anderson localization of light and ultrasound bart...
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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
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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
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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
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I.I. Introduction to Introduction to Anderson Anderson LocalizationLocalization
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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 !
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cc
PhysicsPhysics TodayToday , August 2009, August 200950 50 yearsyears of Anderson of Anderson LocalizationLocalization
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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
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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
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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
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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
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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
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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 »
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V(r)
r
Dimension = 3
MobilityMobilityedgeedge
« metal »
«insulator»
Classical percolation threshold
« sea level »
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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 ??»»
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II. II. TheoryTheory of of LocalizationLocalization
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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
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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’
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IIaIIa RandomRandom MatrixMatrix TheoryTheory
Chaotic cavity
EigenvaluesEigenvalues of Transmission of Transmission matrixmatrix TTabab
Bi-modal transmission !
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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)
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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
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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
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