Preliminary analysis of a recent experiment by F. A. Bovino

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

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:0viXraPreliminaryanalysisofarecentexperimentbyF.A.BovinoEmilioSantosDepartamentodeF´ sica.UniversidaddeCantabria.Santander.SpainFebruary2,2008AbstractAnanalysisismadeoftheresultsofarecentpolarizationcorrela-tionexperimentbyBovino(unpublished)whereabout60,000datahavebeenobtained.Iassumethatthestateofthephotonpairsproducedinthesource(anon-linearcrystal)areina(sightly)non-maximallyentangledstateandthemostrelevantnon-idealitiesoftheset-uparetakenintoaccount.Acomparisonismadebetweenthepredictionsofquantummechanicsandasimplefamilyoflocalhiddenvariablesmodelswiththeresultthattheformerisviolatedbymorethan4standarddeviationswhilstthedataarecompatiblewiththefamilyoflocalmodels.Thepossibilityofmakingatruediscriminationbetweenlocalrealismandquantummechanics,vialoophole-freetestsofBell´sinequality,seemsextremelydi cultorimpossibleasshownbythefa¨ lureofthee ortsmade

duringmorethan40yearsafterBell´swork.Thusthestandardpracticehasbeentotestinequalitiesderivedfromlocalrealismplustheassumptionoffairsampling.TheseinequalitieshavebeenviolatedinmanyHoweverfairsamplingisanassumptiontheoreticallyabsurd(becauseitex-cludesallsensiblehiddenvariablesandithasbeenclaimedtobeempiricallyThus,inordertomakerealprogressweshouldtestinequalitesvalidforrestricted,butsensible,familiesoflocalhiddenvariables(LHV)theoriessuchastheonesderivedinandsumarizedinInthislightIwillanalyzetheresultsofarecentexperimentbyBovino[6].

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An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

IntheBovinoexperimentthedetectione ciencyofphotoncountershasbeen55%andtheoveralle ciencyabout17%[6].Thus,assuminglossesoforder15%in lters[7],Imayestimatethatthereisacollectione ciency(“e ective bercouplingcoe cients”)about36%.Bovinousedtwo-channelpolarizersandhehasrecordedahugeamountofdatawhichcorrespondto4coincidenceratesand4singleratesfor45anglesofAlice´spolarizerscombinedeachwith45anglesofBob´spolarizers.

Intheidealcasequantummechanicspredictsforthecoincidencerates,R12,asafunctionoftheangle,φ,betweenAlice´sandBob´spolarizers,acosinuscurveoftheform

QR12(φj)= R12 [1+Vcos(2φ)],(1)

exp(φj),withthispredictionThedisagreementoftheexperimentaldata,R12maybemeasuredbythequantity

=

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

thecollectione ciency(“e ective bercouplingcoe cients”)andthe ltersbutnottodetectors,seemstobeviolated.Itpredictsthat shouldbelargerthan0.04,aconstraintnotful lledinabouthalfthecases(thedocumentSantos3.xlssentbyBovinoreportsthat -labelledD(eta)there-isabout0.12butmycalculationgivesavalueabout1/3thatofBovino).Inanycasetheviolationmightbeattributedtothenon-idealitiesoftheexperimentalset-up.

Insummarytheprevious(ratherpoor)analysisoftheexperimentseemstoimplythatitiscompatiblewiththefamilyLHV1(andthereforeLHV0)butdisagreeswithboththefamilyLHV2andquantummechanics.Inordertogetmoreinterestinginformationabetteranalysisofthedataisrequired,whichismadeinthefollowing.

Thenon-idealitiesoftheexperimentalset-uparequiteimportant,asisshownbythefactthatthe4singleratesareverydi erent,rangingfromabout70000to110000countsinthe(unspeci ed)timewindow.Furthermore,fora xedpositionofBob´sanalyzerAlice´ssingleratesdependontheangleofAlice´sanalyzers,withavariationupto7%betweenthemaximumandtheminimumvalue.Similarvariationsexistforthesinglerateswhenweconsiderdi erentvaluesofBob´sanalyzerswithAlice´sanalyzers xed.(Nosigni cantvariationexistsinthesinglecountsofAlice(Bob)whenBob(Alice)polarizerisrotated,asisexpectedbythe“no-signallingprinciple”whichforbidssendinginformationatadistance.)Ishallmakethequantum-mechanicalanalysisoftheexperimentbystudyingthestateproducedinthenon-linearcrystalandhowthisstateevolvesinthetravelofphotonsuntilthedetectors,asfollows.

1.Ishouldassumethatinthenonlinearcrystalthephotonsareproducedalwaysinpairs(nosingle-photonproductions)andthatthetwophotonsinapairareinsomepure,entangled,quantum-mechanicalstate.Theentangle-mentmaynotbemaximal,althoughclosetomaximal.ThusIwillassumethatthestateisoftheform

|ψ>=1

1+(1+γ)2[|H>1 |V>2 (1+γ)|V>1 |H>2],(3)

whereγisarealnumbersuchthat|γ|<<1.

2.Onlyafractionµa(µb)ofthephotonsgoingtoAlice(Bob)arecol-lected,sothatthestateofthephotonsintheoptical bersbecomesasta-tisticalmixtureof

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An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

a)theinitialtwo-photonstate(3)withweightµaµb,

b)asingle-photonstatehorizontallypolarizedforAlicewithweightµa(1 µb)/(2+2γ+γ2),

c)asingle-photonstateverticallypolarizedforAlicewithweightµa(1 µb)(1+γ)2/(2+2γ+γ2)),

d)asingle-photonstatehorizontallypolarizedforBobwithweightµb(1 µa)(1+γ)2/(2+2γ+γ2)),

e)asingle-photonstateverticallypolarizedforBobwithweightµb(1 µa)/(2+2γ+γ2),

f)thevacuumstatewithweight1 µa µb+µaµb.

3.ThetransmittancesofthepolarizationanalyzersofAliceforhorizontalorverticalpolarizationaresuchthat,whenabeamoflinearlypolarizedlightwithintensityIinarrivesatthepolarizer,thetransmittedandre ectedintensitiesare,respectively,

1Ia+(θ)=Iin(Ta+ ta+)cos2θ+ta+=

2Iin[(Ta +ta )+(Ta ta )cos2θ],

andsimilarforBobwiththechangesa→b.ThesumT+tisclosetounitywhilst0<t<<1.

4.Thequantume cienciesofthe4detectorsaredi erent,sayζa+andζa forAliceandζb+andζb forBob.

Intheidealsituationthequantumpredictionforthe4coincidenceprob-abilitiesofstate(3)wouldbe[3]

P++=1

[(1+γ)sinαsinβ+cosαcosβ]2,

P +=1+(1+γ)12

1+(1+γ)2[(1+γ)cosαsinβ sinαcosβ]2,(5)

whereα(β)istheangleofAlice´s(Bob´s)polarizationanalyzer.NowIintroducethemostrelevantnon-idealitiesasfollows.Duetolossesand

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An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

absorptions,asexplainedabove,thereisaglobalfactor

µa(Ta++ta+)ζa+µb(Tb++tb+)ζb+

infrontofP++andsimilarlyforP+ ,P +andP .Inadditioneverycos2αorsin2αshouldbeprecededbyeitherafactor(Ta+ ta+)/(Ta++ta+)orafactor(Ta ta )/(Ta +ta )(seeeqs.(4))andsimilarly,witha→b,forcosβorsinβ.ThusIget

P++=1

4ηa+ηb [1+V+ cos2φ γ′(cos2β+cos2α) γ′′sin2αsin2β],

P1 +=

4ηa ηb [1 V cos2φ γ′(cos2β cos2α)+γ′′sin2αsin2β(6)],

where

φ≡α β,γ′≡2γ+γ2

2+2γ+γ2,ηa+≡µaTa+ζa+,

andsimilarlyfortheremainingparametersη.ThetermV++isgivenby

V(Ta+ ta+)(Tb+ tb+)++=

1

2ηa+[1 γ′cos2α],Pa =

2ηb+[1+γ′cos2β],Pb =1

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

ThequestioniswhetherallthedataofBovino´sexperimentmaybe ttedtoeqs.(6)and(7)with10

freeparameters,namelyηa+,ηb+,ηa ,ηb ,γ,V++,V+ ,V +,V andR0.Theanalysissimpli esalotasfollows.Fromeqs.(7)Igettheglobale cienciesbyaveragingoverangles,thatis

ηa+=2 Pa+ ,ηb =2 Pb ,ηa =2 Pa ,ηb+=2 Pb+ .(8)NowImayeliminatethee cienciesηa+,etc.ineqs.(6)usingeqs.(8)andpassfromprobabilitiestocountnumbersbymultiplicationtimesR0intheappropriateplaces.ThusIget

R0R++

=1+V+ cos2φ γ′(cos2β+cos2α) γ′′sin2αsin2β, Ra+ Rb

R0R +

Ra Rb =1 V cos2φ γ′(cos2β cos2α)+γ′′sin2αsin2β.(9)HenceImayde nethefollowingaveragecoincidencedetectionprobability,P(α,β),

P≡16R0f

f≡R+ (α,β)

Ra Rb+ +R++α+ π

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ThequantumpredictionforPdependsontheanglesαandβonlyviathecombinationα β≡φanditisrathersimple,namely

1PQ(φ)=

(V+++V+ +V ++V 2γ2

,β+4π (10)π,β 4. Ra Rb 4

R++(φ)+R (φ)+R+ (φ)+R +(φ)

6,(12)

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

forwhichquantummechanicspredictsasimpleexpression,namelyVcos(2φ),onlyintheidealcase.

Agood tofthedataintoeqs.(10)and(11),withR0andVasfreeparameters,seemstobeanecessaryconditionforthecompatibilityoftheexperimentwithquantummechanics.However,evenifagood tisnotpossi-ble,stilltheexperimentmaybecompatiblewithquantummechanicsbecausethestateproducedinthesourcemaybedi erentfromeq.(3)and/orthereareadditionalnon-idealitiesnotincludedinthepreviousanalysis.Consequentlyprovinganempiricalviolationofquantummechanicsisextremelydi cultorimpossibleinactualexperiments.Similarlyrefutinglocalrealismisimpos-siblewhenevertheglobaldetectione ciencydoesnotsurpasseabout80%,awellknownfact.

Nevertheless,eveniftheBovinoexperiment[6]doesnotallowarigorousdiscriminationbetweenquantummechanicsandlocalrealism,interestinginformationmaybeobtainedbystudyingtheagreement,ordisagreement,ofthedatawithsomesimpleLHVmodelsdepartingbutslightlyfromthequantumpredictionseqs.(6)or(7).Thisstudywouldrequiretode neasimplefamilyofLHVmodelsandto ndwhetherthedataagreewitheitherthequantumpredictionsorthesaidsimpleLHVmodels.ConstructingLHVmodelsappropriateforacomparisonwiththequantumeqs.(6)or(7)isamostinterestingaimforthenearfuture,butfrommyexperienceImayguessthattheresultswillbeasfollows.Themodelpredictionswillbeoftheformgivenbyeq.(37)of[4],thatis(comparewitheqs.(11))

PLHV(φ)=

13π2sin(πη/2)

η22 2η+

cos(2ε)[π 2ε+tan(2ε)]=V(πη/2)2

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

Eqs.(10)and(13)mayberewrittentakingaccountofthe rsttwotermsofδ(φ)bymeansofsmallchangesintheparametersR0andV,thatPLHV(φ)≡16R′f=1

03π2η2

that,thediscriminationbetweenquantumand |φ| π2LHVpredictions + is.(15)Afterwillconsistofcheckingwhetherthequantumeq.(11)(withR0andVasfreepa-rameters)ortheLHVeq.(15)(withR′0,V′andηasfreeparameters)maybe ttedtothedataoftheexperiment.Inpracticeweshouldmakechi-square tsoftheexperimentaldataintoeq.(15)with3freeparameters,namelyR′0,V′andη.TheparameterεisrelatedtoηandV′byeq.(14).Ifη<<1eq.(14)maybe

ε solvedtoorderε2giving[4]2 1 sin2(πη/2)21

(πη/2)2 ,(16)

+

wherethesecondequalityisvalidforV′closetounity,asisusuallythecase.Iguessthatfairlygood tsexistforsomesetofvaluesofη.Ifagood tispossibleforη=0(inthiscasetheLHVeq.(15)becomeidenticaltothequantumeq.(11))thentheexperimentiscompatiblewithstandardquantummechanics(“standard”meansacceptingtheanalysisleadingtoeqs.(6)and

(7)).Ifgood tsarepossibleforη≥0.17thentheexperimentiscompatiblewiththefamilyoflocalmodelsde nedin[4],afamilylabelledLHV1in

[5].Iftherearegood tsforη≥0.55thentheexperimentwouldbealsocompatiblewiththefamilyde nedasLHV2in[5],whichismorerestrictivethanLHV1.HoweverIdonotthinkthelatterwillbethecaseinviewoftheresultsoftheroughanalysismadeatthebeginningofthispaper.

The tofthedataintotheequationsshouldbemadeforeveryoneofthe46setsofdatacorrespondingtooneBob´spolarizerpositioneach,ratherthana tofthewholesetofdata.ThereasonisthatinBelltestsitisessentialthatthemeasuredratescorrespondtothesameproductionrateinthesource.IsupposethatintheBovinoexperimentitismucheasiertoguaranteetheconstancyoftheproductionrateforeveryBob´spolarizerpositionthanforthewholesetofdata.

InordertomakeapreliminaryanalysisoftheexperimentIhaveusesafewdataof“fiBob”=90o.Asimpleconsequenceofthe(LHV)eq.(13)is,assumingη≤1/2,

ν≡f(0)+f(π/2) 2f(π/4)

4+16ε3/(3πη) 4ε3

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

whilststandardquantummechanicspredictsν=0.ThevalueofV′maybeobtainedfromf(0) f(π/2)V′=

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

References

[1]M.Genovese,Phys.Rep.413,319(2005).

[2]E.Santos,Found.

arXiv:quant-ph/0410193.Phys.34,1643(2004);c-print

[3]G.AdenierandA.Yu.Khrennikov,J.Phys.B:At.Mol.Opt.Phys.40,

131(2007).

[4]E.Santos,Eur.Phys.Jour.

arXiv:quant-ph/0612212.D42,501(2007);c-print

[5]E.Santos,Eur.Phys.Jour.D(2007).Inpress.

[6]F.A.Bovino(2007),unpublished.Privatecommunication.

[7]F.A.Bovinoetal.,c-printarXiv:quant-ph/0303126.

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