# Geometric phase in neutron interferometry

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Volume 148, number 1,2 PHYSICS LETTERS A 6 August 1990

Geometric phase in neutron interferometryApoorva G. Wagh and Veer Chand RakhechaSolid State Physics Division, Bhabha Atomic Research Centre, Bombay 400085, India

Received 15 May 1990; accepted for publication 5 June 1990Communicated by J.P. Vigier

We revisit a recent proposal by Wagh to test the dependence of spinor phase on the orientation of the precession axis. Weidentify the spinor phase shift originating from the misalignment between 7r-flippers in the two branches of a neutron interfero-meter with a geometric phase. We also interpret quantum beats in polarised neutron interferometry observed by Rauchs groupas a manifestation ofa time-dependent geometric phase shift.

Recently, Wagh has proposed [1] a neutron in- the rotation operator appropriate for the waveftinc-terferometric observation of spinor phase depen- tion of a spin-i /2 particle. Thus, ifdence on the orientation of the precession axis. Inthe proposed experiment, a uniform magnetic field F2 I t> = exp(ic) ~>B0 applied over the entire interferometer (fig. 1) in it follows thata direction normal to the plane formed by the beampaths 1 and 2, serves as the guide field and defines F1 It> =exp( ia~&$/2)F2exp(ia~~fl/2)It>the z-direction. Two spin flippers F1 and F2 placed = exp(i8fi/2) exp (it) exp ( ia~~$/2)I ~>in the two beam paths reverse the neutron spin di-rection. F1 and F2 are identical flippers with a rel- = exp(i6fl) exp(it) I 1>ative orientation 8$ about the z-axis. The operator = exp(i6fl)F2 It> . (la)for the flipping action in F1 can therefore be repre-sented by F1 = UF2 U, U=exp ( iaAfl/2) being Similarly

FiI~>=exp(i6p)F2I~>. (lb)Thus a neutron traversing the interferometer picks

~P/2 ~0 up a phase shift 8$ originating from a relative ro-/ tation of the two flippers about the z-axis. We shallI identify this shift with a pure geometric phase [2) ~~kN>/ H 4].

It>! 6p/2 / In addition, the neutron undergoes a purely dy-/ ~ / namical phase shift ~ 60 [1], 60 being the excess spinS/ ~ 2 ~J precession about the guide field on the path SF1 over/ I ~ I~>2 that on the path SF2. This phase thus arises from a/ I relative linear translation of the two flippers along/ the respective beam paths.

Aharonov and Anandan [4] have delineated theFig. 1. Schematic sketch of a neutron interferometnc experimentfor observing geometric phase. S: sp1itter~A: analyser; F1, F2: dynamical and geometnc components of the phaseidentical x-flippers. 0 and H: forward and diffracted outgoing acquired by a spinor during a 2x-precession of its spinbeams respectively, vector over a cone of semi-angle 0 about a homo-

0375-9601 1901$ 03.50 1990 Elsevier Science Publishers B.V. (North-Holland) 17

Volume 148, number 1,2 PHYSICS LETTERS A 6 August 1990

geneous magnetic field. We generalise their analysis several sectors representing successive rotations akfor an arbitrary closed circuit C described by the in magnetic inductions Bk, overconesof polar anglesneutron spin over a sphere (fig. 2a). If C comprises 0k~spanning time intervals tk, k= 1, 2, ..., n, the net

phase qi acquired by the neutron can be written asZ exp(ici~a~/2)x...xexp(ia

2a2/2)Xexp(ia1a~/2)=e~. (2a)

C The phase 1 can be expressed as the sum of a dy-fl~ namical component

2 ~r~D=~ kk1~9 (2b)

and a geometric component

kG=~Q, modulo2it. (2c)Here k=+I~uIB~cos9k,with p=Ij~I~foraneutron, is the expectation value of the magneticHamiltonian over the kth sector and Q is the solid

k angle subtendedby C at the centre ofthe spin sphere,a its sign depending on the sense of traversal of C. It

I ~> is clear that both the phase components change signon a reversal of the spin direction resulting in achange of sign of the total phase J~as expected. Fur-ther if a sector forms an arc of a great circle on thespin sphere (4

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

aligned parallel or antiparallel to Bk (Ok=O or it),the sector reduces to a point on the spin sphere and

1 makes no contributionto the solid angle Q. The phase2 acquired over such a sector is thus purely of dynam-

-.-.~ ical origin.As regards the proposal depicted in fig. 1, the op-

- eration of flipper F2 reversing the neutron spin can

-, be represented by the curve labelled 2 on the spinsphere (fig. 2b). The flipper F1, being identical to F2except fora rotation 6$ about the z-axis, flips the spinalong the curve 1, identical to curve 2 but rotatedby6/3 about the z-axis from it. The phase shift picked

b up by the neutron in traversing the interferometer isI ~ that acquired along curve 1 in the forward direction

followed by curve 2 in the reverse direction, i.e. overthe closed circuit formed by curves 1 and 2. The evo-

Fig. 2. (a) Evolution of phase overa closed circuitC traversed lution of the spinor wavefunction along this closedby the neutron spin vectoron the spin sphere. (b) Geometric . . . .depiction ofthe phase shift arising from a relative orientation ~ circuit is cyclic and the corresponding cyclic phasebetween the flippers F1 and F2. Spin trajectories I and 2 pro- has dynamical and geometric components given byduced by F1 and F2 respectively enclose a solid angle 2bfi. eqs. (2b) and (2c). The dynamical phase contri-

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Volume 148, number 1,2 PHYSICS LETTERS A 6 August 1990

butions along curves 1 and 2 are equal since the two subtending an angle 9= cot with a fixed direction inflippers are identical. The dynamical phase acquired the xy plane. The rf flipper is thus seen to be con-along the closed circuit therefore vanishes. The solid ceptually identical to a dc it-flipper whose axis ro-angle subtended at the centre by the closed circuit is tates uniformly at an angular frequency w about the 26$, leading to a geometric and therefore net phase z-direction. If time t = 0 is selected to be the instantshift of 6$ in accordance with eq. (1 a). The solid when the rotating fields of the two flippers point inangle Q, and therefore the phase shift, changes sign the same direction, it follows that at any time t, theif the incident neutron is in the spin-down state. The flipper axes enclose an angle 6/3= (w1 w2) t. A spin-phase 6/1 arising due to the relative rotation be- up neutron passing through the interferometer at timetween the two flippers is thus purely of geometric t therefore undergoes a geometric phase shiftorigin. 6/3(t) =6w t, which is proportional to t. The inten-

In the light of the foregoing discussion, we shall sity of the outgoing beam from the interferometernow reexamine the quantum beats observed by Bad- consequently oscillates with a frequency 6w/2it. Suchurek et al. [5] in polarised neutron interferometry. conversion of a time-proportional phase into fre-Their experimentemployed two radio-frequency (rf) quency has been described by Maddox [6] in a corn-flippers in the two branches. A small frequency dif- ment on an optical experiment by Simon et al. [7]ference 6 v maintained between the two flippers wherein an evolving geometric phase was used tocaused intensity oscillations in the outgoing beam cause a shift in the frequency of a laser beam. Thewith a macroscopic time period 1/6v. We note first present interpretationof the neutron quantum beatsthat an rf flipperessentially simulates a field rotating [5] as a manifestation of an evolving geometricphasein the xy plane at an angular frequency w super- implies that the neutron analogue of the optical ex-imposed on the guide field B~jover a pathlength 1. periment [7] preceded it by two years!The flipper must satisfy two resonance conditions,viz, for the frequency and amplitude of the If field,to achieve the spin reversal with probability one. The Referencesfrequency resonancecondition stipulates that coequalthe angular frequency ofthe neutron spin precession [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;frame attached to the rotating transverse field, the M.V. Be~,J. Mod. Opt. 34 (1987)1401.neutron experiences only a static transverse field. The [4] Y. Aharonov and I. Anandan, Phys. Rev. Lett. 58 (1987)amplitude resonancecondition then ensuresthat over 1593.the pathlength 1, the neutron spin undergoes exactly [5] 0. Badurek, H. Rauch and D. Tuppinger, Phys. Rev. A 34a precession equal to it about this static field in the [6] J. Maddox, Nature 334 (1988) 99.rotating frame. As seen from the laboratory frame, [7] R. Simon, H.J. Kimble and E.C.G. Sudarshan, Phys. Rev.the spin precession thus takes place about an axis Lett. 61(1988)19.

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