Villar Institut f57596r Experimentalphysik Universit57572t Innsbruck Technikerstrasse 254 6020 Innsbruck Austria and Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences 6020 Innsbruck Austria Received 27 November 2007 a ID: 55897
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TheconversionofphasetoamplitudeßuctuationsofalightbeambyanopticalcavityAlessandroS.VillarInstitutfürExperimentalphysik,UniversitätInnsbruck,Technikerstrasse25/4,6020Innsbruck,AustriaandInstituteforQuantumOpticsandQuantumInformation,AustrianAcademyofSciences,6020Innsbruck,AustriaReceived27November2007;accepted9May2008VerylowintensityandphaseßuctuationsarepresentinabrightlightÞeldsuchasalaserbeam.Thesesubtlequantumßuctuationsmaybeusedtoencodequantuminformation.Althoughintensityiseasilymeasuredwithcommonphotodetectors,accessingthephaseinformationrequiresinterferenceexperiments.Weintroduceonesuchtechnique,therotationofthenoiseellipseoflight, whichgiveusbetterphysicalinsightintotheßuctuations.Theintegrationtimeismuchlongerthanthetypicalßuc-tuationtimescaleofthesystem,andwethereforetaketosimplifythecalculations.ThisFouriertransformisevaluatedaroundtheopticalfrequency,becausephotodetectorscannotresolvethefastos-cillationsofanopticalÞeldandrecordonlyitsaveragein-tensityovermanyopticalcycles.ThemeanÞeldisthusrepresentedbythezerofrequencycomponent,=0,whichiscalledthecarrier.PracticallyalltheenergypresentinthelightÞeldbelongstothecarrier.Itsquantumcharacterisnotconsideredherebecauseoftechnicallimitationsindistin-guishingphotonnumbersatthisintensitylevel,althoughtheymayshowveryinterestingquantumfeatures.fore,itisjustiÞedtoconsideritasaclassicalÞeldforourThefrequencycomponentsaroundtheopticalcarrierarecalledsidebands.Theanalysisfrequencyinwhichtoob-servethemischosenaccordingtotheexperimentradiofrequencies,1Ð100MHz.Inthetimedomaintheyarerelatedtotheintensityandphasemodulationsofthecarrier.Therefore,inthefrequencydomainthequantumstatestobeconsideredbelongtothesidebandregion.Ifnoenergyexistsinthesemodes,theyareinthevacuumstate,whichalsopossessesintensityandphaseuncertaintiesbuthaszeromeanvalues.Thus,lightwithsidebandsinthevacuumstatepresentsamplitudeandphasenoiseÑanotherwayofunderstandingtheshotnoise.Incontrast,whenlightisexternallymodulatedsuchasinthePoundÐDreverÐHall,wemayassociatethisenergywiththeexistenceofphotonsinthesidebands.Similarly,ifweapplyaspeciÞcquantumdynamicstothebeam,thenoiseinonequadraturebecomessmallerthantheshotnoiseattheexpenseofin-creasingtheconjugatenoise.Thissituationrepresentsasqueezedstateoflight,andinthiscasethesidebandsarealsopopulatedwithphotons.Fortypicalexperimentallyobservedsqueezinglevels,themeannumberofphotonsperfrequencyintervalisontheorderofunity.Thesingle-frequencyquadraturecomponentsaregivenbyTheyhavetheformofbeatsbetweenthecarrieropticalfre-quencyandthesidebandssymmetricallylocatedatthefre-arounditnormalizedbythecarrierabsolute.Thus,despitethefactthatthesidebandshaveaverylowphotonnumber,theireffectcanberecordedbynormalphotodetectorsbecauseoftheenormouscarrierpower.Amplitudeandphaseßuctuationsdifferonlybyaphase:TheamplitudequadratureisinphasewiththemeanÞeldcomplexamplitude,andthephasequadratureisinquadra-turewithit.Itispossibletoconvertphasetoamplitudeßuc-tuationsbymanipulatingtherelativephasebetweenthecar-rierandsidebands.ToshowthisconversionweconsiderhowtheamplitudequadratureofEq.wouldchangeifwecouldshiftthecarrierorthesidebandsbyaphasebysomephysicalmeans.Byaddingthisphase,forexample,tothecomplexsideband,theamplitudequadraturegoestoByvaryingfrom0to2,theamplitudequadratureas-sumesthevalueofthephasequadratureforonepossibletheleadingphaseplaysnorole.Asimilareffectoc-cursbyvaryingtheothersidebandphase.Thecarrierphasecouldbevariedaswell,.Inthiscase,accordingtowhichisequaltothephasequadratureforequalto2and2.Homodynedetectionisdescribedbyanidenticalex-pression,withthedifferencethatrepresentsthelocaloscil-latorphase.Therefore,thepossibilityofindependentlyvary-ingthephasesofthecarrierandsidebandsallowscompleteconversionbetweenthequadratures.Thiseffectiswhatanopticalcavityrealizesclosetoresonance.III.OPTICALCAVITYAnopticalcavityisaregionofspacedelimitedbymirrorswherelightisconÞnedforsometime.Onlycertainfrequen-ciesoflightwhichfulÞllaresonanceconditionareabletoprobethisregion.Everyresonancehasadispersivecharacter.Thus,thedifferentfrequencycomponentsofalightbeamreßectedbyanopticalcavityclosetoresonanceexperiencedifferentphaseshifts.ConsiderthesituationdepictedinFig..Theopticalcav-ityhasacouplingmirrorwithintensityreßection,andanoutputmirror,withreßection1representingspuriouslosses.Theampli-tudereßectionandtransmissioncoefÞcientsare,respectively,=1,2.Thecavityresonancefrequencyclosesttotheincidentlightbeamisdenotedby.Threeimportantparameterscharacterizetheopticalcavity:TheÞ- ,thefreespectralrangeisthecavityperimeter,thatis,thedis-tancetraveledbylightinoneroundtrip,andtheresonanceorfullwidthathalfmaximumTheyrepresenttheampliÞcationoftheÞeldinsidethecavitycomparedtotheincidentÞeld,theinverseoftheround-triptimeofaphoton,andtheinverseoftheaveragetimeapho-tonremainsinsidethecavity,respectively.AnopticalÞeldincidentonthecavitycouplingmirrorgeneratesareßectedbeam Fig.3.Linearopticalcavitywithcouplingmirrorshowingthereßectivityandtheoutputmirrorwithreßectivity.ThereßectedÞeldamplitudethesumoftheincidentamplitudewiththevacuumcoupledbytheoutputmirrorandlosses.Am.J.Phys.,Vol.76,No.10,October2008AlessandroS.Villar ßectedbeamamplitudequadrature.Toshowitsexplicitdependenceoncarrierandsidebands,itmaybewrittenas ¯RR+¯R WesubstituteinEq.theresultsofEqs.shouldbecomparedtoEqs.Forsimplicity,theoutputmirrortransmissionissettozerointhefollowingdiscussion=1and,sothatthelasttwotermsofEq.arezero.Itisalsosupposedthatthecavityactsonlyononefrequencycomponentata,asdepictedinFig..OnlyoneofthephasefactorsappearinginEq.isdifferentfromunityinthissituation.WhenonesidebandisclosetothecavityresonanceseeFig.,Eq.takestheformwhichisanalogoustoEq..IfthecavityisresonanttotheseeFig.,thenasinEq..Inthiscasethecarrierplaysthesameroleasthelocaloscillatorinhomodynedetection,butwithoutintro-ducingextranoiseinthebeam.Inconclusion,thehypotheti-calcontrollablephaseconsideredinthediscussionofSec.IIispreciselyInthegeneralcasewhenthesimpliÞcation1doesnotapply,thedephasingsofthethreefrequencycomponentsin-terfere.Bydecreasingtheanalysisfrequencyevenfurther,thelimitingsituationwherethecarrierandsidebandscannotbedistinguishedfromeachotherinsidethecavitybandwidthisreached:Becauseallcomponentsacquirethesamephaseshift,thereisnoquadratureconversion.Thus,thereisaminimumvalueofforwhichthephasenoiseiscompletelyconvertedtoamplitudenoise.Thisvaluecanbeshowntobe 2byimposing2in.If 2,quadratureconversionispartial;if 2,itiscomplete.Thesamefrequency-dependentdephasingisexploitedbythePoundÐDreverÐHalltechnique,butintheoppositesense:ClassicalsidebandsresultingfromexternalphasemodulationserveasreferencestothemeanÞeld.Evenasmalldetuningbetweenthelaserandthecavityconvertsthephasetoadetectableamplitudemodulation,suchthatthelaserintensityßuctuationworksasasensitiveprobeforthedetuning.TheÞnalexpressionforasafunctionoftheinputisobtainedbyinvertingEqs.fortheincidentÞeldandsubstitutingtheresultsin2isthevacuumßuctuation,andarefunctionsdependentonthecavitycharacteristicsandde-ThecoefÞcientsinvolvethedependenceofontheinputÞeldquadratures,andrelatedtothevacuumcontributions.Theeffectofthesespu-riouslossesistoremovephotonsfromtheresonantÞeldcomponent,partiallytakingitintothevacuum:Ifoneside-bandisresonant,thereßectedbeamnoisetendstoshotnoise;inthecaseofthecarrier,onlytheÒlocaloscillatorÓisattenuated,andnolossofnoiseinformationoccursbecausethesidebandsaretotallyreßected.Ifnocavitylossesarepresent,then=0and=1,andwemay=cos=sinisacavity-dependentparametertoshowtheanalogywithhomodynedetection.Themaindifferencebetweenthesetwotechniquesisthefunctionalformofontheparameters.Althoughitvarieslinearlywiththerelativephasebetweenthelocalos-cillatorandthemeasuredÞeldinhomodynedetection,inthecasetreatedhereresultsfromtheinterferenceofthreephaseshifts,eachofwhichvariesasthearctangentofthecavitydetuningrespectivetoitsfrequencycomponent.V.ROTATIONOFTHENOISEELLIPSEThenoiseattheanalysisfrequencyisdeÞnedasthepowerpresentintheßuctuationsofthisfrequencycompo-nent.Itismeasuredrelativetotheshotnoise,deÞnedhereasthequadraturenoiseofalightbeamwithsidebandsinthevacuumstate.Thenoisespectrumofageneralized=exp+expiscalculatedbytheWienerÐKhintchinetheorem,whichresultsintherelationThisidealnoisespectrumisaverygoodapproximationtotheactualmeasuredquantityifthemeasurementintegrationtimeismuchlongerthanthetypicaltimescaleofthesystemvariations,asassumedinEq..ThedeltafunctioninEq.meansthatthetreatmentassumesperfectlydeÞnedfre-quencies;itdisappearsassoonastheÞniteprecisionofthefrequencydeÞnitionisincluded. Fig.6.NoiseellipserepresentationinthecomplexplaneshadedellipseThedottedcirclerepresentstheshotnoise.Theellipsesizecomparedtothemeanvalueisexaggeratedforeaseofvisualization.Am.J.Phys.,Vol.76,No.10,October2008AlessandroS.Villar abitclosertotheshapeofacircle.For1,thecarrierisresonant,whilethesidebandsarefaroffresonance6Ð12.Thecarrierexperiencesa2phaseshiftat0.5,givingrisetotwocompleteconversionsbetweenquadraturesframes7and11.Thevacuumnowcontaminatesonlythecarrierandattenuatesit.Becauseonlythelocaloscillatorisattenuated,thereisnovacuumcontri-butiontothenoisemeasurement,andperfectphasenoiseisobserved.Hence,peaks7and11areslightlyhigherthanpeaks3and15.Thecarrierrotatesbyatexactresonanceframe9,withoutanyaffectonthereßectedbeamnoisespectrum,andweobserveamplitudenoiseoncemore.Thenoiseellipserotatesagainastheothersidebandentersreso-frames14Ð16.ThisnoiseproÞlegivescompletein-formationaboutthenoiseellipse,allowingfortherecon-structionofthesidebandstatebyquantumtomography.Allthreefrequenciesdephasingsinterfereastheanalysisfrequencyisdecreased.Peaks3and7aswellaspeaks11and15approacheachotherandÞnallycollapseinonesinglepeakfor 2.Belowthisanalysisfrequencytheiramplitudesdecrease,becausethecarrierandsidebandscan-notbeperfectlydistinguishedinsidethecavitybandwidth,andapartialellipserotationoccurs.TheseregimescanbebettercharacterizedbyobservingthedetuningsforwhichhaszeroderivativesasisvariedseeFig..Forsymmetryreasonsonly0isconsidered,andaperfectcavityisnowassumed.For 2,thereisonlyonedetuningwhereazeroderivativeoccurs,becausephasetoamplitudeconversionispartialopencircles.Asincreasesabove 2,threepositivede-tuningswithzeroderivativesappearintotal.Twoofthemcorrespondtothefullconversionofphasetoamplitudenoise,andthethirdcorrespondstotheinßectionpointbetweenthetwocompletenoiseconversionsfullcircles.As,thesidebandsandcarrierphaseshiftsnolongerin-terfere.TheÞrstfullconversion,duetothecarrierrotation,occursatfulltriangles,andthesecondone,duetothesidebandrotation,occursatopentriangles,asexpected.Thegraylinesshowthisasymptoticbehavior.Itcanbeshownthatthezeroderivativepointsalsodependonthelosses.Asisvariedforagivenanalysisfrequency,thesmallestdetuningforwhichcompleteconversionoccursgoestozeroas.ForthisreasonthePoundÐDreverÐHallmethodhasaverysteeperrorsignalclosetoexactreso-nanceandismosteffectivewhenVI.CLASSROOMEXPERIMENTAclassroomdemonstrationofthiseffectcanbeperformedwiththesameapparatususedinthePoundÐDreverÐHallanopticalcavity,alaser,aphasemodulator,aphotodetector,andaspectrumanalyzerThecavityshouldbestable,possiblybuiltasarigidme-tallicbody.Theringgeometrywouldbemoreconvenientthanthelinearone,becauseitdoesnotrequireanopticalisolator,waveplate,andpolarizingbeamsplitter.Becausethequantumnoiseisveryweak,itwouldbemorepracticaltomodulatethelaserphasetocreatestrongsidebands.ThecavitybandwidthhastobecompatiblewiththisfrequencyseeFigs..Thebeamreßectedbythecavityismea-suredwithaphotodetector.Thefastcomponentsofthepho-tocurrentarethensenttoaspectrumanalyzersettomeasureatasinglefrequency,thesameusedtomodulatethebeamphase.AsthelaserfrequencyislinearlyscannedaroundthecavityresonancethecavitytransmissionorreßectionproÞlecanbemonitoredtocheckfortheresonancecondition,thenoiseproÞleshouldshowthefeaturesinFig..ItwouldbeinterestingtochangethelasermodulationfrequencyandcheckhowitaffectsthenoiseproÞle.Alternatively,wecouldemployadiodelaser,whichpossessesstrongphasenoisebutlowamplitudenoise,toeliminatethephasemodu-lator.Inthiscasewewoulddirectlymeasurethelaserphasenoise;however,thismeasurementmightrequireustoam-plifythephotocurrentsignal.Itisnaturalthatsuchademon-strationbepartofalectureontheopticalcavityandonthequantumnoiseoflight.VII.CONCLUSIONAdetaileddescriptionofthephysicalprocessthatallowsanemptyopticalcavitytoconvertphasetoamplitudenoiseofabrightlightbeamhasbeenpresented.Wehaveconsid-eredamplitudeandphaseßuctuationsasfrequencyside-bandsaroundthecarrieropticalfrequency.Formeasure-mentsbasedonphotodetectors,onlythelightßuctuationsinphasewiththecarriertheamplitudequadraturecanbemeasured.However,thedispersivecharacterofanopticalcavityresonancecanbeusedtointroducearelativephase 0.00.51.01.52.02.53. 0 0.00.52.5 3 . 0 Analysisfrequency, Fig.9.Detunings0forwhichhaszeroderivativesasafunc-tionoftheanalysisfrequency.Eachsymbolrepresentsadifferentzeroderivativepoint.At 2asingledetuningwithzeroderivativegivesrisetothreesuchpointsasincreases,becausethecompleteconversionofphasetoamplitudenoisebeginstobepossible.Thegraycurveshelptovisualizetheasymptoticbehavior.Lossesareassumedtobezero Fig.10.Schematicoftheexperimentalsetuprequiredtoperformaclass-roomdemonstrationoftheeffect.PM:phasemodulator.Am.J.Phys.,Vol.76,No.10,October2008AlessandroS.Villar betweenthecarrierandthesidebands.Forcertaindetuningstheconversionofphasetoamplitudenoiseinthebeamre-ßectedbythecavityoccurs:Inphasespace,theÞeldcarrierandnoiseellipserotaterelativetoeachother.Thecarrierhasacompletelyanalogousroletothelocaloscillatorusedinthehomodynedetectiontechnique,withtheadvantageofperfectspatialandtemporaloverlapandnoadditionalnoise.Fur-thermore,everyquadratureiseasilyaccessible,asinhomo-dynedetection,givingcompleteinformationabouttheÞeldquantumstate.TherotationofthenoiseellipseoflightcanbeviewedasanapplicationofthePoundÐDreverÐHalltech-niqueintheoppositesense:Insteadofusingclassicalside-bandstomeasurethedetuningbetweencarrierandarefer-encecavity,itemploysthecarrierasalocaloscillatortothequantumsidebandsofbrightlightbeams.TheauthorgratefullythankPauloNussenzveigandMar-celoMartinellifortheiradvicethroughoutmystudentyears,andforposingtomethequestionsdiscussedinthisarticle:Thinkingabouttheirintuitivemeaninghasprovidedmealotofthejoyofphysics.Idedicatethisworktothem.IalsothankKatiusciaN.Cassemiroforvaluablesuggestions,WolfgangSchleichforhisencouragement,andGerdLeuchsforhisinterestinthismanuscript.ThisworkwassupportedbyFundaodeAmparoPesquisadoEstadodeSoPauloFAPESPandtheEuropeanCommissionthroughtheSCALAnetwork.Electronicmail:alessandro.villar@uibk.ac.at;Formeraddress:InstitutodeFsica,UniversidadedeSoPaulo,CaixaPostal66318,05315-970SoPaulo,SP,Brazil.M.A.NielsenandI.L.Chuang,QuantumComputationandQuantumCambridgeU.P.,Cambridge,2000S.L.BraunsteinandP.vanLoock,ÒQuantuminformationwithcontinu-ousvariables,ÓRev.Mod.Phys.,513Ð577G.Grynberg,A.Aspect,andC.Fabre,AnIntroductiontoLasersandQuantumOpticsCambridgeU.P.,Cambridge,2001A.Einstein,B.Podolsky,andN.Rosen,ÒCanquantum-mechanicalde-scriptionofphysicalrealitybeconsideredcomplete?,ÓPhys.Rev.R.JozsaandN.Linden,ÒOntheroleofentanglementinquantum-computationalspeed-up,ÓProc.R.Soc.London,Ser.A,2011Ð2032T.M.Forcer,A.J.G.Hey,D.A.Ross,andP.G.R.Smith,ÒSuperposi-tion,entanglementandquantumcomputation,ÓQuantumInf.Comput.97Ð116P.Nachman,ÒMach-Zehnderinterferometerasaninstructionaltool,ÓAm.J.Phys.,39Ð43H.A.BachorandT.C.Ralph,AGuidetoExperimentsinQuantumWiley-VCH,Weinheim,2004A.S.Villar,L.S.Cruz,K.N.Cassemiro,M.Martinelli,andP.Nussen-zveig,ÒGenerationofbrighttwo-colorcontinuousvariableentangle-ment,ÓPhys.Rev.Lett.,243603-1Ð4E.D.Black,ÒAnintroductiontoPoundÐDreverÐHalllaserfrequencystabilization,ÓAm.J.Phys.,79Ð87R.W.P.Drever,J.L.Hall,F.V.Kowalski,J.Hough,andG.M.Ford,ÒLaserphaseandfrequencystabilizationusinganopticalresonator,ÓAppl.Phys.B,97Ð105M.D.Levenson,R.M.Shelby,A.Aspect,M.Reid,andD.F.Walls,ÒGenerationanddetectionofsqueezedstatesoflightbynondegeneratefour-wavemixinginanopticalÞber,ÓPhys.Rev.A,1550Ð1562P.Galatola,L.A.Lugiato,M.G.Porreca,P.Tombesi,andG.Leuchs,ÒSystemcontrolbyvariationofthesqueezingphase,ÓOpt.Commun.D.F.WallsandG.J.Milburn,QuantumOpticsSpringer-Verlag,Berlin,R.W.HenryandS.C.Glotzer,ÒAsqueezed-stateprimer,ÓAm.J.Phys.,318Ð328A.K.EkertandP.L.Knight,ÒCorrelationsandsqueezingoftwo-modeoscillations,ÓAm.J.Phys.,692Ð697P.G.KwiatandL.Hardy,ÒThemysteryofthequantumcakes,ÓAm.J.,33Ð36C.FabreandS.Reynaud,ÒQuantumnoiseinopticalsystems:Asemi-classicalapproach,ÓinLesHouchesSessionLIII,editedbyJ.Dalibard,J.M.Raimond,andJ.Zinn-JustinElsevier,NewYork,1992,pp.675Ð711.ThisconceptfailsforveryweakÞelds,becauseaphaseoperatorisadifÞcultobjecttodeÞnerigorouslyseeRef.W.SchleichandJ.A.Wheeler,ÒOscillationsinphotondistributionofsqueezedstatesandinterferenceinphasespace,ÓNature574Ð577NotethatcorrespondstotwicethedistancebetweenthemirrorsinthespeciÞccaseofalinearcavity.Thisnotationisemployedtoeasilyallowforanycavitygeometry.L.MandelandE.Wolf,OpticalCoherenceandQuantumOpticsbridgeU.P.,Cambridge,1995C.W.Gardiner,QuantumNoiseSpringer-Verlag,Berlin,1991G.C.Bjorklund,M.D.Levenson,W.Lenth,andC.Ortiz,ÒFrequencyspectroscopy,ÓAppl.Phys.B,145Ð152ThedeÞnitionofthenoisepowerinvolvesthevarianceoftheßuctuationstheellipse.IthasbeentacitlyassumedthattheÞeldstatepossessesaGaussianWignerfunctiontheßuctuationssatisfyGaussianstatisticsbecauseitisthecommonsituationfoundinthelaboratory.Higherordermomentsnecessarytoreconstructthestateinthemoregeneralcaseareinprincipleaccessiblefromthedirectmeasurementoftheßuctuations,asshownbyEq.,giventhatthelinearizationprocedureofEq.applies.ThestatisticsofanyquadraturedirectioninphasespacecanbecompletelytransferreddisregardinglossestothereßectedamplitudeJ.Zhang,T.Zhang,K.Zhang,C.Xie,andK.Peng,ÒQuantumself-homodynetomographywithanemptycavity,ÓJ.Opt.Soc.Am.BA.Zavatta,F.Marin,andG.Giacomelli,ÒQuantum-statereconstructionofasqueezedlaserÞeldbyself-homodynetomography,ÓPhys.Rev.A,043805-1Ð8Am.J.Phys.,Vol.76,No.10,October2008AlessandroS.Villar