observation of geometric and dynamical phases by neutron interferometry

PHYSICAL REVIEW A DECEMBER 1997VOLUME 56, NUMBER 6

Observation of geometric and dynamical phases by neutron interferometry

B. E. Allman,* H. Kaiser, and S. A. WernerResearch Reactor Center and Physics Department, University of Missouri–Columbia, Columbia, Missouri 65211

A. G. Wagh† and V. C. Rakhecha†

Solid State Physics Division, Bhabha Atomic Research Center, Mumbai 400085, India

J. SummhammerAtominstitut der O¨ sterreichischen Universita¨ten, Schu¨ttelstrasse 115, A-1020 Wien, Austria

~Received 9 April 1997!

The total phase acquired during an evolution of a quantal system has two components: the usual dynamicalphase,21/\*H(t)dt, given by the integrated expectation value of the Hamiltonian, and a geometric phaseFG(C) which depends only on the geometry of the curve traced in ray space. We have performed an inter-ference experiment using polarized neutrons which clearly demarcates the separate contributions of the dy-namical and geometric phases to the total. In the experiment, the two phases arise from two distinct physicaloperations, a translation and a rotation of a spin flipper within the interferometer, respectively. This work alsoconstitutes the first direct observation of the Pauli anticommutation; a purely geometric phase shift ofp radiansappears after a reversal of the current in one of the spin flippers. The experiment was carried out at the 10 MWUniversity of Missouri Research Reactor using a skew-symmetric perfect-silicon-crystal neutron interferom-eter. A detailed description of the experiment and its interpretation is given in this paper.@S1050-2947~97!03911-5#

PACS number~s!: 03.65.Bz, 42.25.Hz, 42.25.Ja

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I. INTRODUCTION

The concept of geometric phase has been of increainterest and activity since Berry’s inspired discovery in 19@1#. While considering the behavior of quantum-mechanisystems, Berry found that, for a Hamiltonian, slowly cyclin time T, along a curveC of external parameters, an initiaeigenstate with the energy eigenvalueE(t) returns to theoriginal state, in the adiabatic approximation, multipliedtwo phase factors, prescribed by

uC~T!&5expH 21

\ ETE~ t !dtJ exp$ iFG~C!%uC~0!&.

~1!

The first phase factor is the standard, Hamiltonian-dependynamical phase. The second is a nonintegrable, pdependent phase, which he termed the ‘‘geometric phaThis phase is truly geometric, as it is independent of the rthe Hamiltonian, and the energy eigenstate, and dependclusively on the geometry of the curve traced in ray spaSuch phase shifts of geometric origin were already incluin the standard formulations of quantum mechanics, e.g.,topological Aharonov-Bohm effects, but their true geometnature has only been appreciated recently.

Historically, however, the geometric phase was first oserved by Pancharatnam in his study of the interferencpolarized light @2#. By considering three nonorthogon

*Present address: School of Physics, University of MelbouParkville, Australia 3052.

†Electronic address: [email protected]

561050-2947/97/56~6!/4420~20!/$10.00

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states of polarization represented by three points on the Pcaresphere, he found that if the first and second states‘‘in phase’’ ~a condition defined as a maximum in their interfered intensity!, and the second and third states are ‘phase,’’ then the first and third states are not necessarilphase. Pancharatnam showed that the excess phase othird state over the first is minus half the solid angle sutended by the spherical triangle formed by the three statethe center of the Poincare´ sphere. The ideas of Pancharatnahave since been used to obtain geometric phase@3,4#, arisingin completely general evolutions.

Following its discovery, the geometric phase has begeneralized and found to occur even when the evolutionnonadiabatic@5#, noncyclic @3#, and nonunitary@3#. For aparallel transported quantum state, the geometric phaseconsequence of the curvature of the connection~a rule per-mitting the comparison of their phases! between two neigh-boring rays. Moreover, the geometric phase representsexample of anholonomy in that the parallel transported qutal system fails to return to its original value when alteraround a cycle, and is thus nonintegrable. A completely geral expression in terms of just the state density operatorthe geometric phase brings out its exclusive dependencthe ray space geometry@4#.

The geometric phase has been observed in a broad strum of physical phenomenon from classical@6# to quantumphysics@7–16#. In early experiments to observe the geomric phase, an adiabatic evolution was employed, via spin pcession in various magnetic-field configurations@12,13#. Un-fortunately, this generates a dynamical phase backgrowhich is generally much larger than the geometric comnent. Therefore, an ideal experiment should not effect

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4420 © 1997 The American Physical Society

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56 4421OBSERVATION OF GEOMETRIC AND DYNAMICAL . . .

FIG. 1. Schematic diagram of the experiment to demarcate geometric and dynamical phases. A uniform vertical guide fieldB0z from apair of Helmholtz coils is applied over the Si~220! skew symmetric interferometer. A relative rotationDb between the identical dual flipperF1 andF2 produces a pure geometric phaseFG , predicted to be equal toDb, for the incidentuz&-polarized neutron beam; their relativtranslationdx results in a pure dynamical phaseFD , proportional todx.

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adiabatic evolution but a parallel transport@17# will effect anintrinsically nonadiabatic evolution. However, it is still posible to use a nonparallel transported state provided thenamical phase can be made to disappear@17–19#, as willnow be described.

The basic idea of our experiment involves the interfereof the neutron wave packets traversing the two separate pin a Mach-Zehnder-type neutron interferometer after ewave packet has been individually spin flipped insideinterferometer. The clear demarcation of the geometricdynamical phases occurs in the two physically distincterations, a rotation and a translation, of the two spin flippinside the interferometer. The initial proposal for the expement was made by Wagh and Rakhecha@19#, after their re-alization of the dependence of the spinor phase on the ortation of the precession axis@18#, as is now explained.

The wave function of a spin-12 particle changes sign afte

a 2p precession@20,21#. This 4p spinor symmetry of thespin-12 particle, has been directly verified in both division-oamplitude @22,23# and division-of-wave-front@24# neutroninterferometry experiments. However, the spinor phasedepends on the orientation@18# of the precession axis.

In the present interferometric experiment with polarizneutrons, we perform a pair ofp spin-flip operations, one ineach arm of the interferometer, but with the two flipper axenclosing an angleDb. When the two flippers are identicathe Hamiltonian-dependent dynamical phases within theflippers cancel each other. The dependence of the sp

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phase on the orientation@18# of the precession axis manifesitself here as the Hamiltonian-independent geometric phequal to minus half the solid angleDV of the slice on thespin sphere, enclosed between the curves traced by thetron spin within the two flippers. A relative translation btween the two flippers, on the other hand, leaves the scurves, and hence the geometric phase, unaltered, produa pure dynamical phase due to the ambient guide field.

II. DESCRIPTION OF THE EXPERIMENT

Overview

The experiment is shown schematically in Fig. 1. A colimated, polarized, monochromatic neutron beam is incidon the first blade of a neutron interferometer. The monlithic, perfect-silicon-crystal neutron interferometer of skesymmetric design sits in a region of uniform magnetic gufield B0z. Bragg diffraction by the~220! planes at the firstcrystal plate coherently splits the neutron de Broglie wapacket. Each of the separated subbeams is again diffractthe intermediate blades, and after passing through a spinper in the long arm of each path, overlap and interfere atlast blade. The recombined neutron wave packets aredetected by two3He proportional detectors,C2 andC3 , be-yond the interferometer. The interference appearing aswapping of the detected neutron intensity in the recombibeams as a function of the phase difference between

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4422 56B. E. ALLMAN et al.

separate subbeams. The whole arrangement is akin toMach-Zehnder interferometer of classical optics. For a mdetailed description of neutron interferometry, see Ref.@25#.

A neutron in a magnetic field experiences Larmor precsion. Here, au1z& eigenstate gains dynamical phase asprecesses about the magnetic guide field, which also poin thez direction and leads up to the spin flipper. After beispin flipped, also incorporating a dynamical phase,u2z& eigenstate also gains phase by precessing in the gfield, but of an opposite sense than theu1z& state before thespin flipper. This precession establishes some dynamphase offset for the neutron’s path. The experiment requa spin flipper to change the polarization fromu1z& tou2z&. The simplest form of this flipper is a single horizontfield componentB0y which requires the spin angular momentumS of the initial stateu1z& to precess through aanglep on the spin sphere to the stateu2z&, as illustrated inFig. 2~a!. Rotating~without translating! a pair of such flip-pers inside an interferometer, one in each subbeam, to ative orientation of angleDb between the two, results in thseparate precession paths ofSI andSII enclosing a solid angleV522Db ~an ‘‘orange slice’’!, at the center of the spinsphere@26#. This manifests itself as a pure geometric phashift

FG52V/25Db, ~2!

without altering the dynamical phase offset, i.e., without adynamical phase shift contamination. This phase shift isdependent of the Hamiltonian. In the experiment each flipis actually a dual flipper producing successivep precessionsabout two mutually orthogonal axes,q and p, say, directed45° on either side ofz in the y-z plane. This two-stage process precesses theu1z&-polarized neutron state around thunit sphere of spin directions, to the stateu2z&. It is straight-forward to show that their combined effect on the wave fution of the neutron is equivalent to a precession around thyaxis by an anglep. A second identical dual flipper orienteat Db to the first again results in a pure geometric phaFG52V/25Db, as shown in Fig. 2~b!.

A pure geometric phase can be achieved without anytion of the spin flipper at all. Reversing the current~andhence the direction of the magnetic field! in one dual flipper,is equivalent to affecting a 180° rotation of the precessaxes without any physical motion@4,18#, ensuring the origi-nal offset phases are unchanged. The dual flipper effectssuccessivep precessions aboutq andp, respectively. Its op-eration,

e2 ispp/2e2 isqp/25~2 isp!~2 isq!52spsq , ~3!

brings theu1z& state tou2z&, wheresq and sp representtwo orthogonal components of the Pauli operators along qandp, respectively. Reversing the current in the two coilsthe dual flipper reverses the order of the two fields. Tneutron is now subjected to a field first alongp followed bya field alongq. This reversed flipper then operates as

e2 isqp/2e2 ispp/252sqsp5spsq , ~4!

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FIG. 2. Schematic of the path traced out by the neutron spinit precesses around the spin sphere from spin-up,uz&, to spin-down,u2z&. The top figure,~a!, shows thep precession of the spinsSI

and SII on the separate interferometer paths, each about a sihorizontal field component oriented at an angleDb relative to theother. The lower figure,~b!, shows the precession of the two spinSI andSII about the spin sphere under the action of the dual flippHere the two field components of the individual flipper, oriented145° and245° to N-S on the same great circle~p and q in thetext! result in successivep precessions. Again the second dual fliper is oriented at an angleDb to the first. In both cases, the geometric phaseFG is equal to half the solid angle between the twtrajectories.

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FIG. 3. Overall view of theC-port interferometer setup, including the reactor, beam port, shielding, polarizing supermirror, Helmcoils, interferometer, and Heusler alloy analyzing crystal. The inset shows a detail of the inside of the masonite box.

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again taking the stateu1z& to u2z&, but with a change ofsign compared to the previous situation, sincesp and sqanticommutate, being orthogonal components ofs. This signchange manifests itself as ap phase shift of the neutroninterferogram@4,18#. Observation of thisp phase shift wouldconstitute, to our knowledge, the first direct verificationthe anticommutativity of orthogonal components of the Paspin operator. We emphasize that a polarimetric experimis incapable of detecting a current reversal in the flipper@27#.

Finally, a linear translationdx of one of the spin flippersalong the beam path results in the neutron spending mtime in one spin orientation at the expense of the othwhich changes the original Larmor precession phase dibution according to

DfL52mB0dx/\v, ~5!

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wherem andv are the neutron’s magnetic moment and vlocity, respectively. The translation is responsible forpurely dynamical phase@28#

FD5DfL . ~6!

Unlike the previous situation, the dynamical phase, bedependent onB0 in the Hamiltonian, is generated, whilleaving the spin trajectory on each path, and hence the ogeometric phaseFG , unaltered. A translation of the otheflipper would generate an identical dynamical phase shif

The purpose of this paper is to describe, in some deour neutron-interferometric observation of separate dynacal and geometric phases. Brief accounts of the results ofexperiment have already been reported@29#.

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Experimental setup

The experiment was performed at the beam portC inter-ferometer facility@30# at the University of Missouri ResearcReactor ~MURR! in a Bhabha Atomic Research CentVienna–Missouri collaboration. A schematic of the beastation is shown in Fig. 3. A thermal neutron beam from treactor is incident on a vertically focusing three-crystal prolytic graphite ~002! monochromator. The Bragg angl(uB520.5°) is fixed by the exit tube, and yields a nominamonochromatic beam withl52.349 Å (Dl/l'0.012). A5-cm-thick block of pyrolytic graphite filters the beam re

FIG. 4. Photograph of the single crystal Si skew-symmetricterferometer used in the experiment.

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moving any second-~l/2! and third-order (l/3) neutrons.The neutrons from the monochromator were then polariby reflection from a 50-cm-long, magnetic, Fe-Si multilaysupermirror (uc52uc Ni>10 mrad). The reflected, polarize2-mm-wide beam passes through a 6-mm-high aperturthe entrance to an aluminum box and onto the first bladethe interferometer crystal. The interferometer, a singsilicon crystal with four blades, in skew-symmetric aligment~see Fig. 4!, has been described in several experime@30#. The interferometer~resting in a cradle! and two 3Heproportional detectors are housed in the aluminum box~withclear Plexiglas lid!, providing for an isothermal enclosureThis box is rigidly mounted to an 850-kg black granite slwhich supports a 1.031.231.8 m3 Benelex-70 masonite box~again with Plexiglas lid! enclosing the Al box. The graniteslab rests on four Firestone pneumatic vibration isolatmounted on steel posts embedded in sand enclosures. Hing below the slab is a large Al plate supporting 710 kglead bricks, which are used to lower the center of gravitythe optical table. An all-encompassing Plexiglas greenhoprovides additional environmental isolation against tempeture gradients and microphonics due to air currents.

A uniform vertical (1 z) magnetic guide field is providedover the entire experimental region by a pair of water-cooCu-wire Helmholtz coils wound on horizontal 52-cm squaaluminum formers with a vertical separation of 7 cm~cen-tered in height about the beam path through the cryst!.These are placed around the Al box containing the interometer crystal. To compensate for the return field~along2 z! of the Helmholtz coils, two pairs of small permanemagnets were placed either side of the beam as it enteredexited the Al box. This maintained a uniform guide fie

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FIG. 5. Cross-sectional view of the dual flipper showing the neutron trajectory through the anodized aluminum foil windings of thThe horizontal magnetic field,B0y of the individual coils when added to the vertical environment field,B0z, of the Helmholtz coils resultsin a field directed in theq direction for the first coil and in thep direction for the second directed at2p/4 andp/4 to z, respectively. It isthese directions about which the neutron precesses.

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along thez direction over the entire neutron path. The balayout of the experiment described to this point is identicathat used in a neutron interferometric measurement of mtiphoton exchange amplitudes@31# that had just been completed.

Essential to the operation of the experiment were the sflippers. As already mentioned, spin flipping was achieveda two-stage process, illustrated in Fig. 5. First, theu1z& po-larized neutron state enters a region of magnetic field,&B0q~the vector sum of the guide fieldB0z and the flipper coilfield 2B0y!, oriented at245° in they-z plane perpendiculato the neutrons direction of motionx. The magnitude of thefield in this region is set so that while traversing the coil tneutron spin precesses by an azimuthal anglep, in a planeperpendicular top, the precession axis, becoming orientedthe 2 y direction, half way on its journey tou2z&. Theu2y&-polarized neutron then enters a similar region of mnetic field directed alongp, orthogonal toq, at 145° in they-z plane. The neutron again precesses through an azimuanglep aboutp to theu2z& state. In measuring the geomeric phase, the spin flippers are rotated in a plane perpendlar to z, so thatp and q now lie in they8- z plane, as shownin Fig. 1.

To produce the two precessions responsible for the sflipping, each spin flipper consisted of two back-to-back reangular coils shown in Fig. 5. The two rectangular co25 mm long315 mm high37 mm wide, were placed acrosthe neutron path with the long axis horizontal, so thatneutron beam had a 7-mm path through each of the coils~Win Fig. 5!. The two coils were connected in series, but wewound with opposite orientation producing horizontal ma

FIG. 6. A map of the transverse field componentB0y of dualflipper F1 along the neutron trajectory through the interferometThe circles represent the measured transverse field on the pathe spin flipper. A theoretical prediction is indicated by the linThe squares are the measured field values extending onto the osite interferometer path. The centers of the two beams have asured separation of 22 mm, so there is very little field overbetween the two dual flippers.

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netic fields of the same magnitude but opposite sense.transverse field component of one flipper as measured aeach beam path is shown in Fig. 6. Each coil is wouaround two pegs defining their top and bottom which aattached to a TeCu-145 heat-sink block. An individual cowith self-supporting sides, consisted of seven turns ofmm-wide, 90-mm-thick anodized Al foil. Low-temperatureAl to Cu brazing provided excellent electrical contact btween the coil and the Cu leads. To reduce thermal gradiin the vicinity of the crystal, a thin Cu sheet enclosed tends of the coils. This sheet was clamped between twoplates attached either side of the heat-sink block. The hsink blocks were water cooled by a closed-loop water circwith the temperature of the coils and block monitored amaintained to 0.01 °C accuracy. Under operating conditi~operating current of 7 A!, the water circuit removed aboutW of heat per dual flipper. Each spin flipper was suspenin the long parallel paths of skew-symmetric interferometand rigidly attached to a precision translation-rotatimechanism~built in the Missouri Physics Machine Shop!.High precision was needed in positioning the spin flippersthe nuclear phase shift when rotating the flippers, due tochange in effective length of the neutron path throughvarious materials of the flipper coils and housing, was ordof magnitude larger than the expected geometrical phThe experimental geometry allowed sufficient space to tralate each spin flipper about 10 mm along the beam pathrotate each one to622°. A photograph of the water-coolespin flippers attached to the precision translation-rotatmechanism is shown in Fig. 7.

The spin-flipping efficiency of the flippers could be mesured using an Heusler alloy analyzer crystal placed inundeflected beam beyond the aluminum box. Optimizatof the spin flipper,F1 , in the undeflected arm was achieveby adjusting the resultant magnetic fieldB0 produced by thevertical Helmholtz coil component and the horizontal flippcomponent~each with magnitude'30 G!, in a procedurethat will be explained later.

Experimental strategy

In the neutron interferometer the neutron-counting ratethe recombined beams, detected by the3He countersC2 andC3 are given by

N25N1~a22b2cosDf! ~7!

and

N35N1~a31b3cosDf!, ~8!

whereDf is the phase shift along path II relative to pathandN1 is the counting rate in the monitor counterC1 . Theconstantsa2 , a3 , b2 , andb3 such thata2 /a3'2.5 andb2'b3 , characterize the interferometer. It is found thatN21N3 is independent ofDf, i.e., asDf is varied the neutronsare swapped back and forth betweenC2 andC3 ~as expectedby particle conservation!, showing thatb2 andb3 are equal.

A common practice is to introduce a spin-independphase shifta by inserting a thin aluminum~1.05 mm thick!plate across both beams and rotating it through an angld.

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FIG. 7. Photograph of the precision translation-rotation mechanism~top! and the water-cooled heat-sink blocks~hanging below! con-taining spin-flipping coils. The blocks are then lowered into the long parallel paths of the neutron interferometer so that thesubbeams pass through the Cu windows and Al coils in the lower corner of each block.

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Figure 8 shows an example of just such an interferogrwith a fringe contrast of 64%. The total phase may thenwritten as

Df5a1a01DF tot ,

5a1g, ~9!

mewherea0 is the spin-independent offset phase of the intferometer, andDF tot is the total spin-dependent phase in tpresence of the magnetic field of the Helmholtz coils aspin flippers. The termg5a01DF tot represents the offsephase determined from fitting the interferogram with

N35N1@a31b3cos~a1g!# ~10!

as a function ofa. By switching the spin flippers on and of

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and interleaving data collection, two interferograms arecorded simultaneously. An example of the ‘‘flipper-off’’ an‘‘flipper-on’’ interferograms, with contrasts of 32% an28%, respectively, are shown in Fig. 9. The differencecontrast is mainly due to the change in the neutron’s kinenergy by the amount 2mB0 when being rotated from thespin-up state to the spin-down state, which leads to a slchange in the reflectivity of the Bragg reflections at the ssequent crystal slabs@32#. For the flipper-on case the offsephase is

gon5a01DF toton , ~11!

while for the flipper-off case

goff5a01DF totoff . ~12!

Consequently, the difference

Dg5gon2goff5DF toton2DF tot

off ~13!

corresponds to the total spin-dependent phase shift betwthe flipper-on and flipper-off interferograms. This procedueliminates the effect of time-dependent drifts in the spindependent phasea0 within the interferometer due to various environmental effects. This was found to be less thaper day. The spin-dependent magnetic phase shifts wfound to be very stable.

FIG. 8. Plot of the counts achieved in the3He detectors,C2 andC3, for the empty interferometer, as a function of the spindependent phasea, achieved by rotating the Al phase flag acroboth subbeams by an angled.

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The experiment was performed in three parts. In the fiinstance, as already mentioned, a pure geometric phaseis achieved by a relative rotationDb of the spin flippers, oneby an angle1Db/2 and the other by2Db/2. Interference isthen observed for a range of anglesDb/2 from 220° to120°. In the second case, a geometric phase ofp can beaffected without any physical motion of the flipper. With thspin flippers normal to the beam path, the current to eacturn is reversed by the mere flick of a switch. This corrsponds to rotating the spin flipper through 180°. A flippeon–flipper-off pair of interferograms was made for each pmutation of current directions. Finally, for a measurementa purely dynamical phase shift, the spin flippers remain ppendicular to the beam (Db/250), and first one and then thother is translated in 2-mm steps for'10 mm along the re-spective subbeam neutron paths. Interleaved flipper-flipper-off interferograms are again made at each of thpositions.

III. EXPERIMENTAL RESULTS

The following experimental procedure was used for eapart of the experiment: At each experimental setting a phrotator scan is made with the flippers sequentially switchon and off every 60 000 monitor counts~5 s! for a total of53106 monitor counts per point~7.5 min!. A single inter-ferogram pair took 8 h toperform. The on-off period is muchshorter than the thermal time constants of the dual-flipheat sinks. By recording two interferograms simultaneoua difference measurement could be made that eliminatedspin-independent nuclear phase shifts due to the heatinthe flipper materials and thermal gradients inside the inferometer. An example of the flipper-on and flipper-off iterferogram pair is shown in Fig. 9. The phase differenbetween these two is the spin-dependent phase shift of inest. The difference phasesDg5gon2goff for a series of runsare then plotted after requiring that all the flipper-off inteferograms overlap at the same offset value. The offset vaused as the constant reference corresponded to the exmental conditionsDb/250, F1 at x1539 mm andF2 at x2539 mm from the first blade of the interferometer.

To measure the geometric phase, the spin flippers windividually rotated in opposite directions, to1Db/2 and2Db/2, through the angular rangeDb/25220° to120° in5° steps. This was done so that the nuclear phase shiftquired in passing through the windings of the coils cancenearly identically. Rotation of the dual flipper increased tpath length byW/cos(Db/2), necessitating the appropriareduction in field to maintain precise spin flipping. For eample at 20°,B0 should be reduced by 4%. However, in thexperiment we keptB0 fixed, and later corrected the phasshift for excess spin flip. This correction ranged betweenand 8°. Figure 10 displays examples of the flipper-on gmetric phase interferograms which have all been shifted rtive to the concurrently recorded flipper-off interferogramoverlap. The phase advance of the interferogram is cleseen. The lines are fits based on Eq.~10!. A summary of theentire geometric phase data set is shown in Fig. 11, ploagainstDb, the angle between the two flippers. The actuon-off phase difference is shown on one ordinate, andgeometric phase shift~relative to the reference condition! on

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FIG. 9. Plots of the nuclear neutron interferograms achieved by rotating the Al phase flag with the spin-flipping coils in placinterferometer for the two cases of spin flipper off and on, i.e., when a current is not applied to the coils and when one is. Thdifference between these two interferograms isDg, used to measure the dynamical and geometric phases.

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With the spin flippers oriented perpendicular to thespective subbeams, another measure of the geometric pcould be affected without mechanical rotation, simply byversing the current applied to each dual flipper in turn. Sua current reversal reverses the magnetic field direction ofcoil, and is equivalent to rotating the individual flipper b180°. A plot of the~shifted relative to the flipper-off interferograms! flipper-on interferograms for the four possibcombinations of the flipper fields is shown in Fig. 12. Itseen that each current reversal shifts the interferogram bp.The lines are fits again based on Eq.~10!, and the values ofthe fitted phasesg are shown in the corner of each plot. Thaverage phase shift from the top plot to the bottom182.3°62.4°, and confirms the anticommutativity ofsp andsq . This phase shift is of purely geometric origin, and costitutes the first direct verification of Pauli anticommutatio

To achieve a measure of the dynamical phase, the flip~oriented perpendicular to the beam! are translated in 2-mmsteps on path I and then on path II, with interferograms mat each step. A plot of the dynamical phase advance of aof shifted flipper-on interferograms for each path is shownFig. 13. Summaries of these phase shifts are shown in F14 and 15. The line represents a theoretical fit, as will nbe explained.

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Data analysis

In an ideal experiment, the neutron beam incident oninterferometer would be 100% polarized in theu1z& state,and the action of the spin flipper would have this staflipped perfectly to theu2z& state. Unfortunately, in this experiment, neither the supermirror nor either dual flipper w100% efficient. This meant that the final interferencecluded neutrons initially distributed in the two spin statesome of which were then flipped by the action of the spflipper and some that were not. This added greatly tocomplexity of the data analysis.

First, consider the distribution of neutron intensities. Dfining P as the polarization of the beam incident from tpolarizing supermirror, then

P5I ↑

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0 , ~14!

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0 are the intensities of spin-up and spin-downeutrons after the mirror, and the total incident intensityI 05I ↑

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maxima ~perfect spin flipping! and minima ~no spin flip-ping!, respectively, of the data in Fig. 16~a!, i.e., the neutronintensity measured by the Heusler polarization analyz

a

n

mity

ee

ath

eu-

eu-henotes

re

thrr


crystal as a function of the field strength of the spin flippF1 . These intensities giveP50.85.

Similarly, the combined efficiency of the two coils offlipper, f , may be defined by

FIG. 10. A series of the nuclear interferograms showingadvance of the geometric interference phase over the range oftive angular separationsDb of the two dual flippers. The angulaseparation is achieved by physically rotating the coil in path I toangle1Db/2, and one in path II to2Db/2.

er

f 5I ↑

f 2I ↓f

I ↑f 1I ↓

f , ~15!

whereI ↑f and I ↓

f are the intensities of spin-up and spin-downeutrons detected. The value off , from the actual measuredvalues of the spin flipper and Helmholtz fields, differed fro1 by at most 1% in the dynamical data due to nonuniformin the Helmholtz field as the flipper was translated~see Fig.17!. In the geometric data,f varied by as much as 8% at thextreme setting,Db/2520°. This was a consequence of thdual flipper’s magnetic field not being scaled as the plength increased through the flipper byW/cos(Db/2) as theflipper was rotated. The resultant number of spin-down ntrons detected is given by

I ↓5S f 11

2 D I ↑01S 12 f

2 D I ↓0

5S f 11

2 D S P11

2 D I 01S 12 f

2 D S 12P

2 D I 0 , ~16!

In this expression the first term represents those spin-up ntrons incident on the spin flipper that are flipped, and tsecond those incident spin down neutrons that areflipped. Accounting for the efficiency of the two parts of thindividual dual flipper byf 1 and f 2 , this expression become

I ↓5F S f 111

2 D S f 211

2 D G1/2S P11

2 D I 0

1F S 12 f 1

2 D S 12 f 2

2 D G1/2S 12P

2 D I 0 . ~17!

The same is true for the spin-up neutrons detected, whe

I ↑5F S 12 f 1

2 D S 12 f 2

2 D G1/2S P11

2 D I 0

1F S f 111

2 D S f 211

2 D G1/2S 12P

2 D I 0 . ~18!

eela-

an

in flippers
FIG. 11. Plot of the on-off phase difference for the entire geometric phase data set, as a function of the angle between the spachieved by physically rotating the two spin flippers. The right abscissa shows the equivalent value of the geometric phaseFG . The linerepresents a theoretical fit, as explained in the text.

to

e

ionis

was, itfairnd

re-A

inthe

. 6.coilsi-airnd

in-

in-r-off

rsto


FIG. 12. Series of interferograms showing the 182.3°62.4°phase shift as the current is reversed to each of the dual flippeturn. Thisp phase shift accurately verifies Pauli anticommutationwithin about 1.3%.

ach

The experimental optimization of the current appliedspin flipperF1 ~in the direct beam!, and the current in theHelmholtz coils to flip an incident spin-up neutron to thspin-down state is shown in Figs. 16~a! and 16~c!, respec-tively. The maxima of these plots establishes the conditf 151, perfect spin flipping. The currents appropriate for thcondition wereI H51.549 A andI F1

57.357 A. We were un-able to perform the same optimization forF2 , in the twice-deflected subbeam on path II, as the neutron intensitytoo low to give a meaningful measurement. Consequentlywas assumed that the two dual flippers were identical, aassumption given the uniformity of materials, structure, aproduction.

In the experiment, the currents used wereI H51.55 A andI F1

57.3 A. These currents generated the magnetic fieldsponsible for flipping the neutron spin. A current of 7.3generated a measured magnetic-field strength of 29.9 Gthe upstream coil and 28.4 G in the downstream coil ofdual flipperF1 . A plot of the measured~points! and theoret-ical ~line! field strengths due to the coils in spin flipperF1 asa function of position along the beam path is shown in FigIn F2 , the measured fields were 28.5 G in the upstreamand 29.6 G in the downstream coil. At the reference potions of the dual flippers, the vertical field strength of the pof Helmholtz coils and return field magnets was 30.2 a29.6 G for the upstream and downstream coils ofF1 , respec-tively. The corresponding values forF2 were 30.8 and 29.9G. A field map along the neutron trajectory through theterferometer is shown in Fig. 17.

The experiment measures the difference in the spdependent offset phase between the flipper-on and flippeinterferograms, as determined by a least-squares fit at esetting, namely,

Dg5DF toton2DF tot

off . ~19!

in

f relative
FIG. 13. A series of nuclear interferograms showing the advance of the dynamical interference phase over the range otranslations~from the first interferometer blade! of the dual flippers in the two paths of the interferometer.

esents a


FIG. 14. Plot of the on-off phase difference as a function of translation ofF1 along path I, measured relative to the reference position~39mm! from the first blade of the interferometer. The right axis shows the equivalent value of the dynamical phase. The line reprtheoretical fit, as explained in the text.

n,-

dpin

pi

ter-r-onasee of

by

i-e thent

To first consider the effect of imperfect initial polarizatioset f 51 ~perfect spin flipping!. For spin-up incident neutrons, the interferogram detected inC3 has the general form

N3↑5S P11

2 DN1$a31b3cos~a1a01DF!%, ~20!

where (a1a0) and DF are the total spin-independent antotal spin-dependent phases respectively. Similarly, for sdown neutrons,

N3↓5S 12P

2 DN1$a31b3cos~a1a02DF!%. ~21!

The total interferogram is then the sum due to the two scomponents,

N35N3↑1N3↓5N1$a31b3@P sin~a1a0!sinDF

1cos~a1a0!cosDF#%. ~22!

-

n

This interferogram is of the form

N35N1$a31b38cos~a1a01DF tot!%

5N1$a31b38@cos~a1a0!cosDF tot

2sin~a1a0!sinDF tot#%, ~23!

whereDF tot is the observed spin-dependent phase of inest, that is, the spin-dependent offset phase in the flippeand flipper-off interferograms. Hence, the measured phDF tot is dependent upon the actual spin-dependent phasinterestDF by the relation

tanDF tot5P tanDF. ~24!

The observed amplitude and actual amplitude are related

b3825b3

2@P2sin2DF1cos2DF#. ~25!

Including the effects of imperfect spin flipping complcates these expressions considerably. We now summarizcalculation of the combined corrections of imperfect incide

ionresents a

FIG. 15. Plot of the on-off phase difference as a function of translation ofF2 along path II, measured relative to the reference posit~39 mm! from the first blade of the interferometer. The right axis shows the equivalent value of the dynamical phase. The line reptheoretical fit, as explained in the text.

current,l

of

entual-


FIG. 16. Plot of the intensity of spin-down component of the beam incident on the Heusler analyzer crystal as a function ofapplied~proportional to the horizontal magnetic-field componentB0y8! to the dual flipperF1 , when oriented~a! normal to the neutron beamand~b! with the flipper oriented at 20° from normal incidence. The current corresponding to the maximum in~b!, should, by geometry, equaI MAX 0° /cos(20°). Both of these measurements were made for a Helmholtz coil current of 1.55 A (B0z'30 G). A fit to ~a! was used todetermine the (2mW/\v) factor used to scale the action of the dual flipperF1 , as described in the text. Plot of the analyzed intensityspin-down neutrons as a function of current applied~proportional to the vertical field componentB0z! to the Helmholtz coils,~c! whenF1

is oriented normal to the incident neutron beam, and~d! with the flipper oriented at 20° from normal incidence. Again, the currcorresponding to the maximum in~d!, should, by geometry, equalI MAX 0° /cos(20°). Both of these measurements were made for a dflipper current of 7.3 A (B0y8'30 G). The solid lines to the data represent fits, as explained in the text.

rs

tetin

alshift

n

beam polarization and flipping efficiencies of the flippeThe detailed derivations are given in the Appendix.

To obtain the geometric phase from the measured inferogram phases, we must separately consider the frac(11P)/2 of spin-up incident neutrons and the fractio

enine

cch

n-

.

r-on

(12P)/2 of spin-down incident neutrons. When the duflippers are switched on, the total spin-dependent phaseof the interferogram as a function ofDb, namely,@DF tot

on(Db)#, for spin-up incident neutrons and spin-dowincident neutrons is given by

tan@DF toton~Db!#5P

A~12 f I!~12 f II !sinDF↑↑on~Db!1A~11 f I!~11 f II !sinDF↑↓

on~Db!

A~12 f I!~12 f II !cosDF↑↑on~Db!1A~11 f I!~11 f II !cosDF↑↓

on~Db!. ~26!

pinor-uidetal

sion

~This expression is derived in the Appendix.! In this expres-sion, DF↑↑

on(Db) @52DF↓↓on(Db)# and DF↑↓

on(Db)@52DF↓↑

on(Db)# are the spin-dependent phases for incidspin-up neutrons remaining unflipped and those beflipped, respectively, by the action of the dual flippers, givby Eqs.~A22! and~A23! in the Appendix. The flipping effi-ciencies of the dual flippers are denotedf I and f II for paths Iand II, respectively. The spin-dependent phase differenfor path II minus path I for incident spin-up neutrons whiare not spin flipped isDF↑↑ ; for those spin flipped it isDF↑↓ . Similarly, the phase differences for incident spidown neutrons areDF↓↓ andDF↓↑ .

tgn

es

In the case where there is no current applied to the sflippers, the expression is much simpler. This situation cresponds to the neutron precessing about the vertical gfield while traversing the interferometer, so that the tospin-dependent phase is

tanDF totoff~Db!5P tan~DfA1B

0 1DfC1D0 !. ~27!

@see Eq.~A31! in the Appendix#, whereDfA1B0 is the dy-

namical phase accumulated on path II due to the preces

lues


FIG. 17. Map of the vertical field componentBz due to the Helmholtz coils for all positions inside the interferometer. These field vaare used in the calculation of the spin-flipping efficiency of the two dual flippers.

sheat-icem

t

am

thsin

thts,p

ctsd-ey

will

ateterter

t. Ais

-the

these

r is

perh a

Iinesare

about the vertical Helmholtz coil field, with the flipperturned off, from the first blade of the interferometer to tmiddle of the dual flipper minus the same phase accumulon path I. Similarly,DfC1D

0 is the dynamical phase accumulated on path II due to the precession about the vertHelmholtz coil field, with the flippers turned off, from thmiddle of the dual flipper to the last blade of the interferoeter minus the same phase accumulated on path I. Inexperiment the data actually measured isDg~Db!, the mea-sured phase difference between the on and off interferogras a function ofDb, given by Eq.~19!.

As explained in the Appendix, the constant terms inphase determination were experimentally determined uthe optimization curve of Fig. 16~a!, which was fit as a func-tion of the applied current. Having scaled the action ofspin flipper, by these experimentally measured constanfit was made to the geometric phase data of Fig. 11. A simlinear regression to the data, has a slope of21.3660.04.After correction for incident polarization and nonperfespin-flipping using Eqs.~26! and ~27!, the slope become21.2360.03. The line in Fig. 11 represents this fit. Accoring to Eq.~2!, FG5Db; then, theoretically, the slope of thdata should equal21. However, in the fitting procedure, an

ed

al

-he

s

eg

ea

le

dynamical phase contamination appearing in the resultsbe accounted for in the slope~see the Appendix!. The flippertranslation-rotation mechanism did not allow for an accurpositioning of the rotation axis of each flipper over the cenline of the subbeam. Consequently, rotation of an off-cenflipper would be accompanied by a displacementdx alongthe beam and therefore a dynamical phase componencombined off-centeredness for the two flippers of 0.8 mmsufficient to generate aFD to account for the observed deviation from the expected slope of the data. Note thatgeometric phase measurements achieved by reversingcurrent to the flippers are free from any dynamical phacomponent.

In the dynamical phase measurement, the spin flippeset perpendicular to the incident beam (Db50), and thespin flipper is translated along each beam. As the spin-flipis translated inside the interferometer, it moves througregion of varying magnetic-field strengthB0z, as shown inFig. 17. HereDg is fit as a function ofdx ~as explained in theAppendix! for the two cases of a translation along pathsand II. Results of the fits to the two cases are shown as lin Figs. 14 and 15. The slopes of these two data sets219.4°60.5°/mm and 18.9°60.4°/mm, respectively.

on of the


FIG. 18. Figure showing the four sections of the neutron subbeam path through the interferometer used in the determinatispin-dependent geometric and dynamical phases.

trotndcts,ietro

rst

eothu

ti-in

thata

o0rt

x-anis

ethe

ngsub-r isst,e

a

e

ra-

-as

es-

a-

ra-r-s ise.,

These slopes correspond to field strengths of 31.060.7 and30.360.6 G, respectively, as calculated from Eq.~5!, whichagree to within a few percent with the measured averages~asseen in Fig. 17! over these regions of 29.960.1 and 30.360.1 G.

IV. CONCLUSION

We have performed a polarized neutron interferomeexperiment where, by means of dc spin flippers in bbeams, geometric and dynamical phase shifts could be ividually controlled and observed. This experiment affea clean separation of geometric and dynamical phaseconsequence of the spinor phase dependence on the ortion of the precession axis. The dynamical phase shifproportional to the difference in path length that the neutpassed in a spin-up state and then in a spin-down state intwo arms of the interferometer. A pure dynamical phaseproduced by a relative translation of the two dual flippeThe geometric phase has a true geometric expression inexperimental layout, as it corresponds to the angle betwthe magnetic fields of the spin flippers. A reversalthe current applied to a flipper, equivalent to changingangle between the two flippers by 180°, generates a pgeometric phase ofp, and its observation confirms the ancommutativity of orthogonal components of the Pauli spoperators.

ACKNOWLEDGMENTS

The success of this work depended greatly uponskilled workmanship of Clifford Holmes and his staffthe Physics Machine Shop at Missouri. This work wcarried out under the Austrian Funds~FWF! ProjectNo. P9266-PHY, the NSF–Physics Division, Grant N9603559, and a UM Research Board Grant No. RB-94-0One of us ~A.G.W.! wishes to acknowledge support fotwo months from the Austrian Funds, and for one monfrom the University of Missouri–Columbia during the eperiment. B.E.A. would like to acknowledge support fromAustralian Research Council grant during the writing of thpaper.

yhi-

sa

nta-isntheis.heenfere

e

s

.3.

h

APPENDIX: CORRECTIONS DUE TO THECOMBINED EFFECTS OF NONIDEAL BEAMPOLARIZATION AND FLIPPING EFFICIENCY

In this appendix we give a derivation of the formulas wused to correct the measured interferogram phases forcombined effects of imperfect polarization and spin flippito obtain the geometric and dynamical phases. On eachbeam I and II, the neutron path through the interferometebroken into four sections, as shown in Fig. 18. The fircalled A, from the first blade of the interferometer to thfront edge of the spin flipper, corresponds to a lengthLA ; thesecond,B, the first coil of the dual flipper, corresponds tolengthLB ; the third,C, the second coil, to a lengthLC ; andthe fourth,D, from the back edge of the spin flipper to thlast blade of the interferometer, to a lengthLD . It is possibleto write the effect on the spinor neutron wave function tversing each of these regions in terms of a transfer matrixM .For regionsA andD, in the presence of the vertical Helmholtz field only, such a transfer matrix is diagonal and hthe form

M5Feif 0

0 e2 ifG , ~A1!

wheref is the dynamical phase accumulated due to precsion in the vertical field. In the spin-flipping regions ofB andC, with both horizontal and vertical fields, the transfer mtrix is

M5Fcosf1 i sinf cosjsinf sinjeiDb/2

2sinf sinje2 iDb/2

cosf2 i sinf cosj G . ~A2!

The wave function for a spin-up incident neutron after tversing the four sections on either path I or II of the inteferometer under the action of the spatially dependent fieldgiven by the product of the four transfer matrices, i.MDMCMBMA . Writing out the matrix multiplications, wehave


c5eik0xeiaH FcosfBcosfC2sinfBcosjBsinfCcosjC2sinfBsinjBsinfCsinjC

1 i ~sinfBcosjBcosfC1sinfCcosjCcosfB! Gei ~fA1fD!u↑&

1F ~2cosfBsinfCsinjC2cosfCsinfBsinjB!

1 i ~2sinfBcosjBsinfCsinjC1sinfBsinjBsinfCcosjC!Ge2 iDb/2ei ~fA2fD!u↓&J , ~A3!

in

es

o

rn

,

-eex-

eenis

eachaseI,

II.

is

edre-

where

fA5mBZLA

\v, fB5

mBLB

\v,

fC5mBLc

\v, fD5

mBZLD

\v. ~A4!

The neutron’s magnetic moment and velocity arem and v,respectively. The strength of the magnetic field in the spflipper region is given by

B5ABz21By8

2 , ~A5!

the vector sum of the vertical Helmholtz fieldBZ and thetransverse horizontal flipper coil fieldBy8 . The precessionaxis p or q of a single spin-flipper coil is at an anglej to zsuch that cosj5BZ /B and sinj5By8 /B. Finally we will defineDfA to be the difference phase shift due to Larmor precsion about the guide field in regionA on the two subbeampaths I and II, so thatDfA5fA

II2fAI . Similarly DfB ,

DfC , and DfD are the difference phase shifts in eachregionsB, C, and D respectively,DfB and DfC being ameasure of the phase associated with the spin-flippinggions, and withDfD5fD

II 2fDI the difference in precessio

phase accumulated in regionsD of paths I and II.Equation~A3! may then be written as

c II5eik0xIIeia IIF S 12 f II

2 D 1/2

eix↑IIei ~fA1fD!IIu↑&

2S f II11

2 D 1/2

eix↓IIei ~fA2fD!IIe2 iDb IIu↓&G ~A6!

for path II, and

c I5eik0xIeia IF S 12 f I

2 D 1/2

eix↑Iei ~fA1fD!Iu↑&

2S f I11

2 D 1/2

eix↓Iei ~fA2fD!Ie2 iDb Iu↓&G ~A7!

for path I. Heref I is the flipping efficiency defined aboveand can be written as

-

-

f

e-

f I5F I ↑f 2I ↓

f

I ↑f 1I ↓

f GI

5@bBbCsin2fBsin2fC

2bBaBbCaC~12cos2fB!~12cos2fC!

2~aB21bB

2cos2fB!~aC2 1bC

2 cos2fC!# I , ~A8!

where aB[cosjB and bB[sinjB for the first coil, andaC[cosjC and bC[sinjC for the second coil of the dual flipper, and the bracket@ # I means that the quantities inside thbracket are evaluated along subbeam I. An equivalentpression applies to subbeam II.

The two wave functions in Eqs.~A6! and~A7! produce aninterference pattern as a function of the nuclear phasea,wherea represents the difference in nuclear phase betwpath II and I achieved by rotating the Al phase plate, andgiven bya5a II2a I . In these expressionsk0x refers to thestandard spin-independent phase accumulated alongpath, which is responsible for the interferometer offset pha05k0xII2k0xI . Traversing the dual spin flipper in paththe neutron accumulates a phasex I↑ if it is not flipped, andx I↓ if it is spin flipped. Similar expressions apply to pathThe desired geometric phase appears in the termDb5Db II2Db I .

In either path, spin flipping only occurs when a currentapplied to the spin flipper. In this case~for incident spin-upneutrons!,

eix↑on

5~cosfB1 iaBsinfB!~cosfC1 iaCsinfC!

2bBbCsinfBsinfC ~A9!

and

eix↓on

52bCsinfC~cosfB1 iaBsinfB!

2bBsinfB~cosfC2 iaCsinfC! ~A10!

are the terms in the wave function for the non-spin-flippand spin-flipped neutrons, respectively, after traversinggionsB andC, the spin flippers. Therefore

tanx↑on~Db!5

aCsinfC~Db!cosfB~Db!1aBcosfC~Db!sinfB~Db!

cosfC~Db!cosfB~Db!2aCaBsinfC~Db!sinfB~Db!2bCbBsinfC~Db!sinfB~Db!~A11!

and

tanx↓on~Db!5

~aBbC2aCbB!sinfC~Db!sinfB~Db!

bBcosfC~Db!sinfB~Db!1bCsinfC~Db!cosfB~Db!. ~A12!

el

ifts


The terms that make up this expression~which we willevaluate later! are based on the measured magnetic-fistrengths, so the phase shiftsx↑

on(Db) andx↓on(Db) can be

determined.The general intensity distribution detected inC3 for inci-

dent spin-up neutrons is

N3↑↑5F S 12 f I

2 D S 12 f II

2 D G1/2S P11

2 D3N1$a31b3cos~a1a01DF↑↑

on!% ~A13!

for the nonflipped neutrons and

N3↑↓5F S 11 f I

2 D S 11 f II

2 D G1/2S P11

2 D

d3N1$a31b3cos~a1a01DF↑↓

on!% ~A14!

for flipped neutrons. The total spin-dependent phase share given by

DF↑↑on~Db!5DfA~Db!1DfD~Db!1Dx↑

on~Db!~A15!

and

DF↑↓on~Db!5DfA~Db!2DfD~Db!2Db1Dx↓

on~Db!,~A16!

where Dx↑on5DxII↑

on2DxII↑on and Dx↓

on5DxII↓on2DxI↓

on. The re-sultant intensity for spin-up incident neutrons is then

y

N3up5N3↑↑1N3↑↓

5S P11

2 DN1H F S 12 f I

2 D S 12 f II

2 D G1/2

$a31b3@cos~a1a0!cos~DF↑↑on!2sin~a1a0!sin~DF↑↑

on!#%

1F S f I11

2 D S f II11

2 D G1/2

$a31b3@cos~a1a0!cos~DF↑↓on!2sin~a1a0!sin~DF↑↓

on!#%J . ~A17!

This may be written as

N3up5S P11

2 DN1$a3up1b3upcos~a1a01DFupon!%

5S P11

2 DN1$a3up1b3up@cos~a1a0!cosDFupon2sin~a1a0!sinDFup

on#%, ~A18!

whereDFupon is the observed total spin-dependent phase for spin-up incident neutrons, such that

tan DFupon5

A~12 f I!~12 f II !sinDF↑↑on1A~11 f I!~11 f II !sinDF↑↓

on

A~12 f I!~12 f II !cosDF↑↑on1A~11 f I!~11 f II !cosDF↑↓

on. ~A19!

All of the above equations apply to the fraction (11P)/2 of the incident beam that is spin-up.We must now add to these results the intensities for the fraction (12P)/2 of the incident beam that is spin-down. B

symmetry, it is straightforward to do this. We have

N3↓↓5F S 12 f I

2 D S 12 f II

2 D G1/2S 12P

2 DN1$a31b3cos~a1a01DF↓↓on!% ~A20!

for the nonflipped neutrons and

N3↓↑5F S f I11

2 D S f II11

2 D G1/2S 12P

2 DN1$a31b3cos~a1a01DF↓↑on!% ~A21!

for flipped neutrons. By symmetry, the spin-dependent phase shifts are given by

DF↓↓on52DfA2DFD2Dx↓

on52DF↑↑on ~A22!

and

DF↓↑on52DfA1DfD1Db2Dx↓

on52DF↑↓on , ~A23!

the negative of the spin-up case. The resultant intensity is then

,


N3down5N3↓↓1N3↓↑

5S 12P

2 DN1H F S 12 f I

2 D S 12 f II

2 D G1/2

$a31b3@cos~a1a0!cos~DF↑↑on!1sin~a1a0!sin~DF↑↑

on!#%

1F S f I11

2 D S f II11

2 D G1/2

$a31b3@cos~a1a0!cos~DF↑↓on!1sin~a1a0!sin~DF↑↓

on!#%J ,

~A24!

where this expression has been written in terms of the spin-up phases. This is in the form

N3down5S 12P

2 DN1$a3down1b3downcos~a1a01DFdownon !%, ~A25!

whereDFdownon is again the observed total spin-dependent phase for spin-down incident neutrons, such that

tanDFdownon 5

2A~12 f I!~12 f II !sinDF↑↑on2A~11 f I!~11 f II !sinDF↑↓

on


on52tanDFup

on5tan~2DFupon!, ~A26!

i.e.,DFdownon 52DFup

on, as expected. Therefore, from Eqs.~A17! and~A24!, the observed total spin-dependent phase shiftDF toton

of an interferogram, with a fraction (11P)/2 spin-up incident neutrons and a fraction (12P)/2 spin-down incident neutronsis given by the spin-dependent phasesDF↑↑

on andDF↑↓on of interest by

tanDF toton5P

A~12 f I!~12 f II !sinDF↑↑on1A~11 f I!~11 f II !sinDF↑↓

on


on. ~A27!

ere

at

t

ad

de.

am

as

con-

in

For the case when there is no current to the spin flippthe expression is much simpler. This corresponds to the ntron precessing about the guide field along the entire pthat is

DFupoff52DFdown

off 5DfA1B0 1DfC1D

0 , ~A28!

where the phase accumulated by the spin-up neutrons isnegative of that accumulated by spin-down neutrons:

DfA1B0 5DfA~Db/2!1DfB

off~Db/2! ~A29!

is the phase accumulated from the first interferometer blto the middle of the spin-flipper, and, similarly,

DfC1D0 5DfD~Db/2!1DfC

off~Db/2! ~A30!

is the phase for the remainder of the path to the last blaDfA

0 and DfD0 , except for small changes due to off-cent

rotation as a function ofDb/2, are constant. Again, from Eq~24!,

tanDF totoff5P tan~DfA

01DfD0 !. ~A31!

The difference betweenDF toton obtained from Eq.~A27! and

DF totoff , obtained from Eq.~A31!, is Dg of Eq. ~19!, the mea-

sured phase difference between the on and off interferogrto be fit as a function ofDb.

Geometric phase

The procedure goes as follows: for the geometric phmeasurements, the expressions

s,u-h,

he

e

e.r

s

e

DF↑↑on~Db!5DfA~Db!1DfD~Db!1Dx↑

on~Db!

5DfA02DfB

off~Db!1DfD0 2DfC

off~Db!

1Dx↑on~Db! ~A32!

and

DF↑↓on~Db!5DfA~Db!2DfD~Db!2sDbDb1Dx↓

on~Db!

5DfA02DfB

off~Db!2DfD0 1DfC

off~Db!

2sDbDb1Dx↓on~Db!, ~A33!

representing the spin-dependent phase shifts, containstantsDfA

0, DfD0 , and sDb , and measurablesDx↑

on(Db),Dx↓

on(Db),

DfBoff~Db!52S mBZB

II

\v2

mBZBI

\v DW8 ~A34!

and

DfCoff~Db!52S mBZC

II

\v2

mBZCI

\v DW8, ~A35!

whereW85W/cos(Db/2).The values forfB(Db) and fC(Db), the phase shift

across an individual coil of the dual flipper usedDx↑

on(Db) and Dx↓on(Db) of Eqs. ~A11! and ~A12!, are

given by

fB~Db!52S mBB

\v DW8 ~A36!

c

p

-

of

.

n

on-the


and

fC~Db!52S mBC

\v DW8. ~A37!

From Eq.~16!,

1↓5f 1P11

2I 0 ~A38!

represents the intensity of spin-down neutrons from the indent intensity, wheref 1 , flipper F1’s efficiency, is given byEq. ~A8! which is a function offB andfC . fB andfC , inturn, depend on the total magnetic field@Eq. ~A5!#

B5ABZ21By8

25ABZ

21~gIF1!2, ~A39!

where I F1is the flipper current, andg is a proportionality

constant determined from the measured value of the flipcoil field. Thus

i-

er

f~Db!5S 2mABZ21~gIF1

!2W8

\vD . ~A40!

The value of the constant (2mW/\v), was determined ex-perimentally from a fit to theF1 efficiency scan of Fig. 16~a!plotted as a function ofI F1

. As already explained, optimiza

tion of the flipper efficiency could only be performed forF1 ;the results were assumed to be identical forF2 .

A simple linear regression to the geometric phase dataFig. 11, as a function ofDb, has a slope of21.3660.04. Athree parameter fit forDfA

0, DfD0 , andsDb , based on Eqs

~A27! and ~A31!, which corrects for incident polarizationand nonperfect spin flipping, gives21.2360.03 for theslope of the data,sDb . A fit using these parameters is showin Fig. 11. According to Eq.~2!, FG5Db, then theoreti-cally, sDb51; however, in assuming thatDfA

0 andDfD0 are

constants, any dynamical phase contamination will be ctained insDb . The consequences of this are explained intext.

ns

e

n in Figs.text.

Dynamical phase

For the dynamical phase data as a function of translation along the beam pathdx, the spin-dependent phase shift expressiohave the forms

DF↑↑on~dx!5DfA~dx!1DfD~dx!1Dx↑

on~dx,Db50!

5DfA02DfB

off~dx,Db50!1sdxdx1DfD0 2DfC

off~dx,Db50!2sdxdx1Dx↑on~dx,Db50!

5DfA02DfB

off~dx,Db50!1DfD0 2DfC

off~dx,Db50!1dx↑on~dx,Db50! ~A41!

and

DF↑↓on~dx!5DfA~dx!2DfD~dx!1Dx↓

on~dx,Db50!

5DfA02DfB

off~dx,Db50!1sdxdx2DfD0 1DfC

off~dx,Db50!1sdxdx1Dx↓on~dx,Db50!

5DfA02DfB

off~dx,Db50!2DfD0 1DfC

off~dx,Db50!12sdxdx1Dx↓on~dx,Db50! . ~A42!

Again, DfA0, DfD

0 , andsdx are constants andDx↑on(dx,Db50), Dx↓

on(dx,Db50),

DfBoff~dx,Db50!5FmBB

II~dx!

\v2

mBBI ~dx!

\v GW ~A43!

and

DfCoff~dx,Db50!5FmBC

II ~dx!

\v2

mBCI ~dx!

\v GW ~A44!

are based on measured values. A plot of the Helmholtz coils’ field strengthB0z, as a function of position inside thinterferometer, is shown in Fig. 17. It is these values that are used in the equations forDfB

off, DfCoff, fB, and fC. A

three-parameter fit is made for each of the two cases, translations along paths I and II. Results of these fits are show14 and 15, and givesdxI5219.460.5°/mm andsdxII518.960.4°/mm, respectively. These results are discussed in the

-

.

s,

en,

l

.

by

on

ek,.

-

er,

A.


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observation of geometric and dynamical phases by neutron interferometry

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