Rubidium85DLineDataDanielAdamSteckOregonCenterforOpticsandDepartmentof - PDF document

Rubidium85DLineDataDanielAdamSteckOregonCenterforOpticsandDepartmentofPhysics,UniversityofOregon Copyright©2008,byDanielAdamSteck.Allrightsreserved.ThismaterialmaybedistributedonlysubjecttothetermsandconditionssetforthintheOpenPublicationLicense,v1.0orlater(thelatestversionispresentlyavailableathttp://www.opencontent.org/openpub/).Distributionofsubstantivelymodi�edversionsofthisdocumentisprohibitedwithouttheexplicitpermissionofthecopyrightholder.Distributionoftheworkorderivativeoftheworkinanystandard(paper)bookformisprohibitedunlesspriorpermissionisobtainedfromthecopyrightholder.Originalrevisionposted30April2008.Thisisrevision2.2.1,21November2019.Citethisdocumentas:DanielA.Steck,“Rubidium85DLineData,”availableonlineathttp://steck.us/alkalidata(revision2.2.1,21November2019).Authorcontactinformation:DanielSteckDepartmentofPhysics1274UniversityofOregonEugene,Oregon97403-1274dsteck@uoregon.edu 1Introduction3 1IntroductionInthisreferencewepresentmanyofthephysicalandopticalpropertiesof85Rbthatarerelevanttovariousquantumopticsexperiments.Inparticular,wegiveparametersthatareusefulintreatingthemechanicale�ectsoflighton85Rbatoms.Themeasurednumbersaregivenwiththeiroriginalreferences,andthecalculatednumbersarepresentedwithanoverviewoftheircalculationalongwithreferencestomorecomprehensivediscussionsoftheirunderlyingtheory.Atpresent,thisdocumentisnotacriticalreviewofexperimentaldata,norisitevenguaranteedtobecorrect;foranynumberscriticaltoyourresearch,youshouldconsulttheoriginalreferences.Wealsopresentadetaileddiscussionofthecalculationof�uorescencescatteringrates,becausethistopicisoftennottreatedclearlyintheliterature.MoredetailsandderivationsregardingthetheoreticalformalismheremaybefoundinRef.[1].Thecurrentversionofthisdocumentisavailableathttp://steck.us/alkalidata,alongwith“CesiumDLineData,”“SodiumDLineData,”and“Rubidium87DLineData.”ThisistheonlypermanentURLforthisdocumentatpresent;pleasedonotlinktoanyothers.Pleasesendcomments,corrections,andsuggestionstodsteck@uoregon.edu.2Rubidium85PhysicalandOpticalPropertiesSomeusefulfundamentalphysicalconstantsaregiveninTable1.Thevaluesgivenarethe2006CODATArecommendedvalues,aslistedin[2].Someoftheoverallphysicalpropertiesof85RbaregiveninTable2.Rubidium85has37electrons,onlyoneofwhichisintheoutermostshell.85Rbistheonlystableisotopeofrubidium(although87Rbisonlyveryweaklyunstable,andisthuse�ectivelystable),andistheonlyisotopeweconsiderinthisreference.Themassistakenfromthehigh-precisionmeasurementof[3],andthedensity,meltingpoint,boilingpoint,andheatcapacities(forthenaturallyoccurringformofRb)aretakenfrom[4].Thevaporpressureat25CandthevaporpressurecurveinFig.1aretakenfromthevapor-pressuremodelgivenby[5],whichislog10Pv=2:881+4:857�4215 T(solidphase)log10Pv=2:881+4:312�4040 T(liquidphase);(1)wherePvisthevaporpressureintorr(forPvinatmospheres,simplyomitthe2.881term),andTisthetemperatureinK.Thismodelisspeci�edtohaveanaccuracybetterthan5%from298–550K.Older,andprobablyless-accurate,sourcesofvapor-pressuredataincludeRefs.[6]and[7].Theionizationlimitistheminimumenergyrequiredtoionizea85Rbatom;thisvalueistakenfromRef.[8].Theopticalpropertiesofthe85RbDlinearegiveninTables3and4.ThepropertiesaregivenseparatelyforeachofthetwoD-linecomponents;theD2line(the52S1=2�!52P3=2transition)propertiesaregiveninTable3,andtheopticalpropertiesoftheD1line(the52S1=2�!52P1=2transition)aregiveninTable4.Ofthesetwocomponents,theD2transitionisofmuchmorerelevancetocurrentquantumandatomopticsexperiments,becauseithasacyclingtransitionthatisusedforcoolingandtrapping85Rb.Thefrequency!0oftheD2isacombinationofthe87Rbmeasurementof[9]withtheisotopeshiftquotedin[10],whilethefrequencyoftheD1transitionisanaverageofvaluesgivenby[10]and[11];thevacuumwavelengthsandthewavenumberskLarethendeterminedviathefollowingrelations:=2c !0kL=2 :(2)Duetothedi�erentnuclearmassesofthetwoisotopes85Rband87Rb,thetransitionfrequenciesof87Rbareshiftedslightlyupcomparedtothoseof85Rb.Thisdi�erenceisreportedastheisotopeshift,andthevaluesaretakenfrom[10].(See[11,12]forlessaccuratemeasurements.)Theairwavelengthair==nassumesanindexofrefractionofn=1:000267308(10)fortheD2lineandn=1:000267215(10)fortheD1line,corresponding 42Rubidium85PhysicalandOpticalProperties totypicallaboratoryconditions(1atmpressure,23Ctemperature,and40%relativehumidity).Theindexofrefractioniscalculatedfromthe1993revision[13]oftheEdlénformula[14]:nair=1+"8342:54+2406147 130�2+15998 38:9�2P 96095:431+10�8(0:601�0:00972T)P 1+0:0036610T�f�0:037345�0:0004012
#10�8:(3)Here,PistheairpressureinPa,TisthetemperatureinC,isthevacuumwavenumberkL=2inm�1,andfisthepartialpressureofwatervaporintheair,inPa(whichcanbecomputedfromtherelativehumidityviatheGo�-Gratchequation[15]).Thisformulaisappropriateforlaboratoryconditionsandhasanestimated(3)uncertaintyof310�8from350-650nm.1Thelifetimesareweightedaverages2fromfourrecentmeasurements;the�rstemployedbeam-gas-laserspec-troscopy[19],withlifetimesof27:70(4)nsforthe52P1=2stateand26:24(4)nsforthe52P3=2state,thesecondusedtime-correlatedsingle-photoncounting[20],withlifetimesof27:64(4)nsforthe52P1=2stateand26:20(9)nsforthe52P3=2state,thethirdusedphotoassociationspectroscopy[21](asquotedby[20]),withalifetimeof26:23(6)nsforthe52P3=2stateonly,andthefourthalsousedphotoassociationspectroscopy[22],withlifetimesof27:75(8)nsforthe52P1=2stateand26:25(8)nsforthe52P3=2state.Notethatatpresentlevelsoftheoretical[23]andexperimentalaccuracy,wedonotdistinguishbetweenlifetimesofthe85Rband87Rbisotopes.Invertingthelifetimegivesthespontaneousdecayrate�(EinsteinAcoe�cient),whichisalsothenatural(homogenous)linewidth(asanangularfrequency)oftheemittedradiation.Thespontaneousemissionrateisameasureoftherelativeintensityofaspectralline.Commonly,therelativeintensityisreportedasanabsorptionoscillatorstrengthf,whichisrelatedtothedecayrateby[24]�=e2!20 20mec32J+1 2J0+1f(4)foraJ�!J0�ne-structuretransition,wheremeistheelectronmass.Therecoilvelocityvristhechangeinthe85Rbatomicvelocitywhenabsorbingoremittingaresonantphoton,andisgivenbyvr=~kL m:(5)Therecoilenergy~!risde�nedasthekineticenergyofanatommovingwithvelocityv=vr,whichis~!r=~2k2L 2m:(6)TheDopplershiftofanincidentlight�eldoffrequency!Lduetomotionoftheatomis
!d=vatom c!L(7)forsmallatomicvelocitiesrelativetoc.Foranatomicvelocityvatom=vr,theDopplershiftissimply2!r.Finally,ifonewishestocreateastandingwavethatismovingwithrespecttothelabframe,thetwotraveling-wavecomponentsmusthaveafrequencydi�erencedeterminedbytherelationvsw=
!sw 2 2;(8) 1Anuncertaintyof1wasusedinthecalculationsinthisreference,butthereal-lifeuncertaintyalsohasimportantcontributionsfromtheuncertaintiesoftemperature,humidity,andpressure,whicharenotincludedhere.Formoredetails,seethediscussioninRef.[16].2Weightedmeanswerecomputedaccordingto=(Pjxjwj)=(Pjwj),wheretheweightswjweretakentobetheinversevariancesofeachmeasurement,wj=1=2j.Thevarianceoftheweightedmeanwasestimatedaccordingto2=(Pjwj(xj�)2)=[(n�1)Pjwj],andtheuncertaintyintheweightedmeanisthesquarerootofthisvariance.SeeRefs.[17,18]formoredetails. 3HyperfineStructure5 because
!sw=2isthebeatfrequencyofthetwowaves,and=2isthespatialperiodicityofthestandingwave.Forastandingwavevelocityofvr,Eq.(8)gives
!sw=4!r.Twotemperaturesthatareusefulincoolingandtrappingexperimentsarealsogivenhere.Therecoiltemperatureisthetemperaturecorrespondingtoanensemblewithaone-dimensionalrmsmomentumofonephotonrecoil~kL:Tr=~2k2L mkB:(9)TheDopplertemperature,TD=~� 2kB;(10)isthelowesttemperaturetowhichoneexpectstobeabletocooltwo-levelatomsinopticalmolasses,duetoabalanceofDopplercoolingandrecoilheating[25].Ofcourse,inZeeman-degenerateatoms,sub-Dopplercoolingmechanismspermittemperaturessubstantiallybelowthislimit[26].3HyperneStructure3.1EnergyLevelSplittingsThe52S1=2�!52P3=2and52S1=2�!52P1=2transitionsarethecomponentsofa�ne-structuredoublet,andeachofthesetransitionsadditionallyhavehyper�nestructure.The�nestructureisaresultofthecouplingbetweentheorbitalangularmomentumLoftheouterelectronanditsspinangularmomentumS.ThetotalelectronangularmomentumisthengivenbyJ=L+S;(11)andthecorrespondingquantumnumberJmustlieintherangejL�SjJL+S:(12)(HereweusetheconventionthatthemagnitudeofJisp J(J+1)~,andtheeigenvalueofJzismJ~.)Forthegroundstatein85Rb,L=0andS=1=2,soJ=1=2;forthe�rstexcitedstate,L=1,soJ=1=2orJ=3=2.TheenergyofanyparticularlevelisshiftedaccordingtothevalueofJ,sotheL=0�!L=1(Dline)transitionissplitintotwocomponents,theD1line(52S1=2�!52P1=2)andtheD2line(52S1=2�!52P3=2).Themeaningoftheenergylevellabelsisasfollows:the�rstnumberistheprincipalquantumnumberoftheouterelectron,thesuperscriptis2S+1,theletterreferstoL(i.e.,S$L=0,P$L=1,etc.),andthesubscriptgivesthevalueofJ.Thehyper�nestructureisaresultofthecouplingofJwiththetotalnuclearangularmomentumI.ThetotalatomicangularmomentumFisthengivenbyF=J+I:(13)Asbefore,themagnitudeofFcantakethevaluesjJ�IjFJ+I:(14)Forthe85Rbgroundstate,J=1=2andI=5=2,soF=2orF=3.FortheexcitedstateoftheD2line(52P3=2),Fcantakeanyofthevalues1,2,3,or4,andfortheD1excitedstate(52P1=2),Fiseither2or3.Again,theatomicenergylevelsareshiftedaccordingtothevalueofF.Becausethe�nestructuresplittingin85Rbislargeenoughtoberesolvedbymanylasers(15nm),thetwoD-linecomponentsaregenerallytreatedseparately.Thehyper�nesplittings,however,aremuchsmaller,anditisusefultohavesomeformalismtodescribetheenergyshifts.TheHamiltonianthatdescribesthehyper�ne 63.2InteractionwithStaticExternalFields structureforeachoftheD-linecomponentsis[24,27–29]Hhfs=AhfsI
J+Bhfs3(I
J)2+3 2(I
J)�I(I+1)J(J+1) 2I(2I�1)J(2J�1)+Chfs10(I
J)3+20(I
J)2+2(I
J)[I(I+1)+J(J+1)+3]�3I(I+1)J(J+1)�5I(I+1)J(J+1) I(I�1)(2I�1)J(J�1)(2J�1);(15)whichleadstoahyper�neenergyshiftof
Ehfs=1 2AhfsK+Bhfs3 2K(K+1)�2I(I+1)J(J+1) 4I(2I�1)J(2J�1)+Chfs5K2(K=4+1)+K[I(I+1)+J(J+1)+3�3I(I+1)J(J+1)]�5I(I+1)J(J+1) I(I�1)(2I�1)J(J�1)(2J�1);(16)whereK=F(F+1)�I(I+1)�J(J+1);(17)Ahfsisthemagneticdipoleconstant,Bhfsistheelectricquadrupoleconstant,andChfsisthemagneticoctupoleconstant(althoughthetermswithBhfsandChfsapplyonlytotheexcitedmanifoldoftheD2transitionandnottothelevelswithJ=1=2).Theseconstantsforthe85RbDlinearelistedinTable5.ThevalueforthegroundstateAhfsconstantisfrom[27],whiletheconstantslistedforthe52P3=2manifoldareaveragesofthevaluesfrom[27]and[10].TheAhfsconstantforthe52P1=2manifoldistheaveragefromtherecentmeasurementsof[10]and[11].Thesemeasurementsarenotyetsu�cientlyprecisetohaveprovidedanonzerovalueforChfs,andthusitisnotlisted.Theenergyshiftgivenby(16)isrelativetotheunshiftedvalue(the“centerofgravity”)listedinTable3.Thehyper�nestructureof85Rb,alongwiththeenergysplittingvalues,isdiagrammedinFigs.2and3.3.2InteractionwithStaticExternalFields3.2.1MagneticFieldsEachofthehyper�ne(F)energylevelscontains2F+1magneticsublevelsthatdeterminetheangulardistributionoftheelectronwavefunction.Intheabsenceofexternalmagnetic�elds,thesesublevelsaredegenerate.However,whenanexternalmagnetic�eldisapplied,theirdegeneracyisbroken.TheHamiltoniandescribingtheatomicinteractionwiththemagnetic�eldisHB=B ~(gSS+gLL+gII)
B=B ~(gSSz+gLLz+gIIz)Bz;(18)ifwetakethemagnetic�eldtobealongthez-direction(i.e.,alongtheatomicquantizationaxis).InthisHamilto-nian,thequantitiesgS,gL,andgIarerespectivelytheelectronspin,electronorbital,andnuclear“g-factors”thataccountforvariousmodi�cationstothecorrespondingmagneticdipolemoments.ThevaluesforthesefactorsarelistedinTable6,withthesignconventionof[27].ThevalueforgShasbeenmeasuredveryprecisely,andthevaluegivenistheCODATArecommendedvalue.ThevalueforgLisapproximately1,buttoaccountforthe�nitenuclearmass,thequotedvalueisgivenbygL=1�me mnuc;(19)whichiscorrecttolowestorderinme=mnuc,wheremeistheelectronmassandmnucisthenuclearmass[30].ThenuclearfactorgIaccountsfortheentirecomplexstructureofthenucleus,andsothequotedvalueisanexperimentalmeasurement[27]. 3.2InteractionwithStaticExternalFields7 Iftheenergyshiftduetothemagnetic�eldissmallcomparedtothe�ne-structuresplitting,thenJisagoodquantumnumberandtheinteractionHamiltoniancanbewrittenasHB=B ~(gJJz+gIIz)Bz:(20)Here,theLandéfactorgJisgivenby[30]gJ=gLJ(J+1)�S(S+1)+L(L+1) 2J(J+1)+gSJ(J+1)+S(S+1)�L(L+1) 2J(J+1)'1+J(J+1)+S(S+1)�L(L+1) 2J(J+1);(21)wherethesecond,approximateexpressioncomesfromtakingtheapproximatevaluesgS'2andgL'1.Theexpressionheredoesnotincludecorrectionsduetothecomplicatedmultielectronstructureof85Rb[30]andQEDe�ects[31],sothevaluesofgJgiveninTable6areexperimentalmeasurements[27](exceptforthe52P1=2statevalue,forwhichtherehasapparentlybeennoexperimentalmeasurement).Iftheenergyshiftduetothemagnetic�eldissmallcomparedtothehyper�nesplittings,thensimilarlyFisagoodquantumnumber,sotheinteractionHamiltonianbecomes[32]HB=BgFFz ~Bz;(22)wherethehyper�neLandég-factorisgivenbygF=gJF(F+1)�I(I+1)+J(J+1) 2F(F+1)+gIF(F+1)+I(I+1)�J(J+1) 2F(F+1)'gJF(F+1)�I(I+1)+J(J+1) 2F(F+1):(23)Thesecond,approximateexpressionhereneglectsthenuclearterm,whichisacorrectionatthelevelof0.1%,sincegIismuchsmallerthangJ.Forweakmagnetic�elds,theinteractionHamiltonianHBperturbsthezero-�eldeigenstatesofHhfs.Tolowestorder,thelevelssplitlinearlyaccordingto[24]
EjFmFi=BgFmFBz:(24)TheapproximategFfactorscomputedfromEq.(23)andthecorrespondingsplittingsbetweenadjacentmagneticsublevelsaregiveninFigs.2and3.ThesplittinginthisregimeiscalledtheZeemane�ect.Forstrong�eldswheretheappropriateinteractionisdescribedbyEq.(20),theinteractiontermdominatesthehyper�neenergies,sothatthehyper�neHamiltonianperturbsthestrong-�eldeigenstatesjJmJImIi.Theenergiesarethengiventolowestorderby[1]EjJmJ;ImIiAhfsmImJ+Bhfs9(mImJ)2�3J(J+1)m2I�3I(I+1)m2J+I(I+1)J(J+1) 4J(2J�1)I(2I�1)+B(gJmJ+gImI)B:(25)TheenergyshiftinthisregimeiscalledthePaschen-Backe�ect.Forintermediate�elds,theenergyshiftismoredi�culttocalculate,andingeneralonemustnumericallydiagonalizeHhfs+HB.AnotableexceptionistheBreit-Rabiformula[24,32,33],whichappliestotheground-statemanifoldoftheDtransition:EjJ=1=2mJImIi=�
Ehfs 2(2I+1)+gIBmB
Ehfs 21+4mx 2I+1+x21=2:(26) 83.2InteractionwithStaticExternalFields Inthisformula,
Ehfs=Ahfs(I+1=2)isthehyper�nesplitting,m=mImJ=mI1=2(wherethesignistakentobethesameasin(26)),andx=(gJ�gI)BB
Ehfs:(27)Inordertoavoidasignambiguityinevaluating(26),themoredirectformulaEjJ=1=2mJImIi=
EhfsI 2I+11 2(gJ+2IgI)BB(28)canbeusedforthetwostatesm=(I+1=2).TheBreit-Rabiformulaisusefulin�ndingthesmall-�eldshiftofthe“clocktransition”betweenthemF=0sublevelsofthetwohyper�negroundstates,whichhasno�rst-orderZeemanshift.Usingm=mFforsmallmagnetic�elds,weobtain
!clock=(gJ�gI)22B 2~
EhfsB2(29)tosecondorderinthe�eldstrength.Ifthemagnetic�eldissu�cientlystrongthatthehyper�neHamiltonianisnegligiblecomparedtotheinter-actionHamiltonian,thenthee�ectistermedthenormalZeemane�ectforhyper�nestructure.Forevenstronger�elds,therearePaschen-BackandnormalZeemanregimesforthe�nestructure,wherestateswithdi�erentJcanmix,andtheappropriateformoftheinteractionenergyisEq.(18).Yetstronger�eldsinduceotherbehaviors,suchasthequadraticZeemane�ect[32],whicharebeyondthescopeofthepresentdiscussion.Thelevelstructureof85Rbinthepresenceofamagnetic�eldisshowninFigs.4-6intheweak-�eld(Zeeman)regimethroughthehyper�nePaschen-Backregime.3.2.2ElectricFieldsAnanalogouse�ect,thedcStarke�ect,occursinthepresenceofastaticexternalelectric�eld.TheinteractionHamiltonianinthiscaseis[28,34,35]HE=�1 20E2z�1 22E2z3J2z�J(J+1) J(2J�1);(30)wherewehavetakentheelectric�eldtobealongthez-direction,0and2arerespectivelytermedthescalarandtensorpolarizabilities,andthesecond(2)termisnonvanishingonlyfortheJ=3=2level.The�rsttermshiftsallthesublevelswithagivenJtogether,sothattheStarkshiftfortheJ=1=2statesistrivial.Theonlymechanismforbreakingthedegeneracyofthehyper�nesublevelsin(30)istheJzcontributioninthetensorterm.ThisinteractionsplitsthesublevelssuchthatsublevelswiththesamevalueofjmFjremaindegenerate.Anexpressionforthehyper�neStarkshift,assumingaweakenough�eldthattheshiftissmallcomparedtothehyper�nesplittings,is[28]
EjJIFmFi=�1 20E2z�1 22E2z[3m2F�F(F+1)][3X(X�1)�4F(F+1)J(J+1)] (2F+3)(2F+2)F(2F�1)J(2J�1);(31)whereX=F(F+1)+J(J+1)�I(I+1):(32)Forstronger�elds,whentheStarkinteractionHamiltoniandominatesthehyper�nesplittings,thelevelssplitaccordingtothevalueofjmJj,leadingtoanelectric-�eldanalogtothePaschen-Backe�ectformagnetic�elds.Thestaticpolarizabilityisalsousefulinthecontextofopticaltrapsthatareveryfaro�resonance(i.e.,severaltomanynmawayfromresonance,wheretherotating-waveapproximationisinvalid),sincetheopticalpotentialisgivenintermsoftheground-statepolarizabilityasV=�1=20E2,whereEistheamplitudeoftheoptical�eld. 3.3ReductionoftheDipoleOperator9 Aslightlymoreaccurateexpressionforthefar-o�resonantpotentialarisesbyreplacingthestaticpolarizabilitywiththefrequency-dependentpolarizability[36]0(!)=!200 !20�!2;(33)where!0istheresonantfrequencyofthelowest-energytransition(i.e.,theD1resonance);thisapproximateexpressionisvalidforlighttunedfartotheredoftheD1line.The85RbpolarizabilitiesaretabulatedinTable6.Noticethatthedi�erencesintheexcitedstateandgroundstatescalarpolarizabilitiesaregiven,ratherthantheexcitedstatepolarizabilities,sincethesearethequantitiesthatwereactuallymeasuredexperimentally.ThepolarizabilitiesgivenhereareinSIunits,althoughtheyareoftengivenincgsunits(unitsofcm3)oratomicunits(unitsofa30,wheretheBohrradiusa0isgiveninTable1).TheSIvaluescanbeconvertedtocgsunitsvia[cm3]=(100
h=40)(=h)[Hz/(V/cm)2]=5:95521379(30)10�22(=h)[Hz/(V/cm)2](see[36]fordiscussionofunits),andsubsequentlytheconversiontoatomicunitsisstraightforward.Thelevelstructureof85Rbinthepresenceofanexternaldcelectric�eldisshowninFig.7intheweak-�eldregimethroughtheelectrichyper�nePaschen-Backregime.3.3ReductionoftheDipoleOperatorThestrengthoftheinteractionbetween85Rbandnearly-resonantopticalradiationischaracterizedbythedipolematrixelements.Speci�cally,hFmFjerjF0m0Fidenotesthematrixelementthatcouplesthetwohyper�nesublevelsjFmFiandjF0m0Fi(wheretheprimedvariablesrefertotheexcitedstatesandtheunprimedvariablesrefertothegroundstates).Tocalculatethesematrixelements,itisusefultofactorouttheangulardependenceandwritethematrixelementasaproductofaClebsch-Gordancoe�cientandareducedmatrixelement,usingtheWigner-Eckarttheorem[37]:hFmFjerqjF0m0Fi=hFkerkF0ihFmFjF01m0Fqi:(34)Here,qisanindexlabelingthecomponentofrinthesphericalbasis,andthedoubledbarsindicatethatthematrixelementisreduced.Wecanalsowrite(34)intermsofaWigner3-jsymbolashFmFjerqjF0m0Fi=hFkerkF0i(�1)F0�1+mFp 2F+1F01Fm0Fq�mF:(35)Noticethatthe3-jsymbol(or,equivalently,theClebsch-Gordancoe�cient)vanishesunlessthesublevelssatisfymF=m0F+q.Thisreducedmatrixelementcanbefurthersimpli�edbyfactoringouttheFandF0dependenceintoaWigner6-jsymbol,leavingafurtherreducedmatrixelementthatdependsonlyontheL,S,andJquantumnumbers[37]:hFkerkF0ihJIFkerkJ0I0F0i=hJkerkJ0i(�1)F0+J+1+Ip (2F0+1)(2J+1)JJ01F0FI:(36)Again,thisnewmatrixelementcanbefurtherfactoredintoanother6-jsymbolandareducedmatrixelementinvolvingonlytheLquantumnumber:hJkerkJ0ihLSJkerkL0S0J0i=hLkerkL0i(�1)J0+L+1+Sp (2J0+1)(2L+1)LL01J0JS:(37)ThenumericalvalueofthehJ=1=2kerkJ0=3=2i(D2)andthehJ=1=2kerkJ0=1=2i(D1)matrixelementsaregiveninTable7.Thesevalueswerecalculatedfromthelifetimeviatheexpression[38]1 =!30 30~c32J+1 2J0+1jhJkerkJ0ij2:(38) 104ResonanceFluorescence Wetakethevaluesofthesematrixelementstoberealandpositive,withtherelativesigndeterminedbyEq.(37).NotethatalltheequationswehavepresentedhereassumethenormalizationconventionXM0jhJMjerjJ0M0ij2=XM0qjhJMjerqjJ0M0ij2=jhJkerkJ0ij2:(39)Thereis,however,anothercommonconvention(usedinRef.[39])thatisrelatedtotheconventionusedhereby(JkerkJ0)=p 2J+1hJkerkJ0i.Also,wehaveusedthestandardphaseconventionfortheClebsch-Gordancoe�cientsasgiveninRef.[37],whereformulaeforthecomputationoftheWigner3-j(equivalently,Clebsch-Gordan)and6-j(equivalently,Racah)coe�cientsmayalsobefound.Thedipolematrixelementsforspeci�cjFmFi�!jF0m0FitransitionsarelistedinTables9-20asmultiplesofhJkerkJ0i.Thetablesareseparatedbytheground-stateFnumber(2or3)andthepolarizationofthetransition(where+-polarizedlightcouplesmF�!m0F=mF+1,-polarizedlightcouplesmF�!m0F=mF,and�-polarizedlightcouplesmF�!m0F=mF�1).4ResonanceFluorescence4.1SymmetriesoftheDipoleOperatorAlthoughthehyper�nestructureof85Rbisquitecomplicated,itispossibletotakeadvantageofsomesymmetriesofthedipoleoperatorinordertoobtainrelativelysimpleexpressionsforthephotonscatteringratesduetoresonance�uorescence.InthespiritoftreatingtheD1andD2linesseparately,wewilldiscussthesymmetriesinthissectionimplicitlyassumingthatthelightisinteractingwithonlyoneofthe�ne-structurecomponentsatatime.First,noticethatthematrixelementsthatcoupletoanysingleexcitedstatesubleveljF0m0Fiadduptoafactorthatisindependentoftheparticularsublevelchosen,XqFjhF(m0F+q)jerqjF0m0Fij2=2J+1 2J0+1jhJkerkJ0ij2;(40)ascanbeveri�edfromthedipolematrixelementtables.Thedegeneracy-ratiofactorof(2J+1)=(2J0+1)(whichis1fortheD1lineor1=2fortheD2line)isthesamefactorthatappearsinEq.(38),andisaconsequenceofthenormalizationconvention(39).Theinterpretationofthissymmetryissimplythatalltheexcitedstatesublevelsdecayatthesamerate�,andthedecayingpopulation“branches”intovariousgroundstatesublevels.Anothersymmetryarisesfromsummingthematrixelementsfromasingleground-statesubleveltothelevelsinaparticularF0energylevel:SFF0:=Xq(2F0+1)(2J+1)JJ01F0FI2jhFmFjF01(mF�q)qij2=(2F0+1)(2J+1)JJ01F0FI2:(41)ThissumSFF0isindependentoftheparticulargroundstatesublevelchosen,andalsoobeysthesumruleXF0SFF0=1:(42)Theinterpretationofthissymmetryisthatforanisotropicpump�eld(i.e.,apumping�eldwithequalcomponentsinallthreepossiblepolarizations),thecouplingtotheatomisindependentofhowthepopulationisdistributedamongthesublevels.ThesefactorsSFF0(whicharelistedinTable8)provideameasureoftherelativestrengthofeachoftheF�!F0transitions.InthecasewheretheincidentlightisisotropicandcouplestwooftheFlevels,theatomcanbetreatedasatwo-levelatom,withane�ectivedipolemomentgivenbyjdiso;e(F�!F0)j2=1 3SFF0jhJjjerjjJ0ij2:(43) 4.2ResonanceFluorescenceinaTwo-LevelAtom11 Thefactorof1=3inthisexpressioncomesfromthefactthatanygivenpolarizationofthe�eldonlyinteractswithone(ofthree)componentsofthedipolemoment,sothatitisappropriatetoaverageoverthecouplingsratherthansumoverthecouplingsasin(41).Whenthelightisdetunedfarfromtheatomicresonance(
�),thelightinteractswithseveralhyper�nelevels.Ifthedetuningislargecomparedtotheexcited-statefrequencysplittings,thentheappropriatedipolestrengthcomesfromchoosinganygroundstatesubleveljFmFiandsummingoveritscouplingstotheexcitedstates.Inthecaseof-polarizedlight,thesumisindependentoftheparticularsublevelchosen:XF0(2F0+1)(2J+1)JJ01F0FI2jhFmFjF01mF0ij2=1 3:(44)Thissumleadstoane�ectivedipolemomentforfardetunedradiationgivenbyjddet;ej2=1 3jhJjjerjjJ0ij2:(45)Theinterpretationofthisfactorisalsostraightforward.Becausetheradiationisfardetuned,itinteractswiththefullJ�!J0transition;however,becausethelightislinearlypolarized,itinteractswithonlyonecomponentofthedipoleoperator.Then,becauseofsphericalsymmetry,j^dj2je^rj2=e2(j^xj2+j^yj2+j^zj2)=3e2j^zj2.Notethatthisfactorof1=3alsoappearsforlight,butonlywhenthesublevelsareuniformlypopulated(which,ofcourse,isnottheequilibriumcon�gurationforthesepolarizations).Thee�ectivedipolemomentsforthiscaseandthecaseofisotropicpumpingaregiveninTable7.4.2ResonanceFluorescenceinaTwo-LevelAtomInthesetwocases,wherewehaveane�ectivedipolemoment,theatomsbehavelikesimpletwo-levelatoms.Atwo-levelatominteractingwithamonochromatic�eldisdescribedbytheopticalBlochequations[38],_gg=i 2(~ge�~eg)+�ee_ee=�i 2(~ge�~eg)��ee_~ge=�( +i
)~ge�i 2(ee�gg);(46)wheretheijarethematrixelementsofthedensityoperator:=j ih j, :=�d
E0=~istheresonantRabifrequency,disthedipoleoperator,E0istheelectric�eldamplitude(E=E0cos!Lt),
:=!L�!0isthedetuningofthelaser�eldfromtheatomicresonance,�=1=isthenaturaldecayrateoftheexcitedstate, :=�=2+ cisthe“transverse”decayrate(where cisaphenomenologicaldecayratethatmodelscollisions),~ge:=geexp(�i
t)isa“slowlyvaryingcoherence,”and~ge=~eg.Inwritingdowntheseequations,wehavemadetherotating-waveapproximationandusedamaster-equationapproachtomodelspontaneousemission.Additionally,wehaveignoredanye�ectsduetothemotionoftheatomanddecaysorcouplingstootherauxiliarystates.Inthecaseofpurelyradiativedamping( =�=2),theexcitedstatepopulationsettlestothesteadystatesolutionee(t!1)=( =�)2 1+4(
=�)2+2( =�)2:(47)The(steadystate)totalphotonscatteringrate(integratedoveralldirectionsandfrequencies)isthengivenby�ee(t!1):Rsc=� 2(I=Isat) 1+4(
=�)2+(I=Isat):(48)Inwritingdownthisexpression,wehavede�nedthesaturationintensityIsatsuchthatI Isat=2 �2;(49) 124.3OpticalPumping whichgives(withI=(1=2)c0E20)Isat=c0�2~2 4j^
dj2;(50)where^istheunitpolarizationvectorofthelight�eld,anddistheatomicdipolemoment.WithIsatde�nedinthisway,theon-resonancescatteringcrosssection,whichisproportionaltoRsc(
=0)=I,dropsto1=2ofitsweaklypumpedvalue0whenI=Isat.Moreprecisely,wecande�nethescatteringcrosssectionasthepowerradiatedbytheatomdividedbytheincidentenergy�ux(i.e.,sothatthescatteredpowerisI),whichfromEq.(48)becomes=0 1+4(
=�)2+(I=Isat);(51)wheretheon-resonancecrosssectionisde�nedby0=~!� 2Isat:(52)Additionally,thesaturationintensity(andthusthescatteringcrosssection)dependsonthepolarizationofthepumpinglightaswellastheatomicalignment,althoughthesmallestsaturationintensity(Isat(mF=3!m0F=4),discussedbelow)isoftenquotedasarepresentativevalue.SomesaturationintensitiesandscatteringcrosssectionscorrespondingtothediscussionsinSection4.1aregiveninTable7.Amoredetaileddiscussionoftheresonance�uorescencefromatwo-levelatom,includingthespectraldistributionoftheemittedradiation,canbefoundinRef.[38].4.3OpticalPumpingIfnoneofthespecialsituationsinSection4.1appliestothe�uorescenceproblemofinterest,thenthee�ectsofopticalpumpingmustbeaccountedfor.Adiscussionofthee�ectsofopticalpumpinginanatomicvaporonthesaturationintensityusingarate-equationapproachcanbefoundinRef.[40].Here,however,wewillcarryoutananalysisbasedonthegeneralizationoftheopticalBlochequations(46)tothedegeneratelevelstructureofalkaliatoms.TheappropriatemasterequationforthedensitymatrixofaFg!Fehyper�netransitionis[1,41–43]@ @t~m;m=�i 224eXmg (m;mg)~gmg;m�gXme (me;m)~m;eme+gXme (me;m)~eme;m�eXmg (m;mg)~m;gmg359������=������;(pumpeld)�ee�~m;m�eg� 2~m;m�ge� 2~m;m+gg�1Xq=�1h~e(m+q);e(m+q)hFe(m+q)jFg1mqihFe(m+q)jFg1mqii9���������������=���������������;(dissipation)+i(eg�ge)
~m;m)(freeevolution)(53) 4.3OpticalPumping13 where (me;mg)=hFgmgjFe1me�(me�mg)i �(me�mg)=(�1)Fe�Fg+me�mgr 2Fg+1 2Fe+1hFemejFg1mg(me�mg)i �(me�mg)(54)istheRabifrequencybetweentwomagneticsublevels, q=2hFejjerjjFgiE(+)q ~(55)istheoverallRabifrequencywithpolarizationq(E(+)qisthe�eldamplitudeassociatedwiththepositive-rotatingcomponent,withpolarizationqinthesphericalbasis),andistheKroneckerdeltasymbol.ThismasterequationignorescouplingtoFlevelsotherthantheground(g)andexcited(e)levels;hence,thisequationisappropriateforacyclingtransitionsuchas.Additionally,thismasterequationassumespurelyradiativedampingand,asbefore,doesnotdescribethemotionoftheatom.TocalculatethescatteringratefromaZeeman-degenerateatom,itisnecessarytosolvethemasterequationforthesteady-statepopulations.Then,thetotalscatteringrateisgivenbyRsc=�Pe=�Xmeeme;eme;(56)wherePeisthetotalpopulationintheexcitedstate.Inaddition,byincludingthebranchingratiosofthespontaneousdecay,itispossibletoaccountforthepolarizationoftheemittedradiation.De�ningthescatteringrateRsc;�qforthepolarization(�q),wehaveRsc;�q=XmemgjhFemejFg1mgqij2eme;eme;(57)where,asbefore,theonlynonzeroClebsch-Gordancoe�cientsoccurforme=mg+q.Aswehavede�nedithere,q=1correspondsto-polarizedradiation,andq=0correspondsto-polarizedradiation.Theangulardistributionforthescatteredlightissimplytheclassicalradiationpatternforarotatingdipole,fsc(;)=3 16(1+cos2);(58)andtheangulardistributionforthe-scatteredlightistheclassicalradiationpatternforanoscillatingdipole,f0sc(;)=3 8sin2:(59)Thenetangularpatternwillresultfromtheinterferenceofthesethreedistributions.Ingeneral,thismasterequationisdi�culttotreatanalytically,andevenanumericalsolutionofthetime-dependentequationscanbetime-consumingifalargenumberofdegeneratestatesareinvolved.Inthefollowingdiscussions,wewillonlyconsidersomesimplelightcon�gurationsinteractingwiththeF=3�!F0=4cyclingtransitionthatcanbetreatedanalytically.DiscussionsofZeeman-degenerateatomsandtheirspectracanbefoundinRefs.[43–47].4.3.1Circularly()PolarizedLightThecaseswheretheatomisdrivenbyeither+or�light(i.e.circularlypolarizedlightwiththeatomicquantizationaxisalignedwiththelightpropagationdirection)arestraightforwardtoanalyze.Inthesecases,thelighttransfersitsangularmomentumtotheatom,andthustheatomicpopulationistransferredtothestatewiththelargestcorrespondingangularmomentum.InthecaseoftheF=3�!F0=4cyclingtransition,a+driving�eldwilltransferalltheatomicpopulationintothejF=3;mF=3i�!jF0=4;m0F=4icyclingtransition, 144.3OpticalPumping anda�driving�eldwilltransferallthepopulationintothejF=3;mF=�3i�!jF0=4;m0F=�4icyclingtransition.Inbothcases,thedipolemoment,satisfyingjd(mF=3!mF=4)j2=2J+1 2J0+1jhJ=1=2kerkJ0=3=2ij2;(60)isgiveninTable7.Also,inthiscase,thesaturationintensityreducestoIsat=~!3� 12c2;(61)andthescatteringcrosssectionreducesto0=32 2:(62)Notethatthesevaluesareonlyvalidinsteadystate.Ifthepumping�eldisweak,the“settlingtime”oftheatomtoitssteadystatecanbelong,resultinginatime-dependente�ectivedipolemoment(andsaturationintensity).Forexample,beginningwithauniformsublevelpopulationintheF=3groundlevel,thesaturationintensitywillbeginat3:8951(11)mW/cm2andequilibrateat1:66932(49)mW/cm2foracircularlypolarizedpump.Also,ifthereareany“remixing”e�ectssuchascollisionsormagnetic�eldsnotalignedwiththeaxisofquantization,thesystemmaycometoequilibriuminsomeothercon�guration.4.3.2Linearly()PolarizedLightIfthelightis-polarized(linearlypolarizedalongthequantizationaxis),theequilibriumpopulationdistributionismorecomplicated.Inthiscase,theatomstendtoaccumulateinthesublevelsnearm=0.Gao[43]hasderivedanalyticexpressionsfortheequilibriumpopulationsofeachsublevelandshowedthattheequilibriumexcited-statepopulationisgivenbyEq.(47)if 2isreplacedbygP(2Fg+1)j 0j2;(63)where 0istheonlynonzerocomponentoftheRabi-frequencyvector(calculatedwithrespecttothereduceddipolemomentjhFjjerjjF0ij2=SFF0jhJjjerjjJ0ij2),andgPisa(constant)geometricfactorthataccountsfortheopticalpumping.Forthe85RbF=3�!F0=4cyclingtransition,thisfactorhasthevaluegP=572=96510:05927,leadingtoasteady-statesaturationintensityofIsat=3:12950(92)mW/cm2.4.3.3One-Dimensional+��OpticalMolassesWenowconsidertheimportantcaseofanopticalmolassesinonedimensionformedbyone+andone��eld(e.g.,bytworight-circularlypolarized,counterpropagatinglaser�elds).These�eldsinterferetoforma�eldthatislinearlypolarized,wherethepolarizationvectortracesoutahelixinspace.Becausethelightislinearlypolarizedeverywhere,andthesteady-statepopulationsareindependentofthepolarizationdirection(intheplaneorthogonaltotheaxisofquantization),theanalysisoftheprevioussectionapplies.Whenweapplytheformula(48)tocalculatethescatteringrate,then,wesimplyusethesaturationintensitycalculatedintheprevioussection,andusethetotalintensity(twicethesingle-beamintensity)forIintheformula.Ofcourse,thissteady-statetreatmentisonlystrictlyvalidforastationaryatom,sinceamovingatomwillseeachangingpolarizationandwillthusbeslightlyoutofequilibrium,leadingtosub-Dopplercoolingmechanism[26].4.3.4Three-DimensionalOpticalMolassesFinally,weconsideranopticalmolassesinthreedimensions,composedofsixcircularlypolarizedbeams.Thisopticalcon�gurationisfoundinthecommonlyusedsix-beammagneto-optictrap(MOT).However,asweshallsee,thisopticalcon�gurationisquitecomplicated,andwewillonlybeabletoestimatethetotalrateof�uorescence. 4.3OpticalPumping15 First,wewillderiveanexpressionfortheelectric�eldandintensityofthelight.AtypicalMOTisformedwithtwocounterpropagating,right-circularlypolarizedbeamsalongthez-axisandtwopairsofcounterpropagating,left-circularlypolarizedbeamsalongthex-andy-axes.Thus,thenetelectric�eldisgivenbyE(r;t)=E0 2e�i!teikz^x�i^y p 2+e�ikz^x+i^y p 2+eikx^y+i^z p 2+e�ikx^y�i^z p 2+eiky^z+i^x p 2+e�iky^z�i^x p 2+c:c:=p 2E0cos!th(coskz�sinky)^x+(sinkz+coskx)^y+(cosky�sinkx)^zi:(64)Thepolarizationislineareverywhereforthischoiceofphases,buttheorientationofthepolarizationvectorisstronglyposition-dependent.ThecorrespondingintensityisgivenbyI(r)=I0h6�4(coskzsinky+coskysinkx�sinkzcoskx)i;(65)whereI0:=(1=2)c0E20istheintensityofasinglebeam.Thesixbeamsformanintensitylatticeinspace,withanaverageintensityof6I0andadiscretesetofpointswithzerointensity.Note,however,thattheformofthisinterferencepatternisspeci�ctothesetofphaseschosenhere,sincetherearemorethantheminimalnumberofbeamsneededtodeterminethelatticepattern.Itisclearthatthissituationisquitecomplicated,becauseanatommovinginthismolasseswillexperiencebothachangingintensityandpolarizationdirection.Thesituationbecomesevenmorecomplicatedwhenthemagnetic�eldgradientfromtheMOTistakenintoaccount.However,wecanestimatethescatteringrateifweignorethemagnetic�eldandassumethattheatomsdonotremainlocalizedinthelattice,sothattheyare,ontheaverage,illuminatedbyallpolarizationswithintensity6I0.Inthiscase,thescatteringrateisgivenbythetwo-levelatomexpression(48),withthesaturationintensitycorrespondingtoanisotropicpump�eldOfcourse,thisisalmostcertainlyanoverestimateofthee�ectivesaturationintensity,sincesub-Dopplercoolingmechanismswillleadtoopticalpumpingandlocalizationinthelightmaxima[48].Thesee�ectscanbeminimized,forexample,byusingaverylargeintensitytooperateinthesaturatedlimit,wherethescatteringrateapproaches�=2.Thisestimateofthescatteringrateisquiteusefulsinceitcanbeusedtocalculatethenumberofatomsinanopticalmolassesfromameasurementoftheopticalscatteringrate.Forexample,iftheatomsareimagedbyaCCDcamera,thenthenumberofatomsNatomsisgivenbyNatoms=8h1+4(
=�)2+(6I0=Isat)i �(6I0=Isat)texpcountd Ncounts;(66)whereI0istheintensityofoneofthesixbeams,NcountsistheintegratednumberofcountsrecordedontheCCDchip,texpistheCCDexposuretime,countistheCCDcamerae�ciency(incounts/photon),andd isthesolidangleofthelightcollectedbythecamera.Anexpressionforthesolidangleisd = 4f (f=#)d02;(67)wherefisthefocallengthoftheimaginglens,d0istheobjectdistance(fromtheMOTtothelensaperture),andf=#isthef-numberoftheimagingsystem. 165DataTables 5DataTablesTable1:FundamentalPhysicalConstants(2006CODATArecommendedvalues[2]) SpeedofLight c 2:99792458108m/s(exact) PermeabilityofVacuum 0 410�7N/A2(exact) PermittivityofVacuum 0 (0c2)�1(exact)=8:854187817:::10�12F/m Planck’sConstant h 6:62606896(33)10�34J
s4:13566733(10)10�15eV
s ~ 1:054571628(53)10�34J
s6:58211899(16)10�16eV
s ElementaryCharge e 1:602176487(40)10�19C BohrMagneton B 9:27400915(23)10�24J/Th
1:399624604(35)MHz/G AtomicMassUnit u 1:660538782(83)10�27kg ElectronMass me 5:4857990943(23)10�4u9:10938215(45)10�31kg BohrRadius a0 0:52917720859(36)10�10m Boltzmann’sConstant kB 1:3806504(24)10�23J/K Table2:Rubidium85PhysicalProperties. AtomicNumber Z 37 TotalNucleons Z+N 85 RelativeNaturalAbundance (85Rb) 72:17(2)% [4] NuclearLifetime n (stable) [4] AtomicMass m 84:911789732(14)u1:409993199(70)10�25kg [3] Densityat25C m 1:53g/cm3 [4] MeltingPoint TM 39:30C [4] BoilingPoint TB 688C [4] Speci�cHeatCapacity cp 0:363J/g
K [4] MolarHeatCapacity Cp 31:060J/mol
K [4] VaporPressureat25C Pv 3:92(20)10�7torr [5] NuclearSpin I 5=2 IonizationLimit EI 33690:79890(20)cm�14:17712633(10)eV [49] 5DataTables17 Table3:Rubidium85D2(52S1=2�!52P3=2)TransitionOpticalProperties. Frequency !0 2
384:230406373(14)THz [9,10] TransitionEnergy ~!0 1:589049139(38)eV Wavelength(Vacuum)  780:241368271(27)nm Wavelength(Air) air 780:0328595(78)nm WaveNumber(Vacuum) kL=2 12816:54678496(45)cm�1 Isotopeshift !0(87Rb)�!0(85Rb) 2
78:095(12)MHz [10] Lifetime  26:2348(77)ns [19–22] DecayRate/NaturalLineWidth(FWHM) � 38:117(11)106s�12
6:0666(18)MHz Absorptionoscillatorstrength f 0:69577(20) RecoilVelocity vr 6:0230mm/s RecoilEnergy !r 2
3:8597kHz RecoilTemperature Tr 370:47nK DopplerShift(vatom=vr)
!d(vatom=vr) 2
7:7194kHz DopplerTemperature TD 145:57K Frequencyshiftforstandingwavemovingwithvsw=vr
!sw(vsw=vr) 2
15:4387kHz Table4:Rubidium85D1(52S1=2�!52P1=2)TransitionOpticalProperties. Frequency !0 2
377:107385690(46)THz [10,11] TransitionEnergy ~!0 1:559590695(38)eV Wavelength(Vacuum)  794:979014933(96)nm Wavelength(Air) air 794:7666414(79)nm WaveNumber(Vacuum) kL=2 12578:9483900(15)cm�1 Isotopeshift !0(87Rb)�!0(85Rb) 2
77:583(12)MHz [10] Lifetime  27:679(27)ns [19,20,22] DecayRate/NaturalLineWidth(FWHM) � 36:129(35)106s�12
5:7500(56)MHz Absorptionoscillatorstrength f 0:34231(33) RecoilVelocity vr 5:9113mm/s RecoilEnergy !r 2
3:7179kHz RecoilTemperature Tr 356:86nK DopplerShift(vatom=vr)
!d(vatom=vr) 2
7:4358kHz Frequencyshiftforstandingwavemovingwithvsw=vr
!sw(vsw=vr) 2
14:8716kHz 185DataTables Table5:Rubidium85DTransitionHyper�neStructureConstants. MagneticDipoleConstant,52S1=2 A52S1=2 h
1:0119108130(20)GHz [27] MagneticDipoleConstant,52P1=2 A52P1=2 h
120:527(56)MHz [10,11] MagneticDipoleConstant,52P3=2 A52P3=2 h
25:0020(99)MHz [11,27] ElectricQuadrupoleConstant,52P3=2 B52P3=2 h
25:790(93)MHz [11,27] Table6:Rubidium85DTransitionMagneticandElectricFieldInteractionParameters. Electronsping-factor gS 2:0023193043622(15) [2] Electronorbitalg-factor gL 0:99999354 FinestructureLandég-factor gJ(52S1=2) 2:00233113(20) [27] gJ(52P1=2) 0:666 gJ(52P3=2) 1.3362(13) [27] Nuclearg-factor gI �0:00029364000(60) [27] ClocktransitionZeemanshift
!clock/B2 2
1293:98Hz/G2 Ground-statepolarizability 0(52S1=2) h
0:0794(16)Hz/(V/cm)2 [36] D1scalarpolarizability 0(52P1=2)�0(52S1=2) h
0:122306(16)Hz/(V/cm)2 [50] D2scalarpolarizability 0(52P3=2)�0(52S1=2) h
0:1340(8)Hz/(V/cm)2 [51] D2tensorpolarizability 2(52P3=2) h
�0:0406(8)Hz/(V/cm)2 [51] 5DataTables19 Table7:Rubidium85DipoleMatrixElements,SaturationIntensities,andResonantScatteringCrossSections. D2(52S1=2�!52P3=2)TransitionDipoleMatrixElement hJ=1=2kerkJ0=3=2i 4:22753(62)ea03:58425(52)10�29C
m E�ectiveDipoleMoment,SaturationIntensity,andResonantCrossSection(F=3!F0=4)(isotropiclightpolarization) diso;e(F=3!F0=4) 1:95696(29)ea0 1:65918(24)10�29C
m Isat(iso;e)(F=3!F0=4) 3:8951(11)mW/cm2 0(iso;e)(F=3!F0=4) 1:245726051126(88)10�9cm2 E�ectiveFar-DetunedDipoleMoment,SaturationIntensity,andResonantCrossSection(D2line,-polarizedlight) ddet;e;D2 2:44076(36)ea0 2:06937(30)10�29C
m Isat(det;e;D2) 2:50399(73)mW/cm2 0(det;e;D2) 1:93779607953(14)10�9cm2 DipoleMoment,SaturationIntensity,andResonantCrossSectionjF=3;mF=3i!jF0=4;m0F=4icyclingtransition(-polarizedlight) d(mF=3!m0F=4) 2:98931(44)ea0 2:53444(37)10�29C
m Isat(mF=3!m0F=4) 1:66932(49)mW/cm2 0(mF=3!m0F=4) 2:90669411930(20)10�9cm2 D1(52S1=2�!52P1=2)TransitionDipoleMatrixElement hJ=1=2kerkJ0=1=2i 2:9931(14)ea02:5377(12)10�29C
m E�ectiveFar-DetunedDipoleMoment,SaturationIntensity,andResonantCrossSection(D1line,-polarizedlight) ddet;e;D1 1:72809(84)ea0 1:46513(71)10�29C
m Isat(det;e;D1) 4:4876(43)mW/cm2 0(det;e;D1) 1:00584592573(24)10�9cm2 Table8:Rubidium85RelativeHyper�neTransitionStrengthFactorsSFF0[fromEq.(41)]. D2(52S1=2�!52P3=2)transition S34 9=14 S23 14=45 S33 5=18 S22 7=18 S32 5=63 S21 3=10 D1(52S1=2�!52P1=2)transition S33 4=9 S23 7=9 S32 5=9 S22 2=9 205DataTables Table9:Rubidium85D2(52S1=2�!52P3=2)DipoleMatrixElementsfor+transitions(F=3;mF�!F0,m0F=mF+1),expressedasmultiplesofhJ=1=2jjerjjJ0=3=2i. mF=�3 mF=�2 mF=�1 mF=0 mF=1 mF=2 mF=3 F0=4 r 1 56 r 3 56 r 3 28 r 5 28 r 15 56 r 3 8 r 1 2 F0=3 r 5 72 r 25 216 r 5 36 r 5 36 r 25 216 r 5 72 F0=2 r 5 63 r 10 189 r 2 63 r 1 63 r 1 189 Table10:Rubidium85D2(52S1=2�!52P3=2)DipoleMatrixElementsfortransitions(F=3;mF�!F0,m0F=mF),expressedasmultiplesofhJ=1=2jjerjjJ0=3=2i. mF=�3 mF=�2 mF=�1 mF=0 mF=1 mF=2 mF=3 F0=4 �r 1 8 �r 3 14 �r 15 56 �r 2 7 �r 15 56 �r 3 14 �r 1 8 F0=3 �r 5 24 �r 5 54 �r 5 216 0 r 5 216 r 5 54 r 5 24 F0=2 r 5 189 r 8 189 r 1 21 r 8 189 r 5 189 Table11:Rubidium85D2(52S1=2�!52P3=2)DipoleMatrixElementsfor�transitions(F=3;mF�!F0,m0F=mF�1),expressedasmultiplesofhJ=1=2jjerjjJ0=3=2i. mF=�3 mF=�2 mF=�1 mF=0 mF=1 mF=2 mF=3 F0=4 r 1 2 r 3 8 r 15 56 r 5 28 r 3 28 r 3 56 r 1 56 F0=3 �r 5 72 �r 25 216 �r 5 36 �r 5 36 �r 25 216 �r 5 72 F0=2 r 1 189 r 1 63 r 2 63 r 10 189 r 5 63 5DataTables21 Table12:Rubidium85D2(52S1=2�!52P3=2)DipoleMatrixElementsfor+transitions(F=2;mF�!F0,m0F=mF+1),expressedasmultiplesofhJ=1=2jjerjjJ0=3=2i. mF=�2 mF=�1 mF=0 mF=1 mF=2 F0=3 r 2 135 r 2 45 r 4 45 r 4 27 r 2 9 F0=2 r 7 54 r 7 36 r 7 36 r 7 54 F0=1 r 3 10 r 3 20 r 1 20 Table13:Rubidium85D2(52S1=2�!52P3=2)DipoleMatrixElementsfortransitions(F=2;mF�!F0,m0F=mF),expressedasmultiplesofhJ=1=2jjerjjJ0=3=2i. mF=�2 mF=�1 mF=0 mF=1 mF=2 F0=3 �r 2 27 �r 16 135 �r 2 15 �r 16 135 �r 2 27 F0=2 �r 7 27 �r 7 108 0 r 7 108 r 7 27 F0=1 r 3 20 r 1 5 r 3 20 Table14:Rubidium85D2(52S1=2�!52P3=2)DipoleMatrixElementsfor�transitions(F=2;mF�!F0,m0F=mF�1),expressedasmultiplesofhJ=1=2jjerjjJ0=3=2i. mF=�2 mF=�1 mF=0 mF=1 mF=2 F0=3 r 2 9 r 4 27 r 4 45 r 2 45 r 2 135 F0=2 �r 7 54 �r 7 36 �r 7 36 �r 7 54 F0=1 r 1 20 r 3 20 r 3 10 225DataTables Table15:Rubidium85D1(52S1=2�!52P1=2)DipoleMatrixElementsfor+transitions(F=3;mF�!F0,m0F=mF+1),expressedasmultiplesofhJ=1=2jjerjjJ0=1=2i. mF=�3 mF=�2 mF=�1 mF=0 mF=1 mF=2 mF=3 F0=3 r 1 9 r 5 27 r 2 9 r 2 9 r 5 27 r 1 9 F0=2 r 5 9 r 10 27 r 2 9 r 1 9 r 1 27 Table16:Rubidium85D1(52S1=2�!52P1=2)DipoleMatrixElementsfortransitions(F=3;mF�!F0,m0F=mF),expressedasmultiplesofhJ=1=2jjerjjJ0=1=2i. mF=�3 mF=�2 mF=�1 mF=0 mF=1 mF=2 mF=3 F0=3 �r 1 3 �r 4 27 �r 1 27 0 r 1 27 r 4 27 r 1 3 F0=2 r 5 27 r 8 27 r 1 3 r 8 27 r 5 27 Table17:Rubidium85D1(52S1=2�!52P1=2)DipoleMatrixElementsfor�transitions(F=3;mF�!F0,m0F=mF�1),expressedasmultiplesofhJ=1=2jjerjjJ0=1=2i. mF=�3 mF=�2 mF=�1 mF=0 mF=1 mF=2 mF=3 F0=3 �r 1 9 �r 5 27 �r 2 9 �r 2 9 �r 5 27 �r 1 9 F0=2 r 1 27 r 1 9 r 2 9 r 10 27 r 5 9 5DataTables23 Table18:Rubidium85D1(52S1=2�!52P1=2)DipoleMatrixElementsfor+transitions(F=2;mF�!F0,m0F=mF+1),expressedasmultiplesofhJ=1=2jjerjjJ0=1=2i. mF=�2 mF=�1 mF=0 mF=1 mF=2 F0=3 �r 1 27 �r 1 9 �r 2 9 �r 10 27 �r 5 9 F0=2 �r 2 27 �r 1 9 �r 1 9 �r 2 27 Table19:Rubidium85D1(52S1=2�!52P1=2)DipoleMatrixElementsfortransitions(F=2;mF�!F0,m0F=mF),expressedasmultiplesofhJ=1=2jjerjjJ0=1=2i. mF=�2 mF=�1 mF=0 mF=1 mF=2 F0=3 r 5 27 r 8 27 r 1 3 r 8 27 r 5 27 F0=2 r 4 27 r 1 27 0 �r 1 27 �r 4 27 Table20:Rubidium85D1(52S1=2�!52P1=2)DipoleMatrixElementsfor�transitions(F=2;mF�!F0,m0F=mF�1),expressedasmultiplesofhJ=1=2jjerjjJ0=1=2i. mF=�2 mF=�1 mF=0 mF=1 mF=2 F0=3 �r 5 9 �r 10 27 �r 2 9 �r 1 9 �r 1 27 F0=2 r 2 27 r 1 9 r 1 9 r 2 27 245DataTables Figure1:VaporpressureofrubidiumfromthemodelofEqs.(1).Theverticallineindicatesthemeltingpoint. 5DataTables25 Figure2:Rubidium85D2transitionhyper�nestructure,withfrequencysplittingsbetweenthehyper�neenergylevels.Theexcited-statevaluesaretakenfrom[11,27],andtheground-statevaluesarefrom[27].Therelativehyper�neshiftsareshowntoscalewithineachhyper�nemanifold(butvisualspacingsshouldnotbecomparedbetweenmanifoldsortotheopticalsplitting).TheapproximateLandégF-factorsforeachlevelarealsogiven,withthecorrespondingZeemansplittingsbetweenadjacentmagneticsublevels. 265DataTables Figure3:Rubidium85D1transitionhyper�nestructure,withfrequencysplittingsbetweenthehyper�neenergylevels.Theexcited-statevaluesaretakenfrom[10,11],andtheground-statevaluesarefrom[27].Therelativehyper�neshiftsareshowntoscalewithineachhyper�nemanifold(butvisualspacingsshouldnotbecomparedbetweenmanifoldsortotheopticalsplitting).TheapproximateLandégF-factorsforeachlevelarealsogiven,withthecorrespondingZeemansplittingsbetweenadjacentmagneticsublevels. 5DataTables27 Figure4:Rubidium8552S1=2(ground)levelhyper�nestructureinanexternalmagnetic�eld.ThelevelsaregroupedaccordingtothevalueofFinthelow-�eld(Zeeman)regimeandmJinthestrong-�eld(hyper�nePaschen-Back)regime. Figure5:Rubidium8552P1=2(D1excited)levelhyper�nestructureinanexternalmagnetic�eld.ThelevelsaregroupedaccordingtothevalueofFinthelow-�eld(Zeeman)regimeandmJinthestrong-�eld(hyper�nePaschen-Back)regime. 285DataTables Figure6:Rubidium8552P3=2(D2excited)levelhyper�nestructureinanexternalmagnetic�eld.ThelevelsaregroupedaccordingtothevalueofFinthelow-�eld(Zeeman)regimeandmJinthestrong-�eld(hyper�nePaschen-Back)regime. Figure7:Rubidium8552P3=2(D2excited)levelhyper�nestructureinaconstant,externalelectric�eld.ThelevelsaregroupedaccordingtothevalueofFinthelow-�eld(Zeeman)regimeandjmJjinthestrong-�eld(“electric”hyper�nePaschen-Back)regime.LevelswiththesamevaluesofFandjmFj(foraweak�eld)aredegenerate. 6Acknowledgements29 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