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School on Interaction of Light with Cold Atoms, Sept. 16-27, 2019, Sao Paulo, Brazil, ICTP-SAIFR/IFT-UNESP.

Eric Akkermans

Based on Mesoscopic physics of electrons and photons, by Eric Akkermans and Gilles Montambaux, Cambridge University Press, 2007

Mesoscopic Physics of Photons

Part 2• Introduction to mesoscopic physics

• The Aharonov-Bohm effect in disordered conductors.• Phase coherence and effect of disorder. • Average coherence: effect and coherent

backscattering.• Phase coherence and self-averaging: universal

fluctuations.• Classical probability and quantum crossings.

�2

Sharvin2

The tools (some of them)

Incoherent propagation !

�3

Quantum probability for electron diffusion between two points

P(r,r ') = ai∗(r,r ') aj (r,r ')

i, j∑

r

r'

P(r,r ') = aj (r,r ')2

j∑ + ai

∗(r,r ')aj (r,r ')i≠ j∑

vanishes on average

Do coherent effects survive disorder average?

aj

ai∗

δ i − δ j ≫ 1

ai∗ aj = ai aj e

i δi −δ j( )


�4


P(r,r ') = ai∗(r,r ') aj (r,r ')

i, j∑

r

r'

P(r,r ') = aj (r,r ')2

j∑ + ai

∗(r,r ')aj (r,r ')i≠ j∑

vanishes on average


aj

ai∗

δ i − δ j ≫ 1

ai∗ aj = ai aj e

i δi −δ j( )


�5


P(r,r ') = ai∗(r,r ') aj (r,r ')

i, j∑

r

r'

P(r,r ') = aj (r,r ')2

j∑ + ai

∗(r,r ')aj (r,r ')i≠ j∑

vanishes on average


aj

ai∗

δ i − δ j ≫ 1

ai∗ aj = ai aj e

i δi −δ j( )


�6


P(r,r ') = ai∗(r,r ') aj (r,r ')

i, j∑

r

r'

P(r,r ') = aj (r,r ')2

j∑ + ai

∗(r,r ')aj (r,r ')i≠ j∑

vanishes on average


aj

ai∗

δ i − δ j ≫ 1

ai∗ aj = ai aj e

i δi −δ j( )


�7


P(r,r ') = ai∗(r,r ') aj (r,r ')

i, j∑

r

r'

P(r,r ') = aj (r,r ')2

j∑ + ai

∗(r,r ')aj (r,r ')i≠ j∑

vanishes on average


aj

ai∗

δ i − δ j ≫ 1

ai∗ aj = ai aj e

i δi −δ j( )


�8


P(r,r ') = ai∗(r,r ') aj (r,r ')

i, j∑

r

r'

P(r,r ') = aj (r,r ')2

j∑ + ai

∗(r,r ')aj (r,r ')i≠ j∑

vanishes on average


aj

ai∗

δ i − δ j ≫ 1

ai∗ aj = ai aj e

i δi −δ j( )

aj

ai*r r'

r'r

(a)

(b)

Before averaging : speckle pattern (full coherence)Configuration average: most of the contributions vanish because of large phase differences.

Diffuson Pcl(r, r′) =

!

j

|Aj(r, r′)|2

Ai

A∗

jVanishes upon averaging

Ai

A∗

j

A new design !

aj

ai*r r'

r'r

(a)

(b)



!

j

|Aj(r, r′)|2

Ai

A∗


Ai

A∗

j

A new design !

aj

ai*r r'

r'r

(a)

(b)



!

j

|Aj(r, r′)|2

Ai

A∗


Ai

A∗

j

A new design !

To a good approximation, the incoherent contribution obeys a classical diffusion equation

∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)

Incoherent electrons diffuse in the conductor with a diffusion coefficient D

L

l

r2 = 2d Dt

space dimensionality

l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1

with D =vgl3


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1

with D =vgl3


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1

with D =vgl3


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1

t

τe τD

τφ

ballistic

diffusive ergodic

mesoscopic limit classical limit


∂∂t

− DΔ⎛⎝⎜

⎞⎠⎟P(r,r ',t) = δ (r − r ')δ (t)


L

l

r2 = 2d Dt


l ≪ L

L2 = DτD

Thouless time

t ≪ τD t ≫ τD

⇔ −iω + Dq2( )P(q,ω ) = 1

t

τe τD

τφ

ballistic

diffusive ergodic

mesoscopic limit classical limit

?What is the first correction i.e., with the

smallest phase shift ? When amplitude paths cross

Example :

Classical diffusion

quantum crossing

Exchange of amplitudes

Coherent effects



Example :

Classical diffusion

quantum crossing


Coherent effects



Example :

Classical diffusion

quantum crossing


Coherent effects



Example :

Classical diffusion

quantum crossing


Coherent effects

Occurrence of a quantum crossing after a time t for a photon diffusing in a volume Ld

p× (t) =λd−1ctLd

The time spent by a diffusing photon is so that τD = L2

D

p× (τD ) =λd−1cτDLd

≡1g

g = Dcλ d−1 L

d−2




D


≡1g

g = Dcλ d−1 L

d−2




D


≡1g

g = Dcλ d−1 L

d−2




D


≡1g

g = Dcλ d−1 L

d−2

λd−1 lVolume

Quantum crossings decrease the diffusion coefficient D : weak localization

λ : wavelength

Physical meaning of this parameter ?

g = Dcλ d−1 L

d−2

A metal can be modeled as a quantum gas of electrons scattered by an elastic disorder.

Classically, the conductance of a cubic sample of size is given by Ohm’s law: where is the conductivity. G = σL

d−2

Ld

σ

g =le

3λd−1Ld−2 = Gcl/(e2/h)

is the classical electrical conductance so that

Gcl/(e2/h) ≫ 1

Gcl

Electrical conductance of a metal



d−2

Ld

σ

g =le



Gcl/(e2/h) ≫ 1

Gcl




d−2

Ld

σ

g =le



Gcl/(e2/h) ≫ 1

Gcl




d−2

Ld

σ

g =le



Gcl/(e2/h) ≫ 1

Gcl


Classical transport : Gcl = g ×e2

hwith g ≫ 1

Quantum corrections: ∆G = Gcl ×1

g

so that ∆G ≃

e2

h

A direct consequence: quantum corrections to electrical transport


hwith g ≫ 1


g

so that ∆G ≃

e2

h



hwith g ≫ 1


g


so that is universal∆G ≃

e2

h

Independent of the microscopic (and often unknown) disorder - Depends only on the geometry


hwith g ≫ 1


g


so that is universal#

Not that simple ! We wish to obtain precise

numbers... Need to sum up Feynman diagrams.


hwith g ≫ 1


g


so that is universal#

Not that simple ! We wish to obtain precise

numbers... Need to sum up Feynman diagrams.

+= +

= + +

==

=

b) diffusion anisotrope

a) diffusion isotrope

c) =

+= +

= + +

==

=

b) diffusion anisotrope

a) diffusion isotrope

c) =

(a) (b) (c)

(d) (e)

(d) (e)

An intermezzo based on our understanding of coherent effects

�41

Expansion in powers of quantum crossings allows to calculate quantum corrections to physical quantities.

This singular perturbation expansion is not a simple coincidence but an expression of scaling

A renormalization of D(L) changes also g(L):

1 g

The diffusion coefficient D is reduced (weak localization) and becomes size dependent :

g(L) = D(L)cλd−1 L

d−2 ≈N⊥2 (L)N

A quantum phase transition: Anderson localization

D(L) = D 1− 1πgln L

l( ) + 1πgln L

l( )⎛⎝⎜

⎞⎠⎟

2

+ ....⎛

⎝⎜

⎞

⎠⎟ (d = 2)

�42




1 g



d−2 ≈N⊥2 (L)N



l( ) + 1πgln L

l( )⎛⎝⎜

⎞⎠⎟

2

+ ....⎛

⎝⎜

⎞

⎠⎟ (d = 2)

�43




1 g



d−2 ≈N⊥2 (L)N



l( ) + 1πgln L

l( )⎛⎝⎜

⎞⎠⎟

2

+ ....⎛

⎝⎜

⎞

⎠⎟ (d = 2)

�44




1 g



d−2 ≈N⊥2 (L)N



l( ) + 1πgln L

l( )⎛⎝⎜

⎞⎠⎟

2

+ ....⎛

⎝⎜

⎞

⎠⎟ (d = 2)

�45




1 g



d−2 ≈N⊥2 (L)N



l( ) + 1πgln L

l( )⎛⎝⎜

⎞⎠⎟

2

+ ....⎛

⎝⎜

⎞

⎠⎟ (d = 2)

�46

Scaling and its meaning :

If we know , we know it at any scale :

g (L(1 + ϵ)) = g(L)!

1 + ϵβ(g) + O(g−5)"

β(g) =d ln g

d lnL

Expanding, we have

with (Gell-Mann - Low function)

Scaling behavior :

(P.W. Anderson et al.,1979)

g(L)

g L(1+ ε)( )= f g(L),ε( )

ξ(W ) is the localization length

g(L,W ) = f Lξ(W )( )

�47



g (L(1 + ϵ)) = g(L)!

1 + ϵβ(g) + O(g−5)"

β(g) =d ln g

d lnL

Expanding, we have


Scaling behavior :


g(L)

g L(1+ ε)( )= f g(L),ε( )


g(L,W ) = f Lξ(W )( )

�48



g (L(1 + ϵ)) = g(L)!

1 + ϵβ(g) + O(g−5)"

β(g) =d ln g

d lnL

Expanding, we have


Scaling behavior :


g(L)

g L(1+ ε)( )= f g(L),ε( )


g(L,W ) = f Lξ(W )( )

�49



g (L(1 + ϵ)) = g(L)!

1 + ϵβ(g) + O(g−5)"

β(g) =d ln g

d lnL

Expanding, we have


Scaling behavior :


g(L)

g L(1+ ε)( )= f g(L),ε( )


g(L,W ) = f Lξ(W )( )

g(L,W )

Lξ(W )

d = 3

Anderson phase transition

d = 2

B.Kramer, A. McKinnon, 1981

Anderson localization phase transition occurs in d > 2

Numerical calculations on the (universal) Anderson Hamiltonian

End of the intermezzo based on our understanding of coherent effects

Weak disorder limit:

Probability of a crossing is small: phase coherent corrections to the classical limit are small.

Quantum crossings modify the classical probability (i.e. the Diffuson). Due to its long range behavior, the Diffuson propagates (localized) coherent effects over large distances.

Weak disorder physics

Quantum crossings are independently distributed : We can generate higher order corrections to the Diffuson as an expansion in powers of 1 / g

∝1 g( )

λ<< l ⇒ g >> 1






∝1 g( )

λ<< l ⇒ g >> 1






∝1 g( )

λ<< l ⇒ g >> 1



Quantum crossings modify the classical probability (i.e. the Diffuson). Due to its long range behaviour, the Diffuson propagates (localized) coherent effects over large distances.



∝1 g( )

λ<< l ⇒ g >> 1

To the classical probability corresponds the Drude conductance Gcl

First correction involves one quantum crossing and the probability to have a closed loop:

(∝1 / g)

Return probability

quantum correction decreases the conductance: weak localization

L

Weak localization- Electronic transport

τD = L2 D

Z(t) =

!

drPint(r, r, t) =" τD

4πt

#d/2

�

po (τD )

�

ΔGGcl

=− po (τD )

�

po (τD ) =1g

Z(t) dtτD0

τ D

∫



(∝1 / g)

Return probability


L


τD = L2 D

Z(t) =

!


4πt

#d/2

�

po (τD )

�

ΔGGcl

=− po (τD )

�

po (τD ) =1g

Z(t) dtτD0

τ D

∫



(∝1 / g)

Return probability


L


τD = L2 D

Z(t) =

!


4πt

#d/2

�

po (τD )

�

ΔGGcl

=− po (τD )

�

po (τD ) =1g

Z(t) dtτD0

τ D

∫

|A(k,k′)|2 =!

r1,r2

|f(r1, r2)|2"

1 + ei(k+k′).(r1−r2)#

Generally, the interference term vanishes due to the sum over , except for two notable cases:r1 and r2

k + k′≃ 0 : Coherent backscattering

r1 − r2 ≃ 0 : closed loops, weak localization and periodicity of the Sharvin effect.

φ0/2

-100 0 100 200 300

0.8

1.2

1.6

2.0

Scal

ed In

tens

ity

Angle (mrad)

-5 0 51.8

1.9

2.0

Coherent backscattering

A reminder !

|A(k,k′)|2 =!

r1,r2

|f(r1, r2)|2"

1 + ei(k+k′).(r1−r2)#




φ0/2

-100 0 100 200 300

0.8

1.2

1.6

2.0

Scal

ed In

tens

ity

Angle (mrad)

-5 0 51.8

1.9

2.0

Coherent backscattering

|A(k,k′)|2 =!

r1,r2

|f(r1, r2)|2"

1 + ei(k+k′).(r1−r2)#




φ0/2

this case

In the presence of a dephasing mechanism that breaks time coherence, only trajectories with contribute.

In the presence of an Aharonov-Bohm flux, paired amplitudes in the Cooperon acquire opposite phases:

φ2πφ/φ0 −2πφ/φ0 the phase difference becomes: 4πφ/φ0

t < τφ

Cooperon

φ0/2 periodicity of the Sharvin effect

is obtained from the covariant diffusion equationPint(r, r′, t)

!

1

τφ+

∂

∂t− D

"

∇r′ + i2e

hA(r′)

#2$

Pint(r, r′, t) = δ(r − r′)δ(t)

effective charge 2e




t < τφ

Cooperon



!

1

τφ+

∂

∂t− D

"

∇r′ + i2e

hA(r′)

#2$


effective charge 2e




t < τφ

Cooperon



!

1

τφ+

∂

∂t− D

"

∇r′ + i2e

hA(r′)

#2$


effective charge 2e

Back to coherent effects for light

An analogous problem: Speckle patterns in opticsConsider the elastic multiple scattering of light transmitted through a fixed disorder configuration.

Outgoing light builds a speckle pattern i.e., an interference picture:L

saa sb

sa' sb'

An analogous problem: Speckle patterns in opticsConsider the elastic multiple scattering of light transmitted through a fixed disorder configuration.

Outgoing light builds a speckle pattern i.e., an interference picture:L

saa sb

sa' sb'

Elastic disorder is not related to decoherence : disorder does not destroy phase coherence and does not introduce irreversibility.

What about speckle patterns ?

Averaging over disorder does not produce incoherent intensity only, but also an angular dependent part, the coherent backscattering, which is a coherence effect. We may conclude:

Tab = tab2

Caba 'b ' =δTabδTa 'b 'TabT ′a ′b

Slab geometry

transmission coefficient

Correlation function of the transmission coefficient :

correlations involve the product of 4 complex amplitudes with or without quantum crossingsa

a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'

Fluctuations and correlations

Slab geometry

L

saa sb

sa' sb'

Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


12.3 Average transmission coefficient 433

the following sections, we evaluate these various contributions, corresponding respectivelyto zero, one and two crossings.

12.3 Average transmission coefficient

In order to calculate the average of the transmission and reflection coefficients, we use thediffusion approximation to calculate the product of amplitudes depicted in Figure 12.7. Inreflection, the average outgoing intensity is given by (8.6), and the reflection coefficientis given by the albedo (8.13) (see Exercise 8.2 for the slab geometry). For the averagetransmission coefficient, we obtain an analogous relation,5 namely

T ab = 4πR2

cI0S

!dr1 dr2 |ψa(r1)|2#(r1, r2)|GR

(r2, r)|2 (12.12)

where #(r1, r2) is the structure factor taken at zero frequency, and ψa(r1) accounts foran incident plane wave attenuated over the elastic mean free path while entering into thescattering medium (relation 8.7):

ψa(r1) ="

cI0

4πe−|r1−r|/2le eik sa ·r1 . (12.13)

r is a point on the interface (Figure 12.7(b)). The average Green function GR(r2, R)

accounts for the propagation of the wave between the last scattering event up to a pointR far away from the scattering medium. In this so-called Fraunhoffer limit (relation 8.9),we have

GR(r2, R) = e−|r′−r2|/2le e−ik sb·r2

eikR

4πR, (12.14)

r1 r2

r'r

(a) (b)

sa

sa

sa

sb

sb

sb

Figure 12.7 Schematic representation (a) of the product of two amplitudes ψ corresponding to twoincident plane waves along the directions sa emergent along sb, (b) of the Diffuson that results from thepairing of these two amplitudes and that represents the main contribution to the average transmissioncoefficient Tab. r and r′ are points on the interface plane whereas (r1, r2) are the extremities of amultiple scattering trajectory.

5 We assume that the difference in refraction index between the two media is negligible.

Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


12.3 Average transmission coefficient 433

the following sections, we evaluate these various contributions, corresponding respectivelyto zero, one and two crossings.

12.3 Average transmission coefficient

In order to calculate the average of the transmission and reflection coefficients, we use thediffusion approximation to calculate the product of amplitudes depicted in Figure 12.7. Inreflection, the average outgoing intensity is given by (8.6), and the reflection coefficientis given by the albedo (8.13) (see Exercise 8.2 for the slab geometry). For the averagetransmission coefficient, we obtain an analogous relation,5 namely

T ab = 4πR2

cI0S

!dr1 dr2 |ψa(r1)|2#(r1, r2)|GR

(r2, r)|2 (12.12)

where #(r1, r2) is the structure factor taken at zero frequency, and ψa(r1) accounts foran incident plane wave attenuated over the elastic mean free path while entering into thescattering medium (relation 8.7):

ψa(r1) ="

cI0

4πe−|r1−r|/2le eik sa ·r1 . (12.13)

r is a point on the interface (Figure 12.7(b)). The average Green function GR(r2, R)

accounts for the propagation of the wave between the last scattering event up to a pointR far away from the scattering medium. In this so-called Fraunhoffer limit (relation 8.9),we have

GR(r2, R) = e−|r′−r2|/2le e−ik sb·r2

eikR

4πR, (12.14)

r1 r2

r'r

(a) (b)

sa

sa

sa

sb

sb

sb

Figure 12.7 Schematic representation (a) of the product of two amplitudes ψ corresponding to twoincident plane waves along the directions sa emergent along sb, (b) of the Diffuson that results from thepairing of these two amplitudes and that represents the main contribution to the average transmissioncoefficient Tab. r and r′ are points on the interface plane whereas (r1, r2) are the extremities of amultiple scattering trajectory.

5 We assume that the difference in refraction index between the two media is negligible.

Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2


Slab geometry




a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2





a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2





a

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'a'

b'

b

b'

b

a

a

a'a'

b

b

b'b'

+

+

aa

a'a'

b

b

b'

b'


Tab = tab2





430 Correlations of speckle patterns

(a)

ab a

ab

(b)

a' b'

(c) (d)

Figure 12.3 Schematic representation of the various setups designed for measurement of thecorrelations of the transmission coefficients: (a) transmission coefficient Tab, (b) correlation functionCaba′b′ , (c) transmission coefficient Ta for an incident plane wave sa obtained by integration overall emergent directions, (d) transmission coefficient T obtained by integration over all incident andemergent directions.

These various quantities are represented schematically in Figure 12.3. As we shall see,this is the total transmission coefficient T that will play a role analogous to the electricalconductance. This is the physical content of the Landauer formula (7.141).

Just like electrical conductance fluctuations, the correlation function δTabδTa′b′ involvesthe disorder average of a product of four amplitudes. In order to exhibit the variouscontributions to this correlation function, we follow the strategy developed in section 9.2.The complex scattering amplitude (12.3) takes the form

ψab ∝!

dr dr′ eik(sa ·r−sb·r′)"

CEC , (12.7)

where the amplitude EC corresponds to a given multiple scattering trajectory C. We thusobtain for the correlation function the average product

"

C1,C2,C3,C4

Ea,bC1

E∗a,bC2

Ea′,b′C3

E∗a′,b′C4

, (12.8)

where the notation used is defined in Figure 12.4. The non-vanishing contributions to thisaverage value of four amplitudes involve two Diffusons. We thus retain in the summation(12.8) the two terms such that C1 = C2 and C3 = C4, or C1 = C4 and C3 = C2. Thesetwo combinations of amplitudes are depicted schematically in Figure 12.5. The first isnothing but the product of the average values of the transmission coefficient. The secondcombination gives

δTabδTa′b′(1) =

#4πR2

cSI0

$2 %%%ψabψ∗a′b′

%%%2

(12.9)

and the corresponding term in the correlation function Caba′b′ is denoted by C(1)aba′b′ . For the

specific combination a = a′ and b = b′, which corresponds to a speckle pattern generated

Measure different configurations

Tab = tab2






(a)

ab a

ab

(b)

a' b'

(c) (d)




ψab ∝!


CEC , (12.7)


"

C1,C2,C3,C4

Ea,bC1

E∗a,bC2

Ea′,b′C3

E∗a′,b′C4

, (12.8)



#4πR2

cSI0


%%%2

(12.9)





(c)

(b)

(a)

Figure 12.8 Right: speckle patterns measured in transmission for different directions of the incidentbeam. The arrow is a guide to show the evolution of a particular feature. The first picture is the referencepattern, corresponding to a given direction of the incident beam. By slowly tilting the incident beam(10 mdeg, then 20 mdeg), we notice a corresponding shift of the speckle pattern which retains some“memory” of the reference pattern. If the tilt angle becomes too large, the speckle is distorted. Left:corresponding behavior of the correlation Caba′b′ as a function of the angle !sb. The upper curve (a)is the autocorrelation (i.e., for a = a′) of the reference speckle pattern. The width is of order 1/kW .The two other figures (b, c) display the correlation of the second and of the third speckle patterns withthe reference pattern (i.e., a = a′). The correlation is maximum for !sb = !sa , and this maximumis F1(kL|!sa|). It vanishes for angles larger than |!sa| ≃ 1/kL [303].

Exercise 12.1: correlation of a speckle pattern measured in reflectionWe denote by Rab the reflection coefficient (i.e., the albedo defined in Chapter 8).Using a modification of (12.22), show that the angular correlation of Rab is

δRabδRa′b′ =!

1

(4π)2S

"dr1 dr2 eik[!sa ·r1 − !sb·r2] e − z1/le e − z2/le $(r1, r2)

#2(12.35)

or equivalently,

δRabδRa′b′ = δ!sa ,!sb

!c

4π l2e

" ∞

0dz1 dz2 e − z1/le e − z2/le Pd (qa , z1, z2)

#2(12.36)

with qa = k|!sa|.

Tab = tab2






(a)

ab a

ab

(b)

a' b'

(c) (d)




ψab ∝!


CEC , (12.7)


"

C1,C2,C3,C4

Ea,bC1

E∗a,bC2

Ea′,b′C3

E∗a′,b′C4

, (12.8)



#4πR2

cSI0


%%%2

(12.9)




Universal conductance fluctuations

0 crossing: 1 crossing: vanishes due to the summation over the channels.2 crossings: correction universal

(very different from the classical self-averaging limit )

G2= Gcl

2 = e2 h( )2 g2Landauer description : G=

e2

hTab

ab∑

δG2 ∝G2/ g2 = (e2 / h)2

�

δG2 ∝Ld −4




G2= Gcl


e2

hTab

ab∑

δG2 ∝G2/ g2 = (e2 / h)2

�

δG2 ∝Ld −4




G2= Gcl


e2

hTab

ab∑

δG2 ∝G2/ g2 = (e2 / h)2

�

δG2 ∝Ld −4




G2= Gcl


e2

hTab

ab∑

δG2 ∝G2/ g2 = (e2 / h)2

�

δG2 ∝Ld −4


Dephasing and decoherence

aa

bb

a'a' b'

b'

(K )1

aa

bb

b'b'a'

a'

(K )2

aa

a'a'

bb

b'b'

(K )d3

aa

a'a'

bb

b'b'

(K )c3

cd

2 Diffusons

2 Cooperons

sensitive to an applied Aharonov-Bohm magnetic flux

φ

φ φ

φ

φ

Different contributions either sensitive or not to dephasing

k

k'

r1

rb

ry

r2

ra

rz

k

k'

r1

rb

ry

r2

ra

rz

(b)

(a)

r1 → ra → rb · · · → ry → rz → r2

r2 → rz → ry · · · → rb → ra → r1

The total average intensity is:

|A(k,k′)|2 =!

r1,r2

|f(r1, r2)|2"

1 + ei(k+k′).(r1−r2)#

incoherent classical term

interference term

Reciprocity theorem: If I see you, then you see me.

A reminder !

k

k'

r1

rb

ry

r2

ra

rz

k

k'

r1

rb

ry

r2

ra

rz

(b)

(a)

r1 → ra → rb · · · → ry → rz → r2

r2 → rz → ry · · · → rb → ra → r1

The total average intensity is:

|A(k,k′)|2 =!

r1,r2

|f(r1, r2)|2"

1 + ei(k+k′).(r1−r2)#

incoherent classical term

interference term

Reciprocity theorem: If I see you, then you see me.




t < τφ

Cooperon



!

1

τφ+

∂

∂t− D

"

∇r′ + i2e

hA(r′)

#2$


effective charge 2e A reminder !



aa

bb

a'a' b'

b'

(K )1

aa

bb

b'b'a'

a'

(K )2

aa

a'a'

bb

b'b'

(K )d3

aa

a'a'

bb

b'b'

(K )c3

cd



aa

bb

a'a' b'

b'

(K )1

aa

bb

b'b'a'

a'

(K )2

aa

a'a'

bb

b'b'

(K )d3

aa

a'a'

bb

b'b'

(K )c3

cd

2 Diffusons

2 Cooperons

sensitive to an applied Aharonov-Bohm magnetic flux

φ

φ φ

φ

φ

δG2 δG2

2

φ

1.5

vanishing of the weak localization correction for the same magnetic field

In the presence of incoherent processes : L > Lφ

δG2→ 0

46 Si-doped GaAs samples at 45 mK

δG2

�

G

(Mailly-Sanquer)

We expect the conductance fluctuations to be reduced by a factor 2

δG2 δG2

2

φ

1.5



δG2→ 0


δG2

�

G

(Mailly-Sanquer)


δG2 δG2

2

φ

1.5In the presence of incoherent processes : L > Lφ

δG2→ 0


δG2

�

G

(Mailly-Sanquer)


δG2 δG2

2

φ

1.5In the presence of incoherent processes : L > Lφ

δG2→ 0


δG2

�

G

(Mailly-Sanquer)


δG2 δG2

2

φ

1.5



δG2→ 0


δG2

�

G

(Mailly-Sanquer)


δG2 δG2

2

φ

1.5



δG2→ 0


δG2

�

G

(Mailly-Sanquer)

Thank you for your attention.

Based on Mesoscopic physics of electrons and photons, by Eric Akkermans and Gilles Montambaux, Cambridge University Press, 2007

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