slow light and superluminal propagation · 2009. 1. 10. · outlook • group velocity, case of a...

Slow Light and Superluminal Propagation

M. A. Bouchene

Laboratoire « Collisions, Agrégats, Réactivité », Université Paul Sabatier, Toulouse, France

Interest in Slow and Fast light

• Fundamental aspect in optical physics• How can we stop (reversibly) the light ?• Can light travels faster than c?

• Applications: optical storage, optical information, telecom (buffering)

Outlook• Group velocity, case of a resonant system

• (Ultra) Slow light: - Electromagnetic Induced Transparency- Coherent Population Oscillations- Zeeman Coherence Oscillations. Coherent control

of the optical response

• Superluminal propagation: - Relativity implications- Different velocities for a light pulse- Backward propagation

• Conclusion

Group Velocity( ) ( )

0

0 0( ) 1 ( )

( )

⎛ ⎞≠ + −⎜ ⎟⎝ ⎠

=

dnn nd

cteω

ω ω ω ωω

α ω

z0( )

0

0

( , ) ( / v ) e

v

i kz tgE t z G E t z

k

ω

ϕ

ω

−= −

=

( ) 1;( ) 0

==

n ωα ω

0 0( )0

00

( , ) ( / ) ei k z tE t z E t z c

kc

ω

ω

−= −

=

0v / ( ) :c nϕ ω=

0

0 0

v :( )

gc

dnnd ω

ω ωω

=⎛ ⎞+ ⎜ ⎟⎝ ⎠

Phase velocity

Group velocity

PROPOGATION WITHOUT DISTORTION UNLESS RESHAPING EFFECTS(higher orders contributions)

c

c vϕ

vg


0

0 0( ) 1 ( )

( )

⎛ ⎞≠ + −⎜ ⎟⎝ ⎠

=

dnn nd

cteω

ω ω ω ωω

α ω

z0( )

0

0

( , ) ( / v ) e

v


k

ω

ϕ

ω

−= −

=

( ) 1;( ) 0

==

n ωα ω

0 0( )0

00

( , ) ( / ) ei k z tE t z E t z c

kc

ω

ω

−= −

=

c

c vϕ

vg


0

0 0( ) 1 ( )

( )

⎛ ⎞≠ + −⎜ ⎟⎝ ⎠

=

dnn nd

cteω

ω ω ω ωω

α ω

z0( )

0

0

( , ) ( / v ) e

v


k

ω

ϕ

ω

−= −

=

( ) 1;( ) 0

==

n ωα ω

0 0( )0

00

( , ) ( / ) ei k z tE t z E t z c

kc

ω

ω

−= −

=

0v / ( ) :c nϕ ω= Phase velocity

c

c vϕ

vg


0

0 0( ) 1 ( )

( )

⎛ ⎞≠ + −⎜ ⎟⎝ ⎠

=

dnn nd

cteω

ω ω ω ωω

α ω

z0( )

0

0

( , ) ( / v ) e

v


k

ω

ϕ

ω

−= −

=

( ) 1;( ) 0

==

n ωα ω

0 0( )0

00

( , ) ( / ) ei k z tE t z E t z c

kc

ω

ω

−= −

=

0v / ( ) :c nϕ ω=

0

0 0

v :( )

gc

dnnd ω

ω ωω

=⎛ ⎞+ ⎜ ⎟⎝ ⎠

Phase velocity

Group velocity

c

c vϕ

vg


0

0 0( ) 1 ( )

( )

⎛ ⎞≠ + −⎜ ⎟⎝ ⎠

=

dnn nd

cteω

ω ω ω ωω

α ω

z0( )

0

0

( , ) ( / v ) e

v


k

ω

ϕ

ω

−= −

=

( ) 1;( ) 0

==

n ωα ω

0 0( )0

00

( , ) ( / ) ei k z tE t z E t z c

kc

ω

ω

−= −

=

0v / ( ) :c nϕ ω=

0

0 0

v :( )

gc

dnnd ω

ω ωω

=⎛ ⎞+ ⎜ ⎟⎝ ⎠

Phase velocity

Group velocity

linear-dispersive medium with constant spectral gain/abs: PROPAGATION WITHOUT DISTORTIONHigh orders contributions: RESHAPING EFFECTS

c

c vϕ

vg

( )/ ,i c nL Le eω α

; :g gg

dnn nn d

Group indexcV ωω

= = +

• Dilute medium but vg may be very different from c 1n

; :g gg

dnn nn d

Group indexcV ωω

= = +


vg

ωω

dnd-1

Vg=c

0

; :g gg

dnn nn d

Group indexcV ωω

= = +


vg

ωω

dnd-1

Vg=c

Vg<0

Backward propagation

Vg>0

SLOW LIGHTVg<c

SUPERLUMINAL

Vg>c

0

; :g gg

dnn nn d

Group indexcV ωω

= = +

NORMAL DISPERSIONANOMALOUS DISPERSION

Forward propagation


vg

-1

Vg=c

Vg<0

Backward propagation

Vg>0

SLOW LIGHTVg<c

SUPERLUMINAL

Vg>c

0

; :g gg

dnn nn d

Group indexcV ωω

= = +

NORMAL DISPERSIONANOMALOUS DISPERSION

ωω

dnd

Forward propagation

• Ultra slow light needs w dn/dw >>1How to achieve ??

• vg> c : What about relativity?

• vg< 0 : Propagation of energy?

What happens near a resonant transition?

N 0ωω

0E i n ll c0E e e

ω−α


( )

2

0 2 20

:=− +

absorptioncoefficientγα αω ω γ

N 0ωω

20

00

: , :N absorption at resonance frequency relaxation ratec

μ ωα γε γ

=

0E i n ll c0E e e

ω−α

Increasing ng is possible but absorption also increases!

NORMAL DISPERSION

ANOMALOUS DISPERSION


( )0 0

2 20 0

21 :−= +

− +refracn ction indexα γ ω ω

ω ω ω γ

N 0ωω

20

00

: , :N absorption at resonance frequency relaxation ratec

μ ωα γε γ

=

0E i n ll c0E e e

ω−α

Increasing ng is possible but absorption also increases!

NORMAL DISPERSION


Increasing ng is possible but absorption also increases!Difficult to observe experimentally

NORMAL DISPERSION


I- How to produce slow-fast light?:g n

c

nd

V dωω

=+

Kramers-Kronigs relations2 20

2

2 20

( ')1 ''

4 1 ''

( )

( ')( )

∞

∞

− = ℘−

−= − ℘

−

∫

∫

c d

dc

n

n

α ωω

ωα ω

ωπ ω ωω ωπ ω ω

A narrow spectral hole produce a strong normal dispersion

• Electromagnetic Induced Transparency

• Coherent Population Oscillations

• Zeeman Coherence Oscillations

Slow light

II-a Electromagnetic Induced Transparency

0

1

2

Control field

Propagating fieldC laser intensityΩ ∝

Interpretation

γ

0

1

2

1 2:

2

i teadiabatic states

ω−±± =

Control field

Propagating fieldC laser intensityΩ ∝

Interpretation

γ

0

1

2

1 2:

2


ω−±± =

0

+

−Control field

Propagating field Propagating field

C laser intensityΩ ∝

CΩ

Interpretation

γ

∝ γ

0

1

2

1 2:

2


ω−±± =

0

+

−Control field



CΩ } Autler-TownesSplittingDynamical Starkeffect

Interpretation

γ

∝ γ

0

1

2

1 2:

2


ω−±± =

0

+

−Control field



CΩ

:C γΩ >> Propagating field is non-resonant

} Autler-TownesSplittingDynamical Stark effect

Interpretation

γ

∝ γ

propag. 10ω − ω

AB

SOR

PTIO

N C

OEF

FIC

IEN

T

1 2:

2


ω−±± =

0

+

−

Propagating field

CΩ

Interpretation

Ω ≤C γ

1 2:

2


ω−±± =

0

+

−

Propagating field

CΩ

Interpretation

Ω ≤C γ

Interference between quantum paths 0 and 0→ + → −

RESULT ?

propag. 10ω − ω

AB

SOR

PTIO

N C

OEF

FIC

IEN

T

TOTAL DESTRUCTIF INTERFERENCEpropag. 10ω − ω

AB

SOR

PTIO

N C

OEF

FIC

IEN

T

Coherent Population Trapping

0

1

2

CΩ

PΩ

Bright stateB sin 0 cos 2 :

D sin 2 cos 0 : Dark state

= ϕ + ϕ

= − ϕ + ϕ

p

c

tanΩ

ϕ =Ω

γ

1

BD

No coupling


p 10ω = ω

1

2 2P CΩ = Ω + Ω

BD

No coupling


p 10ω = ω

1

B

D


p 10ω = ω


1

B

D

1

B

D


ALL THE ATOMS ARE IMMUNE TO INTERACTION: perfect transparency

CPT

Autler-Townes splitting

NATURE, 397, 594, (1999)

g 7v10 !!

c−=

Vg decreases with CΩ


Is it possible to stop the light ?



YES



YESWhat means a stopped light ??

CΩ

E(Z 0)=E(Z L)=

CΩ

E(Z 0)=E(Z L)=

STORAGE

CΩ

E(Z 0)=E(Z L)=

STORAGEWhat kind of

storage?

bcE(z, t)(z, t) cos (t) sin (t)g Nψ = θ − σθ

2 2

(t)cos(t) g NΩ

θ =Ω +

Polariton (mixture field-atom)

2ccos (t) (z, t) 0t z

∂ ∂⎛ ⎞+ θ ψ =⎜ ⎟∂ ∂⎝ ⎠

Shape-preserved propagation with2

gv (t) ccos (t)= θ

a

b

1ω2ω

Bichromatic field :

δ

II-b Coherent Population Oscillations

a

b

1ω2ω

Bichromatic field : ( )1 11 2 22I I I 2 I I cos t= + − ω+ ω

BEATINGδ


a

b

1ω2ω

Bichromatic field : ( )1 21 2 12I I I 2 I I cos t= + − ω+ ω

BEATING

TEMPORAL GRATING

(0) ( ) i t ( ) i tba b a ba ba ban n n n n e n e+ δ − − δ= − = + +

δ


2ω 12ω + δ = ω

δ

Reinforcement of the emission at 1ω

2ω − δ

2ω


2ω12ω + δ = ω

δ

δ0

Abs

orpt

ion


2ω12ω + δ = ω

δ

δ0

Abs

orpt

ion


1POPULATION 1 OSCILLATIONt T t −Δ = ≤ Δ = δ

12T −

11T −

II-c Coherent Zeeman Oscillations

z

x

yprop. axes

π polarized control field

σ polarized probe field

Δ

Control fieldpolarizedπ −

A double two level system

F=½

F=½

a b

c d

mF=½ mF=-½

πProbe fieldpolarizedσ −

Bichromatic field, with strong resonant pulse and detuned pulse

π

σ

π σ

( ) δΦ = Δ −, .r t t k r

( )− Φ= +

,i r tT

Ee e e eE

σπ σ

π

( )Φ ,r t

0 2π π 3 2π 2π

πe

σe

polarization producesa time /space grating

Linear ellipt. L linear ellip. R linear


grating imprints on Zeeman coherences

( )1 2

1

1

δρ ρ ρ ρ=

Δ −

=−

= + = ∑ .( )n

in t k rnZ z z

ne

z2ρ

z1ρ


( ) ( ) 2σ

σ π σρ

ρ θ ρ θ − Φ Γ= − − − −

,Im i r tZ e gi e n n

= =Γ Γ

;dE dEπ σπ σθ θ

z2ρ

z1ρ

( )1 2

1

1

δρ ρ ρ ρ=

Δ −

=−

= + = ∑ .( )n

in t k rnZ z z

ne


Absorption bypopulation

= =Γ Γ

;dE dEπ σπ σθ θ

( ) ( ) 2σ

σ π σρ

ρ θ ρ θ − Φ Γ= − − − −


en

gn

( )1 2

1

1

δρ ρ ρ ρ=

Δ −

=−

= + = ∑ .( )n

in t k rnZ z z

ne


Diffraction of strong fieldby Zeeman coherence

π σ1n =

( ) ( ) 2σ

σ π σρ

ρ θ ρ θ − Φ Γ= − − − −


z2ρ

z1ρ

( )1 2

1

1

δρ ρ ρ ρ=

Δ −

=−

= + = ∑ .( )n

in t k rnZ z z

ne


Diffraction of strong fieldby Zeeman coherence

Absorption bypopulation

Cancel exactly

at

π σ1n =

= 0Δ

( ) ( ) 2σ

σ π σρ

ρ θ ρ θ − Φ Γ= − − − −


z2ρ

z1ρ

( )1 2

1

1

δρ ρ ρ ρ=

Δ −

=−

= + = ∑ .( )n

in t k rnZ z z

ne

F. A. Hashmi and M. A. Bouchene Phy. Rev. A 77, 051803(R) 2008

-2 -1 0 1 2

-0.4

-0.2

0.0

0.2

0.4-2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

Sus

cept

ibili

ty (i

n un

its o

f 2α

0k-1

)

Δ (in units of Γ)

Re(χ)

Ωπ= 0 Ωπ= 0.1 Γ Ωπ= 0.2 Γ

Im(χ)


( )2/Width π∝ χ Γ

• Pure Non-linear effect• No dark state• Hybrid properties between CPO and EIT


• Giant Non-linearity:

IId- Coherent Control of the Susceptibility

(2) 2 (3) 3 ...linP E E Eχ χ χ= + + +

If E is strong : ionisation

If resonance : absorption

Increase NL effects increase E or χ

Slow light techniques: reduction of absorption AND resonance

Effects of giant non linearities observable

-2 -1 0 1 2

-0.4

-0.2

0.0

0.2

0.4-2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

Sus

cept

ibili

ty (i

n un

its o

f 2α

0k-1

)

Δ (in units of Γ)

Re(χ)

Ωπ= 0 Ωπ= 0.1 Γ Ωπ= 0.2 Γ

Im(χ)

0= :Δ

Optical response canceled in the direction Optical response determined by other emitted waves

kπ

kσ

kδ

kσ2k kπ σ−

II-d Coherent Control of Susceptibility

z

x

yprop. axes

π polarized control field

σ polarized probe field

0Δ

π σ

F=½

F=½

,k kσ π

a b

c d

( ) :Coherent Controlda cbσ σρ ρ ρ ρ ϕ= + =

0 0= =, kδΔ

( , ) .r t t k rδ ϕ ϕΦ = Δ − + =

da cbσρ ρ ρ= +

a b

c d

Absorptionpath

daρ =


kσ

da cbσρ ρ ρ= +

a b a b

c d c d

π π+

Absorptionpath

Cross- Kerr type path

daρ =


( )k k k kπ σ π σ+ − =kσ

da cbσρ ρ ρ= +

a b a b

c d c d

π π+

Absorptionpath


daρ =


Cancel each other


0=Δ

da cbσρ ρ ρ= +

a b a b a b

c d c d c d

π π π π+ +

Absorptionpath


Phase Conjugate type path

daρ =


Cancel each otherDefine the optical response

( ) 2k k k k k

k

π σ π π σ

σ

− − = −

=


0=Δ

0 1if and L ( thin sample ) :π αΩ << Γ <<


a b

c d

π π

0 1

i

linearProblem equivalent to propagation of fieif

ltwo leve

de

and L (

in

thin s

an assembly of at

am

l

ple ) :

oms :ϕσ

π αΩ

Ω −

<< Γ <<


a b

c d

( ) ( ) ( )2

0 1

i

i i ilin lin

Problem equivalent to propagation of filineartwo lev

if and L ( thin sameld

e in an assembly of atoms :

ple ) :

e

l

e

e

P e

ϕσ

ϕ ϕ ϕσ σ σ

π

σρ χ

α

χ −

Ω

∝ Ω =

Ω < Γ

Ω

< <

−

<


a b

c d

( ) ( ) ( )2

0 1

i

i i ilin lin




ple ) :

e

l

e

e

P e

ϕσ

ϕ ϕ ϕσ σ σ

π

σρ χ

α

χ −

Ω

∝ Ω =

Ω < Γ

Ω

< <

−

<


2≈ !ieff line

ϕχ χ

Independent from pump field characteristics!

a b

c d

( ) ( ) ( )2

0 1

i

i i ilin lin




ple ) :

e

l

e

e

P e

ϕσ

ϕ ϕ ϕσ σ σ

π

σρ χ

α

χ −

Ω

∝ Ω =

Ω < Γ

Ω

< <

−

<


2≈ !ieff line

ϕχ χ

-The system behaves as a linear medium with a modified(linear) suceptibility controlled by the phase shift-Giant non-linearity

Independent from pump field characteristics!

a b

c d

F. A. Hashmi and M. A. Bouchene Phys. Rev. Letters 101, 21306 (2008)

2 :ieff line gain dispersion couplingϕχ χ≈ −

-4 -2 0 2 4-0.5

0.0

0.5

1.0

ϕ ϕ

ϕϕ

Re χeff

Im χeff

=0

=3π/4 =π/2

=π/4

Ωπ=100 Ω

σ= 0.1Γ; Γze=2Γd=Γ; α0L=0.2

-4 -2 0 2 4-1.0

-0.5

0.0

0.5

-4 -2 0 2 4-1.0

-0.5

0.0

0.5

χ eff i

n un

its o

f 2α

0k-1

Δ0 in units of Γ

-4 -2 0 2 4-0.5

0.0

0.5

1.0


0 0ω ωΔ = −

AbsorberAnomalous dispersion

AmplifierNormal dispersion

Anomalous «dispersion-like»gain

«absorption-like»dispersion

Normal «dispersion-like»gain

«gain-like »dispersion

F. A. Hashmi and M. A. Bouchene, Phys. Rev. Letters 101, 21306 (2008)

Superluminal propagation

CAUSALITY AND RELATIVITY

(1)

( )x, t

(2)

( )x x, t t+ Δ + Δ

(R)

(R’) v

( ) ( )

2

22 2

vt x t uv xct ' 1 ; uc t1 v / c 1 v / c

Δ − Δ Δ Δ⎛ ⎞Δ = = − =⎜ ⎟ Δ⎝ ⎠− −

CAUSALITY t ' t 0Δ Δ ≥ u c≤v c<

u: signal velocity

RELATIVITY (implicit)

v c<

WHAT IS A SIGNAL?

WHAT IS A SIGNAL?

In Optics :Wave packet with sharp front (Sommerfeld 1910)

WHAT IS A SIGNAL?


z

WHAT IS A SIGNAL?


C exactly!

z

WHAT IS A SIGNAL?


C exactly!

z

- Envelope faster than c ! (Vg > c), Vg still meaningful-Pulse distortion occurs if the back part of the pulse catch up the sharp front

Extension: non analyticity (Chiao et al. 2000)

>c or <c

Five velocities for a light pulse:Phase velocity

v / ( )c n c or cϕ = > <

v / ( )g gc n c or c= > <

Five velocities for a light pulse:Group velocity

Meaningful in a linear dispersive medium with constant spectral gain

Five velocities for a light pulse:Signal velocity

v :s speed of non analytical point c=

vs

Five velocities for a light pulse:Signal velocity

v :s speed of non analytical point c=

vs

« detectable Information »

3 2 20

30

( ),2 2

FG F

F

r u d r E Br c or c uu d r

εμ

= > < = +∫∫

Five velocities for a light pulse:centroid of field energy

G


G

2 20

.0

v ( ),2 2Field En G FE Br c or c u

tε

μ∂

= > < = +∂


2 20

.0


tε

μ∂

= > < = +∂

G

Field En.v :" "watched velocity in pulse propagation experiment

Field En g. gv v vv( / ) if dispersivemedium with cte gainα ω= + ∂ ∂ =


G

2 20

.0


tε

μ∂

= > < = +∂




3

3≠ ∫

∫Field. En.v

F

S d r

u d rG

2 20

.0


tε

μ∂

= > < = +∂



• Can the energy go faster than c ???

• relativity: energy matter and v < c!!

How to solve the paradox ??

≡

Energy has to be defined as the total energy

3 2 20

. 30

v ( ),2 2

FField En F

F

r u d r E Bc or c ut u d r

εμ

∂= > < = +

∂∫∫


G

3

3≠ ∫

∫Field. En.v

F

S d r

u d r

« Open » energy

Field En.v :" "watched velocity in pulse propagation experimentField En g. gv v vv( / ) if dispersivemedium with cte gainα ω= + ∂ ∂ =

3 2 20

. . 30

v , ' ( )2 2

t

Tot En

r ud r E B Pu E dt u tt tud r

εμ −∞

∂ ∂= = + + + → −∞

∂ ∂∫ ∫∫

Five velocities for a light pulse:centroid of total energy

G« Closed » energy

3 2 20

. . 30

v , ' ( )2 2

t

Tot En


εμ −∞

∂ ∂= = + + + → −∞

∂ ∂∫ ∫∫


G

. . :<Tot En Subluminal transport of the totalv c energy

3 2 20

. . 30

v , ' ( )2 2

t

Tot En


εμ −∞

∂ ∂= = + + + → −∞

∂ ∂∫ ∫∫


3

. . 3v = ∫

∫Tot En

S d r

ud rG

. . :<Tot En Subluminal transport of the totalv c energy

J. Peatross et al.PRL 84, 2370 (2000)

S. Glasgow et al.PRE 64, 046610 (2001)

NATURE, 406 (2000)

Supraluminal propagation: Backward propagation.PROPAGATION THROUGH NEGATIVE MEDIUM

(ng=-2)

SCIENCE, 312 (2006)

Energy flow in the forward direction

SCIENCE, 312 (2006)

S k∝E

Bg Tot. En.v and v not only differ by their magnitude but also by their

P

aradox:sign!!

Pulse envelope displacement: spatial pulse shaping

Alternative description

amplified absorbed

Any saturable absorber may re-shape (distort) the pulseHere (constant spectral gain), the re-shaping is symmetric!, Vg meaningful

Alternative description

amplified absorbed

Any saturable absorber may re-shape (distort) the pulseHere (constant spectral gain), the re-shaping is symmetric!, Vg meaningful

Construc. interf Destruc. interf

Conclusion• Fast light: better understanding of relativity implications and

light pulse velocities.• Relativity implications on signal and information propagation:

*Signal: non analyticity *transmission of (detectable) Information faster than c possible!

• Relativity implications on energy propagation:*Group velocity characterizes the displacement of the envelope but is not representative of the (total) energy flux!*Group velocity is significative in a constant spectral gain medium: displacement of the EM energy, watched velocity.*Superluminal pulse envelope propagation is the result of both temporal and spatial amplitude pulse shaping by the medium.*Superluminal propagation appears when truncated balance of energy

• Challenge in slow light: extension to short pulses (higher bandwidth): SRS, SBS techniques.

• Development of quantum optical memories

slow light and superluminal propagation · 2009. 1. 10. · outlook • group velocity, case of a...

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