IOPPOURNALOFB:ATOMIC,OLECULARAND - PDF document

IOPPOURNALOFB:ATOMIC,OLECULARAND J.Phys.B:At.Mol.Opt.Phys.(2008)035001(12pp)High-precisionmeasurementofhyperÞnestructureinthelinesofalkaliatomsDipankarDasandVasantNatarajanDepartmentofPhysics,IndianInstituteofScience,Bangalore560012,Indiavasant@physics.iisc.ernet.in 2 K(KI(I)J(J F(FI(I J.Phys.B:At.Mol.Opt.Phys.(2008)035001DDasandVNatarajan TUNABLELASER spectrometer controlLIALIA spectrometerDemod Figure1.Schematicoftheexperiment.FigurekeyÑAOM:acousto-opticmodulator,LIA:lock-inampliÞer,BS:beamsplitter,M:mirror.splitsintotwohyperÞnelevelswithenergyshiftsdeterminedbythemagnetic-dipoleinteraction.Theelectric-quadrupoleinteractionispresentonlyforI,J1,andhencedoesnotaffectthisstate.Ontheotherhand,thestate(splitsintofourhyperÞnelevelswithenergyshiftsdeterminedbybothinteractions.InthecaseofCs,wewillseelaterthatthereisevidenceofasmalladditionalcontributionfromamagnetic-octupolehyperÞneinteraction.2.ExperimentaltechniqueTheexperimentalschematictomeasurehyperÞneintervalsisshowninÞgure,andisessentiallythesameforallatoms.Theoutputfromafrequency-stabilizedtunablelaserissplitintotwoparts.TheÞrstpartgoesintoanatomicspectrometer,thesignalfromwhichisusedtolockthelasertoaparticularhyperÞnetransition.ThesecondpartisfrequencyshiftedthroughanAOMandthensentintoasecondatomicspectrometer.ThefrequencyoftheAOMisadjustedsothattheshiftedbeamisonaneighbouringhyperÞnetransition.TheerrorsignalfromthispeakisfedbacktotheAOMdrivertolockitsfrequency.ThustheAOMfrequencygivesadirectmeasurementofthehyperÞneinterval.ForintervalsthataretoosmallortoolargetobemeasuredwithasingleAOM,weuseanadditionalAOMwithaÞxedfrequencyoffset.linesinsodiumareinthevisible(at589nm);hencethetunablelaserisaringdyelaser(Coherent699-21).Thelaserisfrequencystabilizedtoareferencecavitythatgivesitaninstantaneouslinewidthof1MHz.Thelinesinalltheotheratomsareinthenearinfraredwhichcanbeaccessedusingdiodelasers.Thediodelaserisfrequencystabilizedusingopticalfeedbackfromapiezo-mountedgratingsothattheresultinglinewidthisabout500kHz.Thelinewidthisreducedconsiderablywhenthelaserislocked.Forlocking,thelaserisfrequencymodulatedat20kHzandthesignalfromthespectrometersisdemodulatedat3togeneratetheerrorsignals.Suchthird-harmonicdetectionprovidesnarrowdispersivesignalsthatareinsensitivetointensityßuctuations,topeakpullingfromneighbouringtransitions,ortoanyresidualDopplerproÞleinthespectrum[InthecaseofLi,thespectroscopyisdoneusingacollimatedatomicbeam.Theatomsareexcitedbyaperpendicularlaserbeam,andtheresultingDoppler-freeßuorescencesignalismeasuredwithaphotomultipliertube(PMT).Inalltheotheratoms,weuseabsorptionspectroscopyofaprobebeamthroughavapourcell.Thecelliseitheratroomtemperature(RbandCs)orheatedto75C(NaandK),tohaveanatomicdensityofabout10.InordertogetDoppler-freeabsorptionspectra,weuseoneoftwotechniques.TheÞrsttechniqueisthenormalsaturated-absorptionspectroscopy(SAS)[],whereastrongcounter-propagatingpumpbeamdecreasestheabsorptionoftheweakprobebeamonresonance.Asiswellknown,theSAStechniqueproducesadditionalcrossoverresonances,whichoccurexactlymidwaybetweentwohyperÞnepeaks.Forcloselyspacedlevels,thelargecrossoverresonancesoftenswampthetruepeaks.ThisisaproblemespeciallyinthelinesofNaandK,wherethelevelspacingisoftheorderofoneortwonaturallinewidths.Toovercomethisproblem,wehaverecentlydevelopedanalternativetechniqueofhigh-resolutionspectroscopy,whichwecallcoherent-controlspectroscopy(CCS)[].Thistechniqueusesco-propagatingcontrolandprobebeams,anddoesnotproducecrossoverresonances.TheprobebeamislockedontoahyperÞnetransition,whilethecontrolbeamisscannedacrossaneighbouringtransition.Whenthecontrolcomesintoresonancewithatransitioncouplingtothesamegroundlevel,itÔcoherentlyÕreducesprobeabsorptionthroughaprocesssimilartoelectromagneticallyinducedtransparency.Sincetheprobeislockedonatransition,itaddressesonlythezero-velocityatoms,andtheresultantlineshaperemainsDoppler-freeeveninhotvapour.Wehavealsodoneadensity-matrixanalysisoftheeffectivethree-levelatominthepresenceoftheprobeandcontrolbeams.Thecalculatedlineshape,afteraccountingforthermalaveraginginhotvapour,describesthemeasuredspectrumquitewell[].Apartfromtheabilitytoresolvecloselyspacedlevels,anotherimportantadvantageoftheCCStechniqueformeasuringhyperÞneintervalsisthatthemeasuredvalueiscompletelyinsensitivetodetuningoftheprobebeam.Anydetuningoftheprobefromresonancewouldimplythatitisresonantwithanon-zerovelocitygroup.Sincethecontrolbeamco-propagateswiththeprobebeam,thespectrumwillshowpeaksonlywhenthecontrolbeamcomesintoresonancewiththesamenon-zerovelocitygroup,whichhappensonlywhentheAOMshiftmatchesthehyperÞneinterval.Thus,astheprobebeamisdetunedfromresonance,themanifoldofpeaks(correspondingtothedifferenthyperÞnelevels)willshiftwithintheDopplerproÞle,buttheirrelativeseparationwillremainthesame.3.ErroranalysisThedifferentsourcesoferrorinthemeasurementhavebeendiscussedextensivelyinourearlierpublications,andarereviewedhereforcompleteness.3.1.StatisticalerrorsTheprimarysourcesofstatisticalerroraretheßuctuationsinthelockpointofthelaserandtheAOM.Tominimizethese J.Phys.B:At.Mol.Opt.Phys.(2008)035001DDasandVNatarajan 0.8 0.6 0.4 0.2 0.0Fluorescence 300 200 100 0Frequency scan (MHz)



(a)(b) 1.0 0.8 0.6 0.4 0.2 0.0Fluorescence 1000 800 600 400 200 0Frequency scan (MHz) Li5: 2 6: 2 7: 1 8: 1 1.0 0.8 0.6 0.4 0.2 0.0Fluorescence -10 -5 0 5 AOM scan (MHz) -0.1 0.0 0.1Residuals Li3/2
3/2 1.0 0.8 0.6 0.4 0.2 0.0Fluorescence 10 5 0 -5 -10 AOM scan (MHz) -0.1 0.0 0.1Residuals 7Li2
Figure2.linespectrainthetwoisotopesofLiareshownin(a)and(b).ThedifferenthyperÞnetransitionsareclearlyresolved,andlabelledasshown.ThereisalargeincreaseinthegainofthePMTfor(a)comparedto(b)toaccountforthelowabundanceofLi.Thesmallpeakbetweenpeaks5and6isfromthenearbylineinLi.Close-upsofpeaks2and6(normalized)areshownin(c)and(d),respectively.ThecirclesarethemeasuredspectraandthesolidlinesareLorentzianÞts.TheÞtresidualsgiveanideaofthesignal-to-noiseratio.ovencontainingLimetalto300C.Theovenwasplacedinsideavacuumchambermaintainedatapressurebelow5torrwitha20lsionpump.Theatomicbeamwasexcitedwithaperpendicularlaserbeamgeneratedfromahome-builtdiodelasersystemoperatingat670nm.Thelaserbeamwaslinearlypolarizedandtheinteractionregionwasshieldedwithatwo-layermagneticshield.TheresultingDoppler-freeßuorescencesignalwasdetectedbyaPMT(HamamatsuWeobtainedspectrafromthelinesofbothisotopes.Intheline,theindividualhyperÞnetransitionsarewellresolvedandwecanlocktoeachpeakseparately.However,inline,thedifferenttransitionsareonlypartiallyresolvedbecausethelevelspacingislessthanthenaturallinewidth.Inrecentworkwherewemeasuredtheabsolutefrequenciesofthesetransitions[],weshowedthatamultipeakÞttothepartiallyresolvedspectrumyieldsthehyperÞneconstantsinthe2statewithaprecisionofabout50kHz.Thisislessthantheprecisionwithwhichtheyareknownfromearlierexperiments.Therefore,inthisworkwereporthyperÞnestructureonlyinthe2TypicalspectraforthelineinthetwoisotopesofLiareshowninÞgures(a)and(b).ThefourhyperÞnetransitionsineachisotopeareclearlyresolved.Close-upscans(normalized)ofarepresentativepeakineachisotopeareshownin(c)and(d),respectively.ThesolidlinesareLorentzianÞtsshowingtheexcellentÞtwithfeaturelessresiduals.Thesizeoftheresidualsgivesanideaoftheoverallsignal-to-noiseratio.TheÞtlinewidthis925MHz(comparedtothenaturallinewidthof5.87MHz),indicatingthattheatomicbeamiswellcollimatedbecausethespreadintransversevelocityincreasesthetotallinewidthbyonlyabout50%.ThelaserlinewidthdoesnotcontributesigniÞcantlytothisbroadening.AnimportantconsiderationinatomicbeamexperimentsisthesystematicDopplershiftoflinecentrethatcanoccurifthelaserbeamisnotperfectlyorthogonal.Intherecentworkonabsolutefrequencymeasurementsofthelines[],weusedthesamespectrometertomeasurethefrequencieswithlaserbeamstraversingtheatomicbeaminoppositedirections.SincetheDopplershiftisoppositeforthetwocases,theirdifferencegivesameasureofthemisalignmentfromperpendicularity.OurmeasurementsshowthattheDopplershiftofthetransitionfrequencyisonly150kHz,correspondingtoamisalignmentangleof0.1mrad.ForthehyperÞneinterval,thisimpliesanegligibleerroroflessthan1Hz.InordertomeasurethehyperÞneintervals,boththeunshiftedlaserbeamandtheAOM-shiftedbeamweresentacrosstheatomicbeam.ThediodelaserandtheAOMweremodulatedatdifferentfrequencies,andthesignalfromthePMTwasindependentlydemodulatedatthetwofrequencies.TheÞrsterrorsignalwasusedtolockthelaseronagivenhyperÞnetransition,whiletheseconderrorsignalwasusedtolocktheAOMtothehyperÞneinterval.AstringentcheckonourerrorestimateintableistouseourtechniquetomeasurethegroundhyperÞneinterval.Asmentionedinsection,thisintervalisalreadyknowninallalkaliatomswithrelativeprecision.However,ourtechniquerequirestheintervaltobeaccessiblewithanAOM, J.Phys.B:At.Mol.Opt.Phys.(2008)035001DDasandVNatarajan 17.45 17.40 17.35 17.30 17.25MHz Li,A (2P1/2) 46.2 46.1 46.0 45.9 45.8MHz Li,A (2P1/2) Figure4.Comparisonofthevalueofinthe2stateofLiobtainedinthisworktoearliervalues.Alsoshownistherecommendedvaluefrom[AlsolistedistherecommendedvaluefromthereviewworkofArimondo,InguscioandViolino[].Theresultiniscompletelyconsistentwithpreviousmeasurements,asseenfromthedeviationplotinÞgure,buttheaccuracyisimprovedbyafactorof5.InLi,ourvalueisslightlynon-overlappingwiththerecommendedvalue.However,itisconsistentwitharecentmeasurement[],whichhasthesameaccuracyastherecommendedvalue,andanearlierlessprecisemeasurementmeasurement13].Ouraccuracyisalmostanorderofmagnitudebetter.ThelastcolumninthetableliststheresultofarecentcalculationLi[],whichagreeswithourvalueatthe1.7%level.5.SodiumSodiumhasonestableisotope,Na,with2.Boththelinesarenear589nm,andareaccessedusingacwring-dyelaser(Coherent699-21)operatingwithRhodamine-6Gdye.Naspectroscopyisdoneina100mmlongvapourcell.Thecellisheatedtoatemperatureof75Candhasamagneticshieldaroundit.ThehyperÞnelevelsinthe3statearefarapartandtheindividualtransitionsinthelinearewellresolved.Hence,wehaveusedthestandardSAStechniqueforspectroscopytolockthelaserandtheAOM.AsinthecaseofLi,intensity-dependenterrorsarecheckedbyrepeatingthemeasurementsatthreelaserpowers.Theaveragevalueoftheintervalis 94.4 94.3 94.2 94.1 1/2) Figure5.Comparisonofthevalueofinthe3stateofobtainedinthisworktoearliervalues.ThevalueofthehyperÞneconstantiscomparedtotherecommendedvalueandanotherpublishedresult[]in.Ourvalueisconsistentwiththeseresultsbuttheaccuracyisimprovedbyafactorof20.Asmentionedearlier,thehyperÞnelevelsinthe3stateofNaaretooclosefortheindividualtransitionsinthetoberesolved.Hence,wehaveusedthetechniqueofCCStoresolvethesetransitions.Theresultsofourmeasurementshavebeenpublishedpreviously[].TheaveragevaluesoftheintervalsareInÞgure,wecompareourvaluesofthehyperÞneconstantstotherecommendedvalueandtwosubsequentmeasurements,oneusingtime-resolvedhyperÞnequantum-beatspectroscopy[]andthesecondusingpolarizationquantum-beatspectroscopy[].Ourvaluesareconsistentwiththeearliervaluesbuttheaccuracyisimprovedbyafactorof5.6.PotassiumPotassiumhastwostableisotopes:Kwith2andnaturalabundanceof93.3%,andKwith2andnaturalabundanceof6.7%.Thelineisat770nmandthelineisat767nm.ItisdifÞculttoobtainreliablelaserdiodesatthesewavelengths;henceourexperimentsweredonesometimeswithafrequency-stabilizeddiodeandsometimeswitharing-cavityTi:sapphirelaser(Coherent899-21)similartotheringdyelaserusedforthesodiumexperiments.Potassiumspectroscopywasdoneina50mmlongvapourcell.Thecellwasheatedtoatemperatureof75Candhadamagneticshieldaroundit.DuetothelownaturalabundanceofK,itisdifÞculttogetagoodabsorptionsignalforthisisotope;hencewehavemeasuredhyperÞnestructureonlyinThehyperÞnelevelsinthe4stateareseparatedbyabout58MHzcomparedtothenaturallinewidthof6MHz.Thusthe J.Phys.B:At.Mol.Opt.Phys.(2008)035001DDasandVNatarajan 18.7 18.6 18.5 Na, A (3P3/2) 3.1 3.0 2.9 2.8 2.7 2.6 Figure6.Comparisonofthevaluesofinthe3stateofNaobtainedinthisworktoearliervalues.individualtransitionsinthelinearefairlywellresolved.InaÞrstsetofexperiments,wemeasuredtheintervalusingnormalSAStolockthelaserandtheAOM.Notethatthecrossoverresonanceisonly29MHzaway,whichisnotveryfargiventhattheexperimentallyobtainedlinewidthinSASisaround10MHz.Theresultfortheintervalfromthesemeasurementswas57MHz.Theslightlylargeerrorisduetothenearbycrossoverresonancewhichcausessomepeakpulling.Inanearliermeasurementfromourlaboratory[],wehadmeasuredtheabsolutefrequenciesofindividualtransitionsinthelineandhencedeterminedthehyperÞneintervalindirectly.Thedirectmeasurementoftheintervalwasconsistentwiththisearlierresult.Inaddition,itagreedwiththerecommendedvaluein[](fromameasurementbyBuckandRabi[]usingtheatomicbeammagneticresonancetechnique).However,therehassincebeenareportofabsolutefrequencymeasurementsonthesamelineusingafemtosecondcomb[].Thevalueoftheintervaldisagreesfromourvalue(by25combined)andtherecommendedvalue(by3combined),toolargetobeaccountedforbyanyknownsources 0.8 0.6 0.4 0.2 0.0Probe transmission 70 65 60 55 50 45Control detuning (MHz) 0.03 0.03 Figure7.spectrumofK.Theopencirclesshowtheprobetransmissionspectrum(normalized)usingtheCCStechniqueobtainedwiththeprobebeamlockedtothe2transitionandthecontrolbeamscanningacrossthe1transition.ThesolidcurvesareÞtswithresidualsshownontop.ThelowerresidualsarefromaLorentzianÞtwhiletheupperonesarefromadensity-matrixanalysis.TheÞtsyieldavalueof57.713(42)MHzfortheinterval.Thedashedcurveisthespectrumthatwouldbeobtainediftheintervalwere55.500(84)MHzasmeasuredbyFalkeetalal20].oferror.WehavethereforerepeatedthemeasurementoftheintervalbutthistimeusingtheCCStechnique.Asmentionedearlier,thistechniquedoesnotproducecrossoverresonancesandthemeasuredintervalisinsensitivetodetuningoftheprobelaser.InÞgure,weshowatypicalprobetransmissionspectrumKobtainedwiththeprobelockedtothetransitionandthecontrolscanningacrossthetransition.ThescanningisdonebyvaryingthefrequencyoftheAOMinthepathofthecontrol,sothatthescanaxisisproperlycalibratedanditslinearityisguaranteedbythelinearityofthevoltage-controlledoscillator(vco)drivingtheAOM.Notethatdirectscanningofthelaserusingintracavityelementsusuallyresultsinanonlinearscan.TheAOMisdoublepassedtoensurethatthedirectionofthecontrolbeamremainsunchangedwhiletheAOMisscanned,andtheoutputbeamintensityisstabilizedbyfeedbackcontroloftherfpowerexcitingtheAOM.TheopencirclesintheÞgurerepresentthemeasuredspectrumwhilethesolidcurvesareÞts.WedidtwokindsofÞttingtothespectrum.TheÞrstÞtistoaLorentzianlineshapewiththepeakcentreandlinewidthasÞtparameters.ThesecondÞtistoadensity-matrixcalculationoftheeffectivethree-levelsystemtakingintoaccountthethermalvelocitydistributioninhotvapour,asdescribedinourearlierworkork6].ThehyperÞneinterval(whichdeterminesthecontroldetuningatthepeak)andthecontrolRabifrequencyarefreeÞtparameters.TheresidualsfromthetwoÞtsarealsoshownintheÞgure,withtheLorentzianÞtyieldingslightlysmallerresiduals.However,thelocationofthepeakcentrefrombothÞtsisthesame.AfterÞttingtoseveralindependentspectra(totakecareofdriftsinthevcodrivingtheAOM),weobtainavalueof57.713(42)MHzfortheinterval,closetoourearliermeasurementandtheonebyBuckandRabi.Thedashedcurveshowswhatthespectrumwouldlooklikeiftheintervalwere55.500MHzasmeasuredbyFalkeetalAlthoughthespectrumobtainedbyscanningtheAOMyieldstheinterval,thelockingtechniquehashigheraccuracybecausetheAOMfrequencyismeasuredwhileitislocked,andisnotsusceptibletodriftsofthevco.ThenewresultfortheintervalbylockingtheAOMiswhichisagainconsistentwiththeabovemeasurementsandourearlierpublishedvalue. J.Phys.B:At.Mol.Opt.Phys.(2008)035001DDasandVNatarajan 28.5 28.0 27.5 27.0 K, A (4P1/2) Figure8.Comparisonofthevalueofinthe4stateofobtainedinthisworktoearliervalues.InÞgure,wecomparethisvalueofwithotherpublishedvalues.Ascanbeseen,therearetwosetsofvaluesthatareincompletedisagreementwitheachother.Therecommendedvalueandourtworesultsareclusteredononeside,whiletherecentresultofFalkeetalal20]andtwooldermeasurements(whicharebothfromthesamegroup[areclusteredtogether.TheexperimentofFalkeetalal20]hasalsobeendonewithgreatcareandtakesintoaccountmanypotentialsourcesoferror;hencetheirmeasurementseemstobereliablewithinthestatedaccuracy.Wedonothaveasatisfactoryexplanationforsuchalargediscrepancy.Asmentionedbefore,thehyperÞnelevelsinthe4stateofKaretooclosetoberesolvedcompletely.HencewehaveusedourtechniqueofCCStogetapartiallyresolvedspectrum.However,thepeaksarestillsoclosethatthelockpointwillgetpulledbythenearbytransition.Therefore,asdescribedaboveforthespectrum,wemeasuretheintervalsbyscanningtheAOMinthepathofthecontrolbeam.TheAOMisagaindoublepassedandtheoutputbeamisintensitystabilized.Atypicalprobetransmissionspectrumofthelinewiththeprobelockedonthe3transitionisshowninÞgure.WesawaboveinthecaseofthelinethatthelineshapeisdescribedquitewellbyaLorentziancurve,henceweextractthelocationsoftheindividualhyperÞnetransitionsbydoingamultipeakLorentzianÞttothespectrum.AsseenfromtheÞgure,thefeaturelessresidualsarelessthan1%ofthepeak.TheÞtlinewidthis8.4MHz,oronly45%largerthanthenaturallinewidthof5.8MHz.AfterÞttingto25spectra 6.10 6.05 6.00 5.95 3.0 2.9 2.8 2.7 K, B (4P3/2) Figure10.Comparisonofthevaluesofinthe4stateofKobtainedinthisworktoearliervalues. 0.8 0.6 0.4 0.2 0.0 -40 -30 -20 -10 20 -0.02 0.00 0.02Residuals
F' ' = 1 Figure9.spectrum(normalized)ofKwithoutcrossoverresonancesusingthecoherent-controltechnique.Thecontroldetuningismeasuredfromthe3peak.Theopencirclesaretheobservedspectrum,whilethesolidcurveisathree-peakLorentzianÞtwhichyieldsthehyperÞneintervals.weobtaintheaveragevaluesoftheintervalsasTheslightlylargererrorsareduetothedriftofthevcofromshottoshot.ThedifÞcultyinmeasuringhyperÞnestructureinthisstateisevidencedbythefactthattherehasonlybeenoneothermeasurementinrecenttimes[].OurvaluesofthehyperÞnearecomparedtothismeasurementandtherecommendedvaluesinÞgure.Thethreesetsareincompleteagreementwitheachother,withourvalueshavingthesmallestuncertainty.Notethattherecentvaluesin[]arefromthesameworkwhichdifferedsosigniÞcantlyforthe7.RubidiumRubidiumhastwostableisotopes:Rbwith2andnaturalabundanceof72.2%andRbwith2andnaturalabundanceof27.8%.Thelineat795nmandtheat780nmarebothaccessedusingfrequency-stabilizeddiodelasers.Rbspectroscopyisdoneina50mmlongvapourcellcontainingbothisotopes.Thecellisatroomtemperatureandhasamagneticshieldaroundit.