22 May 2000

Ž .Physics Letters A 270 2000 10–13www.elsevier.nlrlocaterpla

Anandan–Aharonov–Casher oscillations in a simplyconnected region

Erik Sjoqvist a,b¨a Department of Quantum Chemistry, Uppsala UniÕersity, Box 518, Se-751 20 Uppsala, Sweden

b ¨Atominstitut der Osterreichischen UniÕersitaten, Stadionallee 2, A-1020 Vienna, Austria¨

Received 15 March 2000; accepted 13 April 2000Communicated by P.R. Holland

Abstract

1An entanglement effect between the spin and spatial degrees of freedom of an electrically neutral spin- particle in a2

planar electric field is predicted. This nonclassical effect is manifested as spatial interference oscillations in a force freesimply connected region of space. It is suggested that these oscillations provide an experimentally accessible analogue of

w Ž . xAnandan’s ‘‘quantum solenoid effect’’ Phys. Lett. A 164 1992 369 . The effect could be tested with spin polarised atoms.q 2000 Published by Elsevier Science B.V. All rights reserved.

PACS: 03.65.Bz

1. Introduction

w x Ž .Aharonov and Bohm 1 AB were first to pointout the existence of nonlocal topological phase ef-fects in quantum-mechanical systems. They demon-strated that a charged particle in a force free multiplyconnected region of space, produced by a shieldedvery long magnetic solenoid, acquires such a phaseshift. Interference effects for electrically neutral par-ticles in electric fields were first predicted by Anan-

w xdan 2 and further developed by Aharonov and1w xCasher 3 , who showed that a neutral spin- particle2

experiences a topological effect in the vicinity of acharged line parallel to the particle’s magnetic dipolemoment. Later it was realised that Anandan–

Ž .Aharonov–Casher AAC does not require spatially

Ž .E-mail address: eriks@kvac.uu.se E. Sjoqvist .¨

w xseparated beams 4;5 and that it could be explainedas a local exchange of angular momentum between

w xthe electromagnetic field and the particle 6 . Thebasic origin of these peculiarities of AAC is that it

w xdepends nontrivially on the particle’s spin 7;8 .The standard AB effect may take place only in a

multiply connected region, where the charged parti-cle’s wave function encircles the classical magnetic

w x Žflux line. It was pointed out by Anandan 9 see alsow x.Ref. 10 that this need not be true if the flux is

treated quantum-mechanically: a solenoid being ob-served in a coherent superposition of states withdifferent magnetic fluxes could give rise to an inter-ference effect on a charged particle in a simplyconnected region of space, although the field strengthin this region due to each of the flux states vanishesso that no force acts on the particle. Unfortunatelythis remarkable effect seems hard to test experimen-

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 00 00282-6

( )E. SjoqÕistrPhysics Letters A 270 2000 10–13¨ 11

tally due to the difficulty to produce coherent fluxsuperpositions. The purpose of the present work is toshow that the nonclassical spin dependence of AACcould be used to overcome this difficulty and pro-vide an experimentally feasible analogue of this‘‘quantum solenoid effect’’.

In the next section we review Anandan’s originalw xanalysis 9 . Section 3 contains the AAC analogue in

a planar electric field produced by a charged capaci-tor plate. Such a configuration is suitable to obtain

w xlarge field strengths 4;5;11–13 . In Section 4, anexplicit experimental set up to verify the entangle-ment effect is sketched.

2. Magnetic quantum solenoid

Consider a charged particle confined to a simplyconnected region U being brought with a smallconstant velocity towards a quantum-mechanicalmagnetic solenoid in a coherent superposition of N

� < :4Nflux states F . These states are eigenvectors ofn ns1ˆ ˆ < : < :the magnetic flux operator F , i.e. F F sF F ,n n n

which defines the quantised solenoidal vector poten-ˆ ˆ< : < : Ž .tial A: A F sA F with A sF e r 2p r ,n n n n n u

say. Here e is a unit vector in the direction of theu

polar angle u and r is the radial distance from thesolenoid. The resulting entangled state of thesolenoid-particle system reads

N

< : < : < :C s a F c , 1Ž .Ý n n nns1

� 4Nwhere a is a set of nonvanishing complexn ns1

coefficients.In order to see interference effects between the

Ž .components of the entangled state in Eq. 1 it isnecessary to make a measurement on both thesolenoid and the particle. First a filtering measure-ment is made on the solenoid of an observablewhose eigenvectors can be expressed as linear com-

< :binations of the magnetic flux states F . The staten< :C is thereby projected onto a new post-selected

< : N < :solenoidal state S sÝ b F , say, correspond-ns1 n n

ing to the outcome of this measurement, yielding thespatial state

N

< : ² < : < :c s S C s a b c . 2Ž .ÝS n n nns1

Using conservation of the kinematical momentum ofthe particle prior to the flux measurement it follows

Ž .that each component c in Eq. 2 acquires a localn

Dirac phase proportional to the corresponding F .n

We thus obtain the spatial probability distribution ofŽthe particle in the simply connected region U SI

.units are used throughout

2 2 2Ž0.< < < < < <c x ,t s c x ,t a bŽ . Ž . ÝS n nn

< <q2 a b a bÝ n n m mn)m

=e F yFŽ .n m

cos uqd 3Ž .nmž /h

with e the unit of elementary charge, hs2p "

Planck’s constant, and d the relative phase be-nm

tween the products a b and a b . The unperturbedn n m m

c Ž0. is the wave function of the particle without thesolenoid and x , at which we have chosen us0, is0

some fixed point inside U. The additional oscillatoryŽ .modulation factor on the right-hand side of Eq. 3 is

independent of the path of integration within Ubetween x and x as the field strength vanishes0

there. It is also gauge invariant as all A undergo then

same transformation under a local phase change ofŽ . < < < < < < < <c x,t . For Ns2 with a , a , b , b )0 the ad-1 2 1 2

Ž . Ž . Ž .ditional factor in Eq. 3 has e F yF r hr os-1 2

cillations per unit length in the angular direction atdistance r from the solenoid. The interference effectis a manifestation of the entanglement in the com-bined solenoid-particle system, being analogous tothat of the Einstein–Podolsky–Rosen gedanken ex-periment.

3. AAC analogue

In this section we argue that suitable spin states inan AAC set up could be used to provide an experi-mentally accessible analogue of the flux superposi-tion and the concomitant entanglement effect. Con-

1 Žsider an electrically neutral spin- particle such as a2

.neutron or an atom with magnetic dipole momentstrength m in the coherent spin superposition

< : < : < :P sa qz qa yz , 4Ž .0 1 2

( )E. SjoqÕistrPhysics Letters A 270 2000 10–13¨12

< < < <where a , a )0. Its spatial wave function is con-1 2

fined to a simply connected region U, which isbrought with a small constant velocity towards acharged capacitor plate that produces a uniform elec-tric field Esß e re , where ß is the surface chargex 0

density of the plate, e is a unit vector in the xx

direction, and e the vacuum permittivity. The torque0

on the particle could be made to vanish by choosingits velocity z parallel to the lines of E so that theeffective magnetic field z=E, felt by the particle inits rest frame, vanishes. Thus the equations of mo-

Žtion of the magnetic dipole moment mAm= z=˙.E reduce to ms0. Furthermore the effective˙

w x Ž w Ž 2 .x.‘‘Lorentz force’’ 7 mz= == m=Er mcvanishes as the electric field is uniform. Hence thecapacitor does not expose the particle to any spatialforce.

Quantum-mechanically the particle experiencesw xthe quantised solenoidal vector potential 7

1 ßA ' m=E s e s , 5Ž . Ž .ˆ ˆxyyAAC y z2 2mc e c0

where s is the z component of the usual Pauliˆz

matrices. The corresponding ‘flux’ eigenstates are< : AAC"z associated with the ‘flux’ eigenvalues F "

ˆ 2² < < : Ž .sE "z A "z Pdrs"lr e c , where theAAC 0

line integral in the x–y plane is taken around thecapacitor plate and l is the linear charge density of

< AAC:the plate in the z direction. By introducing F "

< :' "z , the total state may be written as

< : < AAC: < : < AAC: < :C sa F c qa F c , 6Ž .1 q q 2 y y

where

xmˆ² < : ² <c x ,t s x c sexp i "z AŽ . H" " AACž " x0

=X X Ž .0< :x "z Pdx c x ,t 7Ž . Ž . Ž ./

with c Ž0. the unperturbed spatial wave function ofŽ .the particle. Eq. 6 is the AAC analogue of the

Ž .quantum solenoid state Eq. 1 . Note that c are"

single valued within U, which implies that the corre-Ž .sponding energies of the components in Eq. 6 are

AAC Ž 2 .equal and independent of F slr e c . Now0

proceeding analogously as in the AB case we select< : < : < :the spin state P sb qz qb yz , say, where1 2

< < < < < :b , b )0. After the projection onto P the origi-1 2< Ž0.Ž . < 2nal interference pattern c x,t has changed into

< < 2c x ,tŽ .P

2 2 2Ž0.< < < < < <s c x ,t a b q a bŽ . 1 1 2 2

AACm2F ß< <q2 a b a b cos yqDw ,2 2 1 1 ž /" l

8Ž .

where the additional modulation factor on the right-AAC Ž .hand side has m2F ßr "l oscillations per unit

length in the y direction. We have chosen ys0 atx . Dw is the angle between the projections of P0

and P onto the x–y plane. The largest contrast of0< < < < < <the oscillations is obtained for a s a s b s1 2 1'< <b s1r 2 . Without spin selection, contribution2

< : < :from projection onto the state yP syb qz q2< : < :b yz , being orthogonal to P , should be added1

< Ž . < 2incoherently to c x,t yielding the flux indepen-P< Ž . < 2 < Ž . < 2dent interference pattern c x,t q c x,t sP yP

< Ž0.Ž . < 2c x,t .

4. Experimental set up

In Fig. 1 we sketch a possible single-beam experi-mental set up for testing the interference oscillations

Ž .in Eq. 8 . A localised wave packet is incident fromthe left and moves directly towards the charged platecapacitor Q. Any spatio-temporal influence on theentanglement effect due to AAC induced torque onthe particle is eliminated by preparing a wave packetwith a small extension in the y direction, so that awell defined spin over the spatial support of theparticle is achieved, and by aligning the particle’svelocity with the electric field lines, so that the initialspin polarization P is unchanged prior to the spin0

selection. The polarizer P selects the spin state P atsome point along the beam path. Note that the torqueon the particle due to this filtering measurement isindependent of the electric field generated by Q. It ispreferable to choose both P and P in the x–y0

plane in order to achieve maximal visibility of the

( )E. SjoqÕistrPhysics Letters A 270 2000 10–13¨ 13

Fig. 1. Sketch of a single-beam experimental set up to test theAAC induced oscillations. A localised neutron or atom wavepacket with spin polarization P and moving in the x direction is0

incident from the left on a polarizer P. The correlation betweenthe spin and spatial motion of the particle induced by the chargedcapacitor plate Q, is observed by selecting the polarization P withP and observing the interference oscillations on the screen S.

interference fringes, which are observed on the screenS. With this choice the intensity is reduced roughly

2Ž .by a factor cos Dwr2 due to the filtering. Thespatial interference oscillations could be magnified,e.g., by diffraction of the beam through a narrow slitplaced between P and S. Note that there is no needto enclose Q by coherent splitting of the wavepacket as the desired effect takes place in a simplyconnected region of space.

Let us assume a strong electric field in the orderof 107 Vrm and a spatial interference pattern of sizeD y;10y2 m. With these numbers we obtain a

Ž .modulation in the interference pattern Eq. 8 thatoscillates in the order of 10y2 mrad for neutrons and10 mrad for atoms. Thus an experiment to test the‘‘quantum solenoid effect’’ for spin polarised atomsseems within reach with present day technology.However, the shift for neutrons is probably too smallto be observed.

One could imagine entanglement effects betweenthe spin and spatial degrees of freedom in the case of

1neutral spin- particles moving in spatially nonuni-2

form magnetic fields. Such effects are with necessityŽdispersive they are associated with a nonvanishing

w Ž .xforce = mPB x , B being the magnetic fieldx.strength , in contrast to the wave length indepen-

w xdence of AAC 14 . Clearly it is important that thenondispersive character of the present AAC set up isverified experimentally by changing the particle ve-locity in order to distinguish it from the magneticfield case and demonstrate the absence of force onthe particle.

In conclusion, we have discussed the possibilityof using nonclassical spin states in AAC to providean experimentally accessible analogue of Anandan’s

w x‘‘quantum solenoid effect’’ 9 . By careful choice ofthe experimental arrangement we have argued thatthe torque and force acting on the particle prior tothe spin selection vanish, bringing out the nonclassi-cal origin of the effect. The testability of this newAAC prediction has been discussed in terms of anexplicit single-beam set up for neutrons or atoms.

Acknowledgements

I wish to thank Y. Hasegawa for valuable discus-sions. This work was supported by the European

Ž .PECNO ‘Perfect Crystal Neutron Optics’ TMR-network, Grant No. ERB-FMRX-CT96-0057.

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