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AnAnalysisofThreePathInterference

FrankRioux

Quantummechanicsteachesthatifthereismorethanonepathtoaparticulardestinationinterference

effectsarelikely.Insuchcasestheprobabilityofarrivalatalocationisthesquareofthemagnitudeofthe

sumoftheprobabilityamplitudesforeachpathtothatlocation.Forexample,inatripleslitdiffraction

experimenttheprobabilityofarrivingatxonthedetectionscreeninDiracnotationis,

2 2

1233 3 2 2 1 1P x S x S x S x S

Thesinglephotoninterferometershownbelow[Franson,Science329,396(2010)]isaclosecousinofthe

tripleslitexperimentbecauseitprovidesthreepathstothedetector.Initiallywewillignoretheothertwo

outputchannelsAandB.

ThreePathInterference

ProbabilityAmplitudes

Inthisanalysistheprobabilityamplitudesarecalculateonthefollowingconventions.

Assume5050beamsplittersandassign/2(i)phaseshifttoreflection(usualconvention).

Transmissionatabeamsplitter: T1

2

Reflectionatabeamsplitter: Ri

2

Reflectionatamirror: M 1

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FirstwecalculatetheprobabilitythatthephotonwillarriveattheDetector(3paths):

R M T T T R R T T T M R 2 0.5625

ToestablishthatprobabilityisconservedtheprobabilitiesforarrivalatAandBarecalculated.CalculatetheprobabilitythatthephotonwillarriveatA(2paths):

T R T R M R 2 0.375

CalculatetheprobabilitythatthephotonwillarriveatB(3paths):

R M T R T R R R T T M T 2 0.0625

Demonstratethatprobabilityisconserved: 0.5625 0.375 .0625 1

Whilethistraditionalanalysispostulatesthreepathinterferenceforthearrivalprobabilityatthe

detector,Sinhaetal.[Science329,418(2010)]arguethattrueinterferenceonlyoccursbetweenpairsof

paths.Inthiscasebetweenpaths1&2,1&3,and2&3.Fransonsummarizedthisviewasfollows:

Quantuminterferencebetweenmanydifferentpathwaysissimplythesumoftheeffectsfromallpairs

ofpathways.

Theprobabilityexpressionforaneventinvolvingtwoequivalentpathsis

2 22 * *

ij i j i j i j j i i j ijP P P I

whereIijistheinterferencetermandP

iisdefinedastheprobabilitywhenonlytheithpathisopen.Itis

myopinionthatthislatterdesignationisnotstrictlyvalid.However,acceptingitforthetimebeing

leadstothefollowingdefinitionfortwopathinterference.

ij ij i jI P P P

Bysimilarargumentstheprobabilityexpressionforaneventinvolvingthreeequivalentpathsis,

123 1 2 3 12 13 23P P P P I I I

Usingthisequationtodefinethirdorderinterferenceyields,

123 123 1 2 3 12 13 23I P P P P I I I

UsingthedefinitionforIijweobtain,

123 123 1 2 3 12 13 13I P P P P P P P

FromaboveP123=0.5625.Theremainingtermsinthisequationarecalculatebelow:

P1

R M T T 2 0.125 P2

T R R T 2 0.0625 P3

T T M R 2 0.125

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P12

P13

P23

R M T T T R R T 2 R M T T T T M R 2 T R R T T T M R 2 0.875

Thesecalculationsshowthatthreepathinterferenceiszero.

I123 0.5625 0.125 0.0625 0.125 0.875 0

This,ofcourse,isatheoreticalresultbasedonthedefinitionsofPiandIijgivenabove.The

experimentalresultsreportedbySinhaetal.fortherelatedtripleslitdiffractionexperimentareinagreementwiththetheoreticalexpectationtowithinexperimentalerror

ObjectiontotheDefinitions

Sinhaetal.writethedoubleslitwavefunction(unnormalized)asalinearsuperpositionofthephotontakingtwopaths(AandB)tothedetector.Thesquaremodulusofthewavefunctiongivesthe

probabilityexpression.

2 2 2

A B A B B A

Obviouslythisisjusttraditionalquantummechanicalmathematicalprocedure.TheproblemIhaveis

withtheinterpretationorpartitionofthisequationthatcomesnext.Theauthorswritetheprobability

expressionas

2

A B ABP P I

wherePA

(PB

)istheprobabilitythatonlyslitA(B)isopenandtheremainingtermsrepresenttheactual

interference.

Idonotbelievethissortofpartitioningofthetermsof||2isquantummechanicallylegitimate.Ifonly

slitAisopenthenthewavefunctionis=Aandnot=A+B.Itisspeciousreasoningtosay,in

thedoubleslitexperiment,that|A|2istheprobabilitythatthephotongoesthroughslitA.

AsRoyGlauberhaswritten[AJP63,12(1995)], Thethingsthatinterfereinquantummechanics...are

theprobabilityamplitudesforcertainevents. Thetripleslitexperimentinvolvestheinterferenceof

threeprobabilityamplitudesbecausetherearethreepathsfromthesource(S)toeachposition(x)on

thedetectionscreen.Accordingtoquantumfundamentals,eachphotontakesallthreepaths

simultaneously.

1233 3 2 2 1 1x S x S x S x S

InChapter3ofvolumeIIIofTheFeynmanLecturesonPhysics,Feynmandiscussesadiffractionexperimentinwhichtherearesixpathstoanypositiononthedetectionscreen.Feynmansanalysisleadstothe

followingprobabilityamplitudefortheparticletobedetectedatx.

1 1 1 1 1 1 2 2 2 2 2 2x S x a a S x b b S x c c S x a a S x b b S x c c S

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ThesesuperpositionsofprobabilityamplitudesareexamplesofFeynmanssumoverhistoriesapproach

toquantummechanics,ofwhichthedoubleslitexperimentisthesimplestexample.Itisclearthat

pairwiseinterferenceisnotsuggestedinthisapproach,exceptforthedoubleslitexperimentandthen

onlybecauseinthatcasethereareonlytwopaths.Feynmansruleis,sumtheamplitudesforallpaths

toallowforinterferencebeforesquaringtheabsolutemagnitudeofthesumtoobtaintheprobability.

FreemanDysonhadthefollowingreactionwhenhefirstheardFeynmansnovelapproachtoquantum

mechanics.Thirty

one

years

ago,

Dick

Feynman

told

me

about

his

sum

over

histories

version

of

quantummechanics. Theelectrondoesanythingitlikes, hesaid. Itjustgoesinanydirection,atany

speed,forwardorbackwardintime,howeveritlikes,andthenyouadduptheamplitudesanditgives

youthewavefunction. Isaidtohim Yourecrazy. Butheisnt.