Comment on Neutron Interferometric Observation of Noncyclic Phase
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arXiv:quant-ph/9810030v1 12 Oct 1998Commenton“Neutroninterferometric
observationofnoncyclicphase”
RajendraBhandari
RamanResearchInstitute,Bangalore560080,India.
email:bhandari@rri.ernet.in
PACSNOS.03.65.Bz,42.25.Hz,42.25.Ja
———————————————————————–19September1998
1Itwasfirstpointedoutinref.[1]thatthewellknownneutroninterfer-ometerexperiments[2,3]demonstratingthesignchangeofthewavefunctionofoddhalf-integerspinparticlesunder2πrotations,donewithunpolarized
neutrons,donotconstitutemeasurementofthephaseshiftassociatedwithagivenspinstateandthatsuchameasurementwouldrequireanexperiment
withpolarizedneutrons.Itwasshown[1]thatthecontinuouslymonitoredphaseshiftofaspinstaterotatingaboutthepolaraxis,asgivenbythePan-
charatnamcriterion,withtheinitialstatetakenasthereferencestate(the
noncyclicphase),hastheoppositesignforstateslyingintheupperandthelowerhemispheresandhasadiscontinuousjumpequalto±πforastatelying
ontheequator.Thiswasverifiedexperimentallyinopticalinterferenceex-perimentsusingthepolarizationstatesoflightasatwo-statesystemwhichis
isomorphictothespin-1/2system[1,4].ThephasejumpsinSU(2)evolutionoccurringatpointsintheparameterspacewherethetwointerferingstates
becomeorthogonal,hadearlierbeenpredicted[5,6]andexplainedinterms
ofjumpsinthegeometricpartofthephase.Theoriginofsuchphasediscon-tinuitiesintheexistenceofDiracsingularitiesintheparameterspaceofthe
SU(2)transformationwasdemonstrated,theoreticallyandexperimentally,forthecaseoftwo-statesystemoflightpolarization[7,8]andtheoccurrenceof
similareffectsinneutroninterferencewaspredicted[1,7,4,9].Inref[10],Waghet.al.reportanexperimentaimedatmeasuringthe
noncyclicphaseforaspin-1/2systemusingneutroninterferometry.Iwishto
pointoutthat,contrarytotheimpressiononegetsfromthepaper,themostimportantpartofthephysicsofthenoncyclicphase,namelythedifferentsign
ofthephaseshiftforstatesintheupperandthelowerhemispheres,isnotverifiedinthisexperiment.Thereasonliesinthefactthatthisexperiment
2measures,forwhateverreason,notthenoncyclicphaseitself,butaquantity
derivedfromit,namelythedifferenceofthephaseshiftacquiredbyagivenstateandthelinearphaseshiftacquiredbythestateθ=180◦orθ=0◦,lying
intheotherhemisphere.Forexample,considerthecurvescorrespondingtothestatesθ=70.5◦andθ=109.5◦infigure2ofref.[10].Forthesakeof
thisargument,letusignorethesmallwiggleinthecurvesandconsiderthemasstraightlines.Whatneedstobeshownbymeasurementisthatthelinear
partofthenoncyclicphaseforstatesintheupperhemisphereisa=φ/2and
thatforthelowerhemisphereisb=−φ/2,whereφistheprecessionangle.Insteadofmeasuringaandb,theexperimentshowsbymeasurementthat
(a−b)=φand(b−a)=−φ.Thetwocurves,therefore,do-notrepresentindependentquantitiesandnoneofthetwoimplieswhatneedstobeshown.
Itisalsonoteworthythatthephaseshiftsplottedinfigure2equal±2πfor2πrotationsonthesphereandinthissensecontaininformationequivalentto
thatinapolarimetricexperiment.Thequantities(a−b)and(b−a)infact
representtheanglesofrotationsofthestatesonthesphere.Inmeasuringthesedifferences,therefore,onehassacrificedthetrueadvantageofaninterference
experimentinthecontextofspinorphases.Thecurveforθ=90◦infigure2alsodoesnotrepresentmeasuredquanti-
ties.Thereasonis,thepointscorrespondingtoθ=90◦andprecessionangles±180◦arephasesingularities,wherethephaseshiftbecomesundefinedand
oneisnotjustifiedinmakingadefinitechoiceforthesignoftheπphasejumps
ashasbeendone.Thechoiceofsigninfigure2,whichcorrespondstochoosingθ=90◦+ǫ,ǫbeingasmallpositivequantity,isarbitraryanddoesnotfollow
frommeasureddata.Anotherproblemwiththeanalysisofref.[10]istheuseofequation(6)to
3convertthephaseshiftsforpartiallypolarizedneutronstothecaseoffully
polarizedneutrons.Anelementaryderivationofthisequation,notreproducedhere,showsthatthisisvalidonlyforθ=0◦andθ=180◦andnotinthe
nonlinearregime.Toconclude,themeasurementofthenoncyclicphaseofanevolvingspinor
statealongwithitssignandademonstrationoftheassociatedsingularityinthecaseofquantumsystems,assuggestedbytheresultsoftheopticalpolarization
experiments,remainsanopenquestion.
4References
[1]R.Bhandari,Phys.Lett.A180,15(1993).
[2]H.Rauch,A.Zeilinger,G.Badurek,A.Wilfing,W.BauspiessandU.Bonse,Phys.Lett.A54,425(1975).
[3]S.A.Werner,R.Colella,A.W.OverhauserandC.F.Eagen,Phys.Rev.
Lett.35,1053(1975).
[4]R.Bhandari,Ind.J.PureandAppl.Phys.33,575(1995).
[5]R.Bhandari,Phys.Lett.A157,221(1991).
[6]R.Bhandari,PhysicaB175,111(1991).
[7]R.Bhandari,Phys.Lett.A171,262,267(1992).
[8]R.Bhandari,Phys.Lett.A180,21(1993).
[9]R.Bhandari,Phys.Rep.281,1(1997).
[10]A.G.Waghet.al.,Phys.Rev.Lett.81,1992(1998).
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