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doi:10.1038/nature09933
SupplementaryInformationfor“ElectromagneticallyInducedTransparencyandSlow
LightwithOptomechanics”
A.H.Safavi-Naeini,
1,
T.P.MayerAlegre,
1,
J.Chan,
1
M.
Eichenfield,
1
Q.Lin,
1
J.T.Hill,
1
D.E.Chang,
1
andO.Painter
1
1
ThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,CA91125
(Dated:February14,2011)
THEORYOFOPTOMECHANICALEIT,EIAANDPARAMETRICAMPLIFICATION
HereweprovideatheoreticaltreatmentofsomeofthemainaspectsofEIT[1–4],EIA[5]andparametricamplifi-
cation[6–8]inoptomechanicalsystems.ModelingtheoptomechanicalsystemwiththeHamiltonian
ˆ
H
= ̄
o
ˆ
a
ˆ
a
+ ̄
m
ˆ
b
ˆ
b
+ ̄
hg
(
ˆ
b
+
ˆ
b
a
ˆ
a
+
i
̄
h
κ
ex
2
α
in
,
0
e
c
t
a
ˆ
a
)
,
(S1)
itispossibletolinearizetheoperationofthesystem,undertheinfluenceofacontrollaserat
ω
c
,aboutaparticular
steady-stategivenbyintracavityphotonamplitude
α
0
andastaticphononshift
β
0
.Theinteractionofthemechanics
andpumpphotonsat
ω
c
withsecondary“probe”photonsat
ω
s
=
ω
c
±
∆withtwo-photondetuning∆canthenbe
modeledbymakingthesubstitutions
ˆ
a
α
0
e
c
t
+(
α
e
i
(
ω
c
+∆)
t
+
α
+
e
i
(
ω
c
∆)
t
)
,
ˆ
b
β
0
+
β
e
i
t
.
(S2)
Assumingthatthepumpismuchlargerthantheprobe,
|
α
0
||
α
±
|
,thepumpamplitudeisleftunaffectedandthe
equationsforeachsidebandamplitude
α
±
arefoundtobe
±
iωα
±
=
(
i
OC
+
κ
2
)
α
±
igα
0
β
±
κ
ex
2
α
in
,
±
,
(S3)
iωβ
=
(
m
+
γ
i
2
)
β
ig
(
α
0
α
+
α
0
α
+
)
γ
i
β
in
,
.
(S4)
Wehavedefined∆
OC
=
ω
o
ω
c
asthepumpdetuningfromtheopticalcavity(includingthestaticoptomechanical
shift,
ω
o
),and
β
+
=
β
.Inthesesituationsitistypicaltodefine
G
=
0
,astheeffectiveoptomechanicalcoupling
ratebetweenasidebandandthemechanicalsubsystem,mediatedbythepump.
Red-detunedpump:ElectromagneticallyInducedTransparency
Withthepumpdetunedfromthecavitybyatwo-photondetuning∆,thespectralselectivityoftheopticalcavity
causesthesidebandpopulationstobeskewedinadrasticfashion.Itisthenanacceptableapproximationtoneglect
oneofthesesidebands,dependingonwhetherthepumpisontheredorbluesideofthecavity.Whenthepump
residesontheredside(∆
OC
>
0),the
α
+
isreducedandcanbeneglected.Thisistherotatingwaveapproximation
(RWA)andisvalidsolongas∆

κ
.
ThenEqs.(S3-S4)maybesolvedforthereflectionandtransmissioncoefficients
r
(
ω
s
)and
t
(
ω
s
)oftheside-coupled
cavitysystem.Wefindthat
r
(∆)=
κ
ex
/
2
i
(∆
OC
∆)+
κ/
2+
|
G
|
2
i
(
ω
m
∆)+
γ
i
/
2
(S5)
t
(∆)=1
κ
ex
/
2
i
(∆
OC
∆)+
κ/
2+
|
G
|
2
i
(
ω
m
∆)+
γ
i
/
2
.
(S6)
TheseequationsareplottedinFigs.S1andS2.
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5
10
-2
0
2
0
0.2
0.4
0
5
10
-2
0
2
0
0.5
1
0
5
10
-5
0
5
0
0.2
0.4
0
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10
-5
0
5
0
0.5
1
Cooperativity
|r|
2
(
Δ
-
ω
m
)/
κ
Cooperativity
(
Δ
-
ω
m
)/
κ
Cooperativity
(
Δ
-
ω
m
)/
γ
i
Cooperativity
(
Δ
-
ω
m
)/
γ
i
|t|
2
|r|
2
|t|
2
a
b
d
c
FIG.S1:
ElectromagneticallyInducedTransparencySpectra.a,c,
Thereflectedsignalamplitude,asafunctionof
two-photondetuning∆forthecasewhere∆
OC
=
ω
m
.In
b,d,
thecorrespondingplotsfortransmissionareshown.The
broadeningofthetransmissionwindow,andthesaturationofthetransmissionpeak,andreflectiondipareevidentin
c,d
respectively.
GroupDelay
Forthered-detunedsystem,theexistenceofaneffectivetransparencyontransmissionmakesthegroupdelay
impartedonthepulseaninterestingquantity.Tocalculatethereflectionandtransmissiongroupdelaysweconsider
apulse
f
(
t
o
)=
0
f
(
ω
)
e
iωt
o
d
ω,
(S7)
wheremostofthespectrumisconfinedtoasmallwindow(
<
4
G
2
)aboutacentralsignalfrequency
ω
s
.Thenthe
transmittedsignal
f
(T)
(
t
o
)maybewrittenas
f
(T)
(
t
o
)=
0
t
(
ω
)
f
(
ω
)
e
iωt
o
d
ω
=
e
s
t
o
−∞
t
(
ω
s
+
δ
)
f
(
ω
s
+
δ
)
e
iδt
o
d
δ
=
e
s
t
o
−∞
t
(
ω
s
)
(
1+
1
t
(
ω
s
)
d
t
d
ω
ω
s
δ
+
o
(
δ
2
)
)
f
(
ω
s
+
δ
)
e
iδt
o
d
δ
e
s
t
o
−∞
t
(
ω
s
)
f
(
ω
s
+
δ
)
e
(
t
o
τ
(T)
)
d
δ
(S8)
Thelastlineimpliesthat
f
(T)
(
t
o
)
f
(
t
o
τ
(T)
),where
τ
(T)
=
R
{
i
t
(
ω
s
)
d
t
d
ω
}
.
(S9)
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−10
−5
0
5
10
−2
−1
0
1
2
0
1
ω
/
γ
i
(
Δ
OC
−ω
m
)/
κ
(
Δ
OC
−ω
m
)/
κ
a
c
d
e
f
g
-10
0
10
0
0.2
0.4
0.6
0.8
1
ω
/
γ
i
h
-2
0
2
0
0.5
1
-2
0
2
-2
0
2
-2
0
2
-2
0
2
-2
0
2
Normalized
Reflection/Transmission
Normalized
Reflection/Transmission
a
b
c
d
e
f
g
h
FIG.S2:
a
,normalizedreflection(solidblueline)andtransmission(dot-dashredline)signalfor∆
OC
=
ω
m
versusthe
normalizedtwo-photondetuningfrequency(∆
i
).
b
normalizedreflectionsignalmapasafunctionofthenormalizedpump
detuning((∆
OC
ω
m
)
)andthenormalizedtwo-photondetuningfrequency.Eachdashedlinecorrespondstothecurvesshown
in
a,c-h
.
c-h
normalizedreflection(solidblueline)andtransmission(dot-dashredline)signalsasafunctionofnormalized
pumpdetuning.Eachcurveagaincorrespondstoaspecifictwo-photondetuning,asdesignatedinb.For
f
,∆
ω
m
,thenthe
reflectedsignalispraticallyzeroonthevicinityoftheresonancecondition∆
OC
=
ω
m
.
Thereflectiongroupdelaymayalsobedefinedanalogously,
τ
(R)
=
R
{
i
r
(
ω
s
)
d
r
d
ω
}
.
(S10)
Withthesignalsentatatwo-photondetuning∆=
ω
m
,wefind
τ
(T)
|
∆=
ω
m
=
2
γ
i
(
κ
e
)
C
(1+
C
)(1
(
κ
e
)+
C
)
,
(S11)
wherethecooperativity
C
=4
G
2
/κγ
i
isameasureofthecouplingbetweenthemechanicaloscillatorandtheoptical
bath.Underthesameconditionswefindthatgroupdelayforreflectionisgivenby
τ
(R)
|
∆=
ω
m
=
2
γ
i
C
1+
C
,
(S12)
resultinginthelimit
C

1
τ
(T)
|
∆=
ω
m
2
γ
i
κ
e
κ
1
C
and
τ
(R)
|
∆=
ω
m
→−
2
γ
i
.
(S13)
Aquantityofinterest,thedelay-bandwidthproductcanbecalculatedforthetransmittedsignal,bytakingthe
productofthesignaldelay
|
τ
(T)
|
max
,andthebandwidth∆
ω
=
γ
i
C
,togiveus∆
ω
·
t
d
=2(
κ
e
).
UsingequationsS11andS5wecanestimatethemaximumdelayforoursystem.Thereflectionandtransmission
coefficientsatresonance(∆=∆
OC
=
ω
m
))aregivenby
r
max
=
(
κ
e
)
1+
C
and
t
max
=
1
(
κ
e
)+
C
1+
C
.
(S14)
Forthecasewhereintrinsicopticallossesarenegligible,i.e.
κ
e
=
κ
,theequationsfordelayandtransmissioncoefficient
contrastcanbewrittenas
t
max
=
C
1+
C
(T)
max
=
2
γ
i
1
(1+
C
)
,
(S15)
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0
1.0
-1.0
0.4
1.0
1.2
group
delay
(
μ
s)
0
1.0
0
80
-0.01
0.00
0.01
-40
phase
(
π
)
phase
(
π
)
normalized
Reection
normalized
Transmission
group
delay
(ns)
0
5
10
-5
-10
Cooperativity
0
2
4
6
8
10
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
|t|
2
γ τ
(T)
/2
κ
e
=
κ
κ
e
= 0.75
κ
κ
e
= 0.5
κ
κ
e
= 0.2
κ
(
Δ
-
ω
m
)/
γ
i
0
a
b
c
d
FIG.S3:
PhaseandGroupDelayinEIT.a,
Normalizedreflection(red)andtransmission(blue)signals,withthephase
b
,andgroupdelay
c
,fortypicalsystemparameters.
d,
Maximumdelay(
τ
(T)
max
),inunitsof
γ
i
/
2,andtransmissioncoefficient
(
|
t
max
|
2
)forthetransmittedsignalasafunctionofthecooperativityfordifferentcavity-waveguidecouplings.
andareplottedinFig.S3.
Blue-detunedpump:ElectromagneticallyInducedAbsorptionandAmplification
Byplacingthepumpatamechanicalfrequencyawayfromcavity,ontheblueside(∆
OC
=
ω
m
)wemayignore
the
α
sidebandoftheintracavityphotons.Thereflectioninthiscaseiscalculatedtobe
r
A
(
ω
s
)=
κ
ex
/
2
i
(∆
OC
+∆)+
κ/
2+
|
G
|
2
i
(
ω
m
∆)
γ
i
/
2
(S16)
and
t
A
(
ω
s
)=1+
r
A
(
ω
s
).Thelinearizationwhichleadstothisequationfromthefulldynamicsofthesystem,only
holdsbelowthephononlasingthreshold,
C
=1,andsowelimitourselvestothecasewhere
C<
1.Theeffective
interactionHamiltonianofthesystemwhichisobtainedaftermakingtherotatingwaveapproximationtoremove
termscounter-rotatingatthemechanicalfrequency,isgivenby[9]
H
int
= ̄
hG
a
ˆ
b
a
ˆ
b
)
.
(S17)
ThisisalsotheHamiltonianofaparametricoscillator,whosequantumtheoryhasbeenknownforsometime[6,8].The
onlydistinctionwithoursystem,isthatweconsiderandmeasuremainlythereflectionandtransmissionproperties
oftheparametricoscillator,asopposedtoitsinternaldynamics.Usingtheexpressionineqn.(S16)wefind
r
max
=
(
κ
e
)
1
C
and
t
max
=
1
(
κ
e
)
C
1
C
.
(S18)
TheseexpressionsareplottedinFigS4a-dforarangeofcooperativities.Theratiobetweenthepowerleavingthe
cavitythroughthewaveguidetotheinputpoweristhesumofthereflectionandtransmissioncoefficientamplitudes,
p
=
|
t
max
|
2
+
|
r
max
|
2
.Asthebase-linevalue,wetake
C
=0forwhichtheemittedpoweris
p
0
=(
κ
e
)
2
+(
κ
i
)
2
.
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0
0.5
1
-5
0
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0
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4
0
0.5
1
-5
0
5
0
0.5
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Cooperativity
|r|
2
(
Δ
-
ω
m
)/
κ
Cooperativity
(
Δ
-
ω
m
)/
κ
Cooperativity
(
Δ
-
ω
m
)/
γ
i
Cooperativity
(
Δ
-
ω
m
)/
γ
i
|t|
2
|r|
2
|t|
2
Cooperativity
|r|
2
|t|
2
p
a
b
d
c
e
f
g
h
i
κ
e
=
κ
κ
e
= 0.5
κ
κ
e
= 0.7
κ
κ
e
= 0.25
κ
κ
e
= 0.2
κ
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
1
2
3
0
amplication
FIG.S4:
ElectromagneticallyInducedAbsorptionandAmplificationSpectra.a,c,
Thereflectedsignalamplitude,
asafunctionoftwo-photondetuning∆forthecasewhere∆
OC
=
ω
m
.In
b,d,
thecorrespondingplotsfortransmissionare
shown.Theincreaseinthereflectedsignalisevidentin
a,c
.In
b,d
weseeareductioninthetransmittedsignalamplitude,
downtozero,andfollowedbyanincrease.
e-i
Theamplitudeofthereflected(blue),transmitted(red)andtotalpower(dashed
black)fromthecavity,forvariouscavity-waveguidecouplingefficiencies.Theregionwhere
|
r
|
2
,p>
1iscalledtheamplification
region,andshaded.Notethatfor
κ
e
<κ/
2,thepowerfromthecavityisatfirstreducedwithhighercooperativity,before
increasingandgoingintotheamplificationregimeat
C>
1
κ
e
.Thiscorrespondstoelectromagneticallyinducedabsorption.
Weak-Coupling:ElectromagneticallyInducedAbsorption
Atsmallcooperativities
C

1andweakcavity-waveguidecoupling
κ
e
i
,thebehaviourofoursystemis
analogoustowhathasbeenobservedinatomicgases,andbeencalledElectromagneticallyInducedAbsorption
(EIA)[5].Undertheseconditions,
p
islessthan
p
0
,andmoreoftheincomingphotonsarenowabsorbedthaninthe
casewith
C
=0.Assuch,thereflectionwillexhibitanabsorptionpeak,andthetransmissionanabsorptiondip.As
longas
κ
e
i
,therewillalwaysbeavalueof
C
suchthatabsorptionisenhanced,asexperimentallydemonstrated