Polarized neutron interferometry: A survey

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<ul><li><p>Physica B 151 (1988) 82-92 North-Holland, Amsterdam </p><p>POLARIZED NEUTRON INTERFEROMETRY: A SURVEY </p><p>G. BADUREK, H. RAUCH and J. SUMMHAMMER Atominstitut der Osterreichischen Universitdten, A-1020 Vienna, Austria </p><p>A brief summarizing review is given of all those neutron interferometric experiments performed hitherto which explicitly use the spin- particle properties of the neutron. It covers topics as the verification of the 4~-periodicity of spinors, the cooperative action of nuclear phase shift and spin-rotation on the neutron wave function, the demonstration of the quantum mechanical principles of fermion spin-state superposition and the more recent double resonance experiments where the two interfering beams propagate through spatially separated oscillatory magnetic fields. Finally a proposal will be presented also for a so-called "late-choice" experiment with polarized neutrons. </p><p>1. Introduction </p><p>Since its very first beginnings [1] neutron inter- ferometry has been established as an almost ideal tool for experimental tests of the founda- tions of quantum mechanics with massive parti- cles. Among its other intrinsic features it is the half-integer spin of the neutron which has no counterpart in classical angular momentum physics and hence is the most interesting one with respect to such fundamental questions. In the following we will give a short survey over all those experiments performed so far which have explicitly used the spin- properties of the neut- ron. In this context the expression "polarized" neutron interferometry does not necessarily mean that any of the neutron beams involved is actually polarized, but rather denotes the fact that the experiments and their results, respec- tively, rely exclusively on the spinor character of each individual neutron. </p><p>In the following we first describe the nowadays already historic direct experimental confirmation of the 4"rr-periodicity of spinor wave functions by means of neutron interferometry which was real- ized originally with unpolarized beams [2, 3]. For completeness a hitherto unpublished result of a similar experiment performed later on with polarized incident neutrons will be reported, too. </p><p>The next topic we will treat is what happens when both nuclear and magnetic dipole interac- </p><p>tions change the wave function of the subbeams within a neutron interferometer. Among other characteristic features here it turns out that ac- cording to theoretical predictions the final beams leaving the interferometer are at least partially polarized even when the interfering ones are completely unpolarized [4, 5]. </p><p>Another fundamental experiment where a high degree of polarization of the incident beam is essential is the demonstration of the quantum mechanical principles of linear superposition of fermion spin states. Here it can be nicely shown that the superposition of two coherent beams of spin -1 particles with mutually orthogonal spin eigenstates does not lead - as one might visualize in a classical picture - to a mere mixture of these states but results in a new state whose polariza- tion is perpendicular to that of the interfering beams [6]. There the resulting interference pat- tern is significantly different if either static or time-dependent magnetic fields are applied in order to prepare the orthogonal spin states [7]. </p><p>A wealth of interesting situations arises if both interfering beams are influenced by high- frequency magnetic fields. The most outstanding one is that where extremely small energy differ- ences of the order of 10-17eV between the two coherent subbeams are established leading to pronounced quantum beat effects on a time scale of several tens of seconds [8]. </p><p>Finally it will be shown that the spin-state </p><p>0378-4363/88/$03.50 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) </p></li><li><p>G. Badurek et al. / Polarized neutron interferometry 83 </p><p>superposition experiment cited above can be modified to realize a so-called "late-choice" interference experiment. </p><p>To understand the various experiments de- scribed here it is not necessary to go deeply into methodical details of neutron interferometry. It is sufficient to consider the intefferometer as a device which splits the incoming neutron wave in two coherent components propagating along dif- ferent trajectories (denoted by I, II) over a distance of several centimeters before they are recombined again. The lateral separation ( -2 - 4 cm) of these trajectories is large enough that the wave functions of the two subbeams within the interferometer can by altered independently from each other by means of either nuclear, electromagnetic or gravitational interaction potentials. Thereby observable modifications of the interference pattern, i.e. the intensity distri- bution of the two neutron beams leaving the interferometer, are induced. As usual in neutron interferometry the beam propagating in forward direction will be denoted as "O-beam" whereas the diffracted one is called "H-beam". </p><p>Rotations of the neutron spin are described by the unitary operator </p><p>U R = exp( - io 'a /2 ) </p><p>= cos(a/2) - itr&amp; s in(a /2) , (1) </p><p>where o" is the Pauli spin operator and a -- a&amp; is the rotation vector. The angle of rotation equals the Larmor precession angle around a magnetic field B </p><p>a=v f Bdt=~,/v f Bds. (2) </p><p>Here y = - 1.833 x 108 rad s- 1 T - 1 is the gyromagnetic ratio of the neutron and v its vel- ocity. The integrations extend over the interac- tion time and length, respectively. </p><p>Now, if by means of a magnetic field such a spin rotation takes place in one arm of the neutron interferometer the wave functions ~, ~/ii of the interfering beams, which for an ideal empty interferometer are initially equal*, differ from each other (1~ II= URI~ I) leading to a mea- sureable variation of intensity </p><p>2. 4~-Periodicity of spinor wave functions loc I~ I + ~bnl 2 oc 1 + cos(a /2) . (3) </p><p>Classically any rotation of an object by an integral multiple of 2"rr rad leaves the object in the same state as it was initially. In quantum theory, however, a rotation by 2"rr rad of half- integer particles, i.e. fermions, which are de- scribed by a spinor wave function is inevitably associated with a reversal of the sign of the wave function. Since only the absolute square of the wave function is observable this change of sign can be detected only by means of an interference experiment. Though some earlier theoretical suggestions did exist to measure this effect [9, 10] its first direct experimental verification was real- ized by means of neutron interferometry almost simultaneously by Rauch et al. [2] and Werner et al. [3]. Further experimental evidence was pro- vided later on also by Klein and Opat [11] and Stoll et al. [12] by means of a Fresnel-diffraction neutron interferometer and nuclear magnetic re- sonance methods, respectively. </p><p>This means that 2rr rotations of the neutron spin give in fact an observable effect on the interfer- ence pattern and that a rotation by an angle of 4rr rad is necessary in order to return to the original state. The most precise experimental verification of this spinor behaviour of neutron wave functions is shown in fig. 1 together with a schematic sketch of the experimental arrange- ment [13]. There two counter-rotating soft mag- netic foils of stray-field-free toroidal shape were used to produce well-defined spin rotation an- gles. As a hitherto unpublished byproduct of subsequent experiments we have repeated the original measurements reported in ref. [2] with polarized incident neutrons and polarization analysis of the emerging beams. A typical result </p><p>* This is true only for the forward ("O") beam. Since all formulas given here refer to this beam the corresponding subscripts are omitted for the sake of clarity. </p></li><li><p>84 G. Badurek et al. I Polarized neutron interferometry </p><p>~4 j ~ </p><p>coil _ 5000 </p><p>'T c E </p><p>\ ' , ooo / i n t~ ' fe r~ter z </p><p>(220) plane </p><p>S~r ~" ''-( - z 3000 </p><p>, . L ~ _ ~ 200( </p><p>Zo ;' ' \zH </p><p>B:749638G </p><p>r ~ z /~/ H-beam </p><p>. </p><p>"%" "',. O-beom - , o </p><p>~ G c m ] 1100 1200 1300 </p><p>I 02 0Z, ~,D (mm) </p><p>Fig. 1. Experimental arrangement and results of the hitherto most precise measurements of the 4~r-periodicity of spinor wave functions (Rauch et al. [13]). </p><p>is shown in fig. 2. The pronounced damping of the interference oscillations at small rotation an- gles (i.e. low magnetic fields) is essentially caused by field-induced neutron wavelength var- iations. </p><p>It has to be mentioned, however, that this kind of spin rotat ion- though yielding the same observable result - is conceptually different from that kind of (proposed) experiment where a fermion wave function is rotated by adiabatic rotation of a magnetic field through 2~r rad [9]. Quite recently Berry [14] has shown that during its adiabatic evolution along a closed curve in parameter space any physical system accumu- lates in addition to its "dynamical" phase (i.e. </p><p>o % </p><p>~'60o o /,- o </p><p>k~4 O0 o % </p><p>b J 200 </p><p>~ = 712.7 ._ 78 I 1 2 3 4 5xlO 3 </p><p>ROTATION ANGLE (deg) Fig. 2. Observed intensity as a function of the magnetic spin rotation angle for polarized incident neutrons. The two cur- ves correspond to parallel (I+) and antiparallel ( I ) orienta- tion of the initial polarization and the magnetization direction of the Heusler-crystal spin direction analyzer. </p><p>the one produced for example by the rotation operator of eq. (1)) a purely topological phase shift which depends only on the solid angle subtended by this cyclic curve. Though verified already by Bitter and Dubbers [15] in a classical experiment with polarized cold neutrons (A 8 A), which propagate along a helical magnetic field and allow for an easy fulfillment of the adiabaticity condition, a neutron interferometric measurement of "Berry's phase" is still missing. Aharonov and Ananandan [16] have generalized Berry's concept to the case of nonadiabatic rota- tions, which probably could simplify the realiza- tion of an interference experiment with polarized thermal neutrons where the magnetic fields with- in both arms of the dynamical phase is equal for both interfering beams and the geometrical phase alone will be observable. </p><p>3. Combined action of nuclear phase shift and magnetic spin rotation </p><p>A standard procedure in neutron interferomet- ry is to record the flux of emerging neutrons as a function of the relative phase shift X between the two coherent subbeams. The most practical way of producing this phase shift is to rotate a plane- parallel slab of material in increments through the interfering beams yielding X =-NbcAAD. Here Nb c is the coherent scattering length densi- </p></li><li><p>G. Badurek et al. / Polarized neutron interferometry 85 </p><p>IRON YOKE / '~ z </p><p>\ t / --", ' , , ' t c--c" I / i , ., , . . </p><p>- - - . ,~- - - - - - - - </p><p>FL/P-COlZ INTERFE/~METER 72201 </p><p>12 </p><p>~o </p><p>N ~o 4- </p><p>~ 12 i--- z lO </p><p>8 </p><p>t0 2 </p><p>-.,. ;~ ' J - - - ~',.xx--/- 0 z_ </p><p>-O2 t I I N </p><p>02 ~ xa ~) </p><p>-0 2 1 I I </p><p>-'rr 0 ~x X(rctd) </p><p>Fig. 3. Experimental setup and measured interference fringes if both nuclear phase shift and magnetic spin rotation act on the neutron wave function (Badurek et al. [5]). </p><p>ty of the phase shifter material and AD the variable geometrical path difference between the two neutron trajectories I and II. </p><p>The combined action of both nuclear phase shift (X) and magnetic spin rotation (a) is de- scribed by the unitary operator </p><p>U = exp(ix)U R = exp(ix)exp(-iora/2). (4) </p><p>The intensity and the polarization of the beams leaving the interferometer can then be calculated according to </p><p>I = I~b' + ~bIII2 = ~b~+(1 + U+)(1 + U) I / / I , (5) </p><p>~I+(1 + U+)tr(1 + U)I// I P = ~bI+(1 + U+)(1 + U)t~[ (6) </p><p>The effects associated with eqs. (5) and (6) have been both treated theoretically [4, 17, 18] and verified experimentally [5]. One predicted con- sequence of the combined action of phase shift and spin rotation is the generation of at least partial polarization of the emerging beams even if the incident neutrons are completely unpolar- ized. In this case the final intensity and polariza- tion of the O-beam should vary according to </p><p>I~ 1 + cos X cos(a/2), (7) </p><p>sin X sin(a/2) P = 1 + cos X cos(a/2) ez (8) </p><p>The corresponding quantities of the H-beam are always complementary to that of the O-beam because of particle number and angular momen- tum conservation. Taking into account the inevitable reduction of the interference contrast due to partial incoherence of the subbeams with- in a real interferometer, the validity of the rela- tions (8) and (9) is evidently demonstrated by the measured interference patterns shown in fig. 3. The coherent intensity oscillations vanish where those of the polarization reach their largest amplitude, and vice versa. </p><p>4. Fermion spin state superposition </p><p>Interferometry with polarized neutrons allows the realization of an old Gedankenexperiment proposed by Wigner [19]. If one prepares the two interfering subbeams in such a way that they are in different spin eigenstates, say "up" and "down" with respect to some quantization axis given by an external magnetic field, quantum mechanics predicts that the combined state is not a mere mixture but a coherent superposition of "up" and "down". This means that the final </p></li><li><p>86 G. Badurek et al. / Polarized neutron interferometry </p><p>state should be polarized perpendicular to the quantization axis and hence the emerging beams behind the interferometer should have properties which- from a classical point of view- neither of their constituents had. Choosing the z-axis as quantization direction and denoting the interfer- ing states in the usual way by [I)= [l'z) and [II)= exp(ig)[$z) the linear superposition can be expressed mathematically as </p><p>lo&gt;= Itz&gt;+= e'X'2{ x x } x/~ cs~l l ' x ) - i s in~ ISx) ' (9) </p><p>which obviously shows that the polarization P = / of the O-beam is confined to the xy-plane although the interfering states are polarized parallel and antiparallel, respectively, to the z-axis. </p><p>Two different cases have to be considered, however, in achieving orthogonality of the inter- fering spin states. Eq. (9) holds only if a DC spin turn device inverts the polarization of one sub- beam by means of well-defined Larmor preces- </p><p>sion around a static magnetic field [6]. The ac- tion of such a device can be represented essen- tially by a rotation operator of the form given in expression (1). Since in that case the interaction between the neutron magnetic moment /t and the field does not explicitly depend on time, i.e. OH/Ot = O( - t tB) /Ot = 0, the total energy of the neutron is conserved during the spinflip process although the potential Zeeman energy changes by the amount 2tzB 0 (B 0 - strength of the mag- netic guide field which is necessary to avoid depolarization of the beam). This implies, how- ever, that a corresponding change of kinetic energy or wavelength takes place: </p><p>AA = 2mtzBoA3/h 2 (~10 -s A!!) (10) </p><p>Although this wavelength shift is extremely small it is sufficient to reduce the measured interfer- ence contrast. </p><p>Alternatively the neut...</p></li></ul>

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