Volume148,number1,2 PHYSICSLETTERSA 6 August 1990

Geometricphasein neutroninterferometry

ApoorvaG. Waghand VeerChandRakhechaSolidStatePhysicsDivision,BhabhaAtomicResearchCentre,Bombay400085,India

Received15 May 1990; acceptedfor publication5 June1990Communicatedby J.P.Vigier

We revisit a recentproposalby Wagh to test thedependenceof spinorphaseon the orientationof the precessionaxis. Weidentify thespinorphaseshift originatingfrom themisalignmentbetween7r-flippersin thetwo branchesof a neutroninterfero-meterwith a geometricphase.We alsointerpretquantumbeatsin polarisedneutroninterferometryobservedby Rauch’sgroupasamanifestationofa time-dependentgeometricphaseshift.

Recently,Waghhas proposed[1] a neutronin- the rotation operatorappropriatefor the waveftinc-

terferometric observationof spinor phasedepen- tion of a spin-i/2 particle.Thus, ifdenceon the orientationof the precessionaxis. Inthe proposedexperiment,a uniform magneticfield F2 I t> = exp(ic) ~>B0 appliedoverthe entireinterferometer(fig. 1) in it follows thata directionnormalto theplaneformedby thebeampaths1 and2, servesasthe guidefield anddefines F1 It> =exp(—ia~&$/2)F2exp(ia~~fl/2)It>the z-direction.Two spin flippers F1 andF2 placed = exp(i8fi/2) exp(it) exp( — ia~~$/2)I ~>in the two beampaths reversethe neutronspindi-rection.F1 andF2 are identical flipperswith a rel- = exp(i6fl) exp(it) I 1>ativeorientation8$ aboutthe z-axis. The operator = exp(i6fl)F2 It> . (la)for the flipping action in F1 canthereforebe repre-sentedby F1 = UF2U’, U=exp( — iaAfl/2) being Similarly

FiI~>=exp(—i6p)F2I~>. (lb)

Thus a neutrontraversingthe interferometerpicks

~P/2 ~0 up a phaseshift ±8$ originatingfrom a relativero-/ tation of thetwo flippers aboutthe z-axis.We shallI identify this shift with a pure geometricphase[2—

) ~~kN>/ ‘H 4].It>! 6p/2 / In addition, the neutronundergoesa purely dy-/ ~ / namicalphaseshift ~ 60 [1], 60 beingtheexcessspin

S/ ~ 2 ~J precessionabouttheguidefield onthepathS—F1 over/ I ~ I~>2 thaton thepath S—F2. Thisphasethusarisesfrom a/ I relativelinear translationof the two flippers along/ the respectivebeampaths.

AharonovandAnandan[4] havedelineatedtheFig. 1. Schematicsketchof aneutroninterferometncexperimentfor observinggeometricphase.S: sp1itter~A: analyser;F1, F2: dynamicalandgeometnccomponentsof the phaseidenticalx-flippers. 0 and H: forward anddiffractedoutgoing acquiredby a spinorduringa2x-precessionof its spinbeamsrespectively, vectorovera coneof semi-angle0 abouta homo-

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Volume148,number1,2 PHYSICSLETTERSA 6 August 1990

geneousmagneticfield. We generalisetheir analysis severalsectorsrepresentingsuccessiverotations ak

for an arbitrary closedcircuit C describedby the in magneticinductionsBk, overconesof polaranglesneutronspin overa sphere(fig. 2a). If C comprises 0k~spanningtime intervals tk, k= 1, 2, ..., n, the net

phaseqi acquiredby the neutroncanbe written as

Z exp(—ici~a~/2)x...xexp(—ia2a2/2)

Xexp(—ia1a~/2)=e’~. (2a)

C The phase1 can be expressedas the sum of a dy-

fl~ namicalcomponent2 ~<H>r

~D=~ kk1~9 (2b)

anda geometriccomponent

‘kG=—~Q, modulo2it. (2c)

Here <H>k=+I~uIB~cos9k,with p=—Ij~I~’foraneutron, is the expectationvalue of the magneticHamiltonianover the kth sectorandQ is the solid

k anglesubtendedby C at thecentreofthespin sphere,a its sign dependingon the senseof traversalof C. It

I ~> is clearthat boththe phasecomponentschangesignon a reversalof the spin direction resulting in achangeof signof thetotalphase‘J~asexpected.Fur-ther if a sectorforms an arcof a greatcircle on thespin sphere (4

9k=it/2), the sectormakesa contri-+ bution only to the geometricphase.On the otherII’> hand, if in a particular sector, the spin vector is

alignedparallelor antiparallelto Bk (Ok=O or it),

the sectorreducesto a pointon the spin sphereand

1 makesno contributionto thesolid angleQ. Thephase2 acquiredoversucha sectoris thuspurelyof dynam-

— — — — -.-.~ ical origin.As regardsthe proposaldepictedin fig. 1, the op-

- erationof flipper F2 reversingtheneutron spincan

-, be representedby the curve labelled2 on the spinsphere(fig. 2b). Theflipper F1,beingidenticalto F2exceptfora rotation6$ aboutthez-axis,flips thespinalongthecurve 1, identicalto curve2 but rotatedby6/3 aboutthe z-axisfrom it. Thephaseshift picked

b up by the neutronin traversingthe interferometerisI ~ thatacquiredalongcurve 1 in the forwarddirection

followedby curve 2 in thereversedirection,i.e. overtheclosedcircuit formedby curves1 and2.Theevo-

Fig. 2. (a) Evolutionof phaseoveraclosedcircuitC traversed lution of the spinor wavefunctionalong this closedby theneutronspin vectoron the spin sphere.(b) Geometric . . . .

depictionofthephaseshift arisingfromarelativeorientation~ circuit is cyclic andthe correspondingcyclic phasebetweenthe flippers F1 and F2. SpintrajectoriesI and2 pro- hasdynamicalandgeometriccomponentsgiven byducedby F1 andF2 respectivelyencloseasolidangle — 2bfi. eqs. (2b) and (2c). The dynamical phasecontri-

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Volume148, number1,2 PHYSICSLETTERSA 6 August 1990

butionsalongcurves 1 and2 areequalsincethetwo subtendingan angle 9=cot with a fixed directioninflippersareidentical.Thedynamicalphaseacquired thex—y plane.Therf flipper is thusseento be con-alongtheclosedcircuit thereforevanishes.Thesolid ceptuallyidentical to a dc it-flipper whoseaxis ro-anglesubtendedat the centreby theclosedcircuit is tatesuniformly at an angularfrequencywaboutthe— 26$,leadingto a geometricandthereforenetphase z-direction.If time t = 0 is selectedto be the instantshift of 6$ in accordancewith eq. (1 a). The solid whenthe rotatingfields of the two flipperspoint inangleQ, andthereforethe phaseshift, changessign the samedirection, it follows that at anytime t, theif the incidentneutronis in thespin-downstate.The flipper axesencloseanangle6/3= (w1— w2) t. A spin-phase±6/1 arising due to the relative rotation be- up neutronpassingthroughtheinterferometerat timetweenthe two flippers is thus purely of geometric t therefore undergoes a geometric phase shiftorigin. 6/3(t) =6w t, which is proportionalto t. The inten-

In the light of the foregoingdiscussion,we shall sity of the outgoingbeamfrom the interferometernowreexaminethequantumbeatsobservedby Bad- consequentlyoscillateswith a frequency6w/2it. Suchureket al. [5] in polarisedneutroninterferometry. conversionof a time-proportionalphaseinto fre-Theirexperimentemployedtwo radio-frequency(rf) quencyhasbeendescribedby Maddox[6] in a corn-flippers in the two branches.A small frequencydif- ment on an optical experimentby Simon et al. [7]ference 6 v maintainedbetweenthe two flippers wherein an evolving geometricphasewas used tocausedintensity oscillationsin the outgoing beam causea shift in the frequencyof a laserbeam.Thewith a macroscopictime period 1/6v. We notefirst presentinterpretationof the neutronquantumbeatsthatanrf flipperessentiallysimulatesa field rotating [5] asa manifestationof anevolving geometricphasein the x—y planeat an angularfrequencyw super- implies that the neutronanalogueof the optical ex-imposedon the guidefield B~jovera pathlength1. periment [7] precededit by two years!The flipper must satisfy two resonanceconditions,viz, for the frequencyandamplitudeof the If field,to achievethespinreversalwith probabilityone.The Referencesfrequencyresonanceconditionstipulatesthatcoequaltheangularfrequencyoftheneutronspinprecession [1] A.G. Wagh,Phys.Lett.A 146 (1990) 369.

[2] M.V. Berry,Proc.R. Soc.A 392 ( 1984)45.about B0z. Given this condition, in a coordinate [3] S. Pancharatnam,Proc.md.Acad. Sci.A 44 (1956)247;frameattachedto the rotating transversefield, the M.V. Be~,J. Mod. Opt. 34 (1987)1401.

neutronexperiencesonly a statictransversefield. The [4] Y. Aharonovand I. Anandan,Phys.Rev. Lett. 58 (1987)

amplituderesonanceconditionthenensuresthatover 1593.

the pathlength1, the neutronspinundergoesexactly [5] 0. Badurek,H. Rauchand D. Tuppinger,Phys.Rev. A 34

a precessionequalto it aboutthis static field in the [6] J. Maddox,Nature334 (1988)99.rotatingframe. As seenfrom the laboratoryframe, [7] R. Simon,H.J. Kimble and E.C.G. Sudarshan,Phys.Rev.

the spin precessionthus takesplaceaboutan axis Lett. 61(1988)19.

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