localizationlocalization

No

Maybe

HKUST april 2010 9

Anderson model on the CaleyTree

Abou-Chacra/Anderson/Thoules

1973

Weak localizationMesoscopic physics!

Sharvin, Lagendijk,Maret, Maynard,…

>1982

Sensitivity to BC: g < 1(Thouless criterion)

Thouless1972

Scaling theory of Quantum Hall effect

Halperin, Pruisken1982

Self-consistent transport theoryGötze, Vollhardt, Wölfle1980

Scaling theory of open media

« gang of four »1980

Nobel PrizeAnderson/Mott1977

Minimum conductivityVariable range hopping

Mott1965

Mott/Ioffe-Regel1960

« vanishing of diffusion »Anderson1958

ππππλλλλ

2≤l

)log(

)(log

L

g

∂∂

HKUST april 2010 10

25 years localization« unrecognizable monster »

Anderson1984

Mathematical proof for 3D Anderson model

Fröhlich & Spencer1983

Localization of Gravitational Wavesin Universe?

Souillard1987

Prediction of Localization of Seismic Waves in layered Earth

Crust

Papanicolaou, Sheng1987

« Theory of white paint »Anderson1986

DMPK equation for wire

interactions

Dorokhorov, Melloetal, Beenakker

Altschuler

1990

Tight binding model and numericalScaling

Kramer, Mackinnon, Economou,

Soukoulis, Schreiber

1986

Prediction of Localization of light in Photonic crystals

John1988

HKUST april 2010 11

2D localization microwaves2D localization of ultrasoundQ1D localization microwaves

2D localization of bending waves

Bell-labsWeaverGenack

Exxon (Sheng etal)

1992

Anomalous dynamic transmission of lightnear mobility edge

Maret2006

Transverse Light Localization in 2D latticesFishman/Segev2007

Statistics in localized regime (exp)Idem (theo)

D(r) in localized regime

GenackBeenakker, …Van Tiggelen/

Lagendijk/Wiersma

2000

Localization of light in BEC cold atomes gazes or the contrary?

BEC community>1995

Observation of laser action in random media(threshold and line narrowing)

Lawandy1991

3D localizationof infrared light

Wiersma, Lagendijk1997

Random lasering from(pre) localized states

Cao, Wiersma etal. Sebbah, …>1998

HKUST april 2010 12

Proof of delocalized states near 2D Landau levels

Germinet, Klein etal(Cergy-Pontoise/ Irvine)

2008

3D localization of ultrasoundWinnipeg/Grenoble2008

50 years of Anderson localization

https://www.andersonlocalization.comCambridge UK

IHP Paris2008

3D dynamical localization of cold atomsLille group/ LKB2008

1D localization of noninteracting cold atomsPalaiseau groupFlorence group

2008

HKUST april 2010 13

3D localization of fermionsUrbana Champaign2011

4D kicked rotor with cold atoms2013

Localization of seismic waves in the Earth crust of La Réunion

2013

Localization of light in BEC2012

3D Localization of cold bosonsPalaiseau group20122011

Non-observation of gravitationalwaves. They are localized!…..

VIRGO2014

Localization of WavesReview « Les Houches »,

http://lpm2c.grenoble.cnrs.fr/Themes/tiggelen/cv.html

HKUST april 2010 14

)()(),(

),(),(gainabs,

2St tS

ttDt rr

rrr −=±∇−∂ δδ

τρρρ

tDt

tt 6

),(

),()(

2

2 ==r

rrr

ρρ

Diffusion of Diffusion of WavesWaves

diffusion constantdiffusion constant

*v31 l=D

Diffusion = Diffusion = randomrandom walkwalk of of waveswaves

HKUST april 2010 15

Multiple Multiple scatteringscattering of of WavesWaves

MeanMean free free pathpath ℓℓ

«« particleparticle languagelanguage»»: : meanmeandistance distance betweenbetweensuccessive successive scatteringsscatterings

«« WaveWave languagelanguage»»

distance to randomise phasedistance to randomise phase

0)exp( =φi

HKUST april 2010 16

nm

mL

739

20

==

λµ

Diffusion works even better than expected!

porous GaP

transmission

reflection )1.2(250

/23**

2

==

=

ll knm

smD

1/t3/2

Diffusion equation

Lagendijk etal, PRE 2003

1 µm

HKUST april 2010 17

What is localization?

HKUST april 2010 18

V(r)

r

Dimension =1

«« TrivialTrivial »»LocalizationLocalization

(most mathematicalproofs)

« sea level »

HKUST april 2010 19

V(r)

r

Dimension =1

«« TrivialTrivial »»LocalizationLocalization

«« Tunnel /percolationTunnel /percolationassistedassisted »»

localizationlocalization(Anderson model)

(most mathematicalproofs)

« sea level »

HKUST april 2010 20

V(r)

r

Dimension =1

«« TrivialTrivial »»LocalizationLocalization

«« genuinegenuine »»LocalizationLocalization

E > E > VVmaxmax

(Classical waves,cold atoms ??)

(Anderson model)

(most mathematicalproofs)

«« Tunnel /percolationTunnel /percolationassistedassisted »»

localizationlocalization« sea level »

HKUST april 2010 21

V(r)

r

Dimension = 3

MobilityMobilityedgeedge

« metal »

«insulator»

Classical percolation threshold

« sea level »

HKUST april 2010 22

V(r)

Localisation is « non trivial »: phase!

r

1D light

0),(),()( 22

20

=∇−∂ ttc t rrr ψψε

=

−=⇒−=

20

2

20

2

)](1[)(

)exp()(

cE

cV

tiω

ωεωψψ

rr

rVE >

Classical waves

Localization of classical wavesis not trivial

HKUST april 2010 23

MottMott minimum minimum conductivityconductivity

••ThoulessThouless criterioncriterion and and scalingscaling theorytheory

••Quantum Hall Quantum Hall effecteffect

MIT and MIT and rolerole of interactionsof interactions

TightTight bindingbinding model and model and loopsloops

Chaos Chaos theorytheory (DMPK (DMPK equationequation))

LocalizationLocalization nearnear band gapsband gaps

Full Full statisticsstatistics of conductance and transmissionof conductance and transmission

RandomRandom laser laser

••Transverse Transverse localizationlocalization

••KickedKicked rotorrotor

«« UnrecognizableUnrecognizable monstermonster ??»»

HKUST april 2010 24

II. II. TheoryTheory of of LocalizationLocalization

HKUST april 2010 25

IIaIIa RandomRandom MatrixMatrix TheoryTheory

ElegantElegant, and , and «« nonperturbationalnonperturbational »» restrictedrestricted to Q1Dto Q1D‘‘full full countingcounting statisticsstatistics ((BeenakkerBeenakker, RMP 1997), RMP 1997)

Chaos theory of S-matrix

with respect to dS measure that respects symmetry

Tnm=TnδnmNxN diagonal transmission matrix

β=1,2,4

UCF

∑=

=N

nnT

h

eG

1

22

Weak localization

HKUST april 2010 26

IIaIIa RandomRandom MatrixMatrix TheoryTheory

DMPKEquation(Dorohhov 1982,Mello, Pereya, Kumar, 1988)

L

N

h

eT

h

eG

N

nn

l2

1

2 22 == ∑=

L < Nℓ: ‘diffuse’

nonOhmic conductance Lognormal distribution

L > Nℓ: ‘localized’

HKUST april 2010 27

IIaIIa RandomRandom MatrixMatrix TheoryTheory

Chaotic cavity

EigenvaluesEigenvalues of Transmission of Transmission matrixmatrix TTabab

Bi-modal transmission !

HKUST april 2010 28

IIaIIa RandomRandom MatrixMatrix TheoryTheory

L < Nℓ: ‘diffuse’

Bi-modal transmission !

Chaotic cavity

MelloMello KumarKumar ,1989,1989

EigenvaluesEigenvalues of Transmission of Transmission matrixmatrix TTabab

Localized regime: necklace states exist with T=1 (Pend ry, 1992)

HKUST april 2010 29

IIbIIb. Ab . Ab initioinitio ((«« exact solution of exact solution of simplifiedsimplified modelmodel »»))

Anderson Anderson TightTight BindingBinding Model (80Model (80--90), 90), RandomRandom DipolesDipoles 2000), 2000), Large Large systemssystems difficultdifficult (N < 5000)(N < 5000)

nnVnntH nnnn ∑∑ += ''

Hopping (t=1) Random Disorder:

Energy E

DisorderW

Band edge

métal

isolant

SC theoryKroha, Wölfle, 1992

HKUST april 2010 30

IIbIIb. Ab . Ab initioinitio ((«« exact solution of exact solution of simplifiedsimplified modelmodel »»))

0:)()()(/4

)(10

220

30

30 ==−

−−− ∑

=iinj

N

jjii G

i

crrrrr ψψ

γωωπωψ

)exp(4

1)( ikr

rrG

π=

Eigenvalues of

43421ijg

ijij crGi )/(1 0020

2

ωδλωω +

+

−

No leakage

Im g/λ0 Re g/λ0

Rusek; Orlowski, 1997

HKUST april 2010 31

IIbIIb. Ab . Ab initioinitio ((«« exact solution of exact solution of simplifiedsimplified modelmodel »»))

RandomRandom electricelectric DipolesDipoles (2000), (2000),

Γ-1

13 =ρλ

603=ρλ

P(Γ)

leakage Γ / leakage of 1 dipole Pinheiro etal PRE 2004

1-constant

)(

Γ=Γ

=Γ

=Γd

dz

dz

dn

d

dnP

)/2exp(

)()(

0

2

0

ζψ

z

zz

−Γ=Γ=Γ

HKUST april 2010 32

IIcIIc. Super . Super symmetricsymmetric fieldfield theorytheory•• works for all dimensionsworks for all dimensions•• nearly equivalent to DMPK in Q1D, also close to transitionnearly equivalent to DMPK in Q1D, also close to transition•• difficult to engineer withdifficult to engineer with……..

Multifractality near mobility edge

Inverse participation ratio

f(x): singularity spectrum

Evers and Mirlin, RMP 2008

f(f(αα) ) isis the fractal dimension of the set of the fractal dimension of the set of thosethose points r points r wherewhere the the eigenfunctioneigenfunction intensityintensity isis ||ψψ22(r)| ~ L(r)| ~ L −α−α..

HKUST april 2010 33

II.cII.c MultifractalityMultifractality of of wavewave functionfunctionEversEvers and and MirlinMirlin , , RevRev. . ModMod . Phys. (2008).. Phys. (2008).

LbId

IdI

d

L

d

bb

d

d <<<<=∫∫ λ

)(

)(

rr

rr

−−

∝bL

Ifd

b

b

L

bIP

/log

log

)(log

GeneralizedGeneralized Inverse Participation RatioInverse Participation Ratio ProbabilityProbability Distribution Distribution FunctionFunction

Boxcounting

x

y

( )q

L

bIP

b

qbq

τ

==∑

HKUST april 2010 34

MultifractalityMultifractality of of wavewave functionfunction

−−

=bL

Ifd

bLb

b

L

b

NIP

/log

log

/

1)(log

( )

( ) αα

λ

λ

λαλ

λλ λ

qf

fdqbb

dqbp

dN

IIdN

INPbI

+−

−−

∫

∫

=

==

log

log1 log

log

( )q

L

bIP

b

qbq

τ

==∑0

log

log ↓=≡L

bI b λλ

α

Anomalous gIPR

Lognormal PDF

Boxcounting

x

y

( )2**)("21*)()( ααααα −+= fff

MethodMethod of of steepeststeepest descenddescend

=

−=

*)('

*)(*

ααατ

fq

fqq

ExtendedExtendedregimeregime ::

qq dqd ∆++−=43421

τ

2)(4

1)()1()( γα

γαγ −−−=⇔−=∆ ddfqqq

HKUST april 2010 35

MultifractalityMultifractality of of wavewave functionfunction

Anomalous gIPR Lognormal PDF

2)(4

1)()1()( γα

γαγ −−−=⇔−=∆ ddfqqq

εε=d=d--22(Wegner, 1987)

+− )1(qd

+− )1(qd

=qτ

HKUST april 2010 36

IIdIId. . SelfconsistentSelfconsistent theorytheory of of localizationlocalization

((VollhardtVollhardt & & WWöölflelfle , 1980), 1980)

•• «« ApproximateApproximate solution of an exact modelsolution of an exact model »»•• GeneralizedGeneralized diffusion diffusion equationequation, , «« simplesimple »», , •• Close to Close to experimentsexperiments, but , but meanmean fieldfield theorytheory

HKUST april 2010 37

( ) ( )

( ) ( )lωρ

δ

EB

G

DD

GD

v

),(11

')',(

rrr

rrrrr

+=

−=∇⋅∇−

reciprocityreciprocity

==

HKUST april 2010 38

III. III. mesoscopicmesoscopic transport transport theorytheory

HKUST april 2010 39

)',,('0)(

2

1)',,( 2 kkr

rrrr EG

iVm

pE

EG ⇒

+−−=

),(2

:),()',,( 22'

'

kEm

kVE

EGEG

Σ−−><−==

h

kkkkkkk

δδ

2

2

2

)(2)(

/2),()(),(

kE

iEk

mkEGEkE

−

+

=⇒Σ=Σ

l

h

),(Im1

),( kEGkEAπ

−= ∫ =1),( kEAdE

Dyson Green function

Self energy

Mean free path

Spectral function

),()( kEAEkΣ=ρ Average LDOS

k=1/2ℓ : strongly scattering

),',()',,( kkkk −−= EGEG reciprocity

III.aIII.a Dyson Green Dyson Green functionfunction

HKUST april 2010 40

)'(4)'()(;0 rrrr −== δπδδδ UVVV

Um

1

2

22

= h

l 23

2

2U

mEE B

+<h

Example 1: 3D white noise

UB π4)( =q

Σ−−=

2

1)(

pEpG

UKEiU

pU

ppEU

pEU

K

−Σ−−=

−

+

Σ−−=

Σ−−=Σ ∑∑∑

48476 π

πππ

4/

2222

14

114

14

ppp

)(2

1)( 2

412 UKUEUiKUUE −−−−=Σ

UKUEE B −=≤=Σ 2

41for0Im

222 )2/(

1

41

1),(:

pikEEiUUEEpEGEE

BB

B −+≡

−+−−=≥

lU

EEk B

1=

−=

l

21 UEEk B +≤⇒≤l

HKUST april 2010 41

( ) EUUpE

EUpEA

22

22

41

1),(

+−−=

π

)0,(),()()( EApEApEN ≈Φ=∑ µp

Spectral function

Example 1: white noise

Energy distubution of cold atoms

Fraction of localized atoms

%455

22arctan

2)0,(

2

0

=

−== ∫ πEAdEN

U

loc

µ<<U2

Broad: EBroad: E --3/23/2

HKUST april 2010 42

Example 2: discrete scatterers in d dimensions

[ ] dVc

ic

dV20

2

20

2

)'(exp)(ωδεωδε −≈⋅−

−= ∫ rkkrrkk'

( ) [ ]

[ ] kk'kkkk'

rk"k'k"kk"kk'kk'

rkkr

rkk

δω

ωω

nTiTdn

iVkGVV

=⋅−=

⋅−

++=Σ

∫

∑

)'(exp)(

)'(exp)",( L

( ) ( )

( )

( )1

0

122

240

422

220

22

2

')2(

'

)',(ImImIm

+

+

−

=⇒

≡=

−=

+−=−=Σ−=

∫

∑

d

d

dd

ddd

d

d

cVC

knkc

VnC

kc

Vd

n

GVnTnk

ωδεσ

σωδε

ωδπ

π

ωωω

kk'

kk'k'

kkk

k

k Ll

=

−=

20

2

20

2

)()(

cE

cV

ω

ωδε rr

HKUST april 2010 43

2aQ

πσ=

ElasticElasticCrossCross--sectionsection

StoredStored energyenergy in in spheresphere

Mie Mie spheresphere crosscross--section (3D)section (3D)

PrE ⋅∫3

sphere233

4

1d

Ea inπ

HKUST april 2010 44

Example 3: nematic liquid cystal

)(fB

MeanMean free free pathpath dependsdepends on angle on angle withwith opticaloptical axis and on axis and on polarizationpolarization

De Gennes, 1968De Gennes, 1968

HKUST april 2010 45

)2

'',

2

',

2(*)

2',

2,

2()',,'(*)',,( 2211

qk

qk

qk

qkrrrr ++Ω+−−Ω−⇒

hhEGEGEGEG

'' ),( qqkk q δΩΦE

),(' rkk tEΦ

Two particle Green function

Momentum conservation

Wigner function(looks like phase space distribution)

∑∑ Φ=Φ=Ψ'

''

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

kkkk

kkkk kr

krJkrr St

mtStt

h

Proba density Proba current density

III.bIII.b BetheBethe --SalpeterSalpeter equationequation

1/Ω

time

h/E

HKUST april 2010 46

[ ]),(),(),(),(

),(),(*),(

''''''

21

21

21

21

qq

qqkqkqk

k'kkkk

kk'

kk'

ΩΦΩ+=

ΩΦ−Ω−Σ−+Ω+Σ+⋅+Ω−

∑ EE

E

UkEAkEA

EEii

δ

hh

),(' rkk tEΦ obeyes a generalized transport equation(Bethe-Salpeter equation)

><⊗⊗>><<+>><>=<< **** GGUGGGGGG

Re: effective medium (mass renormalization)

Im: extinction

scattering

propagation

Source(initial condition)

III.bIII.b BetheBethe --SalpeterSalpeter equationequation

HKUST april 2010 47

)exp()()()(0)( 21

21

40

4

xqxrxrrqr ⋅+−== ∫ idc

B klijijklij δεδεωδε

Born approximation: continuous media

Structure factor

)'( pp −= ijklB

III.bIII.b BetheBethe --SalpeterSalpeter equationequation

HKUST april 2010 48

)(),(),( ESii EE =ΩΦ⋅+ΩΦΩ− ∑∑ qkqq kk

kk

),(),(*),( ''''

21

21

21

21 qqkqk kk

k

Ω=−Ω−Σ−+Ω+Σ ∑ EUEE hh

Continuity equation

provided

Ward identityProbability is conserved

)',()',()('

kESkEAES ∑=k

Golden Golden ruleruleEmittedEmitted power of source power of source isis determineddetermined by phase by phase spacespace distributiondistribution

III.bIII.b BetheBethe --SalpeterSalpeter equationequation

HKUST april 2010 49

ФФФФ

k+q/2 k’+q/2

k-q/2 k’-q/2

E+hΩ/2+i0

=

E+hΩ/2-i0

ФФФФ

k+q/2 k’+q/2

k-q/2 k’-q/2

E+hΩ/2+i0

E+hΩ/2+i0

*

**

)0('

)0(*')0('')0(

kk

kkkkkk

iET

iETiETiET

+=

−=−=−

Time reversal Time reversal symmetrysymmetry

Feynman Feynman diagramdiagramReal Real fieldsfields

bottombottom line line «« advancedadvanced »» Green Green functionfunctionG(EG(E--i0)i0)

ScatteringScattering diagramdiagramAll All retardedretarded Green Green functionsfunctions

G(E+i0)G(E+i0)BottomBottom line line isis complexcomplex conjugateconjugate

HKUST april 2010 50

),(),( '' qq kkkk −ΩΦ=ΩΦ −−EE

)',(

),(

2

'

2

'

'

kk

q

qkkqkk

kk

+ΩΦ

=ΩΦ

+−+−E

E

reciprocity

ФФФФ

-(k’+q/2) -(k+q/2)

-(k’-q/2) -(k-q/2)

E+hΩ/2

E-hΩ/2

ФФФФ

k+q/2 k’+q/2

-(k-q/2)-(k’-q/2)

E+hΩ/2

E-hΩ/2

ФФФФ

k+q/2 k’+q/2

k-q/2 k’-q/2

E+hΩ/2+i0

=

2) =

1) =

E+hΩ/2+i0)0()0( ' iETiET +=+ −− kkkk'

HKUST april 2010 51

III.c Diffusion approximation

2)(

),',(),(

)(

2),(

qEDi

EE

EE +Ω−=ΩΦ qkqk,

qkk'

φφρ

π

),(),(),,( kEFikEAE qkqk ⋅−=φ

1th reciprocity

Equipartitionin phase space

Small non-equilibriumcurrent

DOS normalization

)0()',(2)',(),(

]exp[2)(

2

)0,()exp(2

),(

>=Ω−

Ω−Ω=

=ΩΦΩ−Ω=Φ

∑∫

∫∑∫

tkEAi

kEAkEAti

d

E

tid

tdE

E

ππρ

π

π

k

kk'kk'

k

qrr

1(..)2

=∫ πdE

0,0 →→Ω q

Source weigtedby spectral function

Proba of quantum diffusion

HKUST april 2010 52

2)(

),',(),(

)(

2),(

qEDi

EE

EE +Ω−=ΩΦ qkqk,

qkk'

φφρ

π

),(),(),,( kEFikEAE qkqk ⋅−=φ

),()(

1

3)( 2 kEFk

EmED ∑=

kρh

Kubo Greenwood formula

)()(3cos-1

v3

1)( * EEk

mED l

hl ==θ

k+q/2 k’+q/2

k-q/2 k’-q/2

E+hΩ/2

E-hΩ/2

x x x

xxx

Boltzmann approximation

L+−

= ),(Imcos1

2),( 2 pEGkEF

ϑ

Mahan, many particle physics

HKUST april 2010 53

Equipartition of diffuse Equipartition of diffuse seismicseismic waveswaves in phase in phase spacespace

HenninoHennino etaletal 20012001

3,

2

2,

, 2)(

SP

SPSP c

g

πω

ωρ =

HKUST april 2010 54

III.d Inclusion of interference

)',(),(2

'

2

'' kkq qkkqkkkk +ΩΦ=ΩΦ +−+−E

E

=ΩΦ ),( qkk'E

2)')((

1

kk ++Ω− EDi ),('' qkk ΩEU

k+q/2 k’+q/2

k-q/2 k’-q/2

E+hΩ/2

E-hΩ/2

x x x

xxx

+k+q/2 k’+q/2

k-q/2 k’-q/2

E+hΩ/2

E-hΩ/2

x x x

xxx

« ladder »

« most-crossed »

WhatWhat about 2about 2 ndnd reciprocityreciprocity ??

Fully connected diagram: part of

2)(

1

qEDi +Ω−

HKUST april 2010 55

Self-consistent theory of localization

2)(

1

)(v

1

)(

1

qD

C

DDd

B ωωρω ∑+=ql

( )20

2

2

1

mcE

ωh↔

*v1

ld

D =

return quantum probabilityG(r,r)

DOS

Vollhardt and Wölfle, 1980

ll ≠*

III.d Inclusion of interference

HKUST april 2010 56

F

2

v)(

k∝ωρE

2

v)(

k≈ωρ

−=⇒

==−= ∫ <

23

2/1

3

)(v1

1)0(),();'()',(

l

l

ωρC

DD

DqdGGGG

B

qqrrrrrr

3D 3D unboundedunbounded mediummedium

OK OK MottMott: :

Localization of electrons at low energiesLocalization of light at frequencies with low DOS

(near band gaps)

kkℓℓ=1=1Electrons lightωh=:E

HKUST april 2010 57

The erroneous (?) Mott argument for metalThe erroneous (?) Mott argument for metal--insulator insulator transition (1960)transition (1960)

*2

)()()( 12

2 l−== dF

d kh

e

dEDEeE

πγρσ

F

1

vvolumeunitperDOS

h

−

=dF

d

kγ *v1

constantDiffusion lFd=

22:1*if −>> dFh

e kk σl

0:1*if =< σlk

metal

insulator

Quantum phase transition Quantum phase transition isis discontinuousdiscontinuous ??????

2D minimum 2D minimum conductivityconductivity ? ? he2constant=σ

HKUST april 2010 58

SlabSlab withwith periodicperiodic BCBC’’ss

+−≈⇒

+=⇒= ∑∫

≠<

LkDLD

qLnDdGDD

B

nq

l

l

l

2

22220

/1

2

)(

11)(

/

11),()(

πqrrr

OK OK scalingscaling theorytheory

L

HKUST april 2010 59

QuasiQuasi --1D 1D wavewave guideguide

zz

( )

−=⇒+==⇒

=⇒−=∂−⇒≡

ξτ

ξτ

τττττδτττ τ

zDzD

DDdz

d

GGzD

dzd

BB

2exp)(

211:

),(')',()(

2

DMPK: :

NN

ll NAc 2)(2 0 == ωρξ),(21

)(

1),(

1),( zzG

DzDzzG

AG

B ξ+=⇒=rr

)/exp()(

1vdv

100

ξτ LzD

dzT

LL

EE ∝== ∫∫

)2/exp(2/3 ξLLT −∝ −

SCTSCT

DifferenceDifference ??

HKUST april 2010 60

broadeningbroadening22 L

c

L

D l==δωL

c

N

1=∆ω

ξδωω L

N

L ==∆l

>1

LevelLevel spacingspacing

<1

ContinuousContinuous spectrumspectrumTransport Transport theorytheory

QuasiQuasi --discretediscrete modesmodesRandomRandom matrixmatrix theorytheory

QuasiQuasi --1D 1D wavewave guideguide

Courtesy: A.Z. Genack (Queen College, NYC)

HKUST april 2010 61

Yamilov, Skipetrov, etal 2010

source

)(

)()(

xW

xJxD

x

x

∂−≡

QuasiQuasi --1D 1D wavewave guideguide

HKUST april 2010 62

IV. From matter waves to classical waves

HKUST april 2010 63

IV.a analogies and differences

All channel in all channelout (conductance)

Ψ electric field, elastic

dispacement,..Ψ probability amplitude

direct sums AbsorptionDirect productsdecoherence

classical waveselectrons

ψψ

+∇=∂ )(

2

22

rVm

i t

hh 0),(),(

)( 2220

=∇−∂ ttc t rrr ψψε

2ψ∫ rd

∇+∂∫

22

20 2

1)(

2

1 ψψεtc

dr

r

( )*Re ψψ ∇∂= tJ( )*Im ψψ ∇=m

hJ

( ) [ ]

( ) 1

20

2

/10

)(12

)(

+∝→⇒

−=

d

mcV

ωω

εω

l

hrr0)0()( >=⇒ EV lr

probaEnergy

Wave equation

Wave

conserved

scattering

current

Hilbert space LФ Labs

One channel in one channel out (scattering matrix

detection

HKUST april 2010 64

φLLL orabs<<l

1.1. Electrons: Electrons:

≈≈

K)(1m10L

nm1

µµµµφφφφ

l NANONANO

2.2. Photons Photons

≈−≈

cm1-m100L

mm1nm300

µµµµa

l MICROMICRO--MILLIMILLI

3.3. Micro Micro waveswaves

≈≈

cm50L

cm5

a

lCENTICENTI

4.4. SeismicSeismic waveswaves

≈≈

Hz) (1 km150L

km100

a

lKILOKILO

Cold Cold atomsatoms

>≈

mm1L

nm300

φ

l MICROMICRO

IV.b mesoscopic regime

Mesoscopic regime

HKUST april 2010 65

Ω+Ω=−Ω−Σ−+Ω+Σ

∑ iU )(),(

),(*),(

''''

21

21

21

21

ωδωω

ω q

qkqk

kkk

⇒),( ωrV

[ ] )',(),(),()(1 kEAii EE =ΩΦ⋅+ΩΦ+Ω− ∑∑ qkqq kk'k

kk'k

ωδ

IV.c Energy velocity of classical waves

E2 E.Pradiation matter

Dwell time

HKUST april 2010 66

Diffusion approximation for Diffusion approximation for classicalclassicalwaveswaves

2

2*201

),',(),(

)1)((

2

)1(

),',(),(

)(

2),(

Dqi

qv

ci

p

d

+Ω−+=

++Ω−=ΩΦ

qkqk,

qkqk,qkk'

ωφωφδωρ

π

δ

ωφωφωρπ

ω

l

< v >

DOS of radiation + matter

)(1

1v)(

)(1

1

3

1)(

20*

20

ωδωδ +=⇒

+=

pE

p v

cE

v

cED l

HKUST april 2010 67

Mie spheres

Harmonic oscillators

( )( ) 0/

)(1

cnk

n

ωωωραω

=+=

HKUST april 2010 68

Small diffusion constant Small diffusion constant ≠≠ localizationlocalization !!

photonphoton

TrappedTrapped RbRb8585TemperatureTemperatureT = 0,0001 K (v=15 cm/s)T = 0,0001 K (v=15 cm/s)

10101010 atomesatomes

5 mm5 mm

RandomRandom walkwalk of photonsof photons

sec/9.0 23

1 mvD E == l

ℓℓℓℓ

µm8.0=λ

40003.0

1000003.0........ 50

====

ll kmm

cvE δ

LabeyrieLabeyrie, , MiniaturaMiniatura , Kaiser (2005, Nice), Kaiser (2005, Nice)

HKUST april 2010 69

V. Enhanced Backscattering as a precursor V. Enhanced Backscattering as a precursor of localizationof localization

Akkermans, Maynard, Wolf, Maret, Van Albada, Lagendijk1986

HKUST april 2010 70

kkkk −−= ijji TT ''

)(*)(Re2)()()()(222

ABBAABBAABBA →Ψ→Ψ+→Ψ+→Ψ=→Ψ+→Ψ

)12()21( '' →→Ψ=→→Ψ −→−→ LL nn ijji kkkk

Constructive interference of oppositely propagating pa ths at backscattering in equalpolarization channels Factor two enhancement at backscattering

V. a V. a ReciprocityReciprocity

[ ] [ ]

[ ] [ ])exp(

)(exp)(exp)'exp(

)'exp()(exp)(exp)exp(

1

21111'

'111211

112211

11221111

rk

rrkrrkrk

rkrrkrrkrk

kkkkkk

kkkkkk

⋅

−⋅−−⋅−⋅−

⋅−−⋅−⋅⋅

−−−−−−−−

−−

−−

−−

i

TiTiTi

iTiTiTi

iiiinnnijn

njinnniiii

nn

nn

L

L

)12()21( →→Ψ=→→Ψ −→−→ LL nn iiii kkkk

=

HKUST april 2010 71

EnhancedEnhancedBackscatteringBackscatteringas a as a precursorprecursor

0.46 µm Mie spheres, 10 vol-%, wavelength 513 nm

kℓ=250

(Wolf, Maret, 1986)

HKUST april 2010 72

EnhancedEnhancedBackscatteringBackscatteringas a as a precursorprecursor

q=2k sin ϑϑϑϑ/2 z0=2/3 /2 z0=2/3 /2 z0=2/3 /2 z0=2/3 ℓℓℓℓ **** (Akkermans, Maynard, 1986)

halfspace

[ ] ),',(cos1)( 2 ρθα ll ==⋅+∝ ∫ zzQd ρqρ

HKUST april 2010 73

BoundaryBoundary condition for diffuse condition for diffuse propagatorpropagator

[ ][ ] )(),(),()(1)(

2),,(

)()(2

rkrk

rr

QkFkAE

QD

ωωωδωρ

πφ

δ

∇⋅−+

=

=∇−

),(4

v)(v

3

1

2

1

4

v),(ˆˆ *

32*

0ˆ0 ρrrk,zk

zk

ll −≈

∂×−=⋅= ∑>⋅

+ zQQcJ EzE

Eωφ

),(4

v),(ˆˆ *

32

0ˆ0 ρrk,zk

zk

l+≈⋅−= ∑<⋅

− zQcJ Eωφ

Flux to the right:

Flux to the left:

z

J+=0 J-=00)(

0)(*

32

*3

2

=+

=−

l

l

LQ

Q

HKUST april 2010 74

Phase anglePhase angle

Opposition Opposition effecteffect fromfrom SaturnSaturn ’’ssrings: CBS? (M. rings: CBS? (M. MishchenkoMishchenko ))

CoherentCoherent BackscatteringBackscattering in in naturalnatural mediamedia

HKUST april 2010 75

Enhanced Backscattering of ultrasound

Tourin, Derode, Fink, Roux, van Tiggelen, 1997

k/(Dt)1/2

HKUST april 2010 76

D =2.5 m2/s