Direct observation of fermion spin superposition by neutron interferometry

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<ul><li><p>PHYSICAL REVIEW A VOLUME 27, NUMBER S MAY 1983</p><p>Direct observation of fermion spin superposition by neutron interferometry</p><p>J. Summhammer, G. Badurek, and H. RauchAtominstitut der Osterreichischen Uniuersitaten, Schuttelstrasse 115,A-1020 8'ien, Austria</p><p>U. KischkoInstitut Laue-Langeuin, F-38042 Grenoble, France</p><p>and Institut fiir Physik, Uniuersitat Dortmund, D 4600-Dortmund, Federa! Republic of Germany</p><p>A. ZeilingerDepartment ofPhysics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02I39</p><p>(Received 10 November 1982)</p><p>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-</p><p>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.</p><p>I. INTRODUCTION</p><p>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.</p><p>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</p><p>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.</p><p>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.</p><p>27 2523 1983 The American Physical Society</p></li><li><p>2524 SUMMHAMMERBADUREK, RAUCH, KISCHKO, AND ZEILINGER</p><p>ISO DE'BEAht</p><p>EYIAFED (H)BEAhl</p><p>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</p><p>NARD fopBEAN</p><p>iH)= ,i t, )+,e'"e'~e ' i t, )</p><p>FIG. 1. Experimental test of spin superposition: rota-tion of the polarization vectors in the beams leaving theinterferometer by a nuclear phase shift.</p><p>II. THEORETICAL CONSIDERATIONS</p><p>e ix/2cos</p><p>~i&amp;i sin </p><p>~t&amp;</p><p>2 2"</p><p>2</p><p>The polarization of the interferometric beams Iand II before superposition is antiparallel and paral-lel to the z direction, respectively,</p><p>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</p><p>~t, ) state and a</p><p>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</p><p>~0)= , ~t, )+,e' e '~t, )</p><p>=-,'( t, )+-,e") t. &amp;</p><p>e'r" X . . Xcos</p><p>~t&amp; i sin </p><p>~</p><p>t&amp;2 2</p><p>"2</p><p>where oz is the y component of the vector of the</p><p>In contrast, the polarization of the beams in forwardand deviated directions is obtained to be</p><p>PD= &amp;0~</p><p>o~0) =(cosX,sinX, O),</p><p>PH &amp; H~</p><p>o~</p><p>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.</p><p>III. PRINCIPLE OF THE EXPERIMENT</p><p>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(</p><p>~t, ) i experience a different deflection than those</p><p>of the spin-down state (~t, )). With strong labora-</p><p>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-</p></li><li><p>27 DIRECT OBSERVATION OF FERMION SPIN SUPERPOSITION. . . 2525</p><p>MAGNETICPOLARI</p><p>COOLINGWATE</p><p>ETECTOR0 BEAM</p><p>FIG. 2. Schematic of the experiment for spin superposition.</p><p>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. '</p><p>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</p><p>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.</p><p>IV. EXPERIMENTAL SETUP</p><p>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</p></li><li><p>SUMMHAMMER, BADUREK, RAUCH, KISCHKO, AND ZEILINGER</p><p>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.</p><p>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.</p><p>A. Polarization of beams I and II</p><p>FGI' mcasuremcnts of thc polarizations of thc indi-vidual beans phase shifter, accelerator coil, and m/2</p><p>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&amp;FD~ &amp; 0.87 for beaIn I andI'lI =P;B&amp;DpD~ &amp;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&amp; is apossible depolarization along the Aight path, whichis small and thus we sct D&amp; 1. The factor DH ac-counts for a quasidepolarization by the Heusleranajyzer as its reflectivity for the</p><p>~t, ) state was</p><p>&amp; 3% of that of the~</p><p>1', ) state and henceD&amp; &amp;0.94. Finally, the value for beam I containsthe efficiency of the spin Aipper F=25 1 whichwas measured as I&amp; 0,98, FroIQ this 8 value of5-0.1 can be extracted [see Eq. (A9)]. For beam IIa factor DF&amp;0.91 describing a small depolarizingeffect of the stray field of the spin Aipper has to beincluded. Considering 811 disturbing inAucnc...</p></li></ul>