McDonald Joseph Henry Laboratories Princeton University Princeton NJ 08544 Lijun J Wang NEC Research Institute Inc Princeton NJ 08540 Aug 17 2003 1Problem Dirac has written 1 Each photon then interferes only with itself Interference between two di64 ID: 26446
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BunchingofPhotonsWhenTwoBeamsPassThroughaBeamSplitterKirkT.McDonaldJosephHenryLaboratories,PrincetonUniversity,Princeton,NJ08544LijunJ.WangNECResearchInstitute,Inc.,Princeton,NJ08540(Aug.17,2003)1ProblemDirachaswritten[1]Eachphotontheninterferesonlywithitself.Interferencebetweentwodierentphotonsneveroccurs.Indeed,apracticaldenitionisthatclassicaloptics Hint:arelativelyelementaryargumentcanbegivenbyrecallingthatthephaseofareectedphoton(i.e.,ofthereectedwavefromasingleinputbeam)is90dierentfromthatofatransmittedphoton[2].Considerrstthecasesthatoneofiszero.2Solution 2.1ASingleInputBeamWerstconsiderthecaseofasingleinputbeamwith0.Then,ofcourse,=0.Inaclassicalview,theinputbeamwouldhaveenergy,whereistheangularfrequencyofthephotons.Then,theeectofthe50:50beamsplitterwouldbetocreateoutputbeamsofequalenergies,2.Intermsofphotonnumbers,theclassicalviewwouldimplythattheonlypossibilityfortheoutputbeamsisButinfact,thetransmittedbeamcancontainanynumberofphotonsbetween0and,whilethereectedbeamcontainsphotons.Ifthephotonsweredistinguishable,wewouldassignaprobabilityof(1toeachcongurationoftransmittedandreectedphotonsinthe50:50splitter.Butthephotonsareindistinguishable,sothattheprobabilitythatoutofphotonsaretransmittedislargerthan(1bythenumberofwaysthephotonscanbearrangedintoagrouptransmittedandreectedphotonswithoutregardtotheirorder,i.e.,bythebinomialcoecient, (1)Thus,theprobability0)thatoutofphotons(inasingleinputbeam)aretransmittedbythebeamsplitteris,0)= (2)Theresult(2)isalreadyverynonclassical,inthatthereisasmall,butnonzeroprobabilitythattheentireinputbeamistransmitted,orreected.However,inthelimitoflargelargestprobabilityisthatthenumbersofphotonsinthereectedandtransmittedbeamsareverynearlyequal.WeconrmthisbyuseofStirlingsapproximationforlarge n.(3)Forlarge,and=(1+2,wehave, 2k nk21k nnk2=2n +1) n/ (1+(4)Theprobabilityofphotonsoutofbeingtransmitteddropsto1/2thepeakprobabilitywhen .Hence,forlargethenumberdistributionofphotonsinthetransmitted(andreected)beamisessentiallyadeltafunctioncenteredat2,inagreementwiththeclassicalview.Themostdramaticdierencebetweentheclassicalandquantumbehaviorofasinglebeamina50:50beamsplitteroccurswhen=2,0)= 0)= 0)= (5) Inthesubsequentanalysisweshallneedtoconsiderinterferenceeects,sowenotethatthemagnitudeoftheprobabilityamplitudethatoutofphotonsinasinglebeamaretransmittedbya50:50beamsplittercanobtainedbytakingthesquarerootofeq.(2),k,n Cnk1 (6)Theseamplitudeshavetheobvioussymmetries,k,nk,kk,nk,k(7)Wemustalsoconsiderthephasesoftheseamplitudes,oratleasttherelativephases.Thehintisthatwemayconsiderthephaseofareectedphotontobeshiftedwithrespecttothatofatransmittedphotonby90,asfollowsfromaclassicalanalysisofwavesina50:50beamsplitter[2](seealsotheAppendix).Inthisproblem,wedenethephaseofatransmittedphotontobezero,sothattheprobabilityamplitudeshouldincludeafactorof 1foreachreectedphoton.Thus,wehave,k,n0)= Cnk1 (8)k,k0)= Cnk1 (9)k,n Cnk1 (10)k,k Cnk1 (11)2.2TwoInputBeamsWenowcalculatethegeneralprobability)thatoutputphotonsareobservedalongthedirectionofinputbeam1whenthenumberofphotonsintheinputbeamsinandWerstgiveaclassicalwaveanalysis.Theinputwaveshaveamplitudes andareinphaseatthecenterofthebeamsplitter.Theoutputamplitudesarethesumsofthereectedandtransmittedpartsoftheinputamplitudes.Areectedamplitudehasaphaseshiftof90relativetoitscorrespondingtransmittedamplitude,asdiscussedinsec.2.1.Inthe50:50beamsplitter,themagnitudeofboththereectedandtransmittedamplitudesfromasingleinputbeamare1 2timesthemagnitudeoftheamplitudeofthatbeam.Hence,theoutputamplitudesare, (12) (13) Takingtheabsolutesquareofeqs.(12)-(13),wendtheoutputbeamstobedescribedby, =a21+a22 2=n1+n2 (14)Theclassicalviewisthata50:50beamsplittersimplysplitsbothinputbeams,whentheyareinphase.Foraquantumanalysis,weproceedbynotingthatofthephotonsinoutputbeam1,ofthesecouldhavecomebytransmissionfrominputbeam1,andbyreectionfrominputbeam2(solongas).Theprobabilityamplitudethatoutofphotonsaretransmittedfrombeam1whilephotonsarereectedfrombeam2is,towithinaphasefactor,theproductoftheamplitudesforeachofthesecongurationsresultingfromasingleinputbeam,k,Nk,n Cn1kCn2N1k1 (15)referringtoeqs.(8)-(11).Themostdramaticnonclassicalfeaturestobefoundbelowcanbeattributedtothepresenceofthefactor(thatarisesfromthe90phaseshiftbetweenreectedandtransmittedphotons.SincephotonsobeyBosestatistics,wesumthesub-amplitudes(15),weightingeachonebythesquarerootofthenumberofwaysthatoutofthephotonsintherstoutputbeamcanbeassignedtoinputbeam1,namely,timethesquarerootofthenumberofwaysthattheremainingphotonsfrominputbeam1canbeassignedtothephotonsinoutputbeam2,namelytoobtain, k,Nk,n 2(n1+n2)/2k( (16)Whenevaluatingthisexpression,anybinomialcoecientinwhichisnegative,orgreaterthan,shouldbesettozero.Thedesiredprobabilityis,ofcourse,(17)Someexamplesoftheprobabilitydistributionsforsmallnumbersofinputphotonsaregivenbelow. Delicatetojustifynotalsoincludingfactors,and,thesebeingthewaysofassigningphotonstooutputbeam2butthesefactorsarethesameasthosealreadyincluded,andsoshouldnotbecountedtwice... 2.2.1TwoInputPhotonsInput Output( 2|1|0| |2, 41 21 4|1, 201 2|0, 41 21 Wheniszero,theprobabilitydistributionisbinomial,asfoundinsec.2.1.When=1thereiscompletedestructiveinterferencebetweenthecaseswherebothphotonsarereected(combinedphaseshift=180)andwhenbotharetransmitted(combinedphaseshift=0).Thisquantumresultisstrikinglydierentfromtheclassicalexpectationthattherewouldbeonephotonineachoutputbeam.2.2.2ThreeInputPhotonsInput Output( 3|2|1|0| |3, 83 83 81 8|2, 81 81 81 8|1, 81 81 81 8|0, 83 83 81 2.2.3FourInputPhotonsInput Output( 4|4|2|1|0| |4, 43 81 41 3, 41 401 41 4|2, 801 403 8|1, 41 401 41 4|0, 43 81 41 2.2.4SymmetricInputBeams:Inthiscasethereiszeroprobabilityofobservinganoddnumberofphotonsineitheroutputbeam. Toseethis,notethatwhen,themagnitudesofthesubamplitudesareequalforhavingphotonsappearinginoutputbeam1fromeitherinputbeam1orinputbeam2.However,thephasesofthesetwosubamplitudesare180apart,sothattheycancel.Inparticular,whenphotonsaretransmittedintooutputbeam1frominputbeam1,thenphotonsarereectedfrominputbeam2intooutputbeam1;meanwhile,photonsarereectedfrominputbeam1intooutputbeam2.So,theoverallphasefactorofthissubamplitudeis.Whereas,ifphotonsarereectedfrominputbeam2intooutputbeam1,thenphotonsaretransmittedfrominputbeam1intooutputbeam1,andsophotonsarereectedfrominputbeam1intooutputbeam2.So,theoverallphasefactorofthissubamplitudeis.Thephasefactorbetweenthesetwosubamplitudes(whosemagnitudesareequal)iswhichis1forodd,asclaimed.Forthecaseofobservinganevennumberofphotonsintheoutputbeams,aremarkablesimplicationofeq.(16)holds[3].Wehavenotbeenabletoshowthisbyelementarymeans.Itdoesfollowbyinspectionwhen,inwhichcaseeq.(16)containsonlyasinglenonzeroterm.Ingeneral,theindexineq.(16)forn,nrunsfrom0to,orfrom.Thereareanoddnumberofterms,thecentralonehavingindex.Byastrangemiracleofcombinatorics,thesumcollapsestoasimpliedversionofthecentraltermoftheseries....Namely,n,n)=( 2n (18)Therefore,the+1nonvanishingprobabilitiesforsymmetricinputbeamsare.n,n 22nC2mmC2n2mnm1 m nm (19)wheretheapproximationholdsforlargeandlarge.Notethatdx/ Thisprobabilitydistributionpeaksfor=0ori.e.,forallphotonsinoneortheotheroutputbeam,withvalue,n,nn,n (20)Theprobabilityofndingalloutputphotonsinasinglebeamwhentheinputbeamsaresymmetricislargerbyafactorthanwhenthereisonlyasingleinputbeam(ofthesametotalnumberofphotons),becausetherearewaysofassigningthephotonsfrominputbeam1tothe2photonsintheoutputbeam.Thisisanextremeexampleofphotonbunchingcausedbythebeamsplitter.Itisnoteworthythattheresult(19)doesnotagreewiththeclassicalprediction(14)inthelargelimit.Ofcourse,aspointedoutbyGlauber[8],aclassicalwavecorrespondstoaphotonstatewithminimumuncertaintyproducts,whereandaretheelectricandmagneticeldamplitudesofthewave,respectively.Incaseofapulse,weexpectclassicallythatbothitsenergyandphasearewelldened,buttheclosestquantumequivalentisacoherent statewithminimaluncertaintytotheproduct.ThisstateisasuperpositionofstatesofvariousphotonsnumberswhoseexpectationvalueforfollowsaPoissondistributionwith.Forlarge,thevarianceinphotonnumberis Hence,inanexperimentinwhichlargenumbersandofphotonsareobservedatthetwooutputportsofthebeamsplitter,wecansaythatthenumbersandofphotonsattheinputportsobeyed,butwecannotknowandseparately(iftheinputsbeamsareclassical).AllwecanknowarethemeanvaluesandTherefore,weshouldrewritetheprobabilitydistribution(17)as,(21)whereistheamplitudethatinputbeamcontainedphotonswhenthemeannumberofphotonsinthisbeamis.Weconjecturethatadetailedcalculationofeq.(21)wouldagreewiththeclassicalprediction(14),butwehavenotconrmedthis.3Appendix:PhaseShiftinaLosslessBeamSplitterWegiveaclassicalargumentbasedonaMach-Zehnderinterferometer,showninthegurebelow,thatthereisa90phaseshiftbetweenthereectedandtransmittedbeamsinalossless,symmetricbeamsplitter.Then,followingDiracsdictum[1],thisresultappliestoasinglephoton.Abeamoflightofunitamplitudeisincidentontheinterferometerfromtheupperleft.Thereectedandtransmittedamplitudesareand,wheremagnitudesandarerealnumbers.Theconditionofalosslessbeamsplitteristhat,(22)Thereectedandtransmittedbeamsarereectedomirrorsandrecombinedinasecondlosslessbeamsplitter,identicaltotherst. Then,theamplitudefortransmissionattherstbeamsplitter,followedbyreectionatthesecond,istr,eHence,therecombinedbeamthatmovestotherighthasamplitude,rte(23)whiletherecombinedbeamthatmovesdownwardshasamplitude,(24)Theintensityoftherstoutputbeamis,(25)andthatofthesecondoutputbeamis,cos2((26)Forlosslesssplitters,thetotaloutputintensitymustbeunity,=1=([1+cos2((27)Recallingeq.(22),wemusthave,(28)foranyvalueofthesplittingratioTheprecedingargumentdoesnotclarifywherethatphasedierence(28)is90butmoredetailedarguments[2]showthephasedierencetobe.Thatis,+90(29)Furthermore,ifthebeamsplitteristhincomparedtoawavelength,then0andReferences[1]P.A.M.Dirac,ThePrinciplesofQuantumMechanicsed.(ClarendonPress,1958),p.9,http://kirkmcd.princeton.edu/examples/QM/dirac_qm_58.pdf[2]See,forexample,Problem4ofthePrincetonPh501problemset6athttp://kirkmcd.princeton.edu/examples/ph501set6.pdfAnargumentverysimilartothatintheAppendixisgivenbyV.Degiorgio,Phaseshiftbetweenthereectedandtransmittedopticaleldsofasemireectinglosslessmirroris2,Am.J.Phys.,81(1980),http://kirkmcd.princeton.edu/examples/optics/degiorgio_ajp_48_81_80.pdfandalsobyZ.Y.OuandL.Mandel,Derivationofreciprocityrelationsforabeamsplit-terfromenergybalance,Am.J.Phys.,66(1989),http://kirkmcd.princeton.edu/examples/optics/ou_ajp_57_66_89.pdf EarlydiscussionsofphaseshiftsinbeamsplittersincludeG.B.Airy,OnthePhe-nomenonofNewtonsRingswhenformedbetweentwotransparentSubstancesofdif-ferentrefactivePowers,Phil.Mag.,20(1833),http://kirkmcd.princeton.edu/examples/optics/airy_pm_2_30_33.pdfG.G.Stokes,OnthePerfectBlacknessoftheCentralSpotinNewtonsRings,andontheVericationofFresnelsFormulæfortheIntensitiesofReectedandRefractedRaysCamb.andDublinMath.J.,1(1849),http://kirkmcd.princeton.edu/examples/optics/stokes_cdmj_4_1_49.pdfreprintedinMathematicalandPhysicalPapersofG.G.Stokes,V.2(CambridgeU.Press,1883),pp.89-103,Seealsosec.1.6.4ofM.BornandE.Wolf,PrinciplesofOpticsed.(CambridgeU.Press,1999),http://kirkmcd.princeton.edu/examples/EM/born_wolf_7ed.pdf[3]R.A.Campos,B.E.A.SalehandM.C.Teich,Quantum-mechanicallosslessbeamsplit-ter:SU(2)symmetryandphotonstatistics,Phys.Rev.A,1371(1989),http://kirkmcd.princeton.edu/examples/QM/campos_pra_40_1371_89.pdf[4]M.G.A.Paris,HomodynePhotocurrent,SymmetriesinPhotonMixingandNumberStateSynthesis,Int.J.Mod.Phys.B,1913(1997),http://kirkmcd.princeton.edu/examples/optics/paris_ijmp_b11_1913_97.pdf[5]R.P.Feynman,R.B.LeightonandM.Sands,TheFeynmanLecturesonPhysics(Addi-sonWesley,1965),Vol.I,sec.33-6,https://www.feynmanlectures.caltech.edu/I_33.html[6]C.K.Hong,Z.Y.OuandL.Mandel,MeasurementofSubpicosecondTimeIntervalsbetweenTwoPhotonsbyInterference,Phys.Rev.Lett.,2044(1987),http://kirkmcd.princeton.edu/examples/optics/hong_prl_59_2044_87.pdf[7]Z.Y.Ou,J.-K.RheeandL.J.Wang,ObservationofFour-PhotonInterferencewithaBeamSplitterbyPulsedParametricDown-Conversion,Phys.Rev.Lett.,959(1999),http://kirkmcd.princeton.edu/examples/optics/ou_prl_83_959_99.pdfPhotonbunchingandmultiphotoninterferenceinparametricdown-conversion,Phys.Rev.A,593(1999),http://kirkmcd.princeton.edu/examples/optics/ou_pra_60_593_99.pdf[8]R.J.Glauber,PhotonCorrelationshttp://kirkmcd.princeton.edu/examples/QM/glauber_prl_10_84_63.pdfTheQuantumTheoryofOpticalCoherence,Phys.Rev.,2529(1963),http://kirkmcd.princeton.edu/examples/QM/glauber_pr_130_2529_63.pdfCoherentandIncoherentStatesoftheRadiationFieldhttp://kirkmcd.princeton.edu/examples/QM/glauber_pr_131_2766_63.pdf