Quantum Memory for Light

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Quantum Memory for Light

Jean-Louis Le Gouët

To cite this version:

Jean-Louis Le Gouët. Quantum Memory for Light. 3rd cycle. Ecole prédoctorale des Houches, 10-21

septembre 2007, 2007, pp.58. �cel-00258259�

cel-00258259, version 1 - 21 Feb 2008

Quantum Memory for Light

∗

Jean-Louis Le Gouët

Laboratoire Aimé Cotton, CNRS UPR3321, Univ Paris Sud

bâtiment 505, ampus universitaire, 91405 Orsay edex

jean-louis.legouetla .u-psud.fr

Abstra t

Weouline twostrategiesforstorageandre overy ofquantumlight

inanensembleofatoms. Thisseriesofle tureshasbeendevisedasan

elementary introdu tion. Hen e dis ussionis essentially onnedto a

semi- lassi al pi ture. We rst onsider ele tromagneti ally indu ed

transparen y(EIT) andstoppedlight. Therolesof homogeneousand

inhomogeneous broadeningareexamined. Weproposebothtime-and

frequen y-domain des riptions. Then we dis uss the total re all of a

signal after apture by an absorbing material. Rephasing pro esses

are briey reviewed. We refer to various re ent experimental works,

espe iallythose ondu tedinsolidstatemedia. The ourseisintended

to be self ontained andin ludes reminders onsome quantumphysi s

elements su hasthedensity operator and theBlo h ve tor.

Contents

1 Introdu tion 3

2 Two ways of re overing light 5

2.1 Ele tromagneti ally indu ed transparen y

and stopped light . . . 5

2.2 Re overy from anabsorbing medium . . . 7

∗

Thisseriesofle tureswasdeliveredatE olePrédo toraledesHou hes,sessionXXIV,

QuantumOpti s,September10-21,2007. Thesessionwasdire ted byNi olasTrepsand

3.1 Atom ex itationby light . . . 10

3.2 Radiativeresponse . . . 12

4 Three-level

Λ

-system, EIT 15 4.1 Opti alex itation of the

Λ

-system . . . 15

4.2 Solving the Blo h equationswith EIT onditions . . . 17

4.3 EIT wave equation . . . 18

4.4 Storageand retrieval,stopped light . . . 20

4.5 Limitsof the semi- lassi al pi ture . . . 21

4.6 Single photonstorage and retrieval: experiment . . . 22

5 EIT in a solid: inhomogeneous broadening 24 5.1 Line broadening and relaxation . . . 24

5.2 Polarizationand sus eptibility . . . 25

5.3 Wave equation inthe spe tral domain. . . 28

5.4 Memory bandwidth . . . 30

5.5 EIT demonstrationinsolids . . . 31

6 Re overy from an absorbing medium 33 6.1 Polarization ollapse, oheren esurvival . . . 33

6.2 Information re overy by phase reversal . . . 35

7 Pra ti al implementation of phase reversal 39 7.1 Two-pulse photone ho . . . 40

7.2 Tri-level e ho . . . 43

7.3 Controlledreversibleinhomogeneous broadening . . . 46

8 Con lusion 48 A Density operator 48 A.1 statisti almixingand quantum oheren e. . . 48

A.2 Environment and relaxation . . . 49

B The Blo h ve tor 52 B.1 Conne tion with NMR . . . 52

Transport of quantum information is ideally a omplished by light but, at

somestage,amaterialsystemisneeded forpro essingand/orstorage. Many

groupsaroundtheworldstrivetobuildaquantummemorythatshouldstore

thenon- lassi alpropertiesofalightsignal,thentorestoretheoriginalsignal.

Iflong distan e quantum ryptography is ommonlyinvoked tojustify these

resear hes[1℄,aboveallthisisafas inatingquantumphysi sproblem,giving

a new insightin light-matterintera tion.

Quantum information is related to noise. When the u tuations of a

lassi al light sour e are redu ed to the quantum limit,noise is equally

dis-tributed over a pair of onjugated observables su h as photon number and

phase, Stokes ve tor omponents or eld quadratures. A light beam

ar-ries quantum information if the noise ae ting one observable is squeezed

under the standard quantum limit orresponding to equipartition of noise.

Of ourse noise redu tion on one observable entails in reased noise on the

onjugate quantity. Naivelyspeaking, a quantum memoryshould be able to

restore a signalin the tiniest details,beyond the quantum limit.

Resonant ex itation of anatomi transitionprovides appropriate strong

oupling between light and matter. However, intera tion with asingle atom

is not enough to trap the in ident photon with absolute ertainty. One an

in reasethe ouplingbypla ingtheatominsideahighnesse avity. Instead,

in the present ourse, we only onsider trapping of light by a ma ro opi

ensembleof atoms.

We also need interrogate the memory at will, ontrolling the moment

when the signal is restored. This an be a hieved through an auxilliary

opti al transition, oupled to the quantum eld apture transition. Several

proto ols rely on the Lambda three-level system. A ommon upper level

linksthetwotransitionsthatare onne tedtotwosub-levelsoftheele troni

ground state.

The quantum-properties preserving storage of one photon is an unitary

pro ess. Initially, the single ex itationlightstate is ombined with the

ma-terial medium ground state. The ompound system undergoes an unitary

transformtowardsastatewheretheuniqueex itationhasbeentransposedto

matter. Thestored informationis retrieved withthe help of the reverse

uni-tary transform. What makes the pro ess so di ult is pre isely the unitary

transformthat involvesa ma ros opi ensembleofatoms. One an ertainly

a terized by a spatial mode and a spe tro-temporaldistribution. Generally

an in ident photon only transfers its energy tothe absorbing medium. The

photonwillbe reemittedeventually,aftermultiple reabsorptionand

s atter-ing, in a spatial and spe tro-temporal state devoid of any onne tion with

the initialstate.

The reason why energy alone is transferred tothe medium is not so

ob-vious. Whenexposed toopti alex itation,atwo-stateatom, initiallyinthe

groundlevel,ispromotedtoaquantumsuperpositionstate. Quantum

infor-mationthusows fromlighttothe atom. Providedthatatomsarenumerous

enough one thus expe ts that all the in ident light ould be onverted into

quantum atomi ex itation. However one is fa ed with several issues. First,

in general, the mediumdoes not returnto initialstate after readout, a

on-dition to be fullled for total re overy of the quantum state of light. The

re overed eld, propagating along the same waveve tor as the initialsignal,

growsfromzerointheinputside. Thereforetheatoms losetotheinputside

ofthe absorbingmediumarethe moststronglyex itedby thein ominglight

signal, but also undergo the smallest feedba k from the restored eld that

fails to take them ba k tothe groundstate. The atomi state and retrieved

eld mismat hresultsinpartialabsorption andin ompleteextra tionof the

storedinformation. Inadditiontothispropagationissue,oneshouldmention

random redistribution of light by spontaneous emission and quantum state

destru tion by oheren e relaxation. However, in many systems oheren e

lifetimeremains ompatible with the demonstration of quantum storage for

light.

In this series of le tures we shall essentially examine two ways of

e- iently restoring the signal eld, that is to say two ways of addressing the

propagation problem. One approa h is known as Ele tromagneti ally

In-du ed Transparen y. This is a radi al way to deal with absorption. The

storage medium is made transparent to the in oming signal, operating as a

trap that loses on e the quantum eld is inside. The other approa h takes

advantageof rephasingpro edurestooptimizethesignalre onstru tion. We

shallessentiallyrestri tthedis ussiontosemi lassi altheory,assumingthat,

withinthelimitsoflinear onditions,ane ientre overypro eduregenerally

tr

an

sm

is

si

o

n

0

1

in

d

ex

o

f

re

fr

ac

ti

o

n

n

(

ω

)

a

c

b

v

1

dn( )

c

_{n( )}

d

ω

ω ω

ω

=

+

Figure1: Prin ipleof EIT.Allatoms areinitiallyprepared instate

|ai

. The oupling eld, resonant with the

b → c

empty transition, opens a trans-paren y window on the

a → b

transition. The absorption prole distortion goes along with modied dispersion of the index of refra tion. This is

re-e ted ingroup velo ity redu tion within the transparen y window.

2 Two ways of re overing light

2.1 Ele tromagneti ally indu ed transparen y

and stopped light

As noti ed above, the re onstru ted signal tends to be reabsorbed during

propagation through the storage medium. This problem is addressed in a

radi al way by Ele tromagneti ally Indu ed Transparen y (EIT), sin e the

mediumis madetransparentatthe signal inputand output[2,3℄. Withthe

help of an external ontrol, the material opa ity is swit hed on and o at

will.

To ontrolthe opa ity one resorts toanauxilliaryopti altransitionthat

sharesanatomi levelwiththestoragetransition. Hen e,insteadoftwo-level

all the atoms are in

|ai

, whi h makes the medium absorbing on the

a → b

transition. Let us remind that absorption results from the oupling of the

in ident eld with the rea tion of the medium, represented by the

ma ro-s opi polarizationdensity. EIT pre iselypro eeds through the annihilation

ofthepolarizationonthe

a → b

transition. Thisisa omplishedbya ontrol eld that resonantly ex ites the

b → c

auxilliarytransition. When swit hed on, the ontroleld onverts the

a → b

opti al polarizationinto the Raman oheren e of states

|ai

and

|ci

. The opti al polarization vanishing renders the mediumtransparent on

a → b

(see Fig. 1). Sin e

b → c

onne ts empty levels, the medium is transparent on

b → c

too, so that all the atoms ex-perien e the same ontrol eld strength, wherever they are lo ated in the

absorbing medium.

The ontrol eld does not just open a transparen y window. In

a or-dan e with Kramers Krönig relations,the distorsionof absorption prole is

asso iated witha disturban eof the index of refra tion,whi h results inthe

redu tion of the group velo ity

v

. In terms of dispersion of the refra tion index

n(ω)

, the group velo ity

v

an be expressed as:

v

c

=

1

n

(ω) + ω

dn

(ω)

d

ω

(1)

Theeldamplitudeis ontinuousattheva uum-mediuminterfa e.

How-everthe spatialextension of asignal pulseis ompressedalong the dire tion

of propagation be ause of the velo ity group redu tion. The eld envelope

undergoesa

v/c

shrinking. Theenergy arriedby thepulseisredu edby the same ratio, dropping lose to zero when

v << c

. A tually energy transfer from the signal pulse to the ontrol eld omes along with the opti al

po-larization onversion intoRaman oheren e. It is rather intriguingthat the

signalenergyistakenawaybythe ontroleld,whilethespatialand

spe tro-temporalsignal properties keep stored in the medium. Reverse

transforma-tion takes pla eatthe a tive mediumexit. The signaleld then re overs its

initialenergy together with its spatialand spe tro-temporalproperties.

The EIT pro ess has been demonstrated with lassi al light in various

materials ranging from gas to ondensed matter. Light speed redu tion to

17 metres per se ond was observed in an ultra old atomi gas [4℄. Then it

wasrealizedthat light ouldnot onlybesloweddown but even "stopped"in

a

Λ

-system. Indeed, if the ontroleld isswit hed o while the signal pulse is entirely ontained within the a tive medium, the remaining properties

saved in the Raman oheren e. If the ontrol eld is restored before the

Raman oheren e relaxes, the signal eld is rebuilt, resumes its progression

through the medium and nally exits, having preserved most of its initial

hara teristi s[5,6℄.

In the next se tions we analyti ally derive the various operating

ondi-tionsofthememory. Rightnowwe anlistmostofthem. Wealreadynoti ed

thatinformationtransfertothe Raman oheren eissubje ttothe ondition

v << c

. In orderto beentirely ontainedwithin the

L

-thi k materialat the momentof the ontroleld swit hing o,the signalpulsemust exhibit a

du-ration

T

smallerthan

L/v

. Besides the signalbandwidth

∆

must besmaller than the width of the transparen y window. Finally those onditions must

be onsistentwith the time and frequen y Fourier onjugation, a ordingto

whi h

∆T > 1

.

It should be stressed that the ontrol eld, intera ting with a transition

between empty levels, does not ex ite any atoms on its own. As a

onse-quen e this eld does not generate any noise. The signal eld alone onveys

ex itation tothe atomi ensemble.

Finallyitshouldbenoti edthatEIT ongurationimposesthattheweak

signal eld shouldbe isolatedfromthe intense ontroleld. This ould bea

majordrawba k.

2.2 Re overy from an absorbing medium

Instead ofresortingtotheradi alsolutionof indu ingtransparen y,one an

try to retrieve the signal despite of mediumabsorption. Wealready noti ed

that the re overed eld shall be weaker at the input side of the medium,

pre iselyintheregionwherethein omingeldisstronger. Asa onsequen e

the re overed eld is unable to turn the atoms ba k into their initialstate,

whi hhampers orre tinformationretrieval. Inorder toevade this obsta le,

one an tryto makethe restored eld topropagateinthe opposite dire tion

of the in omingsignaleld. This way, buildingup fromthe outputside, the

restored eld gains strength all along the storage medium and is expe ted

to rea h its maximum intensity at the input side and to be intense enough

there to turn the atoms ba k tothe ground state.

Ba k s attering of the signal eld reminds of phase onjugation in

non-linear opti s. Three beams may be appropriate to reverse the dire tion of

 
  
 
   





π

π

π

π

τ

τ

Figure 2: Signal re overy with reversed dire tion of propagation.

Coun-terpropagating

π

-pulses are used to onvert opti al ex itation into Raman oheren e, then ba k to opti al ex itation. Therefore the restored signal

signaltobestoredpropagatingalong

~k

1

ex itesthe

a → b

transition. Thena lightpulsepropagatingalong

~k

2

,resonantwiththe

b → c

transition, onverts the opti al ex itation of

a → b

into the Raman oheren e of states

|ai

and

|ci

. A

π

-pulse an e iently a hieve su h a onversion. The notion of pulse areawillbedenedlater. InformationisstoredintheRaman oheren euntil

another

π

-pulse, propagating along

~k

3

onverts ba k the Raman oheren e intoopti alex itation of

a → b

. In a ordan ewith general phasemat hing onditions, thesignal anbere onstru ted inthe dire tion

~k

3

+ ~k

2

−~k

1

,that is tosay indire tion

−~k

1

provided

~k

3

= −~k

2

(see Fig. 2).

Ifatoms are initiallyprepared instate

|ai

,the medium istransparentto the onversion pulses. In addition,thosepulsesdonot indu eany ex itation

noisesin ethein omingsignaleldalone an onveyex itationtotheatomi

ensemble.

Unfortunately it doesnot work so easily. The pro ess relies onthe time

separation of the dierent steps, namely the apture of the in omingsignal,

the onversion to Raman oheren e, the ba k onversion to opti al

ex ita-tion and the re overed signal emission. In order tobe stored,the data pulse

must be shorter than the

|ai

and

|bi

superposition state lifetime. Equiv-alently, the data pulse bandwidth must ex eed the homogeneous linewidth.

Yet,inanhomogeneouslybroadenedmedium,whereallatomshavethesame

transition frequen y, the storage bandwidth is pre isely limited by the

ho-mogeneous width, given by the inverse duration of the superposition state.

Therefore one is fa ed with ontradi tory onstraints, sin e the in oming

pulse must simultaneously be narrowerthan the absorption prole, inorder

to be aptured, and shorter than the superposition state lifetime, in order

to be stored. In an eort to over ome the ontradi tion, let us onsider an

inhomogeneously broadened medium, where atoms exhibit dierent

transi-tion frequen ies. Thememory bandwidth isnolonger limitedby the inverse

superposition state lifetime and mu h shorter signal pulses an be

onsid-ered. Then one meets another obsta le. The superposition states that are

built indierentatoms evolveatdierentrates, whi h entailsrelativephase

shift. The above des ribed pulse sequen e is unable to rephase the atoms,

a ne essary ondition for signal re overy. We shall see how to solve this

problem.

After the general presentation of the two memory ar hite tures to be

tera tion

3.1 Atom ex itation by light

The sample is illuminated by travelling plane waves. The ele tromagneti

eldisregardedasa lassi alquantity. The omplexamplitudeoftheele tri

eld is given by:

E(~r, t) =

1

2

(E(~r, t) + E

∗

_{(~r, t)) =}

1

2

(A(~r, t)e

iω

L

t−i~k.~

r

+ c.c.)

(2)

The main time and spa e variation is olle ted inthe phase fa tor

e

iω

L

t−i~k.~

r

that hara terizes a wave with entral frequen y

ω

L

, propagating along a wave ve tor

~k

. The envelope

A(~r, t)

little varies on the time and spa e s ales of opti al period and wavelength. The wave ve tor length is dened

as

ω

L

= kc

.

Theterms

E(~r, t)

and

E

∗

_{(~r, t)}

respe tivelystandforthepositiveand

nega-tivefrequen y omponentsoftheeld. Indeedthe time-to-frequen yFourier

transform of

E(~r, t)

,

E(~r, ω) = F[E(~r, t)]

, entered atopti alfrequen y

ω

L

,is lose to 0at

−ω

L

.

Intera tion to the atomi system is des ribed in ele tri dipole

approxi-mation by the hamiltonian:

H

I

= −q ~

R. ~

E

(3) where

q

is the (negative)ele tron harge. Thus

q = −e

, where

e

represents the elementary harge. The transitiondipolematrix elementbetween states

|ii

and

|ji

:

~µ

ij

= hi|e ~

R|ji

(4) is dened with appropriate phase hoi e so that this element isreal.

The atom density matrix equation reads as:











i~ ˙ρ = [H, ρ] +

dρ

dt









relaxation

H

= H

0

− q ~

R · ~

E = H

0

+ e ~

R · ~

E

(5)

This equation ombines theunitaryevolution,driven by theele tromagneti

eld, and the non-unitary evolution aused by oupling with environment.

level atom with the in oming eld. Expanding the density matrix equation

on the set of eigenstates

|ai, |bi

one obtains:







˙ρ

aa

= i(ρ

ab

− ρ

ba

)(Ωe

iω

L

t−i~k.~

r

+ c.c.) + γ

b

ρ

bb

˙ρ

bb

= − ˙ρ

aa

˙ρ

ab

= i(ρ

aa

− ρ

bb

)(Ωe

iω

L

t−i~k.~

r

+ c.c.) + (iω

ab

− γ

ab

)ρ

ab

(6)

where the Rabifrequen y isdened as:

Ω(~r, t) =

µ

ab

A(~r, t)

2~

(7)

If

A(~r, t)

is omplex,theRabifrequen yis omplextoo. Inordertoseparate the fastos illationat opti alfrequen y, one substitutes

ρ

ab

with:

ρ

ab

= ˜

ρ

ab

e

iω

L

t−i~k.~

r

(8)

This is not a swit h to intera tion pi ture. In intera tion representation

one denes the operator

ρ

I

= exp(−

i

~

H

0

t)ρ exp(

i

~

H

0

t)

that involvesa fa tor

exp(−iω

ab

t)

,spe i toea hfrequen y lass. Instead,swit hing tothe frame "rotating" atlaserfrequen y, oneappliesthe sametranform toallfrequen y

lasses. This dieren e willprove important in inhomogeneouslybroadened

media where atomsos illate atvariousfrequen ies.

Then,negle tingallthetermsos illatingatharmoni overtonesof

ω

L

,one obtains the Rotating Wave Approximation of the density matrix equation:







˙ρ

aa

= i(˜

ρ

ab

Ω

∗

− ˜

ρ

ba

Ω) + γ

b

ρ

bb

˙ρ

bb

= − ˙ρ

aa

˙˜ρ

ab

= i(ρ

aa

− ρ

bb

)Ω + (i∆ − γ

ab

)˜

ρ

ab

(9)

where

∆ = ω

ab

− ω

L

. One may formallyintegrate theseequations. One rst integratesthehomogeneousequations. Thenonetakesthenon-homogeneous

termintoa ountbythemethodofvariationoftheparameters. Oneobtains:















n

ab

(t) = 1 + (n

ab

(t

0

) − 1)e

−γ

b

(t−t

0

)

+ 2i

Z

t

t

0

dt

′

_(˜

_ρ

ab

Ω

∗

− ˜

ρ

ba

Ω)e

−γ

b

(t−t

′

₎

˜

ρ

ab

(t) = ˜

ρ

ab

(t

0

)e

(i∆−γ

ab

)(t−t

0

)

+ i

Z

t

t

0

dt

′

Ωn

ab

e

(i∆−γ

ab

)(t−t

′

₎

(10)

Whetherindierentialorintegralforms,theseequationsareknownasopti al

•

intera tionwiththe lassi aleldisdes ribedinele tri dipole approx-imation

•

transitionfrequen ies are onstant parameters

•

relaxationpro esses are des ribed by phenomenologi alde ay rates The density matrix of a two-level atom is omprised of 4 omponents, 2 of

whi hare omplex. Thetra e onservationandthesymmetryproperty

ρ

ab

=

ρ

∗

ba

redu ethenumberofindependentparametersto3,namelythepopulation dieren e and the real and imaginary omponents of the oheren e. Blo h

equations are nothingbut the three linear dierentialequations that ouple

these three quantities.

3.2 Radiative response

When prepared in a superposition of two states linked by an opti al

tran-sition, the atoms behave as os illating dipoles, i.e. as radiatingmi ros opi

antennas. They behave as real sour es of Huyghens wavelets (see Fig. 3).

In the same way as the virtual sour es of Huyghens wavelets, the atoms

a quire the spa e and time phase of the in oming eld. As long as phase

properties are preserved, that is to say as long as the atomi oheren e has

not beenerasedbyhomogeneousrelaxationorphase-shiftbyinhomogeneous

detuning, the atoms radiate as the virtual sour es of Huyghens dira tion

theory. Spe i ally, the spatial oheren e of the sour es makes the wavelets

onstru tively interfere in the dire tion of the in oming wave. Elaborating

theanalysisalittlefurther,one andeterminethedira tionlimitedangular

aperture of the emitted signal.

With this pi ture in mind, let us pro eed to the lo al des ription of the

atomi response,asderivedfromMaxwellequations. Inadiele tri medium,

in the absen e of ele tri harges those equationsread as:

rot( ~

_{E) = −∂}

t

B

~

Faraday law

rot( ~

B) = ∂

t

D

~

Ampère theorem

div( ~

D) = 0

Gauss theorem

(11)

where

D

~

an be expressed in terms of the ma ros opi polarizationdensity

~

P

as:

~

incoming

field

dipole

radiation

absorbing

medium

Figure 3: The oherent atomi response to opti al ex itation an be

under-stood within the frame of Huyghens dira tion theory. The atomi dipoles

behave asreal sour es of Huyghens wavelets.

These equations ombineintothe wave equation with sour es:

∆ ~

_{E − µ}

0

ǫ

0

∂

2

_E

_~

∂t

2

= µ

0

∂

2

_P

_~

∂t

2

−

1

ǫ

0

grad[div( ~

P )]

(13)

The atomi responseis ontained inthe ma ros opi polarizationdensity

P

~

. We assume that the transverse variation of

P

~

is very small on the s ale of the atomi wavelength. Thisenablesustodrop these ondtermontheright

hand side of Eq.13.

Wehave nowtoexpress the ma ros opi polarizationdensity interms of

the opti al Blo h equation solutions. Let us onsider the

N

atoms sitting within an elementary volume

V

. The size of this volume is small enough with respe t to the opti al wave length so that all the atoms intera t with

the same eld. The total dipole moment is expressed as the sum of the

N

individualdipoles. The expe tation value of the orresponding quantum observable reads as:

*

_N

X

i=1

µ

i

+

= Tr

"

_N

X

i=1

µ

i

)

!

ρ

#

(14)

where

ρ

represents

N

-atom density operator. The

N

-atom state is initially fa torizable and is assumed to remain so under semi- lassi al ex itation.

In other words, semi- lassi al ex itation is expe ted not to entangle the

N

atoms. The density operatorthen reads as:

ρ

= ρ

1

⊗ . . . ⊗ ρ

i

⊗ . . . ⊗ ρ

N

(15) Inordertoexpressthetotaldipoleintermsoftheindividualdensitymatri es,

one uses the relation:

Tr

1···N 6=i

(ρ

1

⊗ . . . ⊗ ρ

i

⊗ . . . ⊗ ρ

N

) = ρ

i

(16)

Then the total dipoleexpe tationvalue redu es to:

*

_N

X

i=1

µ

i

+

=

N

X

i=1

Tr

i



Tr

1···N 6=i

(µ

i

ρ)



=

N

X

i=1

Tr

i



µ

i

Tr

1···N 6=i

ρ



=

N

X

i=1

Tr(µ

i

ρ

i

)

(17)

Forthe time being we ignore inhomogeneous broadening. All the atoms

have the same transition frequen y. Then the elementary volume dipole

momentreads as:

N

X

i=1

Tr(µ

i

ρ

i

) = −Nµ

ab

[ρ

ab

(~r, t) + ρ

ba

(~r, t)]

(18)

where the sum runs over all the atoms within the elementary volume, with

ha|µ|ai = hb|µ|bi = 0

. A minus sign appears be ause

µ

ab

has been dened from the elementary harge

e

and not from the ele tron harge

q = −e

. Dividing by the volume

V

, one nally gets the ma ros opi polarization density:

P (~r, t) = −nµ

ab

[ρ

ab

(~r, t) + ρ

ba

(~r, t)]

(19) where

n

denotes the density of a tiveatoms per unit volume.

Inthesamewayastheele tri eld,thepolarizationdensityappearstobe

omprisedofpositiveandnegativefrequen y omponents. Those omponents

do not overlap spe trally, being distant by hundreds of THz, sothey satisfy

un oupledwaveequations. Thepositivefrequen y omponentwaveequation

reads as:

1

2



∆ −

_c

1

₂

∂

2

∂t

2





A(~r, t)e

iω

L

t−i~k.~

r



= −n

_c

µ

₂

ab

_ǫ

0

∂

2

∂t

2



˜

ρ

ab

(~r, t)e

iω

L

t−i~k.~

r



gle tthe ontributions oforder

∂

t

A(~r, t)/[ω

L

A(~r, t)]

and

∇A(~r, t)/[kA(~r, t)]

. The wave equation then redu es to:

 ∂

∂z

+

1

c

∂

∂t



A(~r, t) = ink

µ

_ǫ

ab

0

˜

ρ

ab

(~r, t)

(21)

Substituting

A(~r, t)

with Eq.7 one obtains:

 ∂

∂z

+

1

c

∂

∂t



Ω(~r, t) = ink

µ

2

ab

2~ǫ

0

˜

ρ

ab

(~r, t)

(22)

It is worth expressing this equation of propagationin terms of the resonant

absorption oe ient

α

0

. To rst order in

Ω(~r, t)

the Blo h equation for

˜

ρ

ab

(~r, t)

reads as:

˜

ρ

ab

(~r, t) = i

Z

t

−∞

Ω(~r, t

′

)e

−γ

ab

(t−t

′

)

dt

′

(23)

whi h redu es to

ρ

˜

ab

(~r, t) = iΩ(~r, t

′

_)/γ

ab

if

Ω(~r, t)

little varies on

γ

−1

ab

time

s ale. This onditionsimplymeansthattheeldbandwidthismu hnarrower

than the absorption line, so that the polarization density instantaneously

adjusts to the eld variations. Substituting the expression of

ρ

˜

ab

(~r, t)

in Eq.22 one obtains:

 ∂

∂z

+

1

c

∂

∂t



Ω(~r, t) = −nk

µ

2

ab

2~ǫ

0

γ

ab

Ω(~r, t) = −

α

0

2

Ω(~r, t)

(24)

Finally the wave equation reads as:

 ∂

∂z

+

1

c

∂

∂t



Ω(~r, t) = i

α

0

γ

ab

2

ρ

˜

ab

(~r, t)

(25)

4 Three-level

Λ

-system, EIT

4.1 Opti al ex itation of the

Λ

-system

In a

Λ

-system an upperstate

|bi

is onne ted through opti altransitions to twolowerstates

|ai

and

|ci

. The system isilluminatedby two drivingelds. The

a → b

and

b → c

transitions are respe tively driven at frequen ies

ω

1

and

ω

2

with Rabi frequen ies

Ω

1

and

Ω

2

. Ea h driving eld is assumed to ex ite a single transition. Angular sele tion rules may help to dis riminate

the transitions. Indeed ross-polarizing the light beams may be enough to

separately drive the two transitions when su h sele tionrules apply.

Other-wise, the splitting

ω

ac

must be mu h larger than the homogeneous widths, the Rabifrequen iesandthedetunings

|ω

ab

− ω

1

|

and

|ω

bc

− ω

2

|

. The adjun -tion of a third state signi antly ompli ates the density matrix formalism.

Instead of 3 real independent parameters in a two-level system, one is left

with 8 real parameters in a three-level atom. Those quantities are oupled

by the following dierentiallinear equations:































˙ρ

aa

= i(˜

ρ

ab

Ω

∗

1

− ˜

ρ

ba

Ω

1

) + r

a

γ

b

ρ

bb

˙ρ

cc

= i(˜

ρ

cb

Ω

∗

2

− ˜ρ

bc

Ω

2

) + r

c

γ

b

ρ

bb

˙ρ

bb

= − ˙ρ

aa

− ˙ρ

cc

˙˜ρ

ab

= [i(ω

ab

− ω

1

) − γ

ab

]˜

ρ

ab

+ i(ρ

aa

− ρ

bb

)Ω

1

+ i˜

ρ

ac

Ω

2

˙˜ρ

cb

= [i(ω

bc

− ω

2

) − γ

bc

]˜

ρ

cb

+ i(ρ

cc

− ρ

bb

)Ω

2

+ i˜

ρ

ca

Ω

1

˙˜ρ

ac

= [i(ω

ac

− ω

1

+ ω

2

) − γ

ac

]˜

ρ

ac

+ i(˜

ρ

ab

Ω

∗

2

− ˜

ρ

bc

Ω

1

)

(26)

The system is assumed to be losed. The oe ients

r

a

and

r

c

= 1 − r

a

a ount for the upper level relaxation distribution between the two ground

sublevels. As usual in the rotating wave pi ture, the o-diagonal matrix

elementshavebeen substituted with:

ρ

ab

= ˜

ρ

ab

e

iω

1

t−i ~

k

1

.~

r

ρ

cb

= ˜

ρ

cb

e

iω

2

t−i ~

k

2

.~

r

ρ

ac

= ˜

ρ

ac

e

i(ω

1

−ω

2

)t−i( ~

k

1

− ~

k

2

).~

r

(27)

The rst three lines of Eq.26 express the population evolution. This does

not dierfromthe orresponding two-levelsystem equations. Thelastthree

lines of Eq.26, a ounting for oheren e evolution, are more spe i . First

one observes that oheren e

ρ

ac

is ex ited by the light elds, although no dire t transition onne ts states

|ai

and

|ci

. Besides, oheren es

ρ

ab

and

ρ

bc

are oupled not only to level populations, but also to

ρ

ac

. For instan e, oheren e

ρ

ab

is built not only fromdire t ex itation of state

|ai

population by eld

Ω

1

, but also fromthe ex itation of oheren e

ρ

ac

by eld

Ω

2

.

The system evolution is generally omplex when both elds are applied

simultaneously. One observes phenomena su h as stimulated Raman

adia-bati passage (STIRAP) [7℄, dark resonan e [8℄, or the EIT pro ess we are

However, the ex itationof

ρ

ac

,alsoknown astheRaman oheren e,gives rise to attra tive features even when the elds

Ω

1

and

Ω

2

do not intera t simultaneously with the system. We shall meet su h features within the

frameof signal re onstru tion in anabsorbing medium.

4.2 Solving the Blo h equations with EIT onditions

Inthisse tionwefollowthelinesofRef. [9℄. Withthefollowingassumptions:

•

all the atoms are initiallyprepared instate

|ai

• Ω

2

, known as the " oupling" or" ontrol" eld, isa onstant.

• Ω

1

, arryingthe informationtobestored, has a pulse area

<< 1

the densitymatrixequationsgetmu hsimpler. Torstorderin

Ω

1

,thelevel population does not vary and the term

ρ

˜

bc

Ω

1

an be negle ted. Therefore the equationsof

ρ

ab

and

ρ

ac

turn into:

˙˜ρ

ab

= [i(ω

ab

− ω

1

) − γ

ab

]˜

ρ

ab

+ iΩ

1

+ i˜

ρ

ac

Ω

2

˙˜ρ

ac

= [i(ω

ac

− ω

1

+ ω

2

) − γ

ac

]˜

ρ

ac

+ i˜

ρ

ab

Ω

∗

2

(28)

In additionwe assume the ouplingeld resonantly ex ites the

b → c

tran-sition, and the signal pulse entral frequen y

ω

1

oin ides with

ω

ab

. The equations redu eto:

˙˜ρ

ab

= −γ

ab

ρ

˜

ab

+ i(Ω

1

+ ˜

ρ

ac

Ω

2

)

(29)

˙˜ρ

ac

= −γ

ac

ρ

˜

ac

+ i˜

ρ

ab

Ω

∗

2

(30) Substituting Eq. 30into Eq. 29,one obtains:

˜

ρ

ac

= −

Ω

1

Ω

2

−

i

Ω

2

(∂

t

+ γ

ab

)˜

ρ

ab

= −

Ω

1

Ω

2

−

1

|Ω

2

|

2

(∂

t

+ γ

ab

)(∂

t

+ γ

ac

)˜

ρ

ac

(31)

If

ρ

˜

ac

redu es to the rst term on the right hand side of Eq. 31, then the driving term

Ω

1

+ ˜

ρ

ac

Ω

2

vanishes in Eq. 29. In other words, the Raman oheren e ontribution interferes with single-photon ex itation to prevent

the buildup of

ρ

ab

. The absen e of atomi response to

Ω

1

on the

a → b

transitionis ree ted by the absen e of

Ω

1

absorption.

negle ted, i.e. if:

(∂

t

+ γ

ab

)(∂

t

+ γ

ac

)Ω

1

<< Ω

1

/|Ω

2

|

2

(32) Then

ρ

˜

ac

adiabati ally follows the variations of

Ω

1

. Given that

ρ

aa

∼

= 1

, the solution

ρ

˜

ac

= −Ω

1

/Ω

2

a tually orresponds tothe dark state:

|Di =

Ω

2

pΩ

2

1

+ Ω

2

2

|ai −

Ω

1

pΩ

2

1

+ Ω

2

2

|ci

(33)

This is an important feature of EIT: intera tion with the signal eld

Ω

1

immediatelystarts in the darkstate, unlike what o urs inother three-level

pro esses su h asCoherent PopulationTrapping(CPT)[8 ℄.

Substituting

ρ

˜

ac

intoEq. 30,onenallyobtains theexpression ofopti al oheren e:

˜

ρ

ab

=

i

|Ω

2

|

2

(∂

t

+ γ

ac

)Ω

1

,

(34)

fromwhi hwe an al ulatethe atomi feedba konthe in omingsignaleld

Ω

1

.

4.3 EIT wave equation

Substituting Eq. 34into Eq. 25one obtains:

 ∂

∂z

+

 1

c

+

α

0

γ

ab

2|Ω

2

|

2

 ∂

∂t



Ω

1

(~r, t) = −

α

0

2

γ

ab

γ

ac

|Ω

2

|

2

Ω

1

(~r, t)

(35)

This equation takesthe usual formdes ribing resonant planewave

propaga-tion through an ensemble of two-level atoms inthe linear regime. However,

the propagationparameters are deeply altered:

•

the absorption oe ient is redu ed from

α

0

to:

α

Ω

= α

0

γ

ab

γ

ac

|Ω

2

|

2

(36)

Withtypi al

γ

ab

and

γ

ac

valuesofabout

10

6

_s

−1

and

10

3

_s

−1

respe tively,

an

Ω

2

ontroleldRabifrequen y oforder

3 10

5

_s

−1

isenoughtoredu e

•

the group velo ity is redu ed from to:

v =

 1

c

+

α

0

γ

ab

2|Ω

2

|

2



−1

(37)

Withthesamenumeri alparameters,andwith

α

0

= 10

3

_m

−1

,thegroup

velo ity amounts tono more than

200m/s

!

The wave equation also tells us that, within the transparen y window, an

in omingtravellingwave of the form

Ω

1

(t − z/c)

in freespa e turnsintothe form

Ω

1

(t − z/v)

as it propagates through the a tive medium. The wave preserves its temporal prole, just undergoing spatial ompression by the

fa tor

v/c

. The eld amplitude is also preserved due to ontinuity at the interfa e of free spa e and a tive medium. Therefore neither the in oming

signal durationnor its spe tral width is ae ted by slowing down, provided

that the signal is ontained within the transparen y window. Now we need

larify the notion of transparen ywindow.

The EIT wave equation has been derived within the adiabati ondition

limits. The in oming eld variations have been assumed to be slowenough

so that the Raman oheren e an instantaneously adjust to them. One

ex-pe ts the adiabati ondition to fail if the in oming eld varies too rapidly,

i.e. if itsspe tral widthex eeds some limitingvalue. Letus hara terize the

signal spe tra width by the quantity

Ω

−1

1

∂

t

Ω

1

. Let the signal be narrower than the absorption linewidth

γ

ab

, whi h leads to:

(∂

t

+ γ

ab

)Ω

1

∼

= γ

ab

Ω

1

. Then the adiabati ondition reads as

(∂

t

Ω

1

)/Ω

1

<< |Ω

2

|

2

_/γ

ab

. The trans-paren y width would thus be given by

δ

T

= |Ω

2

|

2

_/γ

ab

. This result need be examined more arefully. The dierential equations we rely on

−

Blo h equation and wave equation

−

only onvey lo al des ription, as illustrated by the linear absorption oe ient. However, we need the overall

transmis-sion through the entire atom ensemble to dene the transparen y window.

Let the absorption oe ient at

∆

fromresonan e be approximated by the fun tion:

α(∆) = α

0

[1 − e

−(∆/δ

T

)

2

]

. Then the transmission fa tor reads as

e

−α(∆)L

∼

_{= e}

−α

0

L(∆/δ

T

)

2

,whi h nallyleads tothe transparen y width:

∆

T

= δ

T

/pα

0

L =

|Ω

2

|

2

γ

ab

√

α

0

L

The energy arried by the in omingsignal an be expressed as:

Z

|Ω

1

(t − z/c)|

2

dz = c

Z

|Ω

1

(t − x)|

2

dx

(39) Ifoneisabletohavetheentire pulsestandingwithinthea tivemedium,the

arried energy be omes,inside the material:

Z

|Ω

1

(t − z/v)|

2

dz =

v

c

Z

|Ω

1

(t − z/c)|

2

dz

(40) whi hrepresents a

v/c

redu tionwithrespe t tothe freespa e value. There-fore most of the energy has been extra ted from the eld if

v << c

. It an be shown that energy has been transferred to the ontrol eld, as soon as

the signal eld rosses the free spa e to materialinterfa e. Nonetheless, the

Raman oheren e is expressed as

Ω

1

/Ω

2

, being proportional to the instan-taneous signal eld. Therefore, a spin wave propagates within the material

along with the signal eld, althoughthe latterdoes not arry any energy.

If one abruptly swit hes o the ontrol eld, the residual signal eld

disappears, being absorbed by the material,whilethe spin wave stops

prop-agating, but survives as long as permitted by de oheren e pro esses. One

improperly says that light is "stopped". A tually one should say that the

signal eld has been split into two parts. On the one hand, its energy has

beenremovedby the ontroleld. Onthe otherhanditsinformation ontent

has been stored inthe Raman oheren e [10℄.

When the ontroleld is turned ba k on, the signal eld is rebuilt from

the Raman oheren e. The restored eld resumes itsprogression, pullingits

ompanion spin wave. Energy is fed ba k to the eld at the output of the

a tive medium.

To "stop" light without losing information, one has to make the entire

signal pulse to stand withinthe boundaries of the a tivemedium. The part

of thesignal enteringthe storagemediumafter ontroleld shutdown islost

by absorption. The spatial extension of a pulse with duration

τ

is

vτ

. This has tobe smallerthan the materialthi kness

L

. Besides the signal spe tral width

∆

must besmallerthanthetransparen y width

∆

T

. Combiningthose two onditions leads to:

With the additional ondition

∆ τ > 1

, be ause of time-frequen y Fourier onjugation, the "stopped" lightstorage requirement readsas:

pα

0

L >> 1

(42) 4.5 Limits of the semi- lassi al pi ture

In a "stopped" light pro ess, a single photon trapping is expe ted to leave

the atomensemblein the following superpositionstate:

|Ψ

1

i =

1

√

N

e

iφ(~

r

1

)

|ca · · · ai + e

iφ(~

r

2

)

|ac · · · ai + · · · + e

iφ(~

r

N

)

|aa · · · ci

(43)

This is a olle tive single ex itation state where the sum runs over all the

atoms intera ting with the eld. All the atoms are onsidered on an equal

footing,whi hdoesnotperfe tlya ountforthenitespatialextensionofthe

stored lightpulse. Howeverthis does not interfere with thegeneral meaning

of the present dis ussion.

The olle tive state appears to be entangled. It annot be fa torized as

a produ t of individual atom states. This is pre isely the type of state that

annot be produ ed in the frameof a semi lassi al pi ture analysis. In the

semi lassi al approa h the atoms ommuni ate with outside world through

a lassi aleld that doesnot onvey any quantum information. As aresult,

olle tiveex itation,withallatoms onsideredonanequalfooting, anonly

build ensemble produ t states su h as the following:

(1 + ǫ

2

)

−N/2

_{|ai + ǫe}

iφ(~

r

1

)

|ci

|ai + ǫe

iφ(~

r

2

)

|ci
· · · |ai + ǫe

iφ(~

r

N

)

|ci

(44)

This state an beexpanded as asum of

n

-ex itationstates:

(1 + ǫ

2

)

−N/2

(

|Ψ

0

i + ǫ

√

N |Ψ

1

i + ǫ

2

r

N(N − 1)

2!

|Ψ

2

i + · · · + ǫ

N

|Ψ

N

i

)

(45)

where

|Ψ

1

i

is dened aboveand where:

|Ψ

0

i

= |aa · · · ai

|Ψ

2

i

=

q

2!

N (N −1)

e

i(φ(~

r

1

)+φ(~

r

2

))

|cca · · · ai + e

i(φ(~

r

1

)+φ(~

r

3

))

|cac · · · ai + · · ·

· · ·

· · · ·

|Ψ

N

i = e

i(φ(~

r

1

)+···+φ(~

r

N

))

|cc · · · ci

(46)

The

1

-ex itation omponent oin ideswiththepreviouslydenedsingle ex i-tationentangledstate

|Ψ

1

i

. Inthe

n

-ex itationstates expansion, theweight of

|Ψ

1

i

, asgiven by

ǫ

2

_{N/(1 + ǫ}

2

₎

−N

∼

_{= ǫ}

2

_Ne

−N ǫ

2

, never ex eeds

1/e

, a value that is rea hed at

ǫ

2

_{N = 1}

and equals the weight of the 0-ex itation state

|Ψ

0

i

. Sin e

ǫ

2

represents state

|ci

population in an individual atom,

ǫ

2

_N

orresponds to the average numberof atoms in

|ci

. Therefore the weight of

|Ψ

1

i

is maximum when the average number of atoms in

|ci

is unity. More generally, one easily he ks that the

n

-ex itation state distribution obeys Poissonstatisti sand is onsistentwith ex itationbya oherent stateof the

eld but isnever onsistent withex itationby aFo k stateof theeld, with

a xed numberof photons.

4.6 Single photon storage and retrieval: experiment

The rst observation of single photon storage and retrieval is published in

De ember 2005 [11℄. A laser- ooled atom loud is used as the storage

ma-terial. The loud ontains about

4 10

9 85

_Rb

atoms, ooled to

100µK

in a magneto-opti trap.

The quantum light signal has to be narrower than the Rubidium D1

line, a few

MHz

-wide. No parametri light sour e is able to generate su h mono hromati single photons. A spe i sour e has to be developed rst.

Another loud,identi altothememoryensemble, playsthisrole. Astrongly

attenuated lassi albeam,dire ted along

~k

1

, illuminatesthis loud(see Fig. 4. One waits for Raman s attering in dire tion

~k

2

. Dete tion of a Raman photoninthisdire tionproje tstheatom loudtothesingleex itationstate:

1

√

N



e

−i(~k

1

−~

k

2

).~

r

1

|ca · · · ai + e

−i(~k

1

−~

k

2

).~

r

2

|ac · · · ai + · · · + e

−i(~k

1

−~

k

2

).~

r

N

|aa · · · ci



(47)

where

a

and

c

refertothegroundsubstatesoftheatoms, onsideredas three-level

Λ

-systems. As soon as a photon is dete ted on PD1, a rather intense pulseisdire tedtothesour e loudalong

−~k

1

. Insyn hrony withthispulse, a single photon is emitted in dire tion

−~k

2

, with probability lose to unity. This emission orresponds tostimulatedRamans attering onthepreviously

prepared single-ex itationensemble superposition state. The radiatedsingle

photon is then dire ted through an opti al ber to the memory loud. The

arrivaltime in the memory isknown fromthe event dete tion onPD1. One


                                   

Figure 4: Singlephoton storageand retrieval[11℄. The single photonsour e

and the memoryare both louds of laser- ooled Rb atoms. PD1, 2, 3

he k the uni ity of the re overed photon, one performs an anti- orrelation

measurement on PD2 and PD3, following the Hanbury Brown and Twiss

pro edure. The memory lifetimeappears to be no more than 10

µs

. This is assigned tomagneti eld inhomogeneity.

5 EIT in a solid: inhomogeneous broadening

5.1 Line broadening and relaxation

The most riti al EIT parameter is the Raman oheren e lifetime, but this

does not restri t the hoi e of material to su h sophisti ated systems as

LCAC.Long oheren elifetime analsobefound insolidmaterialsatliquid

heliumtemperature. Insu hmaterialstheabsen eofmotionkeepsthea tive

enters frommigratingoutsidethe lightbeams,asinLCAC, buteven better

sin e motion is totally absent. One also avoids spatial dephasing that an

ae t superposition states and an be aused by diusion, even in LCAC.

Rare earth iondoped rystals have been onsidered as potentialsolid

mate-rial andidatesforquantummemoryappli ations. Oeringsimilar oheren e

lifetime properties as atomi samples, they dier from LCAC by the large

inhomogeneous broadening of their spe tral lines.

InLCACthe atomsmovesoslowlythatthe Dopplershiftdoesnot ae t

the absorption line prole. In solid materials the absen e of motion of the

absorbing enters ree ts the strength of their intera tion with the rystal.

Intera tionentailsenergylevelshiftand,be ausethe rystalisneverperfe t,

the shift varies from site to site. As a result the transition frequen y is

not unique for all the absorbing enters. Instead the transition frequen y

is distributed over a broad spe tral interval, whose width

W

ab

, named the inhomogeneous width, typi ally ranges from a few

GHz

to several tens of

GHz

.

Beforein orporatinginhomogeneousbroadeninginEITanalysis,weneed

larify dierent aspe ts of intera tion with environment. On the one hand,

the intera tion shifts the energy levels, whi h results in the inhomogeneous

broadening. This represents astati aspe t. Cooling down toa few Kelvins

does not signi antly hange the levelshift. On the other hand, intera tion

also exhibits a dynami al aspe t, orresponding to intera tion u tuations.

width

γ

ab

with respe t to half the population de ay rate

γ

b

/2

. When the sample is ooled down,

γ

ab

de reases and gets loser to

γ

b

/2

.

However homogeneous and inhomogeneous width are not dierent in

essen e. Thisisaquestionofobservationtimes ale. Su hee tthatappears

as a u tuation ata given time s ale, and thus ontributes to homogeneous

broadening, mayberegarded asastati feature onashorter times ale, and

then pertainto inhomogeneous broadening.

Intheabsen e ofinhomogeneousbroadeningwehaveperformedthe

anal-ysis in the vi inity of single-photon resonan e. This is not valid anymore

in ase of large inhomogeneous broadening. Spe tral distan e to

single-photon resonan evaries dramati allyamong the atoms. Instead of

perform-ingtheanalysisintimedomain,wenow onsideraspe traldomainapproa h,

through time-to-frequen y Fouriertransform.

5.2 Polarization and sus eptibility

To a ount for the distribution of transition frequen ies, we rewrite the

ma ro opi polarizationdensity in the form:

P (~r, t) = −µ

ab

Z

dω

ab

G(ω

ab

) [ρ

ab

(~r, t; ω

ab

) + ρ

ba

(~r, t; ω

ab

)]

(48)

where

G(ω

ab

)

standsforthespe tralandspatialdistributionlaw,normalized to the atom density per unit volume

n

as:

R dω

ab

G(ω

ab

) = n

. Time to frequen y Fourier transformleads to:

ˆ

P (~r, ω) = −µ

ab

Z

dω

ab

G(ω

ab

) [ˆ

ρ

ab

(~r, ω; ω

ab

) + ˆ

ρ

ba

(~r, ω; ω

ab

)]

(49)

In linear opti s onditions, whi h apply to our weak signal eld, the

polar-ization an be expressed as:

ˆ

P (~r, ω) = ǫ

0

χ(ω)E(~r, ω)

(50)

where

χ(ω)

denotes the ele tri sus eptibility. This formula, well known in ele trostati s, also applies to ele trodynami s, provided the relevant

quan-tities are expressed in the frequen y domain 1

. Splitting the sus eptibility

1

If

χ(ω)

variesslowlyovertheeldspe tralwidth, thefollowingapproximation:

obtains:

ˆ

P (~r, ω) = ǫ

0

χ

(+)

(ω) + χ

(−)

(ω)

 1

2

h ˆ

E(~r, ω) + ˆ

E

∗

(~r, −ω)

i

(51) The positive (resp. negative) frequen y omponent of the eld vanishes in

the

ω ≈ −ω

1

(resp.

ω ≈ ω

1

) region. Likewise the positive (resp. negative) frequen y omponent of the sus eptibility vanishes in the

ω ≈ −ω

ab

(resp.

ω ≈ ω

ab

)region. Therefore the ross-term

χ

(+)

_{(ω) ˆ}

_E

∗

_{(~r, −ω) + χ}

(−)

_{(ω) ˆ}

_{E(~r, ω)}

vanishesand the polarizationdensity nallyreads as:

ˆ

P (~r, ω) =

1

2

ǫ

0

h

χ

(+)

(ω) ˆ

_{E(~r, ω) + χ}

(−)

(ω) ˆ

_E

∗

_{(~r, −ω)}

i

(52) Inordertodeterminethesus eptibility,letus omeba ktothethree-level

system Blo hequation. The transitionfrequen y isnow distributedoverthe

inhomogeneous width of the absorption line. Westill assumethat:

•

all atoms,whatevertheir transitionfrequen y, initiallysit instate

|ai

•

the signal (resp. the ontrol) eld onlyex ites the

a → b

(resp.

b → c

) transition

As we already noti ed, ross-polarizing the light beams may be enough to

separatelydrivethetwotransitionswhenangularsele tionrulesapply.

How-ever,whenthetwotransitionsonlydierbytheirfrequen y,theyare oupled

toasingle spe i eld onlyifthe groundstatesplittingismu hlargerthan

the homogeneouswidths, the Rabifrequen iesand the transitiondetunings.

This requires that

W

ab

<< ω

ac

. We shall see how to ope pra ti ally with this ondition.

Sin e

Ω

2

is a onstant, the Blo h equations for

ρ

˜

ab

and

ρ

˜

ac

are linear expressions oftime dependentquantitiesand anbesolvedbyFourier

trans-formation. In terms of

ρ

ab

,

E(~r, t)

and the new variable

ζ = ˜

ρ

ac

e

i(ω

1

t−~k

1

.~

r)

,

makesthetimedependentpolarizationdensityproportionaltotheeld,asinthefrequen y

domain. Thisimpliesinstantaneousresponsetoopti alex itationandobs uresthe ausal

hara terofthematerialrea tion. Thegeneralexpression,fully a ountingfor ausality,

readsas:

P (~r, t) = F

h ˆ

P (~r, ω)

i

= ǫ

0

Z

dτ ˇ

χ(τ )E(~r, t − τ)

˙ρ

ab

= [iω

ab

− γ

ab

]ρ

ab

+ i

µ

ab

E(~r, t)

2~

+ iζΩ

2

˙ζ

= [i(ω

ac

+ ω

2

) − γ

ac

]ζ + iρ

ab

Ω

∗

2

(53)

Pro eeding toFourier transformationone gets:

[i(ω − ω

ab

) + γ

ab

]ˆ

ρ

ab

(ω)

= i

µ

ab

E(~r, ω)

ˆ

2~

+ iˆ

ζ(ω)Ω

2

[i(ω − ω

ac

− ω

2

) + γ

ac

]ˆ

ζ(ω) = iˆ

ρ

ab

(ω)Ω

∗

2

(54)

By eliminating

ζ(ω)

ˆ

one nallyobtainsthe opti al oheren e expression 2 :

ˆ

ρ

ab

(ω) = i

µ

ab

E(~r, ω)

ˆ

2~

i(ω − ω

ac

− ω

2

) + γ

ac

[i(ω − ω

ab

) + γ

ab

][i(ω − ω

ac

− ω

2

) + γ

ac

] + |Ω

2

|

2

(55)

This expression depends on both the

ω − ω

ab

detuning of the

a → b

single-photontransitiontothe

E(~r, ω)

ˆ

signaleld omponent,andthe

ω − ω

ac

− ω

2

detuning of the

a → c

two-photon transition to the ompound ex itation by

E(~r, ω)

ˆ

and the ontrol eld at

ω

2

. Let

ω

(0)

ab

represent the enter of the atom spe tral distribution

G(ω

ab

)

. For sake of simpli ity the splitting

ω

ac

is assumed to be the same in all the atoms. In other words, we suppose the

a → c

Ramantransitionisnotinhomogeneouslybroadened. Ingeneralthisis nottrueinasolid,buta ountingforRamanfrequen ydistributionpro eeds

alongthesamelinesasthepresent al ulationand anbeextrapolatedeasily.

2

The oheren e

ρ

ab

(t)

mustsatisfythe ausality ondition. Thus

ρ

ab

(t)

doesnotdepend on

E(~r, t

′

₎

, with

t

′

_{> t}

. This ondition an be translated to the frequen y domain. By

inverseFouriertransformation

ρ

ab

(t)

anbeexpressedas:

ρ

ab

(t) =

i

2π

µ

ab

2~

Z

dt

′

_{E(~r, t}

′

₎

Z

dωe

iω(t−t

′

)

i(ω − ω

ac

− ω

2

) + γ

ac

[i(ω − ω

ab

) + γ

ab

][i(ω − ω

ac

− ω

2

) + γ

ac

] + |Ω

2

|

2

The non- ausal ontribution, arising from

t

′

_{> t}

, is obtained by ontourintegration in

thelower-half omplexplane. Tomakethenon- ausal ontributiontovanish, thesumof

residuesinthelower-halfplanemust an el. However,oneofthetwopolesatleastmust

sit in theupper-half planeto givethe ausal ontribution. Thereforeifapole islo ated

in the lower-half plane, the orresponding residue must vanish. One easily he ks that

i(ω − ω

ac

− ω

2

) + γ

ac

annot vanish at a pole sitting in the lower-half plane. Therefore ausalityimposesthatbothpolessitin theupper-halfplane.

Given the xed

ω

ac

value, the enter of

ω

bc

distribution is lo ated at

ω

(0)

bc

=

ω

_ab

(0)

_{− ω}

ac

. Assuming that the ontrol laser is tuned to resonan e with this entral frequen y, so that

ω

2

= ω

(0)

bc

, substituting Eq. 55 into Eq. 49, and omparing with the sus eptibility denition (Eq.52), one nallyobtains:

χ

(+)

_{(ω) = −i}

µ

2

ab

~

_ǫ

₀

Z

dω

ab

G(ω

ab

)

i(ω − ω

(0)

ab

) + γ

ac

[i(ω − ω

ab

) + γ

ab

][i(ω − ω

ab

(0)

) + γ

ac

] + |Ω

2

|

2

(56)

Theanalyti al al ulation anbe ompletedeasilyiftheatomdistribution

is given the following Lorentzian form[12℄:

G(ω

ab

) =

n

π

W

ab

(ω

ab

− ω

ab

(0)

)

2

+ W

ab

2

(57)

Summation over

ω

ab

is performed by ontour integral. One may noti e that the onlypoleinthe upper-half omplexplaneislo atedat

ω

ab

= ω

(0)

ab

+ iW

ab

. One obtains:

χ

(+)

_{(ω) = −in}

µ

2

ab

~

_ǫ

₀

i(ω − ω

(0)

ab

) + γ

ac

[i(ω − ω

ab

) + W

ab

+ γ

ab

][i(ω − ω

(0)

ab

) + γ

ac

] + |Ω

2

|

2

(58)

Inhomogeneous broadening only results in the substitution of the

homoge-neous width

γ

ab

with the broadened linewidth

W

ab

+ γ

ab

. Without further investigationwe an on lude that the expressions for indu ed transparen y

and redu ed group velo ity, we previously derived in the absen e of

inho-mogeneous broadening,are stillvalidprovided

γ

ab

isrepla ed everywhere by

W

ab

+ γ

ab

. It ould be shown easily that Raman transition inhomogeneous broadening is orre tlydes ribed with substitution of

W

ac

+ γ

ac

to

γ

ac

.

5.3 Wave equation in the spe tral domain

The temporal pi turedeveloped in Se tion 4 is onditionedby an adiabati

approximation. Thepresent spe tralanalysis,not limitedbysu h ondition,

is worth visitinga littlefurther.

In the spe tral domainthe wave equation reads as:

∆ ˆ

E(~r, ω) +

ω

2

c

2

E(~r, ω) = −ω

ˆ

2

_µ

equation for the positivefrequen y eld omponentreads as:

∆ ˆ

_{E(~r, ω) +}

ω

2

c

2

1 + χ

(+)

_(ω)



ˆ

E(~r, ω) = 0

(60) Theeld isassumed tobeaplanewavepropagatingalong

Oz

. Onelooksfor a solutionin the form

E(~r, ω) = E(ω)e

ˆ

−iκz

. The wave equationthen redu es

to:

 ω

2

c

2

1 + χ

(+)

_(ω)

_{ − κ}

2



E(ω) = 0

(61) With

κ = k

′

− iα/2

, the solutionis given by:

k

′2

₋

α

2

(ω)

4

= k

2

h

_{1 + χ}

(+)

r

(ω)

i

α(ω) = −

k

2

k

′2

χ

(+)

im

(ω)

(62) where

χ

(+)

r

(ω)

and

χ

(+)

im

(ω)

respe tivelystandfortherealandimaginarypart of

χ

(+)

_(ω)

. Under the assumption that





χ

(+)

r

(ω)





 << 1

and

α(ω) << ω/c

, the waveve tor k' and the absorption oe ient

α(ω)

read as:

k

′

_{(ω) =}

ω

c

q

1 + χ

(+)

r

(ω)

α(ω) = −kχ

(+)

im

(ω)

(63)

Substituting Eq. 58intoEq. 63, one easilyre overs the previously obtained

expression of opa ity at resonan e. In the same way one an al ulate the

velo ity group atresonan e, given the denition as

v = (dk

′

_/dω)

−1

.

More interestingly,the o-resonan e regime an be explored.

Disregard-inginhomogeneousbroadening,andexpandingsus eptibilitytose ondorder

as a fun tion of detuning, one an express the transmitted power spe trum

I(z = L, ω)

as:

I(z = L, ω) = I(z = 0, ω) exp

(

−α

0

L

γ

ac

γ

ab

|Ω

2

|

2

+

 (ω − ω

ab

)γ

ab

|Ω

2

|

2



2

!)

(64)

whi h leads to a gaussian-shape transparen y window whose width agrees