Volume 121, number3 PHYSICSLETTERSA 13 April 1987

A QUANTUM POTENTIAL APPROACH TO SPIN SUPERPOSITIONIN NEUTRON INTERFEROMETRY

C.DEWDNEYDepartmentofAppliedPhysics,PortsmouthPolytechnic,ParkBuilding, KingHenryIStreet,PortsmouthP012DZ, UK

P.R. HOLLAND andA. KYPRIANIDISInstitut HenriPoincaré,LaboratoiredePhysiqueTheorique,11, rue PierreetMarie Curie, 75231Paris Cedex05, France

Received12 January1987; acceptedfor publication6 February1987

Thecausalinterpretationof thePauliequationis shownto provideanunderstandingof spin superpositionin neutroninterfer-ometryin termsofwell-definedindividual particletrajectorieswith continuouslyvariablespin vectors.

In a recentpaper[1] it wasshownthat the causal depictedin fig. I. A beampolarizedin the z-direc-interpretationof quantummechanics[2] canpro- tion (definedby aguidefield B intothepaper)entersvide a clearunderstandingof thespatialinterference thedeviceandissplit into anupper(II) anda lowerobserved in neutron interferometers.The simple (I) component.Theupperbeamis passedthroughamodeusedto facilitate this descriptionconsistedin devicewhich inverts the spin. The two beamswithreplacingthesetsof crystalplaneswith squarepoten- oppositespin convergeon the lastsemi-transparenttial barriers,in orderto simulatetheactionof semi- surfacewhereeachbeamis split oncemore.Conse-transparentsurfaces,andtheuseof numericalmeth- quently the emergingforward anddeviatedbeamsodsto solvethe appropriatetime-dependentSchrö- both contain a coherentsuperpositionof compo-dingerequation. nentsofoppositespin.Representingtheeffectof the

In this contributionthe aim is to show how the semi-transparentsurfaceby a squarepotential V wecausal interpreationcan also account for the spin maywrite theappropriatePaulihamiltonianassuperpositionexperimentscarriedoutwith neutron H— — (h 2/2m)V2+ V+ 0

interferometers[31which haveconfirmedthewell- — I12

knowntheoreticalresultthatthesuperpositionofspin whereB is the uniform guidefield, andp is theneu-up andspindown beams(in thez-direction)results tron magneticmoment.The initial two-componentin a polarizationvectorlying in thexyplane.Wehave spinor (thatis, justbeforethefinal semi-transparentalreadydemonstrated[4] how the causalinterpre- surface)is a superpositionof two wavepacketsasso-tation of the Pauli equation [5] canbe appliedto ciatedwith thespin up anddownbeams:elementaryspinphenomena.Thesecalculationshave / qi

shownhowquantumphenomenaassociatedwithspin q’0 ( lo

maybe explainedin termsof ensemblesof individ- \ e5

ualparticletrajectorieswithwell-definedbutcontin- Herex is the relativephasedifference(introduceduously variable spin vectors subject to the spin by an aluminium sheet)which may be varied bydependentquantumpotentialandquantumtorque. rotatingthe sheetsothat it presentsdifferent thick-

In orderto calculatethetrajectoriesandspin vec- nessesto eachbeamcomponent,andtor orientationsin the contextof the simplemodelreferredto above,we considerthe interferometeras !Pi~=A exp[ — (x— 0.5)2 /4o~+ ik

0x]

0375-960l/87/$03.50© ElsevierSciencePublishersB.V. 105(North-HollandPhysicsPublishingDivision)

Volume 121, number3 PHYSICSLETTERSA i3 April 1987

yFig. I. Neutron interferometermodelwith onespinrotatingcoil andphaseshifter.

qiulo=A exp[—(x—l)2/4c~—i(k0x+~)], ~h(Th~i/at+cos0oØ/at)+~mv

2+Q+H0+ V=0,

whereA is a constant,a0 is the initial half width, where Q= — (h2/2m)V2R/R is the usual quantum

k~/2misthe incident energyandC1 isa constantfac- potentialandtor introducedtosymmetrizethetwo wavefunctions H

0 = (h2/8m)[(V 0)2 + sin20(V 0)2] — (2pI1~)s~B

with respectto thepotential.To determinetheevolutionof the spinvectorand is a spin-dependentaddition.The particleenergyis

positionof a particlewe first note that the internal givenbydegreesof freedommaybe representedby the Euler — ~h(8~/8t+cos 0 aØ/at)angles0(x, t), Ø(x, t), ~t’(x,t) in termsofwhich thespinorfield at time t takestheform which, like thevelocity (2), containsa spindepend-

ent contribution.

aw/2(

cos(0/2)e”~2‘\

~(x, t) =R e sin(0/2) e_~2) (1) A quantumforcemdv/dt=—V(V+Q)—(2pIh)(VBk)sk

whereR(x, t) is a realamplitudefunction.Thespinvectoris given by — (h2/4mp)ö~[p(V 0 0k° +sin2OVO äkØ)]

qit

0qJ

s=~h k=l,2,3,p

thusactswhich introducesa new form of spin—orbit

= ~h(sin 0 sin0, sin0 cos 0, cos 0) couplinggiving riseto trajectorieswhich dependonthe spin even in the absenceof external magnetic

with p= = R2andthe velocity,derivedfrom the fields.The equationof motionof thespinvectormay

Paulicurrent,by bewritten

v=(h/2m)(V~v+cos0 VØ). (2) ds/dt=T+(2p/h)BXs,

In termsof thevariablesp,0, 0, ~ thePauliequation whereimpliesa continuityequation T= (l/mp)sxôk(pôkS)

op/at+div(pv)=0,is an additionalquantummechanicaltorquewhich

anda Hamilton—Jacobiequation: similarly maygive risetonon-stationaryspinvectors

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Volume 121, number3 PHYSICSLETTERSA 13 April 1987

I I I I

1.05 -

0.45

0 40 I I I I I I I I I0 00 2 00 4 00 6.00 8 00 10.00 12 00 14.00 16 00 15 00

1010

NEUTRON TRAJECTORIES SPIN SUPERPOSITIONFig.2. Particletrajectoriesatthelast semi-transparentsurface.

evenin theabsenceof magneticfields. figuresallowsustoseetheway inwhich thespinori-FromthenumericalsolutionofthePauliequation entationchangesin a continuousmannerduringthe

we may calculatethe spin vectororientationsfrom scatteringfrom the final semi-transparentsurface.(1) andthetrajectoriesfrom (2) by writingv= dx/dt Fig. 4 showstheazimuthalangle0 andin this figureandspecifyingthe initial position (which of course we haveneglectedthe purely classicalprecessionofin practiceis unknown). In fig. 2 ~ we displaythe the spin vector about the guide field. Clearly thetrajectoriesfrom thegaussiandistribution of initial averagevalueof the spin isconserved.Changingthepositionsin eachcomponentbeam.The horizontal relativephasedoesnotaffectthepolaranglebutdoes,lines indicatethe position of the squarepotential. asis shown in fig. 5, causethe final directionof theNoticethatall thetrajectoriesoriginatingin theupper spinvectorin the xyplaneto be rotated.How this(lower) beam inside the interferometerenterthe arisescanbeseenif wewriteupper(lower) beamoutside.This leadsasexpected / R ~ ~

to an equalcountratein thetwo detectorsindepen- ~= ( 1S2/h •~

dentof therelativephase. \R2 e eThe spin orientationfields 0(x, I) andØ(x, t) are .andcomparewith (1). Thisgives

shown in figs. 3 and 4. In the emergingbeamsthespinvectorliesin thexy plane(0=x/2) asshown in 0=2tan~(R2/R1),fig. 3. Superposingthetrajectoriesmentally on these

çti=(S1 +S2)/h+it/2+X,

0 Inthefiguresweputh= 1, m=O.5,~~=O.O5,k0=50E.

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Volume 121, number3 PHYSICSLETTERSA 13 April 1987

1.10 I 1’ 1 1 1 1 ~‘ I~ ~

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2.00 4.00 600 600 1000 1200 1400 1500 1500,ol 0-~NEUTRON INTERFEROMETRY SPINSUP THETA

Fig. 3. Thepolarorientationangleof thespin vector0(x, t) plottedon a meshof pointsassociatedwith fig. 2.

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1I”.—’.I I—Ill ~0,40 II/I—t/I/I’-I,\/1~-~\l-11///////--

2,00 4 00 6 00 6,00 10,00 2,00 14.00 16,00 18,00

,dONEUTRON INTERFEROMETRY SPINSUP PHI

Fig.4. Theazimuthalorientationangleof thespin vectorØ(x, t) associatedwith fig. 2.

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Volume121, number3 PHYSICSLETTERSA 13 April 1987

1.10 lI/’~~—~———~t/—l—\—I—I\—/1

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NEUTRON INTERFEROMETRY SPINSUP 10 PHI

Fig. 5. Theazimuthalorientationangleof thespin vectorwith anadditionalrelativephasex~introducedby rotatingthephaseplate.

0 (S1 S2)/hit/2~. thetimethewave packetpassed.

However, if the spin rotationon beamII is carried By measuringthespinina directionin thexy planeoutwith a time-dependentradiofrequencyspin flip or, as is in fact done,in the z-directionafter a ~tf2coil [7] the situationis ratherdifferent.Theuseof spin turn coil, anintensityvariationwith respecttosucha deviceintroducesanenergyexchangebetween therelativephasemaybeobserved.Sucha measure-the neutronandthe coil, an effect notpresentwith ment could in principle be carried out using astatic magnetic fields, and this raises interesting Stern—Gerlachapparatus,butweshouldguardagainstproblemsof interpretationdiscussedelsewhere[8]. unwarrantedinferencesfreomsuchresults.If thespinIt also producesa time-dependentrelative phase. is measuredin the z-directionoutsidethe interfero-Eachneutronpassingthroughtheapparatuswill suf- meter,in theabsenceof~t/2spinturncoils, theresultfer a differentphaseshift dependingon the phaseof is not relatedto which beampath the neutrontookthe coil as it passes.Thetotal relativephaseis then insidetheinterferometer,asmaybenaivelyexpected

x — w~t. Thedirectionof the spinvectoroutsidethe from the extensionof classicalideasinto this quan-interferometerthusrotatesin thexyplane,requiring tumphenomenon.Accordingtothecalculationspre-a detectionprocesssynchronizedwith the phaseof sentedhereboth resultsspin up andspin down canthecoil torevealthesuperposition.Thesimplemodel be achievedwith neutronsthathavetravelledalongusedheredescribeseventsafterthespinflip hastaken eitherpath,andthe spin measurementprocesscanplace,thusa time-dependentrelativephasecanonly be accountedfor in a deterministicmanner.Eachbesimulatedin a seriesof “stills” eachwith a differ- emergingneutronhasawell-definedspinvectorlyingent relativephase.Of course,eachindividual neu- in the xy planeand a definite position within thetron,apartfrom theprecessionabouttheguidefield, beamas it entersthe Stern—Gerlachapparatus.Thehasa staticspin vectororientationoutsidetheinter- interactionwith the inhomogeneousfield producesferometerwhich dependson the stateof the coil at two separatingbeams,the neutronenteringoneor

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Volume 121, number3 PHYSICSLETTERSA 13 April 1987

the other depending on its initial position at the the Royal Society, the SERCand the French Gov-entranceslit. As the beamsseparatethe quantum ernmentrespectively for grants which made thistorquerotatesthe spinvectorto lie alongor opposed researchpossible,andthe IHP for its hospitality.to thefield gradient.Thuswemaychooseto measurethe spin in the z-direction or in a perpendiculardirection afterthe neutronhasenteredthe appara- Referencestus.Thequentumpotentialapproachshowsthatthereis no needto concludethat the choiceof what to [1] C. Dewdney,Phys.Lett. A 109 (1985) 377.

observehasanyinfluenceon what happenedin the [2] D. Bohm,Phys.Rev.85 (1952) 166, 180;pastanda uniqueoutcomewill follow uniquelyspec- L. deBrogue,Non-linearwavemechanics(Elsevier,Amster-

dam, 1960);ified initial conditions. This is important to bear in

J.P. Vigier,Astron. Nachr. 303 (1982) 55.mind when considering“delayed choice” experi- [3] J. Summhammer,G. Badurek, H. Rauchand U. Kischko.

ments[9] which aredesignedto showthat theact of Phys.Lett.A 90(1982)110;

observationcan affect the set of events occurring G. Badurek,H. Rauch,J. Summhammer,U. KischkoandA.

priorto thatact. Zeilinger,J. Phys.A 16 (1983) 1133;J.Summhammer,G. Badurek,H. Rauch,U. KischkoandA.

In the precedingwe havedemonstratedthat the Zeilinger, Phys. Rev. A 27 (1983) 2523.

phenomenaassociatedwith spin superpositionneu- [4] C. Dewdney,P.R.HollandandA. Kyprianidis,Phys.Lett. A

tron interferometryexperimentscanbe describedin 119 (1986) 259.

the causalinterpretationin termsof individual tra- [5] D. Bohm,R.SchillerandJ. Tiomno, Suppi. Nuovo Cimento

jectorieswith well-definedbutcontinuouslyvariable 1 (1955) 48.[6] T. Takabayasi,Prog.Theor.Phys.14 (1955)283;

spinvectorswithoutambiguityorcontradiction.The T. TakabayasiandJ.P. Vigier, Prog.Theor. Phys.18 (1957)

behaviourof eachneutronin the regionof superpo- 573.

sition, its position, spin vector andenergy,reflects [7] G. Badurek,H. RauchandJ. Summhammer,Phys.Rev.Lett.

theglobalpropertiesof thewholeapparatusthrough 51(1983)1015.

the local propagationof information by the spinor [8] C. Dewdney,A. Garuccio,A. Kypnanidisand J.P. Vigier,Phys.Lett. A 104 (1984) 325.

wave acting through the spin-dependentquantum [9] J.A. Wheeler, in: Mathematicalfoundationsof quantum

potentialandquantumtorque. mechanics,ed. A.R. Marlow (AcademicPress,New York,1978).

TheauthorsC.D.,P.R.H.,andA.K. wishto thank

110