Topological effects in neutron optics

Ph',slca 137B (1986) 2 3 0 - 2 3 4 North-Holland, Amsterdam

TOPOLOGICAL

E F F E C T S IN N E U T R O N O P T I C S

A.G. KLEIN School ~[ Phwws. I,"nn'er.~t(v ~/ .~'Iclbourttc. Parl~rilh'. ~"l~tOrltl.i0.~2. "lln'tra/ta

A class of e x p e r i m e n t s in neutron optics, i~, related to the ,.,.ell-known Aharono,. Bohm e x p e r i m e n t s in electron optics v.ith v. hich the'.' share a c o m m o n topolog3. Several such e x p e r i m e n t s arc briefl', reviev.ed and some other,, of the s a m e t3lpe are proposed.

I. Introduction: The CA effect

2. The A B effec!

The aim of this essay, written for the amusement of Cliff Shull on the occasion of his 70th birthday, is to review a class of experiments in neutron optics and to propose some others of the same type. The experiments in question have in common that thay are related to the wcll-kno,,vn AB (Aharonov and Bohm [1]) experiments in electron optics and exhibit effects which are fundamentalh' of topological origin. These topological phenomena are all variants of the somewhat surprizing CA (Charles Addams [2]) effect, shown in fig. la: the variations involve thc nature of the central object. As will become apparent, scvcral of the experiments to be discussed were actually performed in Cliff Shull's laboratory and cart',' the unmistakenable marks of the master, himself.

It has been known since the fairly early days of quantum mechanics (see, for example the textbook by Kramers [3]) that the Hamiltonian for a charged particle ix] an electromagnetic field involves the vector potential A as well as the scalar potential F: II = ( p -

qA/c)Z/2m

+

ql'.

(1)

Nevertheless a great deal of attention v, as attracted by the paper of Aharonov and Bohm [ll which, as late as 1959. drew a straightforward but surprizing conclusion from the situation sho~n in fig. lb. They showed that the relative phase shift produced when electrons are diffracted by a tube of magnetic flux q~l, is given bv -~e~.~l~ = ( q / h c )

q~l~-

(2 )

B

l o b e of mogr, e:~. flu×

~

f , I li ,

.2_-_

a Fig. 1. (a). The Charles A d d a m s ( ( ' A ) effect. (Drav, m g h v ( ' h a s . (b} The A h a r o n o v - B o h m tAB) effect. 0 3 7 8 - 4 3 6 3 / 8 6 / $ 0 3 . 5 0 " Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Phi.sits Publishing Division)

b / \ d d a m s : ' 1940. 1 9 6 8 . ' I h c New Y o r k e r Magazine. Inc.)

A.G. Klein / Topological effects in neutron opti~

Thus, the position of the interference fringes is shifted even in the case when the electrons are completely excluded from the space containing the magnetic field B, i.e. when the electron paths are completely contained in a region of space where B is zero, though A is finite. In more general terms, this means that potentials, even in the absence of fields applied to the wavefunctions, may have observable consequences. Furthermore, even though the potentials are gauge dependent, the predicted physical effects are gauge invariant. In the case of electrons, the AB effect has been thoroughly investigated and verified. For example, in 1960, Chambers [4] diffracted electrons around the magnetic flux carried by an iron whisker; in 1962 Mt)llenstedt and Bayh [5] used a bi-prism type electron interferometer with current-carrying solenoids of microscopic size, and more recently, in 1983, Tonomura et al. [6] produced remarkable electron holograms of magnetised° permalloy toroids of 6/~m diameter and 50-100 A thickness. In spite of the excellent agreement with theory shown by these experiments, the AB effect continues to intrigue people and to generate continuing contributions to the literature. Some of the more recent papers aim to clarify its deeper topological significance, employing the theory of fibre bundles [7] and the nonlocality of quantum mechanics [8]. Others, on the other hand, have simply disputed its reality. It is probably fair to say that, to the extent that there remains a dispute about the AB effect, it is between those who understand its meaning [9], and those who do not. In the case of neutrons, the relatively recent experimental realisation of interferometry, i.e. of the CA topology, has resulted in a series of experiments, closely related to the AB effect. The most straight-forward of these was suggested by D. Greenberger and carried out by him together with Cliff Shull and his team, making use of a perfect crystal neutron interferometer [10]. They searched for any AB phase shift produced by the magnetic flux of a saturated single crystal of iron included between the separated, coherent neutron beams. Since neutrons carry no nett electric charge, eq. (2) predicts a null result. We know, however (mainly from deep inelastic electron scattering experiments), that neutrons do have a radial charge

231

distribution responsible, among other things, for its magnetic dipole moment. Perhaps a more sophisticated theory, more complicated than the so-called minimal coupling scheme responsible for eqs. (1) and (2), would give a finite AB effect, even for neutrons. As it turned out, the experiment did give a null result, allowing one to place an upper limit of 4.9 x 10-12 on the leading parameter of a more general type of coupling scheme. In its simplest form, therefore, there is no AB effect for neutrons. That, however, does not mean that it was not worth looking for! This might be the appropriate point for a philosophical digression on an important leit-motiv in experimental physics, of particular relevance to neutron optics. It concerns the view-point that may be labelled "naive empiricism". Any new experiment is worth doing, even if current dogma holds it to be unnecessary. This view-point is fully in accord with Cliff Shull's characteristic modesty: We don't know everything, and, even if we did, we should give Nature the chance to surprize us from time to time! I, and 1 am sure many others, are indebted to Cliff Shull for reminding us of this important precept. Now, it is probably true to say that in the brief history of neutron optics the only surprize has been that there were no surprizes! Quantum mechanical theory, in its standard form, was vindicated time after time, in the most extraordinary situations, on a scale of dimensions very many orders of magnitude greater than the atomic scale for which it was conceived. Nevertheless, it is early days: it is bound to yield eventually, if pursued in the right direction [11].

3. The WSC experiment Following the success of the celebrated COW experiment, in which the phase shift of neutrons caused by the Earth's gravitational field was demonstrated [12], Werner, Staudenman and Colella (WSC) carried out a difficult and beautiful experiment showing the effect of the Earth's rotation on the phase shift in a neutron interferometer [13]. This is, in fact, the Sagnac effect and may be attributed to the classical Coriolis force acting on the neutrons. As first pointed out by Page [14]

232

..I.(/.Klein ,." lw~olo~u'al ef.lk'ct~ tn neutrmt opttcs

however, it bears a close resemblance to the AB effect. This is most easily seen by writing down the Hamiltonian for a neutron in a rotating frame of reference:

lt=(p-m~×r)2/2m-(r×~)z/2m.

(3)

By comparing this with eq. (1) and making the appropriate indentifications, it then follows that the Sagnac phase shift is proportional to the '" flux of rotation", / ~ ' A , where A is the area included by the neutron beams, in analogy with the AB formula, eq. (2). In the WSC experiment this flux was varied by changing the orientation of the interferometer relative to the Earth's axis. In a subsequent experiment, carried out by D. Atwood in Cliff Shull's laboratory [15]. the rotation was produced by mounting the whole interferometer on a turntable, thus allowing a much wider range of variation of the rotational flux. Those of us who have carried out experiments with moving objects inside a neutron interferometer recognise this as a real tour-de-force, particularly when we notice that the error bars on the experimental points have been magnified ten times to make them visible over the straight-line fit! This is the kind of "experimentation" (a faw~rite Shull word) that makes us ever conscious of Cliff "looking over our shoulders" as we contemplate our meagre efforts.

4. The AC effect A recent paper by Aharonov and Casher (A(') [16] presents an interesting extension of the AB

effect. It arise,,,, in the first instance, by looking at the AB experiment in the frame of reference of a stationary electron diffracting a moving tube of magnetic flux, rather than the other way around. After adducing the appropriate Hamiltonian, Aharonov and Casher proceed to apply it to the closely related case of a neutral particle (possessing a magnetic moment p.) diffracted by a line of electric charge. For .~ Coulombs per unit length, the,, arrive at the result that there is expected to be a phase shift given by:

J'bAc = I~. ~/ e , h c .

(4)

In order to understand this proposition better, consider fig. 2a, which shows the AB experiment with the tube of magnetic flux replaced by a line of magnetic dipoles. The sense in which the AC effect is the inverse of this is shown in fig. 2b. Here the roles are reversed: a neutral particle possessing a magnetic dipole moment, e.g. a neutron. is diffracted by a line of charge. However, an alternative viewpoint makes the resnhant phaseshift, eq. (4). appear to be hardly unexpected. ('onsider the situation as seen from the frame of reference of the neutron. The line of charge appears as a sheet of electric current which, in that frame, produces a magnetic field given bv B - - ( t , × E ) / c where E is the electric field of the line of charge. The phase shift A~,,c is then the result of the magnetic dipole moment being acted upon by this B field. Pursuing this viewpoint quantitatively arrives at the same result as A('. The practical realisation of this, using a neutron inlerferometer, requires quite large electric fields

A

~

- i n e of mognetlc

~lne of etectrlc

charge

dLpoles ,thl'

a

-

i

b

I:~g. 2. ( a ) A n o t h e r reprc~,entation t~l" the AB effect. ( h ) T h e Aharonov-i'a~hcr (AC) effect.

A.G. Klein / Topological effects in neutron optics

and has yet to be attempted. The outcome may be predicted with confidence; nevertheless, in the spirit of naive empiricism, we propose to attempt such an experiment in the near future. For the sake of completeness, it may be worth remarking that another closely related situation, namely one in which the line of charge is replaced by an electric current, was the original basis of the proposal by Klein and Opat [17], to exhibit the phase-shift produced when neutrons are rotated by 360 °. In the event, the KO version of that experiment was carried out by making use of the Amp~rian current sheet produced by a magnetic domain wall [18].

5. The WY effect Another generalisation of the AB effect was put forward by Wu and Yang (WY) in 1975 [19]. They pointed out that if the gauge particle for the isospin group SU2 were massless, then there ought to exist an AB phaseshift in the appropriate circumstances. The essence of this proposition was grasped by Zeilinger, Horne and Shull [20], in terms of the diagrammatic analogy of fig. 3: The source of gauge field is a rotating neutron-rich rod (viz. uranium) contained inside the beams of a neutron interferometer. Not unexpectedly, the experiment yielded a null result. (If there existed a massless gauge field which coupled baryons together it would have appeared as an additional non-gravitational inverse-square force between pieces of matter and would have shown up long ago in Ertvrs-type experiments.) However, assum-

e •

Electron

233

ing a gauge particle of non-zero mass, the results of the experiment could put an upper limit on the strength of the interaction mediated by it. In a recent paper, Opat [21] makes the observation that a gauge particle having a sufficiently high mass would have too short a range to extend appreciably from the uranium rod to the neutron beams and suggests that this problem can be overcome by allowing the neutrons to pass directly through the rotating matter. He further points out that an experiment of this type has already been carried out [22] in searching for the neutron analog of the Fizeau effect. In that experiment, neutrons diffracted by a double slit passed through the side of a square cross-sectioned rod of quartz rotating about its axis. It was found that the diffraction pattern observed on a distant screen suffered a lateral displacement proportional to the angular velocity of the rod. This displacement agreed, within experimental error, with that predicted by the theory of refraction at moving boundaries, thereby precluding significant additional phase shifts that might arise from the presence of a neutral gauge field of significant strength. In fact, from this experiment, Opat deduces the following lower limit to the ratio of the mass M to the interaction strength ~ of any gauge vector boson: > 2.8 k e V / c 2 where fl is a constant defined to be analogous to the fine structure constant a. He estimates that a newer, more sensitive experiment, along the lines of the one performed by Zeilinger, H o m e and Shull, but with the neutrons penetrating the rotor, could raise this limit to around 1.0 M e V / c 2. Whether the existence of such gauge vector

M/V/-~

current .~

,tilt'

a

/4

Rotating U r o d

b

Fig. 3. (a) Another representation of the AB effect. (b) The Wu-Yang (WY) effect.

234

A (i. Klein / "l'~pob~gwal effect~ m neutron ,pm.~

fields, c o u p l e d to i s o t o p i c s p i n o r to b a r y o n i c c h a r g e w o u l d b e w e l c o m e , o r e v e n a l l o w e d , by' c u r r e n t m o d e l s , is a m o o t p o i n t . I w o u l d s i m p l y c o m m e n t t h a t , s h o u l d N a t u r e s u p r i z e us this t i m e , 1 h a v e a p i o u s f a i t h in t h e i n g e n u i t y o f o u r t h e o r e t ical c o l l e a g u e s in t h e i r a b i l i t y to a c c o m m o d a t e it. F u r t h e r m o r e , 1 w o u l d n o t c o n s i d e r it u n r e a s o n a b l e to e x p e c t t h e f i n e s s e a n d p r e c i s i o n o f n e u t r o n o p t i c a l e x p e r i m e n t s to m a k e u p for t h e b r u t e f o r c e o f h i g h - e n e r g y s c a t t e r i n g , if n o t n o w , t h e n in s o m e more advanced and refined future. But the experim e n t it still w o r t h d o i n g . I h o p e t h a t C l i f f a g r e e s a n d will l o n g c o n t i n u e to l o o k o v e r o u r s h o u l d e r s as w e a t t e m p t it, a n d o t h e r s like it.

References [1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (19591 485. [2] C. Addams, The New Yorker Magazine, January 13, 194(I. p. 13. [3] H.A. Kramers, Die Grundlagen der Quantentheorie (1937), translated as: Quantum Mechanics (Dover. New York. 19641. p. 84. [4] R.G. Chambers, Phys. Rev. Left. 5 (19601 3. [5] G. M611enstedt and W. Bayh, Physik BI. 1~ (1962) 299.

[6] A. Tonomura, tt. Umezaki. N. ()sakabe, .I. Endo and Y. Sugita, Phys. Re','. I,ett. 51 (19831 331. [7] H.J. l:~ern~,teinand A.V. Phillips. Scientific American 245. Jul', lggl, p. 94. [:gl I).M. Greenberger, Phys. Rcv. 1)23 (19811 1460. I9] A. Zeilinger, Left. Nuovo ('imento 51A (1979) 33. [10] I).M. (,;reenberger, D.K...Mwood, J. Arthur. ('.G. Shull and M. Schlenker, Phvs. Re,,. I,ett. 47 (1981)751. [11] A.(L Klein, Phvs. Letl. 151B I19851 275. [12] R. ('olella, A.W. Oserhauser and S.A. Werner, Phxs. Rex. L,ett. 34 (1974) 1472. [13] S.A. Werner, J . - t Slaudeman and R. ('olella, Ph\~,. Re',. l,ett. 42 (1979) 1103. [14J I..A. Page. in: Neutron Interferometry, Bonse and Rauch. eds. (Clarendon. Oxford, 19791 (containing an ackno,,~,ledgement to A. Zeilinger. unpublished ,xork). [15] I).K. Atwcu~d. M.A. Home. (.G. Shull and J. Arthur. Phys. Re','. Lett. 52 (19841 1673. 53 (19841 1300. I16] Y. Aharono~ and A. ('asher, Phys. Re',. l,ett. 53 {19841 319. [17] A.(,;. Klein and G.I. ()pat, Ph~,s. Rev. DI1 (19751 523. [18] A.(J. Klein and G.[. Opal. I'h,.'~,. Rev. geu. 37 (1976) 23S [19] "I'.T. Wu and ('.N. Yang, PIDs. Re,,. [112 (19751 3845. [20] A. Zeilinger. M.A. |[orne and ('.G. Shull, Pro. 1984) p. 289. [21] G.I. ()pat, I_:ni'.'ersit', of Melbourne UM-P-85 '7 (to be published I. [22] A.G. Klein, G.I. ()pat. A. Cmmnno, -~,. Zcilinger. W. ['reimer and R. G~.ihler. Ph',~,. Re',. l.ett. 46 (1'-181) 1551.