WWW.NATURE.COM/NATURE | 1
SUPPLEMENTARY INFORMATION
doi:10.1038/nature12307
SupplementaryInformationfor“Squeezedlightfromasiliconmicromechanical
resonator”
AmirH.Safavi-Naeini,
1,2,
∗
SimonGr ̈oblacher,
1,2,
∗
JeffT.Hill,
1,2,
∗
JasperChan,
1
MarkusAspelmeyer,
3
andOskarPainter
1,2,4,
†
1
KavliNanoscienceInstituteandThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
2
InstituteforQuantumInformationandMatter,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
3
ViennaCenterforQuantumScienceandTechnology,
FacultyofPhysics,UniversityofVienna,A-1090Wien,Austria
4
MaxPlanckInstitutefortheScienceofLight,G ̈unther-Scharowsky-Straße1/Bldg.24,D-91058Erlangen,Germany
CONTENTS
I.Theory
1
A.Approximatequasi-statictheory
2
B.Theeffectofdynamicsandcorrelation
betweenRPSNandposition
3
C.Generalderivationofsqueezing
5
1.Theeffectofimperfectopticalcoupling
andinefficientdetection
5
II.Experiment
6
A.Measurementoflosses
6
B.Datacollectionprocedure
6
C.Relationbetweendetuningandquadrature6
III.SampleFabricationandCharacterization7
A.Fabrication
7
B.OpticalCharacterization
8
C.MechanicalCharacterization
8
D.Mechanicalqualityfactormeasurements8
IV.NoiseSpectroscopyDetails
9
A.Homodynemeasurementwithlasernoise9
B.Measurementandcharacterizationoflaser
noise
10
C.Linearityofdetectorwithlocaloscillator
power
11
D.Detectednoiselevelwithunbalancing11
E.Estimatingaddednoiseintheopticaltrain12
F.Theeffectoflaserphasenoise
12
G.ErrorAnalysis
13
H.Phenomenologicaldispersivenoisemodel:the
effectofstructuraldamping
13
I.Phenomenologicalabsorptivenoisemodel14
J.Comparingmeasuredspectratotheoretically
predictedspectra
14
V.SummaryofNoiseModel
16
VI.MathematicalDefinitions
17
∗
Theseauthorscontributedequallytothiswork.
†
opainter@caltech.edu;http://copilot.caltech.edu
References
18
I.THEORY
Optomechanicalsystemscanbedescribedtheoretically
withtheHamiltonian(seemaintext)
H
= ̄
hω
o
ˆ
a
†
ˆ
a
+ ̄
hω
m0
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
,
(S1)
whereˆ
a
and
ˆ
b
aretheannihilationoperatorsforpho-
tonsandphononsinthesystem,respectively.Generally,
thesystemisdrivenbyintenselaserradiationatafre-
quency
ω
L
,makingitconvenienttoworkinaninteraction
framewhere
ω
o
isreplacedby∆intheaboveHamilto-
nianwith∆=
ω
o
−
ω
L
.Toquantummechanicallyde-
scribethedissipationandnoisefromtheenvironment,we
usethequantum-opticalLangevindifferentialequations
(QLEs)[1–3],
̇
ˆ
a
(
t
)=
−
(
i
∆+
κ
2
)
ˆ
a
−
ig
0
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
−
√
κ
e
ˆ
a
in
(
t
)
−
√
κ
i
ˆ
a
in
,
i
(
t
)
,
̇
ˆ
b
(
t
)=
−
(
iω
m0
+
γ
i
2
)
ˆ
b
−
ig
0
ˆ
a
†
ˆ
a
−
√
γ
i
ˆ
b
in
(
t
)
,
whichaccountforcouplingtothebathwithdissipation
rates
κ
i
,
κ
e
,and
γ
i
fortheintrinsiccavityenergydecay
rate,opticallossestothewaveguidecoupler,andtotal
mechanicallosses,respectively.Thetotalopticallosses
are
κ
=
κ
e
+
κ
i
.Theselossratesarenecessarilyac-
companiedbyrandomfluctuatinginputsˆ
a
in
(
t
),ˆ
a
in
,
i
(
t
),
and
ˆ
b
in
(
t
),foropticalvacuumnoisecomingfromthecou-
pler,opticalvacuumnoisecomingfromotheropticalloss
channels,andmechanicalnoise(includingthermal).
Thestudyofsqueezingisastudyofnoisepropagation
inthesystemofinterestandassuch,adetailedunder-
standingofthenoisepropertiesisrequired.Theequa-
tionsabovearederivedbymakingcertainassumptions
aboutthenoise,andaregenerallytrueforthecaseofan
opticalcavity,wherethermalnoiseisnotpresent,and
whereweareinterestedonlyinabandwidthofroughly
10
8
smallerthantheopticalfrequency(0–40MHzband-
widthofa200THzresonator).Forthemechanicalsys-
tem,whereweoperateatverylargethermalbathoc-
cupancies(
10
3
)andareinterestedinthebroadband
SupplementaryInformationfor“Squeezedlightfromasiliconmicromechanical
resonator”
AmirH.Safavi-Naeini,
1,2,
∗
SimonGr ̈oblacher,
1,2,
∗
JeffT.Hill,
1,2,
∗
JasperChan,
1
MarkusAspelmeyer,
3
andOskarPainter
1,2,4,
†
1
KavliNanoscienceInstituteandThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
2
InstituteforQuantumInformationandMatter,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
3
ViennaCenterforQuantumScienceandTechnology,
FacultyofPhysics,UniversityofVienna,A-1090Wien,Austria
4
MaxPlanckInstitutefortheScienceofLight,G ̈unther-Scharowsky-Straße1/Bldg.24,D-91058Erlangen,Germany
CONTENTS
I.Theory
1
A.Approximatequasi-statictheory
2
B.Theeffectofdynamicsandcorrelation
betweenRPSNandposition
3
C.Generalderivationofsqueezing
5
1.Theeffectofimperfectopticalcoupling
andinefficientdetection
5
II.Experiment
6
A.Measurementoflosses
6
B.Datacollectionprocedure
6
C.Relationbetweendetuningandquadrature6
III.SampleFabricationandCharacterization7
A.Fabrication
7
B.OpticalCharacterization
8
C.MechanicalCharacterization
8
D.Mechanicalqualityfactormeasurements8
IV.NoiseSpectroscopyDetails
9
A.Homodynemeasurementwithlasernoise9
B.Measurementandcharacterizationoflaser
noise
10
C.Linearityofdetectorwithlocaloscillator
power
11
D.Detectednoiselevelwithunbalancing11
E.Estimatingaddednoiseintheopticaltrain12
F.Theeffectoflaserphasenoise
12
G.ErrorAnalysis
13
H.Phenomenologicaldispersivenoisemodel:the
effectofstructuraldamping
13
I.Phenomenologicalabsorptivenoisemodel14
J.Comparingmeasuredspectratotheoretically
predictedspectra
14
V.SummaryofNoiseModel
16
VI.MathematicalDefinitions
17
∗
Theseauthorscontributedequallytothiswork.
†
opainter@caltech.edu;http://copilot.caltech.edu
References
18
I.THEORY
Optomechanicalsystemscanbedescribedtheoretically
withtheHamiltonian(seemaintext)
H
= ̄
hω
o
ˆ
a
†
ˆ
a
+ ̄
hω
m0
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
,
(S1)
whereˆ
a
and
ˆ
b
aretheannihilationoperatorsforpho-
tonsandphononsinthesystem,respectively.Generally,
thesystemisdrivenbyintenselaserradiationatafre-
quency
ω
L
,makingitconvenienttoworkinaninteraction
framewhere
ω
o
isreplacedby∆intheaboveHamilto-
nianwith∆=
ω
o
−
ω
L
.Toquantummechanicallyde-
scribethedissipationandnoisefromtheenvironment,we
usethequantum-opticalLangevindifferentialequations
(QLEs)[1–3],
̇
ˆ
a
(
t
)=
−
(
i
∆+
κ
2
)
ˆ
a
−
ig
0
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
−
√
κ
e
ˆ
a
in
(
t
)
−
√
κ
i
ˆ
a
in
,
i
(
t
)
,
̇
ˆ
b
(
t
)=
−
(
iω
m0
+
γ
i
2
)
ˆ
b
−
ig
0
ˆ
a
†
ˆ
a
−
√
γ
i
ˆ
b
in
(
t
)
,
whichaccountforcouplingtothebathwithdissipation
rates
κ
i
,
κ
e
,and
γ
i
fortheintrinsiccavityenergydecay
rate,opticallossestothewaveguidecoupler,andtotal
mechanicallosses,respectively.Thetotalopticallosses
are
κ
=
κ
e
+
κ
i
.Theselossratesarenecessarilyac-
companiedbyrandomfluctuatinginputsˆ
a
in
(
t
),ˆ
a
in
,
i
(
t
),
and
ˆ
b
in
(
t
),foropticalvacuumnoisecomingfromthecou-
pler,opticalvacuumnoisecomingfromotheropticalloss
channels,andmechanicalnoise(includingthermal).
Thestudyofsqueezingisastudyofnoisepropagation
inthesystemofinterestandassuch,adetailedunder-
standingofthenoisepropertiesisrequired.Theequa-
tionsabovearederivedbymakingcertainassumptions
aboutthenoise,andaregenerallytrueforthecaseofan
opticalcavity,wherethermalnoiseisnotpresent,and
whereweareinterestedonlyinabandwidthofroughly
10
8
smallerthantheopticalfrequency(0–40MHzband-
widthofa200THzresonator).Forthemechanicalsys-
tem,whereweoperateatverylargethermalbathoc-
cupancies(
10
3
)andareinterestedinthebroadband
SupplementaryInformationfor“Squeezedlightfromasiliconmicromechanical
resonator”
AmirH.Safavi-Naeini,
1,2,
∗
SimonGr ̈oblacher,
1,2,
∗
JeffT.Hill,
1,2,
∗
JasperChan,
1
MarkusAspelmeyer,
3
andOskarPainter
1,2,4,
†
1
KavliNanoscienceInstituteandThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
2
InstituteforQuantumInformationandMatter,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
3
ViennaCenterforQuantumScienceandTechnology,
FacultyofPhysics,UniversityofVienna,A-1090Wien,Austria
4
MaxPlanckInstitutefortheScienceofLight,G ̈unther-Scharowsky-Straße1/Bldg.24,D-91058Erlangen,Germany
CONTENTS
I.Theory
1
A.Approximatequasi-statictheory
2
B.Theeffectofdynamicsandcorrelation
betweenRPSNandposition
3
C.Generalderivationofsqueezing
5
1.Theeffectofimperfectopticalcoupling
andinefficientdetection
5
II.Experiment
6
A.Measurementoflosses
6
B.Datacollectionprocedure
6
C.Relationbetweendetuningandquadrature6
III.SampleFabricationandCharacterization7
A.Fabrication
7
B.OpticalCharacterization
8
C.MechanicalCharacterization
8
D.Mechanicalqualityfactormeasurements8
IV.NoiseSpectroscopyDetails
9
A.Homodynemeasurementwithlasernoise9
B.Measurementandcharacterizationoflaser
noise
10
C.Linearityofdetectorwithlocaloscillator
power
11
D.Detectednoiselevelwithunbalancing11
E.Estimatingaddednoiseintheopticaltrain12
F.Theeffectoflaserphasenoise
12
G.ErrorAnalysis
13
H.Phenomenologicaldispersivenoisemodel:the
effectofstructuraldamping
13
I.Phenomenologicalabsorptivenoisemodel14
J.Comparingmeasuredspectratotheoretically
predictedspectra
14
V.SummaryofNoiseModel
16
VI.MathematicalDefinitions
17
∗
Theseauthorscontributedequallytothiswork.
†
opainter@caltech.edu;http://copilot.caltech.edu
References
18
I.THEORY
Optomechanicalsystemscanbedescribedtheoretically
withtheHamiltonian(seemaintext)
H
= ̄
hω
o
ˆ
a
†
ˆ
a
+ ̄
hω
m0
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
,
(S1)
whereˆ
a
and
ˆ
b
aretheannihilationoperatorsforpho-
tonsandphononsinthesystem,respectively.Generally,
thesystemisdrivenbyintenselaserradiationatafre-
quency
ω
L
,makingitconvenienttoworkinaninteraction
framewhere
ω
o
isreplacedby∆intheaboveHamilto-
nianwith∆=
ω
o
−
ω
L
.Toquantummechanicallyde-
scribethedissipationandnoisefromtheenvironment,we
usethequantum-opticalLangevindifferentialequations
(QLEs)[1–3],
̇
ˆ
a
(
t
)=
−
(
i
∆+
κ
2
)
ˆ
a
−
ig
0
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
−
√
κ
e
ˆ
a
in
(
t
)
−
√
κ
i
ˆ
a
in
,
i
(
t
)
,
̇
ˆ
b
(
t
)=
−
(
iω
m0
+
γ
i
2
)
ˆ
b
−
ig
0
ˆ
a
†
ˆ
a
−
√
γ
i
ˆ
b
in
(
t
)
,
whichaccountforcouplingtothebathwithdissipation
rates
κ
i
,
κ
e
,and
γ
i
fortheintrinsiccavityenergydecay
rate,opticallossestothewaveguidecoupler,andtotal
mechanicallosses,respectively.Thetotalopticallosses
are
κ
=
κ
e
+
κ
i
.Theselossratesarenecessarilyac-
companiedbyrandomfluctuatinginputsˆ
a
in
(
t
),ˆ
a
in
,
i
(
t
),
and
ˆ
b
in
(
t
),foropticalvacuumnoisecomingfromthecou-
pler,opticalvacuumnoisecomingfromotheropticalloss
channels,andmechanicalnoise(includingthermal).
Thestudyofsqueezingisastudyofnoisepropagation
inthesystemofinterestandassuch,adetailedunder-
standingofthenoisepropertiesisrequired.Theequa-
tionsabovearederivedbymakingcertainassumptions
aboutthenoise,andaregenerallytrueforthecaseofan
opticalcavity,wherethermalnoiseisnotpresent,and
whereweareinterestedonlyinabandwidthofroughly
10
8
smallerthantheopticalfrequency(0–40MHzband-
widthofa200THzresonator).Forthemechanicalsys-
tem,whereweoperateatverylargethermalbathoc-
cupancies(
10
3
)andareinterestedinthebroadband
SupplementaryInformationfor“Squeezedlightfromasiliconmicromechanical
resonator”
AmirH.Safavi-Naeini,
1,2,
∗
SimonGr ̈oblacher,
1,2,
∗
JeffT.Hill,
1,2,
∗
JasperChan,
1
MarkusAspelmeyer,
3
andOskarPainter
1,2,4,
†
1
KavliNanoscienceInstituteandThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
2
InstituteforQuantumInformationandMatter,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
3
ViennaCenterforQuantumScienceandTechnology,
FacultyofPhysics,UniversityofVienna,A-1090Wien,Austria
4
MaxPlanckInstitutefortheScienceofLight,G ̈unther-Scharowsky-Straße1/Bldg.24,D-91058Erlangen,Germany
CONTENTS
I.Theory
1
A.Approximatequasi-statictheory
2
B.Theeffectofdynamicsandcorrelation
betweenRPSNandposition
3
C.Generalderivationofsqueezing
5
1.Theeffectofimperfectopticalcoupling
andinefficientdetection
5
II.Experiment
6
A.Measurementoflosses
6
B.Datacollectionprocedure
6
C.Relationbetweendetuningandquadrature6
III.SampleFabricationandCharacterization7
A.Fabrication
7
B.OpticalCharacterization
8
C.MechanicalCharacterization
8
D.Mechanicalqualityfactormeasurements8
IV.NoiseSpectroscopyDetails
9
A.Homodynemeasurementwithlasernoise9
B.Measurementandcharacterizationoflaser
noise
10
C.Linearityofdetectorwithlocaloscillator
power
11
D.Detectednoiselevelwithunbalancing11
E.Estimatingaddednoiseintheopticaltrain12
F.Theeffectoflaserphasenoise
12
G.ErrorAnalysis
13
H.Phenomenologicaldispersivenoisemodel:the
effectofstructuraldamping
13
I.Phenomenologicalabsorptivenoisemodel14
J.Comparingmeasuredspectratotheoretically
predictedspectra
14
V.SummaryofNoiseModel
16
VI.MathematicalDefinitions
17
∗
Theseauthorscontributedequallytothiswork.
†
opainter@caltech.edu;http://copilot.caltech.edu
References
18
I.THEORY
Optomechanicalsystemscanbedescribedtheoretically
withtheHamiltonian(seemaintext)
H
= ̄
hω
o
ˆ
a
†
ˆ
a
+ ̄
hω
m0
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
,
(S1)
whereˆ
a
and
ˆ
b
aretheannihilationoperatorsforpho-
tonsandphononsinthesystem,respectively.Generally,
thesystemisdrivenbyintenselaserradiationatafre-
quency
ω
L
,makingitconvenienttoworkinaninteraction
framewhere
ω
o
isreplacedby∆intheaboveHamilto-
nianwith∆=
ω
o
−
ω
L
.Toquantummechanicallyde-
scribethedissipationandnoisefromtheenvironment,we
usethequantum-opticalLangevindifferentialequations
(QLEs)[1–3],
̇
ˆ
a
(
t
)=
−
(
i
∆+
κ
2
)
ˆ
a
−
ig
0
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
−
√
κ
e
ˆ
a
in
(
t
)
−
√
κ
i
ˆ
a
in
,
i
(
t
)
,
̇
ˆ
b
(
t
)=
−
(
iω
m0
+
γ
i
2
)
ˆ
b
−
ig
0
ˆ
a
†
ˆ
a
−
√
γ
i
ˆ
b
in
(
t
)
,
whichaccountforcouplingtothebathwithdissipation
rates
κ
i
,
κ
e
,and
γ
i
fortheintrinsiccavityenergydecay
rate,opticallossestothewaveguidecoupler,andtotal
mechanicallosses,respectively.Thetotalopticallosses
are
κ
=
κ
e
+
κ
i
.Theselossratesarenecessarilyac-
companiedbyrandomfluctuatinginputsˆ
a
in
(
t
),ˆ
a
in
,
i
(
t
),
and
ˆ
b
in
(
t
),foropticalvacuumnoisecomingfromthecou-
pler,opticalvacuumnoisecomingfromotheropticalloss
channels,andmechanicalnoise(includingthermal).
Thestudyofsqueezingisastudyofnoisepropagation
inthesystemofinterestandassuch,adetailedunder-
standingofthenoisepropertiesisrequired.Theequa-
tionsabovearederivedbymakingcertainassumptions
aboutthenoise,andaregenerallytrueforthecaseofan
opticalcavity,wherethermalnoiseisnotpresent,and
whereweareinterestedonlyinabandwidthofroughly
10
8
smallerthantheopticalfrequency(0–40MHzband-
widthofa200THzresonator).Forthemechanicalsys-
tem,whereweoperateatverylargethermalbathoc-
cupancies(
10
3
)andareinterestedinthebroadband
SupplementaryInformationfor“Squeezedlightfromasiliconmicromechanical
resonator”
AmirH.Safavi-Naeini,
1,2,
∗
SimonGr ̈oblacher,
1,2,
∗
JeffT.Hill,
1,2,
∗
JasperChan,
1
MarkusAspelmeyer,
3
andOskarPainter
1,2,4,
†
1
KavliNanoscienceInstituteandThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
2
InstituteforQuantumInformationandMatter,
CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
3
ViennaCenterforQuantumScienceandTechnology,
FacultyofPhysics,UniversityofVienna,A-1090Wien,Austria
4
MaxPlanckInstitutefortheScienceofLight,G ̈unther-Scharowsky-Straße1/Bldg.24,D-91058Erlangen,Germany
CONTENTS
I.Theory
1
A.Approximatequasi-statictheory
2
B.Theeffectofdynamicsandcorrelation
betweenRPSNandposition
3
C.Generalderivationofsqueezing
5
1.Theeffectofimperfectopticalcoupling
andinefficientdetection
5
II.Experiment
6
A.Measurementoflosses
6
B.Datacollectionprocedure
6
C.Relationbetweendetuningandquadrature6
III.SampleFabricationandCharacterization7
A.Fabrication
7
B.OpticalCharacterization
8
C.MechanicalCharacterization
8
D.Mechanicalqualityfactormeasurements8
IV.NoiseSpectroscopyDetails
9
A.Homodynemeasurementwithlasernoise9
B.Measurementandcharacterizationoflaser
noise
10
C.Linearityofdetectorwithlocaloscillator
power
11
D.Detectednoiselevelwithunbalancing11
E.Estimatingaddednoiseintheopticaltrain12
F.Theeffectoflaserphasenoise
12
G.ErrorAnalysis
13
H.Phenomenologicaldispersivenoisemodel:the
effectofstructuraldamping
13
I.Phenomenologicalabsorptivenoisemodel14
J.Comparingmeasuredspectratotheoretically
predictedspectra
14
V.SummaryofNoiseModel
16
VI.MathematicalDefinitions
17
∗
Theseauthorscontributedequallytothiswork.
†
opainter@caltech.edu;http://copilot.caltech.edu
References
18
I.THEORY
Optomechanicalsystemscanbedescribedtheoretically
withtheHamiltonian(seemaintext)
H
= ̄
hω
o
ˆ
a
†
ˆ
a
+ ̄
hω
m0
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
,
(S1)
whereˆ
a
and
ˆ
b
aretheannihilationoperatorsforpho-
tonsandphononsinthesystem,respectively.Generally,
thesystemisdrivenbyintenselaserradiationatafre-
quency
ω
L
,makingitconvenienttoworkinaninteraction
framewhere
ω
o
isreplacedby∆intheaboveHamilto-
nianwith∆=
ω
o
−
ω
L
.Toquantummechanicallyde-
scribethedissipationandnoisefromtheenvironment,we
usethequantum-opticalLangevindifferentialequations
(QLEs)[1–3],
̇
ˆ
a
(
t
)=
−
(
i
∆+
κ
2
)
ˆ
a
−
ig
0
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
−
√
κ
e
ˆ
a
in
(
t
)
−
√
κ
i
ˆ
a
in
,
i
(
t
)
,
̇
ˆ
b
(
t
)=
−
(
iω
m0
+
γ
i
2
)
ˆ
b
−
ig
0
ˆ
a
†
ˆ
a
−
√
γ
i
ˆ
b
in
(
t
)
,
whichaccountforcouplingtothebathwithdissipation
rates
κ
i
,
κ
e
,and
γ
i
fortheintrinsiccavityenergydecay
rate,opticallossestothewaveguidecoupler,andtotal
mechanicallosses,respectively.Thetotalopticallosses
are
κ
=
κ
e
+
κ
i
.Theselossratesarenecessarilyac-
companiedbyrandomfluctuatinginputsˆ
a
in
(
t
),ˆ
a
in
,
i
(
t
),
and
ˆ
b
in
(
t
),foropticalvacuumnoisecomingfromthecou-
pler,opticalvacuumnoisecomingfromotheropticalloss
channels,andmechanicalnoise(includingthermal).
Thestudyofsqueezingisastudyofnoisepropagation
inthesystemofinterestandassuch,adetailedunder-
standingofthenoisepropertiesisrequired.Theequa-
tionsabovearederivedbymakingcertainassumptions
aboutthenoise,andaregenerallytrueforthecaseofan
opticalcavity,wherethermalnoiseisnotpresent,and
whereweareinterestedonlyinabandwidthofroughly
10
8
smallerthantheopticalfrequency(0–40MHzband-
widthofa200THzresonator).Forthemechanicalsys-
tem,whereweoperateatverylargethermalbathoc-
cupancies(
10
3
)andareinterestedinthebroadband
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propertiesofnoisesources(0–40MHzfora30MHz
resonator),amoredetailedunderstandingofthebathis
required,andwillbepresentedinthesectiononthermal
noise.
Atthispoint,welinearizetheequationsassuminga
strongcoherentdrivefield
α
0
,anddisplacetheannihi-
lationoperatorforthephotonsbymakingthetransfor-
mationˆ
a
→
α
0
+ˆ
a
.Thisapproximation,whichneglects
termsoforderˆ
a
2
isvalidforsystemssuchasourswhere
g
0
κ
,i.e.the
vacuumweakcoupling
regime.Weare
thenleftwithaparametricallyenhancedcouplingrate
G
=
g
0
|
α
0
|
.Usingtherelationsgiveninthemathemat-
icaldefinitionssection(VI)ofthisdocument,wewrite
thesolutiontotheQLEsintheFourierdomainas
ˆ
a
(
ω
)=
−
√
κ
e
ˆ
a
in
(
ω
)
−
√
κ
i
ˆ
a
in
,
i
(
ω
)
−
iG
(
ˆ
b
(
ω
)+
ˆ
b
†
(
ω
))
i
(∆
−
ω
)+
κ/
2
,
ˆ
b
(
ω
)=
−
√
γ
i
ˆ
b
in
(
ω
)
i
(
ω
m0
−
ω
)+
γ
i
/
2
−
iG
(ˆ
a
(
ω
)+ˆ
a
†
(
ω
))
i
(
ω
m0
−
ω
)+
γ
i
/
2
.
(S2)
(WeusethenotationdescribedinsectionVIwhere
(
ˆ
A
(
ω
)
)
†
=
ˆ
A
†
(
−
ω
).)
Finallywenotethatbymanipulationoftheseequa-
tions,themechanicalmotioncanbeexpressedasa
(renormalized)responsetotheenvironmentalnoiseand
theopticalvacuumfluctuationsincidentontheoptical
cavitythroughtheoptomechanicalcoupling
ˆ
b
(
ω
)=
−
√
γ
i
ˆ
b
in
(
ω
)
i
(
ω
m
−
ω
)+
γ/
2
+
iG
i
(∆
−
ω
)+
κ/
2
√
κ
e
ˆ
a
in
(
ω
)+
√
κ
i
ˆ
a
in
,
i
(
ω
)
i
(
ω
m
−
ω
)+
γ/
2
+
iG
−
i
(∆+
ω
)+
κ/
2
√
κ
e
ˆ
a
†
in
(
ω
)+
√
κ
i
ˆ
a
†
in
,
i
(
ω
)
i
(
ω
m
−
ω
)+
γ/
2
.
(S3)
Therenormalizedmechanicalfrequencyandlossrate
are
ω
m
=
ω
m0
+
δω
m
,and
γ
=
γ
i
+
γ
OM
,respectively,
with
δω
m
=
|
G
|
2
Im
[
1
i
(∆
−
ω
m
)+
κ/
2
−
1
−
i
(∆+
ω
m
)+
κ/
2
]
,
(S4)
γ
OM
=2
|
G
|
2
Re
[
1
i
(∆
−
ω
m
)+
κ/
2
−
1
−
i
(∆+
ω
m
)+
κ/
2
]
.
(S5)
Itisconvenienttodefineherewhatwemeanbya
quadrature,asitistheobservableofthelightfieldthat
ourmeasurementdevice(thebalancedhomodynedetec-
tor(BHD)setup)issensitiveto:
ˆ
X
(
j
)
θ
(
t
)=ˆ
a
j
(
t
)
e
−
iθ
+ˆ
a
†
j
(
t
)
e
iθ
.j
=in
,
out
,
vac
,...
(S6)
Weareinterestedinthepropertiesof
ˆ
X
(out)
θ
forvarious
quadratureangles
θ
,giventheinfluenceofthemechanical
system.
Themeasurementofthefieldprovidesuswitharecord
ˆ
I
(
t
)=
ˆ
X
(out)
θ
(
t
)foracertain
θ
.Weuseaspectrumana-
lyzertoperformFourieranalysisonthissignalandobtain
asymmetrizedclassicalpowerspectraldensity(PSD)
̄
S
II
(
ω
),asdefinedinthemathematicalappendix(sec-
tionVI).
Foravacuumfieldsuchastheinputfield,themeasured
quadrature
ˆ
X
(vac)
θ
(
t
)willhaveapowerspectraldensity
̄
S
vac
II
(
ω
)=1
.
(S7)
Thisistheshot-noiselevelwhichisduetothequan-
tumfluctuationsoftheelectromagneticfield.Mathemat-
ically,itarisesfromthecorrelator
〈
ˆ
a
vac
(
ω
)ˆ
a
†
vac
(
ω
′
)
〉
=
δ
(
ω
+
ω
′
),withallothercorrelators
〈
ˆ
a
†
vac
(
ω
)ˆ
a
vac
(
ω
′
)
〉
,
〈
ˆ
a
†
vac
(
ω
)ˆ
a
†
vac
(
ω
′
)
〉
,
〈
ˆ
a
vac
(
ω
)ˆ
a
vac
(
ω
′
)
〉
,arisingintheex-
pression
〈
ˆ
I
†
(
ω
)
ˆ
I
(
ω
′
)
〉
equaltozero.
A.Approximatequasi-statictheory
Inthissectionwepresentasimplifiedderivationof
howsqueezingisobtainedinthestudiedoptomechanical
systemtoelucidatetheimportantsystemparametersand
theirroleinsqueezing.Wemakeafewapproximations
tosimplifythederivation:
1.
∆=0:Thelaseristunedexactlytotheoptical
cavityfrequency.
2.
κ
e
=
κ
:Perfectcoupling.
3.
κ
ω
m
:Badcavitylimit.
4.
ω
ω
m
:Weareonlyinterestedinthequasi-static
response,sotheresonantresponseofthemechani-
calresonatordoesnotplayarole.
Undertheseassumptions,equations(S2)and(S3)can
bewrittenas(usingtherelationfortheopticaloutput
fieldˆ
a
out
(
ω
)=ˆ
a
in
(
ω
)+
√
κ
ˆ
a
(
ω
)):
iω
m
ˆ
b
(
ω
)=
−
√
γ
i
ˆ
b
in
(
ω
)+
2
iG
√
κ
(ˆ
a
in
(
ω
)+ˆ
a
†
in
(
ω
))
,
ˆ
a
out
(
ω
)=
−
ˆ
a
in
(
ω
)
−
2
iG
√
κ
(
ˆ
b
(
ω
)+
ˆ
b
†
(
ω
))
.
(S8)
Thefirstequationshowstheresponseofthemechani-
calresonatorsubsystemtothethermalbathfluctuations
(
ˆ
b
in
(
ω
))andtheopticalvacuumnoisefromthemeasure-
mentback-action.WedefineΓ
meas
≡
4
|
G
|
2
/κ
,andinter-
pretitasthemeasurementrate[4],suchthatthefactor
appearinginfrontoftheopticalvacuumnoiseoperators
is
√
Γ
meas
.Thisratealsoappearsinthesecondequation
fortheoutputfield,infrontofthenormalizedposition
operatorˆ
x/x
zpf
=
ˆ
b
(
ω
)+
ˆ
b
†
(
ω
),whichistheobservable
thatisbeingmeasured.
Note,fromtheexpressionforˆ
a
out
(
ω
)itfollows,that
sincethepositionisarealobservablewithanimaginary
prefactor,theeffectsweconsiderdependstronglyonthe
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Frequency (MHz)
10
60
50
40
30
20
Measurement Phase (
π
radians)
-0.5
-0.25
0.25
0.5
0
a
b
c
1
0.5
10
7
10
1
0.9
0.8
0.7
0.6
10
6
10
5
10
4
10
3
10
2
Frequency (MHz)
1
10
S
II
(f )
10
0
10
-1
10
-2
10
1
1
10
Frequency (MHz)
S
II
(f )
10
0
10
-1
10
-2
10
1
FIG.S1.
Squeezingtheory.a
,Densityplotofthepredictedsqueezing
̄
S
out
II
(
ω
)vs.phaseangleandfrequency,normalized
totheshot-noise.Themechanicalmodecanclearlybeseenat
ω
m
/
2
π
=30MHz.Thesolidwhitelinesoutlinetheregion
wherethepowerspectraldensityfallsbelow1(theshot-noiselevel)indicatingthepresenceofsqueezingforthatphaseand
frequency.Thedashedwhitelinesat
θ
=
−
π/
4and
θ
=+
π/
4correspondtoregionswheresqueezingcanbeobtainedbelow
andabovethemechanicalfrequency,respectively,andthecomponentsofthenoisemodelforthesephasesisshownindetail
infigures
b
and
c
.Inthesefiguresthespectraareagainnormalizedtotheshot-noiselevelplottedasagreyline.Thesimple
squeezingmodelwithoutthermalnoise(Eq.(S10))isrepresentedbythedashedgreenlineandthesimplemodelwiththermal
noise(Eq.(S18))isthesolidgreenline.Thesolidblacklineisthefullsqueezingmodel
̄
S
out
II
(
ω
)correspondingto
a
withthe
constituentcomponents:thecontributionfromtheopticalvacuumfluctuations(
̄
S
out
II,a
(
ω
);Eq.(S25))representedbythedashed
blacklineandthethermalnoise(
̄
S
out
II,b
(
ω
);Eq.(S26))representedbythedashedredline.
quadraturebeingprobed,i.e.therealpartoftheexpres-
sion,
ˆ
X
(out)
θ
=0
,willnotbeaffectedbytheoptomechanical
coupling.
Atthispointwecaneasilycalculatethepropertiesof
thedetectedspectrum
̄
S
out
II
(
ω
),bywritingˆ
a
out
interms
ofˆ
a
in
and
ˆ
b
in
forwhichthecorrelatorsareknown:
ˆ
a
out
(
ω
)=
−
ˆ
a
in
(
ω
)
−
2
i
Γ
meas
ω
m
(ˆ
a
in
(
ω
)+ˆ
a
†
in
(
ω
))
+
√
γ
i
Γ
meas
ω
m
(
ˆ
b
in
(
ω
)
−
ˆ
b
†
in
(
ω
))
.
(S9)
Ignoringthermalnoiseforthemoment(
γ
i
=0),and
droppingtermsoforder(Γ
meas
/ω
m
)
2
(assumingΓ
meas
ω
m
)wearriveat:
̄
S
out
II
(
ω
)=
∫
∞
−∞
dω
′
〈
ˆ
X
(out)
θ
(
ω
)
ˆ
X
(out)
θ
(
ω
′
)
〉
=1+4(Γ
meas
/ω
m
)sin(2
θ
)
.
(S10)
Notethatforcertainvaluesof
θ
,thedetectedspectral
densitycanbesmallerthanwhatonewouldexpectfor
avacuumfield.For
θ
=
−
π/
4,weachievethemaximum
squeezingwithanoisefloorof1
−
4(Γ
meas
/ω
m
)which
stronglydependendsontheratioΓ
meas
/ω
m
.
Tounderstandtheeffectofthermalnoise,weas-
sumetheformofthecorrelatortobe
〈
ˆ
b
in
(
ω
)
ˆ
b
†
in
(
ω
′
)
〉
=
( ̄
n
(
ω
)+1)
δ
(
ω
+
ω
′
),
〈
ˆ
b
†
in
(
ω
)
ˆ
b
in
(
ω
′
)
〉
= ̄
n
(
ω
)
δ
(
ω
+
ω
′
),
〈
ˆ
b
†
in
(
ω
)
ˆ
b
†
in
(
ω
′
)
〉
=0,and
〈
ˆ
b
in
(
ω
)
ˆ
b
in
(
ω
′
)
〉
=0(theseex-
pressionsarediscussedinsectionIVH).Thenacalcula-
tionsimilartotheoneleadingtoequation(S10)gives
̄
S
out
II
(
ω
)=1+4(Γ
meas
/ω
m
)sin(2
θ
)
+4
Γ
meas
ω
m
̄
n
(
ω
)
Q
m
(1
−
cos(2
θ
))
,
(S11)
wherewehaveassumed ̄
n
(
ω
),thebathoccupationat
frequency
ω
,tobemuchlargerthanunity.At
θ
=
−
π/
4,
wehave
̄
S
out
II
(
ω
)=1
−
4(Γ
meas
/ω
m
)(1
−
̄
n
(
ω
)
/Q
m
)
.
(S12)
Inthismodel,thereisnosqueezingat
θ
=
−
π/
4and
frequency
ω
if ̄
n
(
ω
)
>Q
m
.Somesqueezingisalways
present,butisshiftedtootherquadraturesandthe
amountofdetectablesqueezingisreducedathighertem-
peratures.Mostofthesqueezing(59%)iswashedout
bythethermalnoiseat ̄
n
(
ω
)=
Q
m
.Thesqueezing
arisesfromthetimeevolutionofthemechanicalres-
onatormaintainingcoherenceoverthetimescaleofthe
fluctuations.Requiringcoherentevolutionovertheme-
chanicalcycleisequivalenttodemandingthattherate
atwhichphononsenterthemechanicalsystemfromthe
bath(
γ
i
̄
n
)tobesmallerthanthemechanicalfrequency
ω
m
.Inconclusion,theimportantrequirementstoachieve
squeezingaretomakeΓ
meas
comparableto
ω
m
andtore-
ducethethermaloccupancyorincreasethemechanical
qualityfactortoachieve ̄
n
(
ω
)
<
∼
Q
m
.
B.Theeffectofdynamicsandcorrelationbetween
RPSNandposition
Asanextstep,wetakeintoaccountthedynamicsof
themechanicalresonatorwhilekeepingtheapproxima-
tionsofthebad-cavitylimit(
κ
ω
m
)andon-resonant
probing(∆=0).Inadditiontofurtherclarifyingsome
oftheobservedfeatures,thistreatement,aspresented
inthemaintext,elucidatestheroleofcorrelationsbe-
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tweenthemechanicalsystem’spositionandtheback-
actionforce.
Theresponseofthemechanicalsystemtoaforceis
capturedbyitssusceptibility:
χ
m
(
ω
)=
1
m
(
ω
2
m
−
ω
2
−
iγ
i
ω
m
)
.
(S13)
Theformofthedampingconsideredhereisthestrongly
sub-ohmicstructuraldampingwhichisobservedinour
measurements[5,6](cf.SectionIVH).Themechanical
systemrespondstorandomnoiseforces
F
T
(
t
)fromthe
thermalbath(whichwetreatedinthelastsectionand
neglecthere),andtothequantumback-actionfromthe
cavity
F
BA
(
t
).
Theback-actionforcefortheresonantcasecansimply
befoundbylinearizingtheexpressionfortheradiation
pressureforce
ˆ
F
RP
(
t
)=
−
̄
hg
0
ˆ
a
†
ˆ
a/x
zpf
.Wefindtheforce
impartedonthemechanicsduetotheshot-noiseofthe
cavityfieldtobe
ˆ
F
BA
(
t
)=
̄
h
·
√
Γ
meas
x
zpf
ˆ
X
(in)
θ
=0
(
t
)
(S14)
forthecaseofresonantdriving.Thefluctuationsim-
partedonthemechanicsarefromtheintensityquadra-
tureofthelight(
θ
=0).Usingequation(S8),wecan
writetheoutputfieldquadratureas:
ˆ
X
(out)
θ
(
t
)=
−
ˆ
X
(in)
θ
(
t
)
−
2
√
Γ
meas
x
zpf
ˆ
x
(
t
)
·
sin(
θ
)
.
(S15)
Wenoteherethatthemechanicalpositionfluctuations
areprimarilyimprintedonthephasequadratureofthe
outputlight,with
θ
=
±
π/
2.Theintensityquadrature
isunmodified(
ˆ
X
(out)
θ
=0
(
t
)=
−
ˆ
X
(in)
θ
=0
(
t
))sincechangesin
thecavityfrequencyarenottransducedaschangesin
intensitywhenthelaserisresonantwiththecavity.
Theoutputofthehomodynedetectornormalized
totheshot-noiselevelisfoundbytakingtheauto-
correlationofeqn.(S15).Thecorrelationsbetweenra-
diationpressureshot-noiseandthemechanicalmotion
areimportantinthiscalculation[7–13]andmustbe
takenintoaccount.Inthetime-domainwefindtheauto-
correlationtobe:
〈
ˆ
X
(out)
θ
(
t
)
ˆ
X
(out)
θ
(
t
′
)
〉
=
δ
(
t
−
t
′
)+4Γ
meas
sin
2
(
θ
)
〈
ˆ
x
(
t
)ˆ
x
(
t
′
)
〉
x
2
zpf
+2 ̄
h
−
1
sin(
θ
)cos(
θ
)
〈
ˆ
F
BA
(
t
)ˆ
x
(
t
′
)+ˆ
x
(
t
)
ˆ
F
BA
(
t
′
)
〉
.
(S16)
Thecos(
θ
)inthelasttermcomesfromthegeneralex-
pressionforaquadrature
ˆ
X
(in)
θ
(
t
)=
ˆ
X
(in)
θ
=0
(
t
)cos(
θ
)+
ˆ
X
(in)
θ
=
π/
2
(
t
)sin(
θ
),andequation(S14).Thekeycompo-
nentsofequation(S16)aretheshot-noiselevel,thether-
malnoise,andthecross-correlationbetweentheback-
actionnoiseforceandmechanicalpositionfluctuations.
Itisonlythelatterwhichcangiverisetosqueezing,
byreducingthefluctuationlevelbelowshot-noise.This
squeezingcanbecalculatedspectrally:
S
sq
(
ω
)= ̄
h
−
1
sin(2
θ
)
×
∫
∞
−∞
d
τ
[
〈
ˆ
F
BA
(
t
)ˆ
x
(
t
−
τ
)
〉
+
〈
ˆ
x
(
t
)
ˆ
F
BA
(
t
−
τ
)
〉
]
e
iωτ
=2 ̄
h
Re
{
χ
m
(
ω
)
}
Γ
meas
/x
2
zpf
sin(2
θ
)
.
(S17)
Thefulldetectedspectraldensityisthen
̄
S
out
II
(
ω
)=1+
4Γ
meas
x
2
zpf
[
̄
S
xx
sin
2
(
θ
)+
̄
h
2
Re
{
χ
m
}
sin(2
θ
)
]
.
(S18)
AttheDCorquasi-staticlimit(
ω
→
0)thesuscepti-
bility
χ
m
→
1
/mω
2
m
canbeusedandwereobtainthe
resultsfromsectionIA(cf.equation(S10)).Weseethat
for
θ<
0,squeezingisobtainedinthislimit.Atfre-
quencieslargerthan
ω
m
,
χ
m
(
ω
)changessign,andwe
expecttoseesqueezingatquadratureangles
θ>
0.
Additionally,since
χ
m
(
ω
)becomeslargeraroundthe
mechanicalfrequency,weexpectthemaximumsqueez-
ingtobeenhanced.Morespecifically,atadetuning
δ
=
ω
m
−
ω
(
|
δ
|
γ
i
)fromthemechanicalresonance,
weexpecttheparametercharacterizingthesqueezing
tobeproportionaltoΓ
meas
/δ
,andthedetectedspec-
trumshowninequation(S18)becomes
̄
S
out
II
(
ω>
0)
≈
1+(2Γ
meas
/δ
)[(
ω
m
/δ
)( ̄
n
(
ω
)
/Q
m
)(1
−
cos(2
θ
))+sin(2
θ
)].
Thesefeaturesareevidentinthespectrapresentedin
Fig.S1.
Itisimportanttonoteherethatintheabsenceofother
nonlinearitiesinthesystem,anyreductionofthenoise
belowthevacuumfluctuationscanonlybecausedbythe
correlationsbetweentheRPSNandthepositionfluctu-
ationsofthesystem.Thismakestheproblemofproving
thecorrelationsbetweenRPSNandmechanicalmotion
equivalenttotheproblemofprovingthatthereflected
lightfromtheoptomechanicalcavityissqueezed.
ConceptuallythisformofprobingtheRPSNissim-
ilartothatcarriedoutbySafavi-Naeinietal.[10,13]
andanalyzedbyKhalilietal.[11].Italsosharesfea-
tureswiththecross-correlationmeasurementsproposed
byHeidmannetal.[7],andBørkjeetal.[9],andrecent
experimentsbyPurdyetal.[14].Thedistinguishingfea-
tureofthistypeofmeasurementisthatthequantumcor-
relationsbetweenthefluctuationsofthepositionandthe
electromagneticvacuummanifestthemselvesassqueezed