VolumeiSA, number3 PHYSICSLETTERS 7 January1980

EXACT SPIN ROTATION BY PRECESSION DURING NEUTRON INTERFEROMETRY

Herbert J. BERNSTEINHampshireCollege,Amherst,MA 01002, USA

and

Anton ZEILINGERMassachusettsInstituteof Technology,Cambridge,MA 02139,USAandAtominstitutder OsterreichischenUniversitdten,A-1020Vienna,Austria

Received6 April 1979Revisedmanuscriptreceived22October1979

Timedependentmagneticfields canbeusedto produceexactrotationsof spinwavefunctions.Thisavoidstheinherentapproximationsencounteredin experimentsperformedwith staticfields to demonstratetheeffect of 21T-rotationsonfermions.Two of theexperimentsproposedexplicitly will use forthcominginterferometersfor ultra cold neutrons.

All of the neutroninterferenceexperimentswhich experimentexactly,i.e. witharbitrarily goodapprox-havebeenperformedto demonstratethe effectof a imation,for finite interactionstrengths(in our casecomplete2ir rotation*1 on fermion wavefunctions for finite rotationangles).haveusedspin precessionin a staticmagneticfield [2]. To demonstratetherealizability of exactspinorro-Sucha field changestheneutronvelocity differently tationsthis letter proposesfeasibleexperiments(asfor thetwo spin states,parallelandantiparallel.This opposedto gedankenexperiments)which use timede-birefringenceresultsin a netphaseshift effectbesides pendentfields to implementa precessionwithout thethespin rotation [3] ,whichgives rise to a measurable correctionsinherentin a static field configuration.Al-polarizationevenfrom an unpolarizedbeam[41,an thoughtheystill representoperationson the spinfunc-effect impossiblefor anypurerotation.Thus,a static tionsalone,anddo notattemptto rotatethe completefield realizesrotationsonly approximately*2, theap- wavefunctionwhichincludesits spatial part, they areproximationbeingbetterthe weakerthe field is, i.e. a vast improvementoverall previousattemptsat im-in the limit of vanishingrotation.We considerit of plementinga true, complete,purerotation.fundamentalimportancewhetheror not a symmetry To theextent thatwe mayproducea time depen-operation(in our casea rotation)canbe realizedinan dent magneticfield of constantdirectioneverywhere

in theregionof nonvanishingprobabilitydensityof

*1 Byrne [11questionedtheadmissibilityof interpretingthe theneutron,thetime evolutionof the neutronwave-existingexperimentsasa direct observationof thesign functionwill producean exactrotation.Fromthereversalof a spinor wavefunctionsubjectedto a 2ir rota- hamiltoniantion. A detailedanalysisof this point involvesa very fun-damentalinvestigationof measurementin quantummecha- 9( = — ~ B (t) = — ji~°~B(t), (1)nics.Thiswill be thetopic ofa future paper[13].

*2 We considerit not relevant,that thedeviationsfrom exact with B (0 < t < T) ~ 0 andB (t ~ 0) B(T ‘~ t) = 0rotationsarevery smallin theexisting experiments,be- we getcausethesedeviationsareof an inherentnaturedifferentfrom usualexperimentalerrors.Furthermore,thedevia-tionswill becomemuchlargerin theinterferometerexperi-mentswith ultra-cold neutronsproposedhere.

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T It is importantto realizethat,in principle, it is not~i (T) = exp~ /l~oPfB (t) dt/h) ~i (0) relevanthow fastthemagneticfield is turnedon and

0 off. The conditionnecessaryfor no spin-flip processes~ T to occur is, that themagneticfield pointsalongthe

= RZ°~I\)f() dt tji(0), (2) samedirectionat all times,with only its strengtho beingvaried. In that casethes~1noperators~corn-

where .t is the neutronmagneticmoment,ands~is mutes at all timeswith thehamiltonianof eq.(1).thespin. ~li(T)is preciselya rotation of ~(0) because The reasonfor rapid switchingin theexperimentistheharniltonianproportionaltos~actsfor thesame simply to achievea largeenoughfield strengthfor atime on bothspin states.For staticfields we must complete4ir-rotation.solve thetime-independentSchrodingerequation[3,4]. The experimentproposedabovecanreadilybeThe result showsthat in that caseLarmor precessionis donewith thermalneutronsusing recenttechniquesunavoidablyconnectedwith momentumchanges*3 re- for fast switchingof magneticfields [6]. With thoseasulting in slightly different interactiontimeswith the magneticfield of 5 X iO—3 T (50 Gauss)canbe obtain-magneticfield for thetwo spin states.Thereforethe ed easilywithin a rise-time of a few ps.To obtainalimitations mentionedaboveare inherentto thestatic spin-rotationby 4ir this field hasto beon for 13.7

1us.field case. If 3 A (0.3nm) neutronswere usedin theexperiment,

In contrast,thereare no momentumchangesto be theywould propagate1.8cm in that timewhich corn-expectedfor the caseof a purelytime dependent paresfavorablywith the pathlengthof existinginter-hamiltonianbecauseit commuteswith themomentum ferometers.operator.Here,Larmorprecessionis associatedwith a Themostconvincingverificationwill be possiblechangenotof thewave vectorbut of the frequencyof onceinterferometersfor ultra-coldneutrons(UCN)the wavefunction.This changeis of equalmagnitude areavailable*5 We mayreasonablycontemplatetheand oppositesign for the two spinstates, developmentof suchinterferometersasthe ultra-cold

Clearly,if we canarrangetheapparatusto intro- neutrontechniquehasalreadyreacheda ratherhighducea magneticfield (whichrisesandreturnsto zero statusof experimentation[8], andbecausetheinstru-all while theneutronis in themagneticfield region) mentationandstability requirementsare similar toin one path of an interferometer,theresulting inter- those usinglight becauseof the longUCN wavelength.ferencewill revealthe effectof anexact rotation*4, In suchan experimentmuchlowermagneticfieldsOur experimentthereforerequiresa long enoughinter- with slowerrise timescould be used,thushavingbet-ferometerpath,slowenoughneutrons,andstrong ter control overthe homogeneitypropertiesof theenoughmagneticfields thatwe may turn on the field, magneticfields.More important,with ultra-coldneu-precesstheneutronspin througha multiple of 2ir ro- trons one caneasilyobtain field strengthswherethetations,and turn the field off again,all in the timeit departuresfrom exact rotationsin the static field casetakestheneutronto passthroughthe apparatus.Evi- are large,but which still cancomfortablybe switched.dently, time-of-flight counting,or beamchopping Thisoffersthe possibilityof directly demonstratingmustbeusedin orderto synchroniseall of theseevents, the differencebetweenthe two casesjust by going

from thestatic to the switchedmode.*3 Thesemomentumchangeshavebeenmeasuredrecently To usan experimentof the kind proposedabove

[5]. providesquite a satisfactoryrealizationof exact rota-*4 Wenote that Maxwell’s equationsrequirethe changing tion of the spinorwavefunction.Somephysicistsmay

magneticfield to propagateat thespeedof light. Thus, the havementalreservationsagainstinterpretingLarmorentireregionof theneutron’swavefunctioncanneverbe . .

subjectto exactlythe samenonstaticfield ata giventime, precessionasa realizationof spin rotation,preferringAdditionally, evena time-changingfield completelyho- anexperiment,wheretherotationis causedby some-mogeneousin spacein thelab’s frameis not homogeneous thing “really” rotatingin space.Realizingthat the in-in themoving neutron’sframebecauseof specialrelativis- terpretationof fundamentalexperimentsmay be atic effects.We mayignoreboth theseeffects sincethespeedof light is so very greatcomparedwith the otherpa- ~ Very recently,someexperimentalschemesfor ultra-coldrametersof theproblem. neutroninterferometryhavebeenproposedexplicitly [7].

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matterof weltanschauung,we proposethefollowing B’ by B, which implies that in this casethe interfer-experimentsusinga rotatingmagneticfield. encewill measureonly the effectof the relativerota-

We considera homogeneousmagneticfield rotating tion exp(.—iwTa~).Wenote thatattaining theadiabaticin spacearoundthez-axis*6, HeretheSchrodinger limit is only desirablefor demonstrativepurposesinequationis order to isolatethe field rotation from the precessioni /2 ~p/~t = pB ai,li (t), (3) effect,becausethecombinedrotation of eq. (4) is al-

waysexact.with B = B (coswt,sin wt,0) for 0 < t ~ T, a is the In principle suchan experimentcould be perform-“vector” of Paulimatricesandw is theangularveloc- ed usingthermalneutronsandperfectcrystal inter-ity of the rotatingmagneticfield. By goingto a ro- ferometers.The rotatingmagneticfield couldbepro-tatingcoordinateframe one canshow that videdby two magneticcoilsarrangedat right angles

to eachotheranddrivenby high frequencyalternating~1i(T)= exp(_~i~Tu~)exp[i(j.zB Va/fl) T] ~0), currentswith relativephaseof ir/2. As presentinter-

for t> T, (4) ferometerbeampathsaretooshort onewould haveto use a two-crystalinterferometer[12] which,in

whereB’ = (B, 0, hci/2p) is theeffectivefield about pinciple, allows muchlongerbeampaths.If e.g. 3 Awhich theneutronprecessesandwT is thegeometric (0.3nm) neutronsare used in theexperimentandifrotationangle(aboutthez-axis) causedby therotating onehasa beampath of about30 cm availableonemagneticfield. The importantpointwhenusingsuch would haveto usemagneticfields of some 10—2 Ta rotating field for therealizationof true rotationsis (100Gauss)oscillatingwith a frequencyof a few kHz.againthat all neutronmomentumchangeshaveto be Suchspecificationsarewithin thereachof presentex-avoided.Such changeswould give riseto the sameprob- perimentation[6].lemspreviouslyencounteredin thestatichomogene- For the samereasonsasabovea more convincingousfield case.This implies that in theexperimentone verification will be possiblewith an interferometerformustturn therotating field on and off while the neu- ultra-coldneutrons.We actuallyproposeto usetwotron is in thefield regionand onemustactuallyusea small rapidly-spinningelectromagnetsplacing one inrotatingmagneticfield ratherthan a staticscrew eacharm of anUCN interferometer,whosehomogene-field *7~Wepointout that evenherethereis no reason ous field we turn on,rotate,and turn off again,all whilein principle for a rapidswitchingof themagneticfield, theneutronis in theelectromagnetregion. In orderThis is dueto theproperty thatany solution of the to perform this experimentwith neutronsof 5 rn/sSchrödingerequationwith a purelytimedependent velocity (E = 1.3 X i07 eV, X 79 nm), a 4ir rotationpotentialof thetypep0 B(t) alwaysrepresentsanex- in 10 cm pathlengthrequiresthat thetwo counter-ro-actspin rotation. tatingmagnetsbe spinningat 3000 rpm. Indeedfor

In the experimentit is desirableto isolate the ef- sucha neutronthe adiabaticconditionp8 ~ /2w is metfect of the physicalfield rotation from the precession at very low fields,namelyB ~ 3.2 X 10—6 T (3.2Xeffect.Oneof the waysto achievethisis to introduce 10—2 Gauss).a compensatingmagneticfield B’ in onearmof the in- The experimentsproposedherewill demonstrate,terferometerwhile the rotatingfield is in theother. A that — at leastwithin non-relativisticquantummecha-moreelegantwayproposedhereisto use in botharms nics — an exact realizationof spinwavefunctionrota-of the interferometerrotatingmagneticfields withop- tion for finite rotation anglesis feasible,in the sensepositerotation directions.In the adiabaticlimit, i.e. that thereareno limitationsof aninherentnature,asfor fields strongenoughthatp8 ~‘ /2w we mayreplace in the staticfield experiments.

*6 This resultwas presentedanddiscussedby Balcar [91in The authorswish to thank ProfessorM.A. Homefororderto demonstrateits closeresemblancewith thesolu- manyusefuldiscussions,andProfessorC.G. Shull for

* tions for a staticscrewfield [101. hishospitality andsupportat MIT wherethis work‘~Thus the claim madeby Mezei[11] that a statichelical . .

field avoidstheproblemsencounteredwith statichomoge- was initiateneousfieldsappearsto bepremature.

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