# Direct observation of fermion spin superposition by neutron interferometry

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PHYSICAL REVIEW A VOLUME 27, NUMBER S MAY 1983

Direct observation of fermion spin superposition by neutron interferometry

J. Summhammer, G. Badurek, and H. RauchAtominstitut der Osterreichischen Uniuersitaten, Schuttelstrasse 115,A-1020 8'ien, Austria

U. KischkoInstitut Laue-Langeuin, F-38042 Grenoble, France

and Institut fiir Physik, Uniuersitat Dortmund, D 4600-Dortmund, Federa! Republic of Germany

A. ZeilingerDepartment ofPhysics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02I39

(Received 10 November 1982)

The coherent superposition of oppositely polarized neutron beams of equal amplitude re-sults in a final beam polarization perpendicular to the polarization of both initial beams.This polarization can be rotated by purely scalar interaction applied to the beams before su-perposition, which is equivalent to an additional Larmor precession applied to the beamafter superposition. We have directly observed these effects in an experiment performed us-

ing the perfect-crystal neutron interferometer at the high-flux reactor at Grenoble. This pa-per gives the experimental results and discusses their theoretical foundation.

I. INTRODUCTION

Single-crystal interferometers for use withthermal neutrons have opened many possibilitiesduring the course of their refinement. ' The relative-ly large separation of the two coherent partial beamsin the order of several centimeters permits the ma-nipulation of one of the beams without much affect-ing the other. Thus, wherever information can beobtained by the influence of a specimen on the phaseof the neutron wave function, the interferometermay be a convenient tool of research. Today the in-terferometer is routinely used for precision measure-ments of coherent neutron-nucleus scatteringlengths. Measurements on solids, liquids, and gaseshave been performed.

Besides these applications another class of experi-ments has become possible where the quantum-mechanical behavior of the neutron itself is investi-gated. Collela et a/. reported on a measurementshowing the influence of the gravitational term inthe Hamiltonian; Rauch et al. ' and Werner et a1.could independently demonstrate the change of signof the neutron wave function when subjected to 2mrotations. Combined effects of nuclear and magnet-ic phase shifts were observed by Badurek et al.These latter experiments exhibit some of the conse-quences of the fact that the neutron is a fermion of

spin ,, and therefore its wave function is a spinor.These experiments could be performed with an un-polarized incident neutron beam. The experimentpresented here belongs to this category, although itrequires a polarized incident beam. The purpose ofour research was a demonstration of the phenomenaencountered when two coherent neutron beams ofopposite spin eigenstates (of polarization directionparallel and antiparallel to the magnetic guide field)are superposed. Quantutn theory predicts that theresulting beam would not be a mixture as one mightintuitively visualize in a classical picture. Instead,one expects the final polarization vector to lie in aplane perpendicular to the initial polarization direc-tions.

The first suggestion for a similar experiment wasmade, although on the gedanken level, by Wigner inan article on the problem of measurement in quan-tum theory. ' More recently, Eder and Zeilingersuggested its realization ' by neutron inter-ferometry and showed that the particular directionof the polarization vector within the plane men-tioned above can be modified by introducing a scalarphase shift between the two constituent beams.Some preliminary results of the measurements havealready been reported. ' Here a detailed descriptionof the experiment, of the theoretical formalism, andof the presentation of the results is given.

27 2523 1983 The American Physical Society

2524 SUMMHAMMERBADUREK, RAUCH, KISCHKO, AND ZEILINGER

ISO DE'BEAht

EYIAFED (H)BEAhl

Pauli spin matrices, o =(o,o~,o., ). One observesfrom Eq. (1) that the polarization of this beam canbe rotated by a purely scalar interaction. Of course,the conservation law of angular momentum is notviolated, since the total wave function includes thedeviated beam, whose spin is always opposite to thatof the forward beam

NARD fopBEAN

iH)= ,i t, )+,e'"e'~e ' i t, )

FIG. 1. Experimental test of spin superposition: rota-tion of the polarization vectors in the beams leaving theinterferometer by a nuclear phase shift.

II. THEORETICAL CONSIDERATIONS

e ix/2cos

~i&i sin

~t&

2 2"

2

The polarization of the interferometric beams Iand II before superposition is antiparallel and paral-lel to the z direction, respectively,

In order to obtain two beams of opposite spinstates we choose to illuminate the interferometerwith polarized neutrons and to invert the spin stateof one of the coherent beams with respect to the oth-er. In addition, a scalar phase shift can be intro-duced. A sketch of the principles of such an experi-ment is seen in Fig. 1. For our purposes the interac-tion of the beam with the interferometer can be con-sidered to be equivalent to one consisting of semi-transparent mirrors, where each reflection adds aphase factor of exp(i~/2) to the reflected wavefunction. The inversion of the spin state in onebeam is formally a rotation of m rad, ~hereas thescalar phase shift gives the phase factor exp(iX).The phase I is given by the index of refraction n,which for purely nuclear interaction is n =(1V/F. )' -1A, Xb, /2m. Thus g= k(1 n)AD= Xk,b,~, where A, is the neutron wavelength, Xis the number of nuclei per unit volume, b, is thecoherent neutron-nucleus scattering length, k is thevacuum wave number of the neutrons, and ~ is theeffective thickness of the phase-shifter plate. As-suming incident neutrons of the

~t, ) state and a

180' rotation around the y axis, the result of the su-perposition in the forward beam after the inter-ferometer can be written (using the notation of Ref.12) as

~0)= , ~t, )+,e' e '~t, )

=-,'( t, )+-,e") t. &

e'r" X . . Xcos

~t& i sin

~

t&2 2

"2

where oz is the y component of the vector of the

In contrast, the polarization of the beams in forwardand deviated directions is obtained to be

PD= &0~

o~0) =(cosX,sinX, O),

PH & H~

o~

H ) =( cosX, sinX, O) .There is no z component in the polarization vectors,but they have length 1 and point in directions in thex-y plane which depend on the scalar phase shift.This is opposed to the properties of a statistical mix-ture, which would render completely unpolarizedbeams. Thus by means of three-dimensional polari-zation analysis the two cases can be distinguished.

III. PRINCIPLE OF THE EXPERIMENT

The experimental setup is sketched in Fig. 2. Amonochromatic but unpolarized beam propagates inthe y direction and passes the air gap of an elec-tromagnet with prism-shaped poles. Owing to theirmagnetic moment the neutrons of the spin-up state(

~t, ) i experience a different deflection than those

of the spin-down state (~t, )). With strong labora-

tory fields an angular separation in the order ofseveral seconds of are, which is larger than the in-trinsic reflection half-width of the nondispersivemonochromator-interferometer arrangement, can beachieved. ' ' Thus by appropriate adjustment of theinterferometer only one of the polarized subbeamscan be made to fulfill the Bragg condition. The oth-er one passes the first and second slab of the inter-ferometer virtually without any reflection and islost. Thus there are neutrons of only one spin direc-tion for use in the interferometer. The axis ofquantization as defined by the direction of the mag-

27 DIRECT OBSERVATION OF FERMION SPIN SUPERPOSITION. . . 2525

MAGNETICPOLARI

COOLINGWATE

ETECTOR0 BEAM

FIG. 2. Schematic of the experiment for spin superposition.

netic field of the prism is kept over the whole exper-imental arrangement by exposing it to a verticalmagnetic guide field (+z direction). In the firstspacing of the interferometer a relative scalar phaseshift X is introduced between the two coherentbeams I and II by means of a plane slab of Al withparallel faces. In the second spacing the polariza-tion direction of beam I is inverted by a miniatur-ized dc spin flipper. '

In the ideal case the wave functions correspondingto interferometric beams I and II represent the twodifferent spin eigenstates of the neutrons in the mag-netic guide field. Therefore as a result of the super-position in the third slab the wave functions of theforward and deviated beam describe no longer eigen-states in the magnetic field. Consequently, theircorresponding polarization vectors precess with theLarmor frequency around the direction of the guidefield. In the experiment only the polarization of the0 beam was investigated. Following its path theneutrons first pass the accelerator coil which pro-duces a variable magnetic field parallel to the guidefield. The Larmor frequency within the coil can bevaried by changing the current. Thus the polariza-tion vector can be turned by an additional angle aand can be made to assume any desired direction inthe x-y plane. In particular, if the polarization vec-tor points in the y direction at the entrance of them/2-spin turn coil, which turns the vector aroundthe x axis by 90', the final polarization points in z

direction. Finally, the Heusler single-crystalanalyzer reflects only that part of the intensitywhich corresponds to a polarization parallel to +z.So in this case there will be a minimum of the inten-sity behind the analyzer. On the other hand, if thecurrent of the accelerator coil is adjusted in such away that the polarization vector behind the m/2 coilpoints in the +z direction, a maximum of the inten-sity occurs. Therefore one wi11 find a sinusoidaldependence of the intensity on the current of the ac-celerator coil. As can be seen from Eq. (4), the po-larization vector can also be turned by a scalar phaseshift introduced between beams I and II. Thisequivalence of scalar phase shift within and magnet-ic spin-dependent phase shift (Larmor precession)behind the interferometer can only be attributed tocoherent spin superposition. A formal descriptionof the experimental scheme is given in the Appen-dix.

IV. EXPERIMENTAL SETUP

The experiment was performed at the neutron in-terferometer setup at the high-flux reactor in Greno-ble. Its layout is shown in Fig. 2. Differing fromthis figure two magnetic prisms were used in series,each having an air gap of 4.5 mm in height and a re-fractive angle of 120'. At saturation magnetizationfields of (0.8 T were measured in the air gapswhich resulted in a total beam separation of 3.9 sec

SUMMHAMMER, BADUREK, RAUCH, KISCHKO, AND ZEILINGER

of Rrc. Thc mean %8vclcngth of thc lncldcnt beamwas determined Rs A.=1.835(2) A. The guide fieldwas produced by a Helmholtz configuration of 620mm in diameter whose center of symmetry coincid-ed with that of the beans inside thc interferometer.The distance of the second magnetic prism wasabout 350 mm from the interferometer. The totalmagnetic field at the site of the crystal was 4 m T.

The accelerator coil and the m/2 coil weresolenoids of 220-mm length, 60-mm width, and 11-mm effective thickness along the beam trajectory.Thc m/2 coil was tilted by an angle of 27' from thevertical, and the current was tuned so that its fieldtogether with that of the Helmholtz coils gave 8 re-sulting field direction within the x-y plane. Thestray field of the coils was not larger than 0.09 mTanywhere on the neutron beam paths. The irnpor-tant spin AlppcI' wlthln thc interferometer %'Rs ofthe dc type. It was 8 solenoid bent to form a "U,"ensuring the desired field at the site of beam I andnegligible stray field along the path of beam II, al-though the maximum separation of the beams wasonly 20 mm. ' Since a homogeneous ficM over thecross section required beam I to pass through wires,and since the coherence properties, i.e., equal nuclearphase shift over the cross section had to be main-tained, the material of the wire had to have an indexof refraction n =1. This condition was met by ahomogeneous alloy of Nbg gV9g 6. Cooling withtemperature-controlled %'Rtcr was Ilcccssary to dissi-pate the 35 %' of heat produced thus directly insidethe interferometer. As the wire was 1 mm in diame-ter and the thickness of each field region was only 5mm, the condition of equal angle of rotation overthe cross section of the beam could not be fulfilledcompletely. We estimate, that this, Rnd dcvlationsof the alloy from the ideal composition, were themain reasons for the reduction of thc contrast of theintensity oscillations in the 0 beam. Furthermore,different effective lengths of the two parts of thespin Aipper along the neutron path and different ef-fective fields in them gave risc to the incompletespin turn (see the Appendix). The entrance slitplaced in front of the interferometer defined 8 crosssection of the incident beam of 2~4 mm . Owingto the Boririjann fan the beams are widened notice-ably after thc first slab (thickness 4.5 mm), so thatfor reasons of space a spin Aip of a wider beamcould Qot have bccn accompllshcd.

A. Polarization of beams I and II

FGI' mcasuremcnts of thc polarizations of thc indi-vidual beans phase shifter, accelerator coil, and m/2

coil were not present. A Cd beamstop was placed ei-ther in the path of beam I or that of beam II rightbefore the second slab. By turning the interferome-ter around 9 first the subbeam with negative, thenthe one with positive polarization fulfills the Braggcondition. Thus the rocking curve gives two peaksin thc I detector (see Fig. 3). In the 0 detector 8different result is obtained because of the Hcusleranalyzer present. If the neutrons have followed thepath of beam I, either the first or the second peakappears depending on %hether the spin Aipper is onor off. For 8 Qonideal spin Aip a small peak at theposition of the supposedly suppressed one is found.If the neutrons have passed via beam II, the rockingcurves should be the same independent of whetherthe spin Aipper is on or off. The resu1ts in Fig. 3show that its stray fields do only have a small inAu-ence on beam II. %ith the interferometer positionedon thc subbcam with posltlvc polarization foi thccase of "Aipper on" a polarization ofP 1 I';D&FD~ & 0.87 for beaIn I andI'lI =P;B&DpD~ &0.81 for beam II was found.Herc P; =0.95 is the theoretical maximum polariza-tion; its value is due to an overlap of the individua1rocking curves of the incident subbeams. D& is apossible depolarization along the Aight path, whichis small and thus we sct D& 1. The factor DH ac-counts for a quasidepolarization by the Heusleranajyzer as its reflectivity for the

~t, ) state was

& 3% of that of the~

1', ) state and henceD& &0.94. Finally, the value for beam I containsthe efficiency of the spin Aipper F=25 1 whichwas measured as I& 0,98, FroIQ this 8 value of5-0.1 can be extracted [see Eq. (A9)]. For beam IIa factor DF&0.91 describing a small depolarizingeffect of the stray field of the spin Aipper has to beincluded. Considering 811 disturbing inAucnccs ltcould therefore be assumed that there are mainlyneutrons of the

~l, ) state in beam I and of the

~t, ) state in beam II.

8. Spin superposition

Since the objective was to measure the intensity as8 function of the scalar phase shift J and the addi-tional Larmor angle A, thc RQgular position of thcphase plate was incrcmented in steps, at each stepdetermining the count rates in the detectors for 8series of currents in thc accelerator coil. Thesescans were performed for both positive and negativeincident polarization. Typical results can be seen inFig. 4. Thc slnusoldal curves werc obtained apply-ing a least-squares fit to 8 set of data points belong-ing to a fixed current in the accelerator coil (fixed{X). ThC fOAYlula

DIRECT OBSERVATION OF FERMION SPIN SUPERPOSITION. . .

BEAM I BEAM ll

200-

FL!IPER OFF, ,!

c[ {a) 200

FLiPeEe O~~

0, 0]

&00

ALII I ER ON

J'2500

I

a7250 ( )

erat)l

0 5

FIG. 3. Rocking curves of the interferometer crystal to determine the polarization of beams I and II in the interferome-ter. Deviated (H) beam: without polarization analysis two peaks occur, corresponding to the two incident subbeams (e).Principal shape of the curve is independent of whether beam I or beam II is blocked off, or whether the flipper is on or off.Forward (0) beam: only that peak is present which is caused by neutrons of the ! t, }state. Beam I [beam II in the inter-ferometer blocked off, (a) and (b)]: with "flipper off" the incident subbeam with neutrons of the ! t, }state gives the peak,whereas with "flipper on" it is the incident subbeam with neutrons of the ! ), ) state, because the spin is inverted. Beam II[beam I in the interferometer blocked off (c) and (d)]: only the incident subbeam with neutrons of the ! t, }state gives apeak. Action of the spin flipper has no influence.

Ic(~)=T, + T2cos[2n(~/T3+ T4)]was used, optimizing the parameters T, . . . , T&.Here T is the mean intensity. It includes also anincoherent part due to imperfections and tempera-ture gradients in the interferometer crystal as well assome small-angle scattering in the wire of the spinflipper. T2 is the amplitude of the intensity oscilla-tions. The quotient Tq/T gives the interferometriccontrast, which is 1 for the 0 beam in an idealized

experiment, but in our experiment it was only0.0S0,1. T3 denotes the A, thickness of the alumi-num plate that causes a relative phase shift of 2m inbetween beams I and II and which is given asT3 2m/NA, b, . Its value was known from manyprevious experiments on scattering lengths as

T3 164.9 pm, and it could thus serve as a control. ~Here typical values of T3 160+9 pm were ob-tained. T& is the phase which was expected to de-pend linearly on the Larmor angle o,'. The dashed

2528 SUMMHAMMER, BADUREK, RAUCH, KISCHKO, AND ZEILINGER 27

500

350

TABLE I. Comparison of theoretical and experimentaldependence of the absolute phase of the intensity oscilla-tions in Fig. 4 on the Larmor angle a. (Measured valueshave been corrected to give the theoretical phase at a =a..)

o"

V)

500

Theoretical phase (rad)

1.372.453.484.43

Measured phase (rad)

1.80+0.562.53+0.553.53w0.694.06+0.75

350

0

d (rad j1acc

250

dO (ym)~X

500

FIG. 4. Results of spin superposition in forward direc-tion. The phase of the intensity oscillations is a functionof the path difference AD in the Al phase shifter of beamsI and II as well as of the current I, in the acceleratorcoil.

VI. DISCUSSION

The results show that the creation and coherentsuperposition of neutron beams of such differentphysical property as opposite spin can be achievedby interferometry. Repeated experiments havedemonstrated a very good agreement of the observedphenomena with the quantum-theoretical predic-tions, despite an error of the relevant phases (2m. T4)of up to (+0.09)X2m (see error bars in Fig. 4).

line in Fig. 4 indicates this relationship as it wasfound in the experiment. The slope of the line is ingood agreement with the theoretical value computedfrom coil parameters and wavelength A, (see Table I).It could also be observed that the sign of the slopechanges if the incident polarization is reversed withthe experimental procedure otherwise remaining thesame.

From Fig. 4 one notices that the mean intensityT&, too, is a periodic function of the additional Lar-mor angle a. This effect is caused by an incompletespin flip (see the Appendix). That its amplitude islarger than T2 is due to the noninterfering part ofthe intensity.

A further verification, that the observed intensityoscillations in the 0 detector as a function of thephase shift X were indeed due to a rotation of thepolarization vector, was done by switching off them. /2 coil. Then the intensity should not depend ong. Results of this experiment are shown in Fig. 5.

2000

Il/'2- COIL OFFP

1500

~ V~ ~ ~ ~ ~ ~ ~0 4 4 Se e e

~ 1000V)u3

00 2000

(ll

Ul

Z 1500~

I I I I I

100 200 300 400 500

'Ff2- COIL ON P

~ ~e

1000

I I I

0 100 200 300 400 500

z D(pm)

FIG. 5. Spin superposition in forward direction. Inten-sity oscillations as a function of the path difference AD(phase p) are only present when the polarization of the ydirection is turned into the z direction, indicating the in-fluence of the nuclear phase shift on the final polariza-tion.

With an improvement of accuracy experiments willbecome feasible that not only demonstrate this spe-cial case of the superposition principle, but permit amore quantitative analysis as well. In one experi-ment, for example, a partial absorber was placed inone beam path inside the interferometer diminishingthe amplitude of the corresponding spinor. Accord-ing to Ref. 12, by changing the effective absorptionand by varying the nuclear phase shift with thealuminum slab one should be able to produce anydirection in three-dimensional space of the polariza-tion vectors of the outgoing beams, although bothcontrolling interactions are spin independent. Ow-

ing to the rather large experimental error these mea-surements did not render decisive results and will be

DIRECT OBSERVATION OF FERMION SPIN SUPERPOSITION. . .

repeated.An additional effect inherent in our experiment

must be mentioned. As is known from dynamicaldiffraction theory, ' ' the direction of propagationof neutrons inside a single crystal is influencedstrongly by changes of wavelength, if the incidentbeam is near the Bragg condition and if Lauegeometry is used, as in the interferometer. Butneutrons of the same total energy and of oppositespin eigenstates in a constant magnetic field havedifferent momentum and consequently differentwavelength. In our experiment this meant that neu-trons of beam I, which experienced a spin Aip sndthus a change of wavelength, had different directionof propagation in the third crystal slab as comparedto the first and second slabs, while for neutrons of

l

beam II no such change occurred. As s conse-quence, the coherence of beams I and II was re-duced, a small effect that could be observed. Itcould not be completely eliminated, ss a guide fieldwas necessary for the definition of the spin states.

A possibility of circumventing wavelength differ-ences is the use of an rf spin flipper instead of thestatic one. The total energy of the neutrons whosespin is inverted is then given by Ep+fuu~, whereAco~ is the energy of the photons emitted or ab-sorbed by the rf coil. * It corresponds to the ener-gy difference of the Zeeman levels of s neutron in amagnetic field, and thus the kinetic energy remainsthe same. Using calculations given in Ref. 25 thestate of the 0 beam behind the interferometer is ob-tained to be with incident neutrons of the

~

1', }state,

i (XmP)/2~0}=,

~t, }+e're

~t, }=

2

r

X~&t I~&I;cos X~ t} i sin

For a magnetic guide field in the order of 10 mTco~/2m. -10 Hz. Because of the explicit timedependence of the resulting intensity, neutron detec-tion must be synchronized with the phase of the rffield. Such measurements will be perforxned in thenear future. Yet without doubt they will show thesame properties of spin superposition as our resultsfor the dc case.

ACKNO%LEDGMENTS

Fruitful discussions with Professor E. Balcar andProfessor G. Eder (Atominstitut Vienna), Dr. F.Mezei (of the Institute Lsue-Langevin at Grenoble),and Professor S. A. %erncr (of Columbia Universi-ty) are gratefully acknowledged. Thc authors wishalso to thank P. Pataki and K. Zeitler (AtominstitutVienna) and notably G. Schmidt {of the InstitutLsue-Langevin at Grenoblc), for technical assis-tance. This work has been supported financially bythe Austrian Fonds zur Forderung der wissenschaft-lichen Forschung (Project No. 4230) and by theBundes minister fur Forschung und Technologie,Bonn {Forschungsbereich No. 03-41E09P).

APPENDIX: FORMAL TREATMENTOF THE EXPERIMENTAL SCHEME

I

There are no magnetic fields acting on it. Thus aspin-independent momentum in vacuum p=kk isdefined, the total energy of the neutrons beingEp (Ak) /2m (where m is the mass of neutron).The magnetic prism separates the incoming unpolar-ized beam in two subbeams polarized parallel andantiparallel to the z axis, respectively. '

It will be assumed that the interferometer is posi-tioned such that only the beam representing the

~t, } state satisfies the Bragg condition. (Here it is

defined as the spin state reAected by the Heuslercrystal. ) The Schrodinger equation of neutrons in sstatic magnetic field is given by

fP 2V p~.B %=Ep% . (A1)2'

Here per =p is the operator of the magnetic mo-ment of the neutrons. In the case of the guide fields8=8pz. As p8p/Ep was of magnitude 10, forthe wave vectors K+ resulting from Eq. (A1) the ap-proximation

r

P&pK+- k 1+p

can be made. Neglecting an arbitrary phase factorand introducing x':pBpk/2Ep, the spinor wavefunction of the incident beam can bc written as

1. Inrident beam eEK'r

For monochromatization the (220) reflection of aperfect silicon crystal in Bragg setting is used. The polarization of the incident beam is

2530 SUMMHAMMER, BADUREK, RAUCH, KISCHKO, AND ZEILINGER 27

P; =W o~; /W%; =(0,0, + 1).

0 pBfU =exp iek 1

2 2Ep

=exp( i'.a/2) . (A4)

2. Coil operators

Similar to spin-echo systems matrix formalismis chosen to describe the action of the dc coils on theneutron state. A rectangular dc coil represents a re-gion of length 1 with a static homogeneous magneticfield BfBfe along the neutron trajectory. Its ef-fect upon the spinor wave function can therefore becalculated solving Eq. (A1) for the case of generalfield direction. As a result a unitary operator U canbe assigned to a dc coil, when the assumption ofzero reflection at the boundaries of the field regionis used. " +, denoting the spinor before and 4'b be-ing the spinor behind the field one gets

0 b U%.with

77+ 6cos

m+ei sin2

P'+ 6i sin2

m. +ecos

2

(A5)

The accelerator coil can be looked at as a regionwhere the strength of the guide field can be varied.Therefore the corresponding operator has the effectof a rotation around +z by some angle a,

eia/2 0

(A6)e'a

Uacc exp I, Oz'2

spin flipper more accurately, as e is some constantangle caused by inexact coil geometry and malad-justment of dc currents. On the other hand, theconsideration becomes more general, showing that toobserve phenomena of spin superposition it sufficesthat the spins in beam I be not parallel to those inbeam II. Thus the operator of the spin flipper readsas

U,"t =exp[ io(n+e)/2]

Here e=(e,ey, e, ) is the unit vector of the fielddirection. In deriving the above operator the sameapproximation as in Eq. (A2) could be made, as thefield strength inside the dc coils was of equal magni-tude as in the guide field. The product in the ex-ponent of Eq. (A4) was transformed using the I.ar-mor frequency coL (2p/R)Bf, the velocity of theincident neutrons u =6k/m, the interval ~=1/u thetime the neutrons need to traverse the field region,and the definition a:ecoL~. Also it was assumedthat the time of flight ~ in the field is the same forneutrons of both spin states: v.+ 1/u+,

=1/u, and ~+-~ -v. Equation (A4) reflectsthe well-known property that the effect of a magnet-ic field region upon the spinor can be interpreted asthat of a rotation of a system with spin ,. Now theindividual operators as represented by the differentcoils shall be calculated. The field region of the spinflipper consists of two parts. Their combined actioncan be understood as a rotation of m. around someaxis in the x-y plane. ' As long as the axis lieswithin this plane, its exact direction is not relevantfor the purpose of inverting the polarization from+z to z. For the sake of simplicity a rotationaround +x is assumed. But instead of the ideal an-gle of m a more realistic one, namely, m. +e will bechosen. By doing this, not an exact transformationfrom the

~t, ) state to the

~t, ) state is obtained,

but the beam after the spin flipper will be represent-ed by a wave function containing both states. Onthe one hand this describes the action of the real

U~2 exp i o./2

1

vz (A7)

3. Wave function

Using the operators presented above the expres-sions for the intensity can be derived in a straight-forward way. The wave functions of beams I and IIimmediately after the first slab of the interferometerhave the form

eI Ky

+u r 0eiky (A8)

Here the transmission and reflection amplitudes of

It should be noted that the guide field along the neu-tron trajectories, too, could be described by a seriesof rotations around +z. As will be seen later, thiswould only give additional phase factors in the ob-served intensity oscillations which, as constantsthroughout the whole experiment, have no influenceon the result. The ~/2 coil causes a rotation around+x (or some axis in the x-y plane) by an angle ofn./2. The operator is

27 DIRECI' OBSERVATION OF FERMION SPIN SUPERPOSITION. . . 2531

the crystal slab are introduced. They obey the rela-tion

~

r~

2+~

t~

2=1, as there is practically no ab-sorption for thermal neutrons in the Si single crys-tal. ' ' Furthermore, the fact that the wave trainshave different direction inside the interferometer isneglected and the consideration is reduced to one di-mension, which we call the y axis and which isparallel to the incident beam. This can be done be-cause the absolute values of the wave numbers arethe same in the transmitted and reflected directions.Then the nuclear phase shift X of beam II relative tobeam I is caused by the aluminum phase plate. Xcan be varied rotating the plate. At the second slabboth beams experience a reflection. Afterwards, %ris subject to the rotation. The spinor behind thespin flipper becomes

5=cos m+e2

(A9)

5 igeiky

j(1 5 ) / e'"y

Here y,f denotes the entrance boundary from theguide field to the spin-flip field. As an experimentalconstant the associated phase factor can be set equalto 1. 5 is the deviation from an exact spin flip thatvanishes if a=0. Taking now into account the prop-erty that the reflectivity on one side of a losslessmirror is the complex conjugate of the reflectivitywhen reflecting on the other side, the spinors justbefore entering the third crystal slab can be writtenas

ei'

Qrr=r r 0iky +iX (A10)

with

i ( 1 52)1/2&iayNow it may be well to look at the polarization ofthese individual beams,

liir &MrPr

&r%

&rr&~rrPrr=

&rrq'rr

25(1 5 )'/ sin2iry 025(1 5 ) '/2cos2icy ~ 0

5~01+25' 1

0(Al 1)

For an ideal spin flip, opposite polarization of the beams can be produced.After superposition in the third slab one obtains for the wave function in forward direction,

(5+~ lg)e J)cg+0 r +I+r+11 rr r

i (1 52)l/2&lay (A12)

Its polarization is given by

(15 )' [5sin2ay+sin(2' X)]P0 (15 )'/ [5cos2ay+cos(2ay I)](1+5cosX)

5 +5cosg

sin(2~y X)cos(2' g)

5~00

(A13)

If one compares Eq. (Al 1) with Eq. (A13) the very different properties of the wave function after superpositionbecome obvious, especially the influence of the phase shift X on the polarization. The subsequent passage ofthe beam through the accelerator and the n./2 coils and the reflection at the Heusler crystal afford an analysisof this polarization. The wave function behind the n/2 coil is obtained by the transformation

(5+&iX)eiay e -ia/2 (1 52)1/2eiay cia/2% 0~Ua/2( Uacc OI I~- tf r . I c ir

)iry' ia/2 ~ ~ ilc2/ir2y ia/2' (A14)

Here the phase factors exp( iscy'), which are determined by the exact location of the m/2 coil, can be set to 1,since their action can formally be incorporated into that of the accelerator coil.

The Heusler crystal reflects only neutrons of the~t, ) state into the detector with a reflectivity q and can

SUMMHAMMBR, BADURBK, RAUCH, KISCHKO, AND ZBILINUBR

therefore be symbolized by an appropriate projection operator. The final intensity in the detector turns out tobe

Io ,)qtr r~

~e' (5+e' )(152)'~~)2

=~

qtr'r~[1+5cosX5(15 )'~ cosa (152)'~'cos(a g)]

~ ~qtr'r ~~[1cos(a g)] .

Here four characteristic contributions can be dis-tinguished. The first term in square brackets is theaverage intensity. The second term shows oscilla-tions stemming from the superposition of com-ponents of equal spin state of beams I and II; it onlydepends on the nuclear phase shift X. The thirdterm accounts for an incomplete spin flip. Comingonly from beam I it cannot depend on the nuclearphase shift, but it depends on the action of the ac-celerator coil (a). It can be imagined as Larmor pre-

cession of the remaining x-y components in beam I.Both the second and third terms disappear if anideal spin flip can be achieved. The fourth term isthe contribution by the superposition of oppositespin states me are looking for. It exhibits anequivalence of scalar phase shift inside (X) and con-trolled I.armor precession (a) behind the interferom-eter. This feature cannot be explained if strict in-coherence (i.e., mixture) of spin states ~ t, ) and

~t, ) is expected.

~utron Interferometry, edited by U. Bonse and H.Rauch (Clarendon, Oxford, 1979).

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H. Rauch, H. Wilfing, A. Zeilinger, W. Bauspiess, andU. Bonse, Z. Phys. B29, 281 (1978),

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Vienna, 1982 (unpublished).'9U. Bonse and WA Graeff, Top. Appl. Phys. 22, 93

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