Polarization, multiple scatteringand the Berry phase

Vincent RossettoLPMMC Grenoble

Col de Porte, January 14th 2010

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phase

Conclusion

Presentation overview

1 PolarizationGeneralitiesGreen’s function

2 Multiple scatteringSingle scatteringBorn expansionDyson equation

3 Berry phaseBringing the Berry phase to lightGeometryApplications

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

1 PolarizationGeneralitiesGreen’s function

2 Multiple scatteringSingle scatteringBorn expansionDyson equation

3 Berry phaseBringing the Berry phase to lightGeometryApplications

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationfor different waves

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationgeneralities

Acoustic waves 1 d.o.f. no polarizationElectromagnetic waves 2 d.o.f. polarizationElastic waves 3 d.o.f. polarization

Polarization depends on relative phases and amplitudes

Linear Circular Elliptical

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationrepresentations

For the field : Jones representationsExEyEz

cartesian

For the intensity : Stokes representationsI

I⊥QUV

with

I = E†E

I⊥ = E†( 1 0 0

0 1 00 0 0

)E

. . .

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationrepresentations

For the field : Jones representationsExEyEz

E+

E−E0

cartesian circular

For the intensity : Stokes representationsI

I⊥QUV

with

I = E†E

I⊥ = E†( 1 0 0

0 1 00 0 0

)E

. . .

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationrepresentations

For the field : Jones representationsExEyEz

E+

E−E0

cartesian circular

For the intensity : Stokes representationsI

I⊥QUV

with

I = E†E

I⊥ = E†( 1 0 0

0 1 00 0 0

)E

. . .

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationframes

EE

E

E

E

x

x

y

y’E

’ θ

E ′xE ′yE ′z

=

cos θ sin θ 0− sin θ cos θ 0

0 0 1

ExEyEz

E ′+

E ′−E ′0

=

e−iθ 0 00 e+iθ 00 0 e0

E+

E−E0

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationframes

EE

E

E

E

x

x

y

y’E

’ θ

E ′xE ′yE ′z

=

cos θ sin θ 0− sin θ cos θ 0

0 0 1

ExEyEz

E ′+

E ′−E ′0

=

e−iθ 0 00 e+iθ 00 0 e0

E+

E−E0

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationframes

EE

E

E

E

x

x

y

y’E

’ θ

E ′xE ′yE ′z

=

cos θ sin θ 0− sin θ cos θ 0

0 0 1

ExEyEz

E ′+

E ′−E ′0

=

e−iθ 0 00 e+iθ 00 0 e0

E+

E−E0

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationmobile frames

z

Reference frame

Mobile frames

^

x

y

r

r’

E(r , t , R)

Es(r , t , RZ(α)) = e−isαEs(r , t , R)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Polarizationmobile frames

z

Reference frame

Mobile frames

^

x

y

r

r’

E(r , t , R)

Es(r , t , RZ(α)) = e−isαEs(r , t , R)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin vacuum

G0 relates amplitudesat positions r and r ′

at times t and t ′

in frames R and R′R

Y

X

Z Z

Y

X

Rr’−rFrames

Pathr r’

Euler angles R = Z(φ)Y(θ)Z(ψ)

∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

r ′ − r = R Dz

G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin vacuum

G0 relates amplitudesat positions r and r ′

at times t and t ′

in frames R and R′R

Y

X

Z Z

Y

X

Rr’−rFrames

Pathr r’

Euler angles R = Z(φ)Y(θ)Z(ψ)

∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

r ′ − r = R Dz

G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin vacuum

G0 relates amplitudesat positions r and r ′

at times t and t ′

in frames R and R′R

Y

X

Z Z

Y

X

Rr’−rFrames

Pathr r’

Euler angles R = Z(φ)Y(θ)Z(ψ)

∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

r ′ − r = R Dz

G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin vacuum

G0 relates amplitudesat positions r and r ′

at times t and t ′

in frames R and R′R

Y

X

Z Z

Y

X

Rr’−rFrames

Pathr r’

Euler angles R = Z(φ)Y(θ)Z(ψ)

∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

r ′ − r = R Dz

G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin vacuum

G0 relates amplitudesat positions r and r ′

at times t and t ′

in frames R and R′R

Y

X

Z Z

Y

X

Rr’−rFrames

Pathr r’

Euler angles R = Z(φ)Y(θ)Z(ψ)

∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

r ′ − r = R Dz

G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin Fourier domain

G0(q, ω, R, R′)∣∣ss′ = δss′

∆(R, R′)(ωc − q · Rz

)2 ei(s′ψ′−sψ)

∫dR∫

dR′ G0(q, ω, R, R′) =1(

ωc

)2 − q2

0 0 00 0 00 0 1

⇒ Directivity is essential to representpolarized waves in mobile frames.

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin Fourier domain

G0(q, ω, R, R′)∣∣ss′ = δss′

∆(R, R′)(ωc − q · Rz

)2 ei(s′ψ′−sψ)

∫dR∫

dR′ G0(q, ω, R, R′) =1(

ωc

)2 − q2

0 0 00 0 00 0 1

⇒ Directivity is essential to representpolarized waves in mobile frames.

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionin Fourier domain

G0(q, ω, R, R′)∣∣ss′ = δss′

∆(R, R′)(ωc − q · Rz

)2 ei(s′ψ′−sψ)

∫dR∫

dR′ G0(q, ω, R, R′) =1(

ωc

)2 − q2

0 0 00 0 00 0 1

⇒ Directivity is essential to representpolarized waves in mobile frames.

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionand medium anisotropies

Absorption (dichroism)

exp (−κsR)

Birefringence

exp(−iω

csR)

Spin flip (for photons)

G0∣∣s,−s 6= 0

Faraday effect (for photons)

q → q − sVB

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionand medium anisotropies

Absorption (dichroism)

exp (−κsR)

Birefringence

exp(−iω

csR)

Spin flip (for photons)

G0∣∣s,−s 6= 0

Faraday effect (for photons)

q → q − sVB

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionand medium anisotropies

Absorption (dichroism)

exp (−κsR)

Birefringence

exp(−iω

csR)

Spin flip (for photons)

G0∣∣s,−s 6= 0

Faraday effect (for photons)

q → q − sVB

Polarization

V. Rossetto

PolarizationGeneralities

Green’s function

Multiplescattering

Berry phase

Conclusion

Green’s functionand medium anisotropies

Absorption (dichroism)

exp (−κsR)

Birefringence

exp(−iω

csR)

Spin flip (for photons)

G0∣∣s,−s 6= 0

Faraday effect (for photons)

q → q − sVB

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

1 PolarizationGeneralitiesGreen’s function

2 Multiple scatteringSingle scatteringBorn expansionDyson equation

3 Berry phaseBringing the Berry phase to lightGeometryApplications

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringseveral systems, many scales

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Single scatteringsimpler things first

θ

−ψ

RR’

R→R

R′

rotation R−1R′ = R = Z(φ)Y(θ)Z(ψ)

Tss′(ω, R, R′) = ei(sφ+s′ψ) fss′(ω, θ)

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Single scatteringsimpler things first

θ

−ψ

RR’

R→R

R′

rotation R−1R′ = R = Z(φ)Y(θ)Z(ψ)

Tss′(ω, R, R′) = ei(sφ+s′ψ) fss′(ω, θ)

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Single scatteringsimpler things first

θ

φ

−ψ

RR’

R→R

R′

rotation R−1R′ = R = Z(φ)Y(θ)Z(ψ)

Tss′(ω, R, R′) = ei(sφ+s′ψ) fss′(ω, θ)

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

r −r

Rr

r

R’

Paths

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringclassical picture

Find G the effective Green’s functionof a medium filled with scatterers.

r

r’

r −r

Rr

r

R’

Paths

G(q, ω, R, R′)

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringBorn expansion

Without correlations : independent scattering approximation

G(r , r ′) = G0(r , r ′) +

∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringBorn expansion

Without correlations : independent scattering approximation

G(r , r ′) = G0(r , r ′) +

∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

r’r = r r’

G G0

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringBorn expansion

Without correlations : independent scattering approximation

G(r , r ′) = G0(r , r ′) +

∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

r’r = r r’

r’+ rx1

G G0

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringBorn expansion

Without correlations : independent scattering approximation

G(r , r ′) = G0(r , r ′) +

∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

r’r = r r’

r’

+ r r’x x1

+ rx1

2

+ ...

G G0

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringBorn expansion

Without correlations : independent scattering approximation

G(r , r ′) = G0(r , r ′) +

∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

r’r = r r’

r’

+ r r’x x1

+ rx1

2

+ ...

G G0

Self-consistent equation

= +

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

= 0

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

= 0

= 0

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

= 0

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

r

r’

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

r

r’

=

++ + +

Σ

++ ...+

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringself-energy

r

r’

r

r’

=

+

...++

Σ

= + Σ

Green−Dyson equation

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringof polarized light

Change scattering definition :

=

∫dR1

∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

Self energy

= lim|q|→∞

∫dR1

∫dR2 T(ω, R2,R′)G(, ω, R1, R2)T(ω, R, R1)

∫dR =⇒ matrix product

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringof polarized light

Change scattering definition :

=

∫dR1

∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

Self energy

= lim|q|→∞

∫dR1

∫dR2 T(ω, R2,R′)G(q, ω, R1, R2)T(ω, R, R1)

∫dR =⇒ matrix product

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringof polarized light

Change scattering definition :

=

∫dR1

∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

Self energy

= lim|q|→∞

∫dR1

∫dR2 T(ω, R2,R′)G(q

↑has a direction, ω, R1, R2)T(ω, R, R1)

∫dR =⇒ matrix product

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringof polarized light

Change scattering definition :

=

∫dR1

∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

Self energy Σ(q, ω, R, R′)

= lim|q|→∞

∫dR1

∫dR2 T(ω, R2,R′)G(q

↑has a direction, ω, R1, R2)T(ω, R, R1)

∫dR =⇒ matrix product

Polarization

V. Rossetto

Polarization

MultiplescatteringSingle scattering

Born expansion

Dyson equation

Berry phase

Conclusion

Multiple scatteringof polarized light

Change scattering definition :

=

∫dR1

∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

Self energy Σ(q, ω, R, R′)

= lim|q|→∞

∫dR1

∫dR2 T(ω, R2,R′)G(q

↑has a direction, ω, R1, R2)T(ω, R, R1)

∫dR =⇒ matrix product

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

1 PolarizationGeneralitiesGreen’s function

2 Multiple scatteringSingle scatteringBorn expansionDyson equation

3 Berry phaseBringing the Berry phase to lightGeometryApplications

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasephase and rotation

EE

E

E

E

x

x

y

y’E

’ θ

Es(r , t , RZ(α)) = e−isαEs(r , t , R)

Tss′(ω, R, R′) = ei(sφ+s′ψ) fss′(ω, θ)

Total phase for a path ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasephase and rotation

EE

E

E

E

x

x

y

y’E

’ θ

Es(r , t , RZ(α)) = e−isαEs(r , t , R)

Tss′(ω, R, R′) = ei(sφ+s′ψ) fss′(ω, θ)

Total phase for a path ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasephase and rotation

EE

E

E

E

x

x

y

y’E

’ θ

Es(r , t , RZ(α)) = e−isαEs(r , t , R)

Tss′(ω, R, R′) = ei(sφ+s′ψ) fss′(ω, θ)

Total phase for a path ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebringing it to light

Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi 1 φi+1 − φi 1

Rotations R−1i Ri+1 = Ri

φi + ψi ' (φi+1 − φi) cos θi + ψi+1 − ψi

sn−1∑i=0

φi + ψi ' −sn−1∑i=0

(1− cos θi)(φi+1 − φi) mod 2π

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebringing it to light

Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi 1 φi+1 − φi 1

Rotations R−1i Ri+1 = Ri

φi + ψi ' (φi+1 − φi) cos θi + ψi+1 − ψi

sn−1∑i=0

φi + ψi ' −sn−1∑i=0

(1− cos θi)(φi+1 − φi) mod 2π

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebringing it to light

Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi 1 φi+1 − φi 1

R0

R1

R2

R3

R4

R5

R6Rotations R−1

i Ri+1 = Ri

φi + ψi ' (φi+1 − φi) cos θi + ψi+1 − ψi

sn−1∑i=0

φi + ψi ' −sn−1∑i=0

(1− cos θi)(φi+1 − φi) mod 2π

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebringing it to light

Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi 1 φi+1 − φi 1

R0

R1

R2

R3

R4

R5

R6Rotations R−1

i Ri+1 = Ri

φi + ψi ' (φi+1 − φi) cos θi + ψi+1 − ψi

sn−1∑i=0

φi + ψi ' −sn−1∑i=0

(1− cos θi)(φi+1 − φi) mod 2π

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebringing it to light

Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi 1 φi+1 − φi 1

R0

R1

R2

R3

R4

R5

R6Rotations R−1

i Ri+1 = Ri

φi + ψi ' (φi+1 − φi) cos θi + ψi+1 − ψi

sn−1∑i=0

φi + ψi ' −sn−1∑i=0

(1− cos θi)(φi+1 − φi) mod 2π

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebringing it to light

Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi 1 φi+1 − φi 1

R0

R1

R2

R3

R4

R5

R6Rotations R−1

i Ri+1 = Ri

φi + ψi ' (φi+1 − φi) cos θi + ψi+1 − ψi

sn−1∑i=0

φi + ψi ' −sn−1∑i=0

(1− cos θi)(φi+1 − φi) mod 2π

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasegeometric interpretation

(1− cos θi)(φi+1 − φi)

θi

φi+1 − φi

Ω = −s

Area enclosedon the sphereby the directionof propagation

n−1∑i=0

(1− cos θi)(φi+1 − φi)

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasegeometric interpretation

(1− cos θi)(φi+1 − φi)

θi

φi+1 − φi

Ω = −s

Area enclosedon the sphereby the directionof propagation

n−1∑i=0

(1− cos θi)(φi+1 − φi)

R0

R1

R2

R3

R4

R5

R6

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasegeometric interpretation

(1− cos θi)(φi+1 − φi)

θi

φi+1 − φi

Ω = −s

Area enclosedon the sphereby the directionof propagation

n−1∑i=0

(1− cos θi)(φi+1 − φi)

R0

R1

R2

R3

R4

R5

R6

^z

^z

^z

^z

^z

^z

^z

1

2

3

4

=0 R6R

R

R

5R

R

R

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasegeometric interpretation

(1− cos θi)(φi+1 − φi)

θi

φi+1 − φi

ΩB = −s

Area enclosedon the sphereby the directionof propagation

n−1∑i=0

(1− cos θi)(φi+1 − φi)

R0

R1

R2

R3

R4

R5

R6

^z

^z

^z

^z

^z

^z

^z

1

2

3

4

=0 R6R

R

R

5R

R

R

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phaseexample

“Parallel transport” of transverse polarization

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phaseexample

“Parallel transport” of transverse polarization

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phaseexample

“Parallel transport” of transverse polarization

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phaseexample

“Parallel transport” of transverse polarization

π/2

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasestatistics

In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

starting at r and ending at r ′

of length c(t ′ − t)with initial and final directions Rz and R′z

Properties of the Berry phase statistics ?

depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasestatistics

In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

starting at r and ending at r ′

of length c(t ′ − t)with initial and final directions Rz and R′z

Properties of the Berry phase statistics ?

depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasestatistics

In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

starting at r and ending at r ′

of length c(t ′ − t)with initial and final directions Rz and R′z

Properties of the Berry phase statistics ?

depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasestatistics

In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

starting at r and ending at r ′

of length c(t ′ − t)with initial and final directions Rz and R′z

Properties of the Berry phase statistics ?

depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasedepolarization

L ∼ `∗

Forward scattering : the directionof propagation remains near z

The trajectory on the sphere is al-most flat

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasedepolarization

L ∼ `∗

Forward scattering : the directionof propagation remains near z

The trajectory on the sphere is al-most flat

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasedepolarization

L ∼ `∗

Forward scattering : the directionof propagation remains near z

The trajectory on the sphere is al-most flat

The distribution of Berry phase is (Levy) :

p(ΩB) =π`∗

L1

cosh2 (2πΩB`∗

L

)

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasedepolarization

L ∼ `∗

Forward scattering : the directionof propagation remains near z

The trajectory on the sphere is al-most flat

The distribution of Berry phase is (Levy) :

p(ΩB) =π`∗

L1

cosh2 (2πΩB`∗

L

)The outgoing polarisation is :

〈cos 2ΩB〉 =L

2`∗1

sinh L2`∗

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasenumerical simulations

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasenumerical simulations

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasenumerical simulations

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasenumerical simulations

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasenumerical simulations

0 500 1000t-t

0

0

5

10

15

20

Ber

ry p

hase

flu

ctua

tions

R=20R=100R=1000R=5000R=10000

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebackscattering experiment

ES

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebackscattering experiment

ES

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phaseIntroduction

Geometry

Applications

Conclusion

The Berry phasebackscattering experiment

ES

S

E

1 2

2dθ

dθ

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phase

Conclusion

Conclusion and outlooks

Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phase

Conclusion

Conclusion and outlooks

Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phase

Conclusion

Conclusion and outlooks

Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phase

Conclusion

Conclusion and outlooks

Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

Polarization

V. Rossetto

Polarization

Multiplescattering

Berry phase

Conclusion

Conclusion and outlooks

Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data