Joint Temporal Density Measurements for Two-Photon State Characterization

We demonstrate a new technique for characterizing two-photon quantum states based on joint temporal correlation measurements using time resolved single photon detection by femtosecond upconversion. We measure for the first time the joint temporal density o

JointTemporalDensityMeasurementsforTwo-PhotonStateCharacterization

OnurKuzucu,1FrancoN.C.Wong,1SunaoKurimura,2andSergeyTovstonog2

1

ResearchLaboratoryofElectronics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA

2

NationalInstituteforMaterialsScience,1-1Namiki,Tsukuba-shi,Ibaraki305-0044,Japan

(Dated:July10,2008)

Wedemonstrateanewtechniqueforcharacterizingtwo-photonquantumstatesbasedonjointtemporalcorrelationmeasurementsusingtime-resolvedsingle-photondetectionbyfemtosecondup-conversion.Wemeasureforthe rsttimethejointtemporaldensityofatwo-photonentangledstate,showingclearlythetimeanti-correlationofthecoincident-frequencyentangledphotonpairgeneratedbyultrafastspontaneousparametricdown-conversionunderextendedphase-matchingconditions.Thenewtechniqueenablesustomanipulatethefrequencyentanglementbyvaryingthedown-conversionpumpbandwidthtoproduceanearlyunentangledtwo-photonstatethatisexpectedtoyieldaheraldedsingle-photonstatewithapurityof0.88.Thetime-domaincorrela-tiontechniquecomplementsexistingfrequency-domainmeasurementmethodsforamorecompletecharacterizationofphotonicentanglementinquantuminformationprocessing.

PACSnumbers:42.50.Dv,42.79.Nv,42.50.Ar,42.65.Lm

arXiv:0807.1573v1 [quant-ph] 10 Jul 2008

Spontaneousparametricdown-conversion(SPDC)isapowerfulmethodforgeneratingtwo-photonstatesforquantuminformationprocessing(QIP).Thejointquan-tumstatecanbeengineeredforspeci cQIPapplicationsbytailoringitspolarization,momentum,andspectralde-greesoffreedom.Ultrafast-pumpedSPDCisofgreatinterestbecauseawellde nedtimeofemissionisdesir-ableinclockedapplicationssuchaslinearopticsquan-tumcomputing(LOQC)[1].InultrafastSPDC,spectralengineeringofthetwo-photonstatecanbeaccomplishedbymanipulatingthecrystalphase-matchingfunctionandthepumpspectralamplitudetoyielduniqueformsoftwo-photonfrequencyentanglement.Forexample,coincident-frequencyentanglementwithstrongpositivecorrelationbetweensignalandidleremissionfrequenciescanbeusedtoimprovetime-of- ightmeasurementsbe-yondthestandardquantumlimit5].Ontheotherhand,onecanutilizeatwo-photonstatewithnegligiblespectralcorrelationstoimplementaheraldedsourceofpure-statesinglephotons,whichcanbeavaluablere-sourceforLOQC[6,7].

Characterizingthespectralcorrelationsofatwo-photonstatecanbedonebymeasuringthejointspectraldensity(JSD)pro lewithtunablenarrowband lteringofthesignalandidler[6,7,8].Hong-Ou-Mandelquan-tuminterference[9]isalsousefulforquantifyingthetwo-photoncoherencebandwidthandtheindistinguishabil-ityofthephotonpair.However,thetwomeasurementsdonotgivethewholepictureofthetwo-photonstate.Bothmeasurementsareinsensitivetothespectralphaseandthereforecannotcapturethetime-domaindynamicsunlessthejointstateisknowntobetransformlimited.Moreover,JSDmeasurementsinwavelengthregionswithlowdetectore ciencyorhighdetectornoisecanbechal-lengingduetolongacquisitiontimesandlowsignal-to-noiseratios.Frequency-domaintechniquesforestimat-ingthespectralphaseexist,buttheyarenotsimpleto

implementinpracticeInultrafastopticsultrashortpulsesareroutinelyana-lyzedspectrallyandtemporally,buttime-domaincharac-terizationtoolsarenoteasytoimplementforsinglepho-tons.Recentlywehaveintroducedatime-resolvedsingle-photonmeasurementtechniquebyuseoffemtosecondupconversion[11].InthisLetterweutilizethissingle-photontime-domaincharacterizationmethodtomeasureforthe rsttimethejointtemporaldensity(JTD)pro- leofatwo-photonquantumstate.Inparticular,wemeasureddirectlythetimecorrelationsofsignal-idlerar-rivaltimesofultrafastpumpedSPDCunderextendedphasematchingconditionsshowingclearlythatthecoincident-frequencyentangledphotonsweretimeanti-correlated.Furthermore,byvaryingtheSPDCpumpspectrum,wewereabletomanipulatethetemporalcor-relationsofthesignalandidler,andobtainanearlyun-entangled(temporally)two-photonstate.Thisnewtech-niquecanbeusedinconjunctionwithfrequency-domainmethodstoprovideamorecompletecharacterizationofsingleandentangledphotons.

Toproperlyde neJTD,we rstexpressthetwo-photonstateintime-domainvariables|Ψ =

dτSdτIA(τS,τI)|τS |τI ,wherethesingle-photonFockstateisde nedas|τj ≡a (τj)|0 ,forj=S,I.Thetem-poralcorrelationsofthesignalandidleraredeterminedbythejointtemporalamplitude,A(τS,τI),andwede-2

netheassociatedprobabilitydensity,(|A(τS,τI)|),asthejointtemporaldensity.Analogoustothefrequency-domainmethods,theJTDcanbemeasuredbyusingnarrowbandtemporal lteringandcoincidencedetection.FortypicalultrafastSPDCexperiments,timingreso-lutionof～100fsisneededformeasuringarrivaltimesofsinglephotons.Currentsingle-photondetectorswithtensofpicosecondstimingresolutionarenotsuitableforthispurpose.Forthetwo-photonJTDmeasurement,weappliedourrecentlydevelopedtime-resolvedsingle-

We demonstrate a new technique for characterizing two-photon quantum states based on joint temporal correlation measurements using time resolved single photon detection by femtosecond upconversion. We measure for the first time the joint temporal density o

FIG.1:(Coloronline)(a)Synchronizedupconversionanddownconversionexperimentdrivenbythesameultrafastpump.(b)Noncollinearphase-matchinggeometryforsingle-photonupconversion.IF:interference lter;DM:dichroicmirror;FPBS: berpolarizingbeamsplitter.

photon

upconversiontechniquewithatemporalresolu-tionof～150fs[11].Anultrafastupconvertingpumppulsewasusedtotime-stampthesignalandidlerar-rivaltimes,andwemappedtheirrelativearrivaltimesbyvaryingtheinputdelaylinesindependentlyandrecordingthecoincidencesbetweenthetwoupconversionchannels.Thecoincidencestatisticsyieldedthetemporalstructureofthetwo-photonstate.

Ourexperimentalsetupforultrafasttype-IIphase-matchedSPDCandsubsequentJTDmeasurementwithtime-resolvedupconversionisshowninFig.1(a).BothSPDCandupconversionwerepumpedsynchronouslywiththesameultrafastsourceat790nmwitha6-nmbandwidthand80MHzrepetitionrate,therebyelimi-natingthepumptimingjitterfortheJTDmeasurement.WeoperatedthePPKTPSPDCcrystalunderextendedphase-matchingconditionstogenerateacoincident-frequencyentangledtwo-photonstate[3,4].ByFourierduality,thispositivefrequencycorrelationcorrespondedtoanti-correlationinthetimedomainwherethesignalandidlerphotonswith～350-fssingle-photoncoherencetimesweresymmetricallylocatedaboutthecenterofa～1.4-pstwo-photoncoherencetimewindow,asmeasuredbyHOMinterference[4].Thesignalandidlerphotonswerecoupledintoapolarization-maintainingsingle-mode berandseparatedata berpolarizingbeamsplitter.Thesignalandidlerdelaylineswereindividuallyad-justedsothattheyarriveattheupconversioncrystalinthesametimeslotasthepumppulse.Finetuningoftherelativetimingcanbeachievedwithtranslationstages.WeusedthesamesetupasinRef.[11]fortime-resolvedsingle-photonupconversion,brie ydescribedhere.AssketchedinFig.1(b),a1-mmlongperiodicallypoledMgO-dopedstoichiometriclithiumtantalate(PP-MgSLT)crystalwitha8.5µmgratingperiodwasusedfornoncollineartype-0phase-matchedsum-frequencygen-eration(1580nm+790nm→526.7nm).Weusedthenoncollineargeometrytoimplementtwoindependentup-

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converterswithasinglecrystal.Thesingle-photonbeamswerealignedparalleltothepumpbeamwith～3mmlat-eraland～1.5mmverticalseparationfromthepumpaxis,andtheywerefocusedintothePPMgSLTcrystal.Thenon-planarfocusingcon gurationallowedustoavoidthesimultaneousdetectionofthenon-phase-matchedpara-metricphotonpairsthatwerebothgeneratedandupcon-vertedbythepumpatthePPMgSLTcrystal.Therefore,evenwitha nitebackgroundforsingles,thecoincidencepro leshowsnegligibleaccidentals[11].Theupconvertedoutputswere lteredbydichroicmirrorsand10-nmpass-bandinterference lters,coupledintosingle-mode bersanddetectedwith ber-coupledSiAPDs.Werecordedthesinglescountsandalsothecoincidencecountsbe-tweenthetwoSiAPDswithina1.8nscoincidencewin-dow.

s

tnuoC dezialmroN-2000-1000010002000

Pump Delay [fs]

s

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Pump Delay [fs]

FIG.2:(Coloronline)Normalizedsingles(a)andcoincidence

(b)histogramsbytime-resolvedupconversion.Thepumppulsewasscannedthroughcollocatedsignalandidlerarrivalwindows.SolidlinesareGaussian tstothedata.

Wemeasuredthesinglesandcoincidencesbyscan-ningtheupconversionpumppulsedelayrelativetothesignalandidlerarrivalwindows,andeachdatapointwasaveragedfor60seconds.ThenormalizedhistogramsareplottedinFig.2withoutanybackgroundsubtrac-tion.Fortheoptimalpumppowerratio(～360mWfordownconversion,～580mWforupconversion)themaxi-mumsingles(coincidence)rateatthecenterofthedistri-butionwas～5300/s(～17/s),includingthebackground.Thebackgroundlevelinsinglescountswere～1900/sfortheoptimalpumppower-ratio,correspondingtoaback-groundprobabilityperpulseof～2.4×10 5.Thetem-poralwidthforsinglesdistributionwas～1.3ps,consis-tentwiththetwo-photoncoherencetimeof～1.4ps[4].Duetothetimeanti-correlatedgenerationofsignalandidler,thecoincidencepro leexhibiteda～165fsFWHMwidth,whichissigni cantlynarrowerthanthesingles

We demonstrate a new technique for characterizing two-photon quantum states based on joint temporal correlation measurements using time resolved single photon detection by femtosecond upconversion. We measure for the first time the joint temporal density o

histograms.Astheupconversionpulsewasscannedthroughthearrivalwindowsofbothphotons,theonlyin-stancewherethetwoupconverterscouldsimultaneouslydetectphotonswasaroundthetimeorigin.Foranup-conversionpumppowerof580mW,the

internalconver-sione ciencywasestimatedtobe25%[11].However,theupconversionprobabilityperpumppulsewasactu-allylowerbecausethepumppulsewasmuchshorterthanthee ectivepulsewidthofthesignalandidler.Inordertomanipulatethejointtemporalamplitudewithouta ectingtheupconversiontimingperformance,wemodi edonlytheSPDCpumpbandwidthbyin-sertinga lterfromasetofinterference lters(3-dBbandwidths:3.6nm,2.1nm,and1.1nm)beforethePP-KTPcrystal.Themeasurednormalizedcoincidencehis-togramsfordi erentSPDCpumpbandwidthsareplot-tedinFig.3.AstheSPDCpumpbandwidthwasre-duced,thesingle-photoncoherencetimeincreasedandconsequentlythecoincidencepeaksbecamewider.Inthesame gure,wealsoshowthetheoreticalpredictionsforthecoincidencehistogramsthatwecalculatebasedonthejointtemporalamplitudewitha nite-durationupconversionpumppulse.Theparametersforthecalcu-lationaretheupconversionanddownconversionpumpbandwidthsandthetwo-photoncoherencebandwidththatwemeasuredwiththeHOMinterference[4].Weassumea atspectralphasepro leinourjointtempo-ralamplitudecalculationleadingtopredictedtemporalcoincidencepro lesthatsuggesttransform-limitedtwo-photonstates.ThegoodagreementinFig.3betweendataandtheoryindicatesthattheSPDCoutputpho-tonpairswereindeedclosetothetransformlimit.Thisobservationisonlypossiblewithtime-domainmeasure-mentsbecausefrequency-domainmethodswouldbein-sensitivetodispersivebroadeningofthephotons.

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Pump Delay [fs]

FIG.3:(Coloronline)NormalizedcoincidencehistogramsforvariousSPDCpump3-dBbandwidths:(6-,3.6-,2.2-,and1.1-nm).Theoreticalcoincidencepro lesareplottedasdashedlines.

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Thetime-resolvedupconversionmethodenabledustomeasurethejointtemporaldensitybyvaryingthesig-nalandidlerrelativedelaysindependently.Wesettheupconversionpumpbandwidthto～6nm,andwemadetheJTDmeasurementsusingoneofthefourSPDCpumpbandwidths.Thecoincidencecountswererecordedoveratwo-dimensionaltimegridwith60-saveragingforeachdatapoint.ForallSPDCpumpbandwidthsex-cept1.1nm,thegridsizeforthetimedelayswassetto2ps×2ps(witha133fsstepsize).Weincreasedthegridsizeto4ps×4ps(266fsstepsize)forthe1.1nmpumpbandwidth.ThenormalizedcoincidencedataforallSPDCpumpbandwidthsareshownassurfaceplotsoverthetwo-dimensionaltimegridsinFig.4(a)-(d).WeseedramaticchangesintheJTDpro lewithachangeoftheSPDCpumpbandwidth.Witha6nmSPDCpumpbandwidththeJTDcoincidencepro leclearlyexhibitstimeanti-correlationthatisindicativeoftwo-photoncoincident-frequencyentanglement[4].Withsmallerpumpbandwidths,theJTDdistributionsbecomemoresymmetric,whichcorrespondstoreducedtemporalandspectralcorrelations.

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FIG.4:(Coloronline)Experimentaljointtemporaldensitiesforvariousdownconversionpump3-dBbandwidths:(a)6nm,(b)3.6nm,(c)2.2nm,(d)1.1nm.

Wecanquantifythetwo-photonfrequencyentangle-mentasafunctionofthepumpbandwidthbasedonthemeasuredJTDdistributionsandbyusingSchmidtde-compositionforcontinuousvariables[12].Inthisformal-ism,thejointtemporalamplitude,A(τs,τi),isexpressedasadiscretesumofthetemporaleigenmodeswitheigen-valuesλn,throughwhichtheentanglemententropybecomputedasS= n

cank=1λklog2λk[12].Figure5showsthecomputedentanglemententropyfromtheex-perimentalJTDdistributionsinFig.4assumingthatthejointstateistransformlimited.Forcomparison,wehave

We demonstrate a new technique for characterizing two-photon quantum states based on joint temporal correlation measurements using time resolved single photon detection by femtosecond upconversion. We measure for the first time the joint temporal density o

alsocalculatedthetheoreticalentropycurvesasafunc-tionoftheSPDCpumpbandwidth,wherethepumpspectrumis

assumedtobeGaussian.TwocurvesareplottedinFig.5,onerepresentingaGaussianandtheotherasincphase-matchingfunction.ForaGaussianphase-matchingfunction,afullyfactorizabletwo-photonstateispredictedwithapumpbandwidthof～1.2nm,andyieldinganentropyofzero.Forthemorerealis-ticsincfunctionforthephasematching,ahighlybutnotcompletelyfactorizabletwo-photonstateisachiev-able.Sincethesinc-typespectralresponsecorrespondstoaboxcarshapeinthetimedomain,itnecessitatestheinclusionofhigherorderSchmidtmodesandhenceincreasestheentanglemententropy.

Figure5showsagoodqualitativeagreementbetweenthetheoreticalentropycurvesandtheentropyvaluesobtainedfromtheJTDdistributions.Theentangle-mententropycorrespondingtotheexperimentalJTDpro lesarelowerthanthetheoreticalcurveforthesincphase-matchingfunction.Thisisreasonableifwetakeintoaccountthattheactualtime-domainpro leofthephase-matchingfunctionissmootherthanaboxcarshapebecauseofgratinginhomogeneity,ascon rmedbythesingleshistogrammeasurementsofFig.2.Therefore,theexperimentalJTDdistributionscanbeexpressedwithasmallernumberofSchmidtmodes,resultinginalowerentanglemententropythanthatofthetheo-reticalofasincfunction.Fora1.1-nmSPDCpumpbandwidth,whichyieldsanoutputthatisnearlyfac-torizable,wehavecomputedthepurityoftheheraldedsingle-photonstate=Tr( ρS)= ∞as～0.88,wherepurityisde nedasp2

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n=0λn[6,12].Thispurityvaluecom-pareswellwiththatofthepure-statesinglephotonsgen-eratedunderSPDCusingadi erentspectralengineeringmethod[7].Webelievethatthepuritycanbefurtherimprovedby nercontroloverthepumpbandwidthandadditionalspectral ltering.Incomparison,theoutputforthecaseofa6-nmSPDCpumpbandwidthyieldsapurityof～0.38,whichisaconsequenceofthehighdegreeofcoincident-frequencyentanglement.

Inconclusion,wehavedevelopedatime-domainmeasurementtechniqueforsinglephotonswithsub-picosecondresolutionthatweusedtomeasurethetwo-photonjointtemporaldensityforthe rsttime.Weappliedthetechniquetoverifyanti-correlationinthearrivaltimesofthesignalandidlerphotonsthatwerecoincident-frequencyentangled.Finally,thenewtoolal-lowedustomonitorthee ectofvaryingtheSPDCpumpbandwidths,leadingtothegenerationofanearlyfactor-izabletwo-photonstate,whichshouldbeofinteresttomanyquantuminformationprocessingapplications.WebelievethattheJTDmeasurementtechniqueisapow-erfultoolforengineeringtemporalandspectralcorrela-tionsofultrafastSPDCphotons.Suchacharacteriza-tiontechniquewouldcomplementthefrequency-domain

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counterpartstoquantifyandmanipulatemulti-photon

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Pump bandwidth [nm]

FIG.5:(Coloronline)Entanglemententropyvaluescalcu-latedfromexperimentalJTDdistributionsforvariousSPDCpumpbandwidthsofFig.4.Thetheoreticalentropyvaria-tionsforGaussian(black)andsinc-type(red)phase-matchingfunctionsaregiveninsolidcurves.

entanglementforquantuminformationprocessingappli-cations.

ThisworkwassupportedinpartbytheHewlett-PackardLaboratoriesandbytheNationalInstituteofInformationandCommunicationsTechnology,Japan.

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