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2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
Supplementaryinformationfor:“Generalizednonreciprocityinanoptomechanical
circuitviasyntheticmagnetismandreservoirengineering”
KejieFang,
1,2
JieLuo,
1,2
MatthewH.Matheny,
1,3
AnjaMetelmann,
4,5
FlorianMarquardt,
6,7
AashishA.Clerk,
4
andOskarPainter
1,2
1
KavliNanoscienceInstitute,CaliforniaInstituteofTechnology,Pasadena,California91125,USA
2
InstituteforQuantumInformationandMatterandThomasJ.Watson,Sr.,
LaboratoryofAppliedPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA
3
DepartmentofPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA
4
DepartmentofPhysics,McGillUniversity,3600rueUniversity,Montr ́eal,QuebecH3A2T8,Canada
5
DepartmentofElectricalEngineering,PrincetonUniversity,Princeton,NewJersey08544,USA
6
MaxPlanckInstitutefortheScienceofLight,G ̈unther-Scharowsky-Straße1/Bau24,D-91058Erlangen,Germany
7
InstituteforTheoreticalPhysics,DepartmentofPhysics,Universit ̈atErlangen-N ̈urnberg,91058Erlangen
I.DEVICEFABRICATIONANDCHARACTERIZATION
A.Devicefabricationandatomicforcemicroscopenano-oxidationtuning
Thedeviceswerefabricatedfromasilicon-on-insulatorwaferwithasilicondevicelayerthicknessof220nmand
buried-oxidelayerthicknessof2
μ
m.Thedevicegeometrywasdefinedbyelectron-beamlithographyfollowedby
inductivelycoupledplasmareactiveionetchingtotransferthepatternthroughthe220nmsilicondevicelayer.The
deviceswerethenundercutusinganHF:H
2
Osolutiontoremovetheburiedoxidelayerandcleanedusingapiranha
etch.
Afterdevicefabrication,weusedanatomicforcemicroscopetodrawnanoscaleoxidepatternsonthesilicondevice
surface.Thisprocessmodifiestheopticalandmechanicalcavityfrequenciesinacontrollableandindependentway
withtheappropriatechoiceofoxidepattern.Thenano-oxidationprocesswascarriedoutusinganAsylumMFP-3D
atomicforcemicroscopeandconductivediamondtips(NaDiaProbes)inanenvironmentwithrelativehumidityof
48%.Thetipwasbiasedatavoltageof
11
.
5V,scannedwithavelocityof100nm/s,andrunintappingmodewith
anamplitudeof10nm.Theunpassivatedsilicondevicesurfacewasgrounded.
B.Opticaltransmissioncoefficientmeasurement
Frequency (GHz)
5.74
5.76
5.78
5.8
5.82
Transmission coefficient (dB)
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
Frequency (GHz)
5.74
5.76
5.78
5.8
5.82
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
b
a
Transmission coefficient (dB)
B
=
0.34
B
=
1.34
FIG.S1.
a
Microwavesignalpowertransmissionthroughtheoptomechanicalcircuitforforward(right-propagation;blue)
andbackward(left-propagation;bluecurve)directions,withfluxsettoΦ
B
=0
.
34
π
andcavityphotonnumber
n
cL
=1000and
n
cR
=1420.
b
Sameas
a
butwithΦ
B
=1
.
34
π
.
Tomeasuretheopticalpowertransmissionthroughtheoptomechanicalcircuitweusedavectornetworkanalyzer
(VNA).TheVNAoutputsamicrowavetonefromport1withfrequency
ω
mod
toanelectro-opticmodulatorwhich
modulatestheopticalpumptogenerateanopticalsidebandcorrespondingtotheopticalprobe.Inthecaseofa
Generalized non-reciprocity in an optomechanical circuit
via synthetic magnetism and reservoir engineering
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DOI: 10.1038/NPHYS4009
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2
blue-detunedpumpfromtheopticalcavityresonance,theprobefieldcorrespondstothelowersideband(selectedby
thefilteringpropertiesofthecavityitself).Boththeopticalprobeandpumparelaunchedintooneoptomechanical
cavityinthecircuit.Attheothercavity,thetransmittedopticalprobecombineswithasecondpumpandthebeating
ofthetwoisdetectedbyahigh-speedphotodetector(boththefirstandsecondpumpbeamsarefromthesame
lasersource,andthusphasecoherent).Thephotocurrentsignalfromthephotodetectorissentintoport2ofthe
VNAtomeasurethemicrowavesignaltransmissioncoefficient
T
μ
.Fig.S1shows
|
T
μ
|
2
forforward(right-propagating;
bluecurve)andbackward(left-propagating;redcurve)directionsthroughtheoptomechanicalcircuitasafunction
ofthemodulationfrequency
ω
mod
.InFig.S1athesyntheticfluxvalueislockedtoΦ
B
=0
.
34
π
whereasinFig.S1b
Φ
B
=1
.
34
π
.Inbothfluxsettingstheopticalpumpinglevelsweresuchthattheleftandrightcavityphotonnumbers
were
n
cL
=1000and
n
cR
=1420,respectively.Thisistherawtransmissiondatacorrespondingtothenormalized
transmissionratioshowninFigs.3band3cofthemaintext.
Whileabsoluteopticaltransmissionisnotdirectlymeasured,theratiooftheopticaltransmissioncoefficientsfor
forwardandbackwardpropagationcanbeobtainedfromthenormalizedmicrowavesignaltransmissioncoefficient
̄
T
μ
,
|
T
L
R
/T
R
L
|
2
=
|
̄
T
μR
/
̄
T
μL
|
2
,
(S1)
where
|
̄
T
μ
|
2
isnormalizedusingthevalueof
|
T
μ
|
2
awayfromallmechanicalresonancestoremovealltheexternal
asymmetryintheexperimentalsetupforleftandrightpropagationpaths.Theseexternalasymmetriesinclude
modulatorefficiency,cable/fiberloss,etc.Inouranalysisthenormalizationlevelistheaveragevalueof
|
T
μ
|
2
inthe
frequencyrangeof5
.
74-5
.
76GHz.Tobeclear,thereasonthiscalibrationisnecessaryisbecausewedon’tactually
physicallyswapthesourceanddetectorinourmeasurements.Rather,fortheleft-to-righttransmissionpathwehave
onemodulatorontheleftsidewhichgeneratestheprobetoneandonedetectorontherightsidewhichmeasuresthe
transmissionthroughtotherightside.Whenwemeasureright-to-lefttransmissionwehaveadifferentmodulatoron
therightsidetogeneratetheprobetoneandadifferentdetectorontheleftsidetodetectthetransmittedprobe.If
themodulatorontheleftsideisdifferentfromthemodulatorontherightside,thenforthesamemicrowavedrivethat
excitesthemodulatorswewouldgetdifferentadifferentopticalprobepowerinthesidebandsofthepump.Similarly
iftheleftandrightdetectorshavedifferentefficienciesthentheywouldproduceadifferentphotocurrentforthe
sametransmittedopticalprobepower.Sincewemeasureinpracticetheratioofthemicrowavedrivetothedetected
microwavephotocurrent,thiscouldcauseaninherentasymmetryinthemeasuredtransmissionforleft-to-rightand
right-to-lefttransmissionevenifthe
optical
transmissionwasperfectlysymmetric.
C.Devicecharacterization
Todeterminethecomponentsofopticalcavityloss(intrinsicdecayrate
κ
i
,externalwaveguide-to-cavitycoupling
κ
e
,totalcavitydecayrate
κ
)ofboththeleftandrightopticalcavitiesweusedapump-probeschemesimilartothat
usedtomeasurethenonreciprocityoftheoptomechanicalcircuit.Thepumpbeaminthiscase,however,issettobe
veryweaksoastonotresonantlyexcitethemechanicsastheprobesignalissweptacrosstheopticalcavityresonance.
ThecavityscansareplottedinFig.S2aandS2bfortheleftandrightcavities,respectively.Wefitthephaseresponse
curvesandget
κ
iL(R)
/
2
π
=0
.
29(0
.
31)GHz,
κ
eL(R)
/
2
π
=0
.
74(0
.
44)GHz,and
κ
L(R)
/
2
π
=1
.
03(0
.
75)GHz.The
intrinsicandexternalopticalcavityratesareusedtodeterminetheintra-cavityphotonnumberforagivenoptical
pumppower(specifiedattheinputtothecavity).
Thermalmechanicalspectraofthetwocavitiesaremeasuredwithaweakblue-detunedopticalpumpsoastoavoid
back-action;asinglepumpisusedforeachoftheleftandrightcavitymeasurements.Thereflectedpumplightfrom
thecavitycontainsmodulationsidebandsfromthethermalmechanicalmotion,whichupondetectionwithahigh-
speedphotodetectorcreatesaphotocurrentwiththethermalmotionofthemechanicalcavitymodesimprintedonit.
Sincethemechanicalmodescanbehybridizedbetweenleft-cavity,right-cavity,andwaveguidemodes,ameasurement
withtheleft-sidepumpproducesalocalmeasurementofthecavitymodesasmeasuredbythelocalizedleftoptical
cavitymode,andsimilarlyfortheright-sidepumpandcavity.Theintrinsicdecayrateofthemechanicalmodesis
inferredfromthelinewidthoftheLorentzianmechanicalspectrum.
Measurementsofthemechanicalmodespectrawereperformedbothbeforeandafterthecavitieswerenano-oxidized
totunetheirlocalizedopticalandmechanicalmodesintoresonance.Measurementspriortonano-oxidationallowedus
todeterminethelocal(leftandright)mechanicalandopticalcavitymodeproperties(i.e.,thebare,uncoupledmode
properties).Knowingtheleftandrightcavitymodepropertiesfromindependentmeasurementsallowedustofitwith
fewerfittingparametersthemeasuredforwardandbackwardtransmissioncurvesofthehybridizedcavitiespresented
inthemainarticletext.Notethatafternano-oxidationtheleftandrightopticalcavitymodeswereonlyveryweakly
hybridizedsoastomaintaintheirleft-cavityandright-cavitycharacter.Themechanicalmodesweretunedtobe
stronglyhybdridizedasevidencedinFig.2fofthemaintext.FiguresS2candS2dshowthemeasuredlinewidthof
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DOI: 10.1038/NPHYS4009