Polarized neutron interferometry: A survey

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  • Physica B 151 (1988) 82-92 North-Holland, Amsterdam

    POLARIZED NEUTRON INTERFEROMETRY: A SURVEY

    G. BADUREK, H. RAUCH and J. SUMMHAMMER Atominstitut der Osterreichischen Universitdten, A-1020 Vienna, Austria

    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.

    1. Introduction

    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.

    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 For completeness a hitherto unpublished result of a similar experiment performed later on with polarized incident neutrons will be reported, too.

    The next topic we will treat is what happens when both nuclear and magnetic dipole interac-

    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].

    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].

    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].

    Finally it will be shown that the spin-state

    0378-4363/88/$03.50 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

  • G. Badurek et al. / Polarized neutron interferometry 83

    superposition experiment cited above can be modified to realize a so-called "late-choice" interference experiment.

    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".

    Rotations of the neutron spin are described by the unitary operator

    U R = exp( - io 'a /2 )

    = cos(a/2) - itr& s in(a /2) , (1)

    where o" is the Pauli spin operator and a -- a& is the rotation vector. The angle of rotation equals the Larmor precession angle around a magnetic field B

    a=v f Bdt=~,/v f Bds. (2)

    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.

    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

    2. 4~-Periodicity of spinor wave functions loc I~ I + ~bnl 2 oc 1 + cos(a /2) . (3)

    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.

    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

    * 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.

  • 84 G. Badurek et al. I Polarized neutron interferometry

    ~4 j ~

    coil _ 5000

    'T c E

    \ ' , ooo / i n t~ ' fe r~ter z

    (220) plane

    S~r ~" ''-( - z 3000

    , . L ~ _ ~ 200(

    Zo ;' ' \zH

    B:749638G

    r ~ z /~/ H-beam

    .

    "%" "',. O-beom - , o

    ~ G c m ] 1100 1200 1300

    I 02 0Z, ~,D (mm)

    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]).

    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.

    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.

    o %

    ~'60o o /,- o

    k~4 O0 o %

    b J 200

    ~ = 712.7 ._ 78 I 1 2 3 4 5xlO 3

    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.

    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.

    3. Combined action of nuclear phase shift and magnetic spin rotation

    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-

  • G. Badurek et al. / Polarized neutron interferometry 85

    IRON YOKE / '~ z

    \ t / --", ' , , ' t c--c" I / i , ., , . .

    - - - . ,~- - - - - - - -

    FL/P-COlZ INTERFE/~METER 72201

    12

    ~o

    N ~o 4-

    ~ 12 i--- z lO

    8

    t0 2

    -.,. ;~ ' J - - - ~',.xx--/- 0 z_

    -O2 t I I N

    02 ~ xa ~)

    -0 2 1 I I

    -'rr 0 ~x X(rctd)

    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]).

    ty of the phase shifter material and AD the variable geometrical path difference between the two neutron trajectories I and II.

    The combined action of both nuclear phase shift (X) and magnetic spin rotation (a) is de- scribed by the unitary operator

    U = exp(ix)U R = exp(ix)exp(-iora/2). (4)

    The intensity and the polarization of the beams leaving the interferometer can then be calculated according to

    I = I~b' + ~bIII2 = ~b~+(1 + U+)(1 + U) I / / I , (5)

    ~I+(1 + U+)tr(1 + U)I// I P = ~bI+(1 + U+)(1 + U)t~[ (6)

    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

    I~ 1 + cos X cos(a/2), (7)

    sin X sin(a/2) P = 1 + cos X cos(a/2) ez (8)

    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.

    4. Fermion spin state superposition

    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

  • 86 G. Badurek et al. / Polarized neutron interferometry

    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

    lo>= Itz>+= e'X'2{ x x } x/~ cs~l l ' x ) - i s in~ ISx) ' (9)

    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.

    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-

    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:

    AA = 2mtzBoA3/h 2 (~10 -s A!!) (10)

    Although this wavelength shift is extremely small it is sufficient to reduce the measured interfer- ence contrast.

    Alternatively the neutron spin state can be inverted also by resonant coupling to a radio- frequency (rf) magnetic field. There the interac- tion Hamiltonian is (at least in principle) purely

    MAGNETIC PRISM POLARIZER

    PHASE SHIFTER (e iX) / / DETECTOR

    H -BEAM

    , ,IST MAGNET YOKE

    WATER i / IFLIP/ METER xT~/ /~ SPIN FLIPPER

    /

    BST ,,

    ~r2-SPIN TURN COIL

    '~ HEUSLER -" CRYSTAL ~rf ANALYZER t < STROBOSCOPIC

    REGISTRATION )

    DETECTOR O- BEAM

    Fig. 4. Schematic sketch of the setup for both static and time-dependent superposition of spinors.

  • G. Badurek et al. / Polarized neutron interferornetry 87

    time-dependent and hence commutes with the momentum operator p = -ihV. Consequently the wavelength of the neutrons is conserved but their total energy is either increased or decreased by AE = +__htorf due to absorption or emission of rf photons (frequency tOrf ) [20]. This means that two states of different energy interfere resulting in a nonstationary fringe pattern which can be detected only by a stroboscopic method [7]. As in the static flipper case it finally turns out that the polarization of the emerging beams has no component along the initial z-direction but now it rotates with frequency tort = AE/h in the xy- plane:

    V(t) = (cos(x - tOrf/), sin(x - to, ft), 0). (11)

    Fig. 4 shows a schematic sketch of the setup to realize the spinor superposition experiments. Its

    details are described in refs. [6] and [7]. Here it shall only be mentioned that the incident neut- rons were polarized by means of birefringent magnetic prisms, exploiting the extremely nar- row angular acceptance ( -2s of arc) of the perfect monolithic interferometer crystal [21]. Though they are not completely avoidable the coherence losses caused by inhomogeneous phase shift and refraction effects of the static flip coil windings were kept as low as possible by using wires of a special NbV alloy whose neutron refractive index is very close to 1. By means of a Heusler crystal and a ~-/2-spin turn coil the polarization of the O-beam could be analyzed either in z- or in y-direction.

    The results of both the static and the strobos- copic measurements are shown in fig. 5. It is clearly seen that the interference contrast van- ishes completely in accordance with the theoreti-

    2O00

    1500 0

    u~

    IOOO

    o

    2000

    "~ 1500

    t lo00

    o

    400

    POL. ANALYSIS IN z-DIRECtION

    t4t t 4, t,44 1 44 "t ttt't t'tTt'tt 'T ;t't '

    ~o 200 ~ ,oo 5~o a O thin)

    ,q

    200

    0

    -2oo

    - 400 -300 -200

    o o o ~~o o o o ov o o~v ~ o o

    a o o o o o

    i i i i

    -tO0 0 I00 2OO a O (pml

    POL. ANALYSIS IN y-DIRECtION

    400

    200

    o i

    t - 2 00

    Yrf: 55. 4kHz

    at = I/2 Vrf

    ~o 200 ~ ,oo 5oo -,oo -300 -200 -100 0 ~ 200 ----- aD (pro) ~ aD (pro)

    Fig. 5. Measured interference patterns with static (left) and radio-frequency (right) spin flipper in one arm of the interferometer. The static data were corrected both for residual phase inhomogeneities of the flip coil wires and the effective flipping efficiency. In the rf case a stoboscopic neutron detection technique had to be applied to obtain a stationary interference pattern. In both cases obviously the final state has no polarization component in the direction of the interfering ones (Summhammer et al. [6]; Badurek et al. [7]).

  • 88 G. Badurek et al. / Polarized neutron interferometry

    cal predictions if the z-component of the polari- zation is measured, whereas the perpendicular component is fully established. It has to be men- tioned that the fringe contrast of the static mea- surements shown in fig. 5 was enhanced by cor- recting the data for the residual phase shift inhomogeneities of the flip coil wires and the effective spinflip efficiency. On the contrary no such correction was necessary for the strobos- copically measured data since there no material has to be traversed by the neutrons and- unlike the static flipper case - a priori no loss of coher- enc_e due to wavelength differences between the interfering states can occur.

    This time-dependent spinor superposition ex- periment raises the question wether the photon exchange can be used to specify which of the two possible trajectories the neutron has taken with- out destroying the interference pattern. One sim- ple argument against the possibility of such a "quantum nondemoliton measurement" is based on the uncertainty relation between the number of photons N and the phase tb of the rf field [7]

    AN2A4, 2 1> ~- (N/> 4). (12)

    This is, of course, not the fully correct form of the particle number-phase uncertainty relation [22, 23] but it suffices for the qualitative explana- tion given here. To realize a stoboscopic neutron detection system which is phase-locked to the rf field the phase of that field has to be known precisely. This is no problem as long as the field, which can be considered as a superposition of coherent Glauber states [24], contains a large number of photons and hence behaves "classical- ly". However, the photon number of such a field obeys Poisson statistics with a minimum unsharp- ness AN- (AN2) 1/2 = N 1/2 which runs into con- flict with the uncertainty relation (12). Reducing AN by weakening the amplitude of the rf field is no way out since then the phase fluctuations Ark 2 will increase correspondingly. The argument- which can be formulated alternatively also in terms of the spectral width of the neutron beam and the energy-time uncertainty relation [7]- gets less trivial, however, if "squeezed states" of the electromagnetic field have to be considered

    [25, 26]. A rigorous approach which is not based on the number-phase uncertainty relation has been given by Namiki et al. [27].

    5. Double-flipper interference experiments

    From the reasoning given in the previous sec- tion we can draw the conclusion that the energy transfer within the resonance flipper actually does not represent a measuring process, since otherwise the simultaneous observation of inter- ference and particle trajectory would be feasible which is in clear contradiction to the Bohr- Heisenberg ("Copenhagen") interpretation of quantum mechanics. Following a suggestion of Vigier and his group [28], who claimed that new insight into the particle-wave duality on the basis of a causal stochastic interpretation of quantum mechanics [29-31] can be gained by our experiments, an arrangement as shown in fig. 6 with two rf resonance coils was realized. The use of a skew-symmetric interferometer, where the subbeams propagate along parallel directions over a distance of several centimeters, instead of the standard three-plate interferome- ter is of mere technical advantage. It turns out that the various types of measurements [8], which have been performed with this setup until now, can be described equally well both in terms of the Copenhagen and the causal stochastic interpretation of quantum mechanics and no de- cision can be made which of them is more adequate.

    Thereby the most interesting experimental situation arises if the two rf coils are driven at

    i n If their slightly different frequencies tort # tO rf. frequency difference is by orders of magnitude smaller than the width (51/2 of the resonance curve of the spinflip process, i.e. if the condition IAtot=ltoIrf-to~l,~61,2 is fulfilled, both coils can be tuned sufficiently close to resonance to yield virtually 100% flipping efficiency. This means that the two interfering beams are in the same spin state, indeed, but they differ in energy by an amount

    AE = hAto (13)

  • G. Badurek et al. I Polarized neutron interferometry 89

    I00~ OFF RESONANCE ' 'H-BEAM' 8I-oOo'~o o ~o ~ ~%%o~ o 60~ o o oO o o

    (~ ~ O l I I I I I I I t

    | ON RESONANCE H-BEAM -~ 1I- o s 01- o 0%00 00% 0 ooo I 6o~ % ~o o oO~ o 0o0

    (*0~ o o %000

    60"~ ~ " 482 " O-BEAM ~" =, , " , . : ' .

    INTERFEROMETER 2f ~" " " " "~ 0 I_ I I I I I I I I / 0 20 t,0 60 80 100 120 lt,0 160

    TIME (sec) Fig. 6. Observed beating of the emerging beam intensities if the two radio-frequency resonance flip coils are driven at frequencies that differ by only about 0.02 Hz. This corresponds to an exceedingly small energy difference between the interfering beams of only 8.6 10 -17 eV. If the resonance condition is destroyed by increasing the static magnetic field the beating effect vanishes completely (Badurek et al. [8]).

    due to the interaction with photons of slightly different energy. As a consequence no stationary interference pattern but a beating of the final intensity

    I(t) 0c 1 + cos(Atot - X) (14)

    with a period

    2~r h r= iA,o---- = iAe---- (15)

    will be observed, as is demonstrated in fig. 6, too. There the frequency splitting between the two rf coils was as small as Av = 0.02 Hz leading to a beating period T = 47.90--+ 0.15 s, which ac- cording to eq. (15) corresponds to an exceeding- ly small energy difference AE=8.6x 10-17___ 2.7 X 10 -19 eV.

    Eq. (14) is valid for polarized incident neut- rons only. It can be shown that in the case of initially unpolarized neutrons the photon fre- quency difference Aco may induce a nonzero degree of beam polarization and that one can suppress the beating effect of either the intensity or the polarization by proper adjustment of the

    nuclear phase shift X [8], in close analogy to the results of section 3.

    Another interesting, though not yet realized, experiment is sketched in fig. 7. Suppose that in

    i / / i I

    i ( 1

    1,,,,H

    ii i i i

    i I

    i

    E",,, / /E

    | ,yt I

    /t \ 0 p ~H

    Fig. 7. Double-coil interference setup with two radio- frequency flippers in the same arm of the interferometer.

  • 90 G. Badurek et al. / Polarized neutron interferometry

    one arm of the interferometer two radio frequen- cy flip coils are placed within the pole shoes of two separated magnet yokes. Both coils are tuned exactly to the same resonance frequency so that the action of the first flipper is completely undone by the second one. Nevertheless a phase shift

    regions but extends homogeneously over the whole interference region. Anyhow, there is a close relationship between this double-coil inter- ference arrangement and famous double-reso- nance techniques of Ramsey and his group with atomic [32] and neutron beams [33].

    AE L L Aq~ = + - + -- (16)

    - - h U - - O)rf U 6. Late-choice experiment

    between the interfering beams will be observ- able, since along the distance L between the two flip-coils the neutron wave propagates with a kinetic (+ total) energy that differs by an amount AE = +htOr f from the energy E = my2~2 in the other trajectory. The signs in eq. (16) refer to the two possible spin states of the inci- dent neutrons and the length of the coils has been assumed to be much smaller than their distance L.

    It is interesting to note that h~0 depends on the spatial separation of the coils although the neut- ron propagates in vacuum. However, the phase shift disappears in spite of the same change of total energy on the passage through the rf flip- pers if the static field is not restricted to the coil

    Fig. 8 shows how the static spin state superpo- sition setup of section 4 can be modified in order to realize a so-called "late-choice" experiment with polarized neutrons which is somewhat dif- ferent from Wheeler's "delayed-choice" con- cepts [34]. All that is necessary is to place a fast neutron chopper in front of the interferometer and to switch the current through the ~/2-spin turn coil in front of the analyzer crystal on and off with some adjustable delay with respect to the chopper opening time.

    As was shown in fig. 5 no interference contrast is measured as long as the w/2-coil is not active. In this case it seems that one knows which of the two possible paths the neutron has taken:

    Provided that the polarization of the incident

    I.FASTCHOPPE R PHASE SHIFTER I //j 0o,L

    INTERFEROMETER ~ ~ - - ^, , ' ^- r ~

    Zzz(t) or

    Iyz(t)

    Fig. 8. Proposed arrangement for the realization of a "late-choice" experiment with polarized neutrons.

  • G. Badurek et al. / Polarized neutron interferometry 91

    beam is parallel to the direction of the analyzer and assuming further 100% efficiency of both the spin flipper within the interferometer and the polarization analyzer then only such neutrons can reach the detector which have not undergone a spinflip process and hence cannot have prop- agated along the trajectory through the flipper [35].

    However, if this path information gets lost completely by turning the "rr/2-coil on and thus analyzing the orthogonal spin components the expected interference effects become observable. Now, with our proposed setup it will be possible to postpone the decision whether the path shall be determined or not until just before the wavepacket enters the -rr/2-coil, which can be arbitrarily long after the recombination of the interferometric beams. Though it is still not yet realized there is no doubt about the result of this proposed experiment: the late choice will have no influence on the observability of the interfer- ence pattern.

    7. Concluding comments

    From the theoretical point of view most of the topics treated here are well understood and hence out of discussion. But whether or not the experiment proposed in the last paragraph can be considered actually as a delayed-choice ex- periment in Wheeler 's terminology surely can be debated controversially. While in Wheeler 's proposal the decision wether to observe the path or the interference pattern is made before the superposition of the wavepackets (for instance by quickly inserting a detector in one path), in our proposal it can be done at any time after- wards. And, strictly speaking, in the Copenhagen interpretation of quantum mech- anics the described procedure of spin direction analysis to determine the neutron path uniquely is meaningless as long as the final state is a coherent superposition of substates. But on the other hand, obviously no experimental possibili- ty can be conveived either to falsify or to confirm the above given interpretation of path determi- nation.

    Acknowledgements

    The cooperat ion within the Dor tmund- 1

    Grenob le -Wien interferometer group and the hospitality of the Institut Laue-Langev in in Gre- noble are gratefully acknowledged.

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    DISCUSSION

    (Q) M. Bozic: You showed that an unpolarized neutron beam is splitted by the magnetic prism. Do you know what is the direction of propagation inside the magnetic field of an incident beam polarized in an arbitrary direction with respect to the direction of the magnetic field in the prism? (A) G. Badurek: Provided the neutrons enter the prism field region nonadiabatically and are not in an eigenstate with respect to the field, the incident beam is split into two subbeams which belong to the two different spin eigenstates and are deflected into opposite directions. (C) J.F. Clauser: We have seen many discussions so far of the separated oscillatory field technique. Credit should be given to its inventor N.F. Ramsey who did this with atomic beams many years ago.

    (A) G. Badurek: Without doubt all experiments of this kind make profit of Ramsey's pioneering work which will be taken into account in the written version of this contribution.

    (Q) S.A. Werner: It seemed to me that there is also a conceptual problem in the absorption of an rf photon having a wavelength of about 300 m by a neutron within the inter- ferometer (of size -1/10 m).

    (A) G. Badurek: I think the wave picture is not really adequate for the description of such localized absorption and/or emission phenomena of photons. According to the wave-particle duality this conceptual problem does not arise if for that specific situation the particle picture is chosen.