experimental separation of geometric and dynamical phases using neutron interferometry

PHYSICAL REVIEW

LETTERS

VOLUME 78 3 FEBRUARY 1997 NUMBER 5

ria

Experimental Separation of Geometric and Dynamical Phases Using Neutron Interferometry

A. G. Wagh* and V. C. Rakhecha*Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India

J. Summhammer, G. Badurek, and H. Weinfurter†

Atominstitut of Austrian Universities and Technical University of Vienna, Schuettelstrasse 115, A-1020 Vienna, Aust

B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. WernerResearch Reactor Center and Physics Department, University of Missouri-Columbia, Columbia, Missouri 65211

(Received 5 February 1996)

We present results of the first experiment clearly demarcating geometric and dynamical phases.These two phases arise from two distinct physical operations, a rotation and a linear translation,respectively, performed on two identical spin flippers in a neutron interferometer. A reversalof the current in one flipper results in a pure geometric phase shift ofp radians. Thisobservation constitutes the first direct verification of Pauli anticommutation, implemented in neutroninterferometry. [S0031-9007(96)02278-8]

PACS numbers: 03.65.Bz, 42.25.Hz, 42.25.Ja

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In general, a quantal system evolving under a timdependent HamiltonianHstd acquires, apart from the dy-namical phase2Re

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Hamiltonian-independent phase component called gemetric phase, which depends only on the geometry of tcurve traced in the ray space. Pancharatnam was theto explicitly recognize geometric phase during his stuies [1] of interference between optical beams of distinpolarizations. However, geometric phase attracted litfurther attention until Berry provided a general quantframework [2] for geometric phase in the context of adiabatic evolutions, triggering an intense activity in this fieldGeometric phase, already included in the standard formlation of quantum mechanics, can arise in any geneevolution, be it nonadiabatic [3], noncyclic [4], or evenonunitary [4]. A completely general ray-space expresion [5,6], in terms of just the pure state density operathas been provided for geometric phase. Geometric phhas since been observed in a broad spectrum [7–15]physical phenomena.

In the first quantal prescription [2] of geometric phasan adiabatic evolution was considered. The early neutrexperiments [10,11] therefore observed geometric pha

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for an eigenstate of a slowly rotating magnetic fielHowever, an adiabatic evolution generates a dynamiphase background which is much larger than the geomric phase signal. An ideal geometric phase experimeshould therefore effectnot an adiabatic evolution, but aparallel transportation [16], an intrinsicallynonadiabaticevolution, that eliminates dynamical phase and yieldspure geometric phase. In an evolution which does nparallel transport the state, it is still possible to generaa pure geometric phase by arranging for a null dynamiphase [16–18] at the end of the evolution.

The wave function of a spin12 particle changes sign[19,20] when the spin precesses through2p radians. This4p spinor symmetry has been directly verified in neutrointerference experiments [21–23]. The spinor phase adepends on theorientation [17] of the precession axis.While elucidating this dependence, Wagh and Rakhecproposed the first experiment effecting a clear sepation [18] of geometric and dynamical phases. Herejzl-polarized neutron beam incident on an interferome(Fig. 1) permeated by a uniform guide fieldB0z splits intosubbeams I and II. The subbeams pass through identspin flippersF1 and F2 which take the neutron state to

© 1997 The American Physical Society 755

VOLUME 78, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 FEBRUARY 1997

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FIG. 1. Schematic diagram of the experiment demarcatigeometric and dynamical phases. A uniform guide fieldB0z,transverse to the plane of the diagram, is applied over the(220) skew symmetric LLL interferometer. A relative rotationdb between the identical dual flippersF1 and F2 produces apure geometric phaseFG , equal todb, for the incidentjzl-polarized neutron beam; their relative translationdx results ina pure dynamical phaseFD , proportional todx.

j2zl. The relevant ray space for the spinor is the unsphere of spin directions. For a relative rotationdb be-tweenF1 andF2 aboutz, the closed spin trajectory tracedduring the evolution subtends a solid angleV 22db

[cf. Fig. 2(b) in [18] ] at the center of the spin sphereyielding a pure geometric phase [18]

FG 2Vy2 db . (1)

FIG. 2. Geometric phaseFG arising from the angledbbetween the flippersF1 andF2. Error bars are not shown sincethey are smaller than the size of points. The solid line is ththeoretical prediction [Eq. (1)], and the dashed line is the fitthe data for anglesdb from 240± to 140± achieved by rotatingthe flippers mechanically. The second point at 0± is the phasemeasured with both flipper currents reversed.

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This relation brings out the geometric nature ofFG , whichis given by just theangle db [24] regardless of theHamiltonian.

A linear translationdx of F2, say, along the respectivsubbeam, on the other hand, changes the precessionabout the guide fieldB0z by df, say, in thejzl state and2df in the j2zl state. The translation thus leaves tspin trajectory and henceFG unaltered, generating a purdynamical phase shift [18,25]

FD df 2jmjB0dxy"y . (2)

Here m and y denote the neutron’s (negative) magnemoment and speed, respectively. A translation ofF1would produce an identical dynamical phase, but wthe opposite sign. In contrast with the geometric pha[cf. (1)], the dynamical phaseFD depends on the fieldB0in the Hamiltonian.

In our experiment, each flipper was a dual flipper [1consisting of two successive rectangular coils produchorizontal magnetic fields of magnitudeB0, in oppositedirections. Along with the guide fieldB0z, they producednet magnetic fields of magnitude

p2B0 along mutually

orthogonal axes,p and q, say, in a vertical planesubtending angles1py4 and 2py4, respectively, withz. The magnitude of these fields was set so that overneutron path length through each coil, the spin precesthrough an azimuthal anglep . The dual flipper thuseffects two successivep precessions about axesp andq. Its operation,

e2isqpy2e2isppy2 s2isqd s2ispd 2sqsp , (3)

brings thej1zl state toj2zl. Heresp andsq representthe components of the Pauli spin operator$s alongp andq, respectively. On reversing the current in the twcoils of a dual flipper, the neutron is subjected to a fiealongq followed by a field alongp, i.e., to the operation

e2isppy2e2isqpy2 2spsq sqsp , (4)

sincesp andsq anticommute, being orthogonal components of the Pauli spin operator. Thus, the reversed flper also takesj1zl to j2zl, but with a change of sign acompared to the original operation (3). Thissign changemanifests itself as ap phase shift[6,17] andcan only beobserved interferometrically,as will be reported below; apolarimetric experiment is incapable [26] of detectingcurrent reversal in the flipper.

The current reversal in a dual flipper is equivaleto a p rotation [6,17] of the flipper aboutz. Thespin trajectory for the reversed current, comprising twsemicones of polar anglespy4 and3py4 aboutq andp,respectively, is thus obtainable from that for the forwacurrent by ap rotation aboutz. These two trajectoriesenclose half the surface of the spin sphere, i.e., a sangle of62p , resulting in a pure geometric [18] phase7p, generated without physically rotating the flippers.

The experiment was carried out at the beam pC interferometry facility [27] at the 10 MW Researc


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Reactor of the University of Missouri (MURR) in aBARC-Vienna-MURR collaboration. A 2.35 Å neutrobeam from a focusing pyrolitic graphite monochromatwas polarized vertically upwards, i.e., alongz, by a re-flection from a Fe-Si magnetic supermirror and passthrough a 2 mm swided 3 6 mm shighd slit to illumi-nate a skew symmetric (220) LLL silicon interferometer (Fig. 1). With a Heusler spin-state analyzing crysdownstream of the interferometer, thejzl fraction in thisbeam was measured to be 0.925, corresponding to alarizationP 0.85. The interferometer was enclosedan aluminum box (providing an isothermal enclosure)side a heavy Benelex-70 box, which rests upon a 550black granite slab, floated on four Firestone air cushioExcluding the polarizing mirror, the entire setup is eclosed within a large Plexiglas box for general enviromental isolation. With this isolation from the ambiemechanical vibrations and thermal variations, phase drtypically less than 5± over a day were achieved.

A pair of water-cooled Helmholtz coils produced a fairuniform guide field. The two rectangular coils of eacdual flipper were connected in series and operated atdissipating about 2 W each. Each coil was made o25 mm wide and 90mm thick anodized Al foil woundon an aluminum fork, which was firmly mounted ontoTeCu-145 heat sink block. A special low temperature ACu brazing technique provided excellent electrical contabetween the coil ends and the Cu current leads. Two 1thick copper sheets screwed on the front and back sidethe heat sink conducted the heat produced in the coils afrom the interferometer. The flippers were suspendedthe 40 mm spaces between the interferometer blades wprecision rotation/translation gadget specially construcin the Missouri Physics Machine shop. Field mappicarried out with each flipper turned on and off, usingprecision Hall probe within the interferometer, revealthat fields produced by the flippers in the relevant regexterior to them were well below 1 G. The temperatuof the flipper heat sinks was maintained within60.01 ±Cof the ambient air temperature with a controller operatithrough a closed-cycle water loop. Special precautionsto be taken to ensure that no vibrations were transmittethe interferometer by the water flow.

The Heusler alloy analyzer crystal was used to ascera p spin flip in the dual flippers by adjusting the fieldsB0

produced by the Helmholtz coils and the flipperF1. Theoptimum B0 was about 30 G, in agreement with the caculated field for 2.35 Å neutrons traversing a path lengof about 7 mm through each coil (at normal incidencBecause of the space constraints within the interferomethe maximum mechanical rotation of each flipper was liited to 622±. Larger anglesdb were therefore achievedelectrically. With the flippers normal to the respectivsubbeams, a reversal of current inF1 sF2d yields db 180± s2180±d. In each of thedb settings, the two flip-pers were oriented symmetrically relative to the neutr

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subbeams, presenting nearly equal pathlengths and hnuclear phases. Any residual difference between theclear phases acquired in traversing the two flipper matals was eliminated as described later (see also [26]).

The interferograms recorded in the He-3 detectorsC2

and C3 were obtained by rotating a 1.05 mm thick Aphase flag in the interferometer (Fig. 1), which varies tnuclear (spin-independent) phase. TheC3 interferencecontrast of 64% for the empty interferometer reducedabout 32% on inserting the water-cooled dual flippeThis contrast dropped slightly to 28% when the currenin the flippers were switched on.

In each run, two interferograms, one with flippers oand the other with flippers off, were recorded simultanously by periodically switching the flippers on and oafter every6 3 104 monitor counts, taking about 5 s, aeach angular setting of the phase flag. Since this peris much shorter than the time constants for thermal vaations and mechanical phase drifts, the phase shiftdif-ferencebetween the on and off interferograms eliminatnuclear phase variations within the flippers and spuriophase drifts. To obtain the desired spin-dependent ph(1) or (2), corrections were applied to this phase diffeence for a small difference in the quantity

RB0 dl over

paths I and II, the incomplete polarizationP of the in-cident beam and, when the flippers were not normalthe subbeams, for precessionspy cossdby2d aboutp andq instead ofp. P corrections to the observed phase lawithin 65±. The correction for the excess spin flip rangebetween 0± and 1±. The present analysis assumes idencal operations in the two flippers and ignores the mesured small differences,1 Gd between the guide fields athe two flippers.

Interferograms for geometric phase were recorded11 fixed anglesdb betweenF1 and F2 held at fixedpositions. The resultant pure geometric phasesFG areshown in Fig. 2. The data points lie close to th

FIG. 3. Pure dynamical phaseFD as a function of thetranslationdx of the flipper F2, obtained by correcting (seetext) the observed phase shift between the scans recordedthe flippers on and off. Error bars are smaller than the sizepoints.

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theoretical (solid) line [Eq. (1)]. The phases fordb

values between240± and140±, obtained by rotating theflippers mechanically, fall on the dashed line which hasslope of1.23 6 0.03. The discrepancy of this slope withthe theoretically expected slope of 1 may have arisen dto a possible dynamical phase contamination. With oflipper mounting gadget, it was not possible to position throtation axis of each flipper accurately on the centerlinof the respective subbeam. A rotation of such an ofcenter flipper would be accompanied by a displacemedx and a consequent dynamical phase [cf. (2)]. An offsof about 1 mm only of one flipper axis can generateFD contamination accounting for the observed deviatioThe phases fordb 2180, 180, and 360 (shown asthe second point at 0) degrees which are free from sua contamination, since they are measured by reversithe current first in one, then in the other, and thenboth flippers, while both flippers remain normal to thesubbeams, agree to within 2% with theory.

The flippers were then made normal to the respectisubbeamssdb 0d and interference patterns recorded aa function of the linear translation of firstF2 and thenF1. Figure 3 displays the consequent variation of thpure dynamical phase obtained by translatingF2. Theslope 18.7 6 0.2 degymm of the best fit corresponds to[cf. Eq. (2)] a guide field of29.9 6 0.3 G which agreeswith its measured value of29.9 6 0.1 G in the vicinity ofF2. The translation ofF1 yielded a straight line forFD

with a negative slope of similar magnitude, as predictedFigure 4 depicts the interference patterns record

with the current in the flipperF1 switched between theforward (F) and reverse (R) directions. On reversinthe current, the pattern just gets reflected about tline representing its average, as expected. The obser

FIG. 4. The interferogram has shifted by180.5± 6 3.0± onswitching the current inF1 from the forward (F) to the reverse(R) sense. This result verifies Pauli anticommutation to withiabout 2%.

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pure geometric phase shift (cf. Fig. 2) equals180± 6 3±,confirming the anticommutivity [(3)–(4)] betweensp andsq. If the current inF2 is also reversed, the interferogramshifts further by 180± becoming identical to the initialinterferogram. Thisp phase shift observed with a mereflick of a switch reversing the flipper current constitutethe first direct verification of Pauli anticommutationaccomplished here interferometrically with neutrons.

In conclusion, we have observed the spinor phadependence on the orientation of the precession axis inpolarized neutron interferometric experiment. This is thfirst experiment effecting a clean separation of geometrand dynamical phases. Here, a relative rotation of twp flippers gives rise to a pure geometric phase; therelative translation produces a pure dynamical phase.reversal of current in a flipper, equivalent to ap rotationof the flipper, generates a pure geometric phase ofp,and its observation has confirmed the anticommutivity oorthogonal components of the Pauli spin operator.

The success of this work depended on the skilleworkmanship of Clifford Holmes and his staff at thephysics machine shop at the University of MissourColumbia. This research was carried out under thAustrian Fonds (FWF) Project No. P9266-PHY, the U.SNSF-Physics Grant No. 9024608 and a UM ResearBoard Grant No. RB-94-003. One of us (A. G. W.) wisheto acknowledge the support for two months from thAustrian Fonds and for one month from the Universitof Missouri-Columbia during the experiment.

*Electronic address: [email protected]†Present address: Institute for Experimental PhysicUniversity of Innsbruck, Technikerstrasse 26, A-602Innsbruck,Austria.

[1] S. Pancharatnam, Proc. Indian Acad. Sci. A44, 247(1956).

[2] M. V. Berry, Proc. R. Soc. London A392, 45 (1984).[3] Y. Aharonov and J. Anandan, Phys. Rev. Lett.58, 1593

(1987).[4] J. Samuel and R. Bhandari, Phys. Rev. Lett.60, 2339

(1988).[5] A. G. Wagh and V. C. Rakhecha, Pramana-J. Phys.41,

L479 (1993).[6] A. G. Wagh, Ind. J. Pure Appl. Phys.33, 566 (1995).[7] A. Tomita and R. Chiao, Phys. Rev. Lett.57, 937 (1986).[8] G. Delacrétaz, E. R. Grant, R. L. Whetten, L. Wöste, an

J. W. Zwanziger, Phys. Rev. Lett.56, 2598 (1986).[9] R. Tycko, Phys. Rev. Lett.58, 2281 (1987).

[10] T. Bitter and B. Dubbers, Phys. Rev. Lett.59, 251 (1987).[11] D. J. Richardson, A. I. Kilvington, K. Green, and S. K.

Lamoreaux, Phys. Rev. Lett.61, 2030 (1988).[12] D. Suter, K. T. Mueller, and A. Pines, Phys. Rev. Lett.60,

1218 (1988).[13] D. M. Bird and A. R. Preston, Phys. Rev. Lett.61, 2863

(1988).[14] Geometric Phases in Physics,edited by A. Shapere and

F. Wilczek (World Scientific, Singapore, 1989).[15] J. Anandan, Nature (London)360, 307 (1992).


.

by

a-

i-

[16] A. G. Wagh and V. C. Rakhecha, Phys. Rev. A48, R1729(1993); Phys. Lett. A170, 71 (1992).

[17] A. G. Wagh, Phys. Lett. A146, 369 (1990).[18] A. G. Wagh and V. C. Rakhecha, Phys. Lett. A148, 17

(1990).[19] H. J. Bernstein, Phys. Rev. Lett.18, 1102 (1967).[20] Y. Aharonov and L. Susskind, Phys. Rev.158, 1237

(1967).[21] H. Rauch, A. Zeilinger, G. Badurek, A. Wilfing,

W. Bauspiess, and U. Bonse, Phys. Lett.54A, 425 (1975).[22] S. A. Werner, R. Collela, A. W. Overhauser, and C. F

Eagen, Phys. Rev. Lett.35, 1053 (1975).[23] A. G. Klein and G. I. Opat, Phys. Rev. Lett.37, 238

(1976).[24] The phaseFGs1d just equals the phase jump caused

the kink db in the spin trajectory [6] at the polej2zl.[25] Equation (2), though not exact, is an excellent approxim

tion (cf. [17]) for thermal neutrons in fields up to,400 G.[26] A. G. Wagh and V. C. Rakhecha, Phys. Lett. A197, 107

(1995);197, 112 (1995).[27] R. Clothier, Ph.D. thesis, University of Missour

Columbia, 1991 (unpublished).

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experimental separation of geometric and dynamical phases using neutron interferometry

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