superposition experiments in neutron interferometry

IL NUOVO CIMENTO VOL. 110 B, N. 5-6 Maggio-Giugno 1995

Superposition Experiments in Neutron Interferometry(*).

H. RAUCH

Atominst i tut der Osterreichischen Universita'ten, Shuttelstrasse 115 - A-1020 Wien, Austr ia

(ricevuto il 19 Gennaio 1995)

Summary. - - Coherent superposition of states is responsible for any interference phenomena. Recent neutron interferometry experiments based on postselection methods renewed the discussion about quantum non-locality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall contrast of an interference experiment vanishes. This indicates a persisting coupling in phase space even in cases of spatially separated SchrSdinger-cat-like situations. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction area becomes impossible in principle, which indicates that the irreversible quantum measuring process starts with the first interaction of the quantum system already.

PACS 03.75.Dg - Atom and neutron interferometry. PACS 03.65.Bz - Foundation, theory of measurement, miscellaneous theories. PACS 42.50.Dv - Nonclassical photon states (including antibunehed, squeezed, sub-Poissonian). PACS 61.12.Gz - Neutron diffraction techniques (e.g., powder, single crystal, energy dispersive, and pulsed neutron source methods). PACS 01.30.Cc - Conference proceedings.

1 . - I n t r o d u c t i o n .

Since perfect-crystal neutron interferometers became operational in 1974[1], many basic quantum experiments have been realized which are summarized in several proceeding books [2, 3] and review articles (e.g., [4-6]). Here, we will focus on more recent experiments which are intended to shed more light on open questions in quantum mechanics such as the quantum measurement process and the related irreversibility properties by using intrinsically accessible but often not measured quantities in many experimental situations.

The neutron is a massive particle with well-defined particle properties, with an internal quark structure and underlies strong, electroweak and gravitational

(*) Paper presented at the International Conference (,Mesoscopic Physics and Fundamental Problems in Quantum Mechanics),, Rome, February 14-17, 1994.

557

5 5 8 H. RAUCH

TABLE I . - Propert ies o f the neutron.

Particle properties Wave properties

m = 1.6748220(15). 10 -2~ kg Connection (thermal neutrons: 2 = 1.8 A, v = 2200 re~s)

8-- x h h 2 de Broglie 2r = -

/z = - 9.6491783(18). 10-27 J /T h m. c 2 3 = - - = 1.319695(20)- 10 -15 m

= 882.6(2.7) s m. v h 2e = - - = 1.8"10-1~

m - y R = 0.7 fm SchrSdinger 1

dc = ---- 10 -8 m a = 12.0(2.5)" 10 - 4 f n l a ~b( r , t ) 2 ~k

H~(r, t) = ifi ~ Llp= V" At ~- 10 -2 m u-d-d-quark structure LId = v" v = 1.942(5)" 106 m

0 ~< Z ~< 2 ~ ( 4 ~ )

m = mass, s = spin, ~ = magne t i c moment. ~ = fl-decay lifetime, R = (magnetic) confinement radius, ~ = electric polarizability; all other measured quantities like electric charge, magnetie-monopole and magnetic-dipole moment are compatible with zero, ~e =Compton wavelength, ~ s = d e Broglie wavelength, LIe =coherence length, /Ip= packet length, ~k = momenttan width, At = chopper opening time, v = group velocity, Z = phase.

interaction and which exists in two well-defined energy levels when a magnetic field is applied. The correspondence principle of quantum mechanics tells us tha t related wave propert ies exist as well which provide the basis for all neutron optical experiments [7,8]. These propert ies are summarized in table I [9].

The perfect-crystal in ter ferometer consists of a monolithic silicon crystal cut to provide coherent beam splitting, reflection and a superposition at three crystal plates (fig. 1). The reflecting lattice planes have to be parallel within a lattice constant. The wave functions composing the beam in forward direction behind the in ter ferometer are transmitted-reflected-reflected and reflected-reflected-transmitted, respectively. Due to symmet ry reasons they are equal in amplitude and phase. Thus a complete beam modulation is expected when a phase shift Z is applied between both coherent beams

(1) Io + q hi 12 : [2(1 -I- cosz ) .

The intensity of the deviated beam follows from particle conservation as

(2) IH + Io = const .

F rom basic laws of physics it follows that the phase shift is determined by the path integral of the canonical momentum along the beam paths (e.g., [10])

(3) X = ~ kcds = Zk + ~nk,

which can be separated into a kinetic par t which arises due to the action of conservative forces which change the momentum of the neutron and a par t arising from non-conservative forces •,k which do not act on the momentum of the particle.

SUPERPOSITION EXPERIMENTS IN NEUTRON INTERFEROMETRY 559

Fig. 1. - Photograph of a perfect-crystal symmetric neutron interferometer.

Many related experiments have been done in the past of which table II shows the related phase shifts and from which one can notice that the non-conservative phase shifts are non-dispersive (wavelength independent) and independent of the area enclosed by the two beams.

The kinetic phase shift can be described by an index of refraction relating the momenta inside and outside the interaction region n = k/ko. This causes a spatial shift of the wave train compared to the wave not being exposed to that interaction

(1 - n ) ~ (4) A = (fr .D ,

which is related to the phase shift as A-k = X. The orientation of the surface and the thickness of the phase shifter (interaction region) are denoted by ~ and D, respectively.

As in ordinary optics, the coherence function can be defined as the autocorrelation function of the overlapping wave functions

(5) F(A) = (~* (0) ~(A)),

where three-dimensional wave packets have to be considered for the description of the wave fields

(6) ~(r) ~ f a (k )exp[ ikr]d3k .

This gives the momentum distribution as

(7) g(k) ~ I~(k)[ 2 ~ l a ( k ) l ~

560

TABLE II. - Neutron interferometric measured phase shifts.

H. RAUCH

Interaction Potential Phase shift

nuclear

magnetic

gravitation

Coriolis

Aharonov-Casher (Schwinger)

magnetic Josephson

Fizeau

geometry (Berry)

2r~ft 2 be ~(r)

m

- p B ( r )

mgr

-hcoe(r x k)

-~(v x F,)/c

- iJB( t )

-Nbc2D

+_ t z Bm )~D 2~h 2

m 2 g2A sin r

2~h 2

2 m o ) e A s i n Ce sin~

h + 2~E. D

hc +- ot

be = coherent-scattering length, N =particle density of the phase shifter, A = An, D = path length of the neutron inside the interaction region, B = magnetic-field strength, g = gravitational acceleration, A = area enclosed in the coherent beams, r = angle between the horizontal and the area A, ~e = 0.727" 10 -4 s -1 = angular rotation velocity of the Earth, Ce = colatitude angle of the experimental place, e = angle between the area A in relation to the cardinal points, E = electric field, ~ = energy transfer due to the ~-ne-dependent field B ( t ) , w~, vx = velocity components of the phase shifter and the neutrons perpendicular to the moving surface of the phase shifter, ~ = solid angle subtended by a closed circuit in parameter space.

and the real part of the coherence function as the Fourier transform of this quantity

(8) I~A)l ~ I Ig(k)exp[ ikA]dak ["

For Gaussian momentum distributions with widths Sk~

g(k) :r exp [ - (k - ko)2/2 ~k 2 ], (9)

one gets

(10) IE(A)I = rI e x p [ - ( A ~ k i ) 2 / 2 ] �9 i = x , y , z

The mean-square distance defined by the coherence function defines the coherence lengths ~ which are directly related to the uncertainty relation Ac ~ki I> 1/2. When wave packets (eq. (6)) are used to calculate the intensity behind the interferometer (eq. (1)) one obtains

(11) Io = 12 ~ (1 + I r ( a ) l c o s z ) .

In practical cases one has to average over different beam paths where the physical situation of the interferometer may be slightly different (slight variation of thicknesses, of distances or of the crystal lattice). This causes that < IP(0) l > ~ 0 which has to be considered in data evaluation. Any beam attenuation effect (cross-section cs) can be described by an imaginary part in the index of refraction and causes an additional reduction of the coherence function as exp [ - N r All these effects


have been observed in the past (e.g., [11,12]) and used to extract the coherence lengths in the different directions from the decreasing visibility of the interference fringes at high interference orders (A>Ae=(2~k)-l) . Typical quantities for the coherence lengths are zl~- 200 A for the longitudinal direction, Ay- 20 ~m for the transverse direction where the dynamical Bragg diffraction determines the (Darwin) momentum width and Az- 100A in the vertical direction. When the absorbing or diffracting objects become comparably sized with the coherence lengths, diffraction phenomena occur which cause a partial labelling of neutrons which, therefore, do not contribute to the interference pattern anymore[13]. These-last mentioned experiments performed in the quantum limit show how properties of the quantum system--e.g., coherence lengths--influence the attenuation and the interference phenomenon due to the intrinsic coupling of quantum objects and measuring apparatus inherent in quantum mechanics.

2. - P o s t s e l e c t i o n o f m o m e n t u m s ta te s .

Interference experiments are performed with beams having a certain spectral spread which determines the number of visible interference fringes. At large phase shifts, the wave packets do not overlap anymore and the interference fringes disappear as was observed in many cases [14-16]. It was known from ordinary optics and proven in the neutron case that the contrast can, at least partly, be restored by postselection of narrow momentum bands by means of an additional analyser crystal[17] (fig. 2). The somehow surprising feature was that the recurrence of the contrast is not simply given by an enlargement of the coherence volume due to additional monochromatization, but by a distinct structuring of the momentum

monochromator

beam front - - - - ~ ~ a reactor 3He detector

slit

m 3He detector

I ~ ! a s e ~ ~ d i r e : i A a l y s e d tJ~ beam C3 beam

8.3 cm "I a n a l y s e r ~ ^ skew-symmetric crystal ~ , ~ neutron interferometer |

3He detector

Fig. 2. - Sketch of the experimental set-up for the postselection of momentum states behind the interferometer with an indication of separated wave packets behind the interferometer (SchrSdinger-cat--like states).

562 H. RAUCH

distribution due to interference effects. This modulation appears at high interference orders and can be calculated from eqs. (1), (6) and (9), as

(12) lo(k) o~ e x p [ - ( k - ko)2/2 ~k 2] 1 + eosxo -k- ,

where Xo denotes the phase shift for the mean k-vector ko. The intensity distribution of the related wave packets in ordinary space reads as

(13) lo (x) = l~b(x)+@(x+/1)12o~exp[-xZ/26x2]+exp[-(X+Ao)2/26x2]+

+2 exp [ - x 2 / 4 ~x2 ] exp [ - ( z + A)2/4 6x2 ]cos Zo .

This shows that a vanishing interference pat tern in ordinary space is accomplished by

8 0 0 0

6 0 0 0

4000

2000 8000

o 6 0 0 0 -

4000

2000 8000

6 0 0 0

4000

2000

s p e c t r a l distribution

0 A ( d e g r e e s ) 2 2 . 4 22.2 22.0 2 1 . 8

I I I I

I I I I

1 cm Bi

I I I I

I I I I

2 cm Bi

I I i I

2.34 2.36 3.38 2.40 ($.)

integrated contrast

4000

2000

| 4000 O O vM

2000 O r

4000

2000

" I i I

I

I I I

I I I

C2

I I I

C3

I I i

I I I

C2

I I I

C3

I I I I

-0.8 0.0 0.8 a (degrees)

Fig. 3. - Observed loss of contrast of the overall and remaining contrast of the filtred beam and the associated modulation of the momentum distribution at high interference order [18].


an appearing interference modulation in momentum space which carries on the coherence information stored in the system. This modulation in momentum space has been observed experimentally (fig. 3, [18]) and it is known from ordinary optics, as we11119-22]. These states--stationary SchrSdinger-cat-like states--are notoriously fragile and sensitive to dephasing effects. The detailed structure of the wave packets in ordinary and momentum space and in terms of the Wigner function has been investigated by computer simulation[23]. They exhibit the quantum squeezing phenomenon and show that the plane-wave bands (eq. (6)) may act outside the packet where the packet intensity vanishes due to interference effects with other wave packet bands.

The results presented here show that interference properties are stored in momentum space when they disappear in ordinary space and that this piece of information can be measured when a proper measuring method is used.

Nevertheless, it should be mentioned that the extraction of that piece of information becomes increasingly more difficult the wider the separation in phase space in at least one parameter exists, i.e. the stronger the interaction causing this separation acts on the system. A topic to be dealt with in more detail in sect. 6.

3. - Contrast retrieval by phase echo.

Phase echo is a similar technique to spin echo, which is routinely used in neutron spectroscopy[24]. A large phase shift (~1 > Jc) can be applied in one arm of the interferometer, which can be compensated by a negative phase shift acting in the

Fig. 4. - Loss of contrast at high interference order and its retrieval by an opposite phase shifter inserted into the same beam [16]. a) open beam, b) 8 mm Bi sample, c) 10 mm Ti sample, d) Bi + Ti samples.

564 H. RAUCH

same arm or by the same phase shift applied to the second beam path[15,25,16]. According to eq. (3), the phase shift is additive and the coherence function depends on the net phase shift only. Thus, the interference pattern can be restored as it is shown schematically and in form of an experimental example in fig. 4. The phase echo method can also be applied behind the intefferometer loop when multiplate interferometers are used [26]. In this case, the situation becomes even more similar to the situation discussed in sect. 2.

These results tell us that infarmation first appearing in a spatial phase shift becomes transferred into a m o m ~ t u m modulation (see sect 2) which can be revived to ordinary space modulation effects again.

Here also a comment has to be made that it becomes intrinsically more difficult to restore the original contrast the wider the separation of the wave packets in ordinary space happened (see sect. 6).

4. - R e q u e s t for p o s t s e l e c t i o n in E P R exper iment s .

The previously discussed neutron experiments have shown us that phase space coupling persists even if the overlap in one parameter space does not exist anymore. The stored information becomes exchanged between parameter spaces and can be measured by a proper experimental method. This has consequences for EPR experiments too. The entangled states (e.g., [27]) of two photons produced by an atomic-decay cascade (fig. 5)

(14) ~ l - k)l tlc)2 + l - k)2lk)l

are correlated due to the energy conservation of the transition

(15) kl + k~ = kol + k0e = const.

This produces a momentum- and space-dependent intensity distribution when the packet structure of the related wave functions is taken into account[28]:

(16) I(kz, k2, r) = [r 2 = 21a(k1)12ia(k2)]2(1 + cos [2(k2 - k l ) r ] ) .

2.00 decaying

- - ,,1 A t t _ atom - % A r r - " Y V " ~" = ~ 1.50

1.oo y k

/ / / /

/ / /

/ ",. / / �9 ' / / ,.

0.50 , ~. '

, / A \ ~' / \ ~ / ~. \ / / /

0.00 " ~ "" ~" ' -5.00 -3.00 -1.00

,, \

\ \ .x \

\

x \ / xxx ~

x \.we ,

\ ~ i x \ / )

1.00 3.00 5.00 /k r

Fig. 5. - Scheme of a typical EPR transition and expected intensity distribution for individual momentum pairs for ]kz- k2 I/~k = O, 0.1, 0.2, 0.4, 0,% 1.2 and 2 (from above to below [28]).


This shows a characteristic intensity modulation for each photon pair (fig. 5) and indicates that individual I k)-states remain interacting even at arbitrarily large spatial separation of the wave packets. For large distances ( r > (28k)-1), the appearance of a momentum distribution modulation follows from eq. (16) too [28]. If one of these photons is registered on one side, its wave function collapses, which instantaneously changes the wave function on the other side to I Ikh 12. This shows again that much more information can be gained than it is usually extracted. Therefore, it is recommended to repeat this experiment with a proper momentum resolution which would show that the right and the left wave fields of the related momentum band (i.e. the partner photons) remain coupled even at arbitrarily large spatial separation of the overall wave packets.

Related experiments will show that this coupling of the I~artner pairs of photons persists independently of its overall spatial separation. That indicates that locality should be treated in phase space rather than in ordinary space only.

Here too, the required momentum resolution becomes more stringent when the packets become wider separated in ordinary space.

5. - P o s t s e l e c t i o n in t h e t i m e d o m a i n .

Instead of measuring the interference pattern by scanning the phase shifter, one can measure additionally the intensity correlation function

(17) /'(2)(A, ~) = (I(0, O)I(A, ~))

which defines the probability of registering a neutron at time z, if there was another one measured at z = 0. This probability to measure a neutron at a time z after another

Io(t) O - d e t e c ~ r p ~ _ ~

reactor / _ _ _

�9 ,. / H-detector" [__J a u x m a r y I H (t) phase-shi f ter

0 0

/ Imax'eXp [-~"Imax]

Imin'eXp [-'t'.Imi n]

3 5 t ime interval (~/~-)

Fig. 6. - Sketch of the experimental set-up for time selection experiments and the pair correlation function for the maximum and minimum of the interference pattern [30].

566 H. RAUCH

aS

o~

0.010

0.00~ 0 10

time interval (v/J)

Fig. 7. - Measured phase sensitivity of an interference pattern for the case of proper- time-correlation measurements in comparison with the phase sensitivity related to the overall beam (dashed line)[30].

r

neutron has arrived, reads for a stationary beam of a thermal (statistical) source as [29]

(18) W(z) = l(zl) exp [ - z. I(A)],

which exhibits an intensity-dependent ,,decay time,, z(A)= [I(/1)] -1 �9 Figure 6 shows the experimental arrangement and the probability of measuring neutron pairs with a time separation v for the case when the overall interference pattern is turned to its maximum or minimum, respectively (in our case the overall contrast was 40%). It becomes visible that the contrast for neutron pairs arriving within short time intervals is higher than the overall contrast. For larger time separations, the contrast vanishes and appears with an opposite sign reaching values of - 100% for timely wide separated pairs. This behaviour has been verified experimentally [30] and it has been shown that remarkably higher phase sensitivities can be achieved by using this new measuring technique (fig. 7).

These results demonstrate that considerably more information can be deduced even from a statistical beam i f the individual arrival times of the neutrons are additionally registered to define the pair correlation function inherent to the quantum system

As in the previous sections, it should be mentioned that measurements with very short and very long time delays become increas'mgly more difficult due to dead-time problems of the counter and low intensities, respectively.

6. - Irrevers ib i l i ty and m e a s u r i n g process .

In the previous sections it has been shown that more information about a quantum system can be extracted when more experimentally accessible parameters are


measured. It becomes obvious that a system remains coupled in phase space even when it becomes separated in any parameter space. Thus, interference properties can be shifted from one parameter space to another one and back again. Related bands of plane-wave components which compose the wave packets (eq. (6)) may be considered as a responsible factor for the understanding of the non-locality phenomenon in quantum mechanics.

The summaries drawn for the different experimental situations discussed in this article are followed by a statement that the retrieval of the interference properties by several postselection procedures become increasingly more difficult the wider the separation of the quantum system happened before. A more detailed view even shows that a complete retrieval is impossible, in principle, as it is shown in fig. 8 for the case of a phase echo system. In a more complete and more accurate measurement, more and more parts of the complete (not approximative) wave functions become visible which contain more and more of the detailed history the quantum system has experienced between the source and the detector. This indicates a basic irreversiblity process not caused by parasitic effects like absorption or incoherent-scattering processes but by the appearance of an infinite number of additional terms in the wave function which indicates that, by no means, the original state can be restored

reversibility-irreversibility

" ~ o

] [iz] . ~oexp

-] I i [-3 E xJ I l l l : oexp

" ~o exp [i Z] +~ aij exp [i (zj+k (r i-rj))] t j

~oexp [ix]+ ~lb~ex p [i (zj+k (r~-rj))]

source) ~ (detector

Fig. 8. - Approximative and complete wave functions behind a compact and a split interaction region.

568 H. RAUCH

completely. Unavoidable fluctuations (even zero-point fluctuations) cause an irreversibility effect which becomes more influential for widely separated SchrSdinger-cat-like states as discussed in sect. 2. All these effects can be described by an increasing entropy inherently associated with any kind of interaction [31]. This also supports the idea that irreversibility is a fundamental property of nature and reversibility an approximation only, as stated by several authors (e.g., [32-34]).

This shows that irreversibility and, therefore, the measurement process, starts with the first interaction the quantum system experiences in the experimental set-up. The assignment of a source and a detector region define the direction of increz~ing entropy.

7. - S u m m a r y .

Recent neutron interference experiments have shown that a quantum system remains coupled in phase space even when it becomes separated in ordinary space. This kind of coupling becomes visible in case of proper postselection measurements. Thus, locality should be defined in phase space rather than in ordinary space only. In this sense, no additional terms have to be added to the SchrSdinger equation to account for locality in phase space where bands of plane-wave components of the wave packet link the whole quantum system independent of its spatial separation.

More complete quantum experiments show that a complete retrieval of all wave components behind an interaction the quantum system experienced becomes impossible, in principle. This implies on a high-accuracy level a basic non- commutativity of operators A . B I ~ B.A[~b) and indicates that the irreversible quantum measuring process starts with the first interaction of the quantum system with the experimental set-up.

The cooperation with many colleagues cited in the literature is gratefully ackowledged, as well as the financial support provided by Fonds zur FSrderung der Wissenschaftlichen F orschung (project P8456) for the experimental part of that research.

REFERENCES

[1] RAUCH H., TREIMER W. and BONSE U., Phys. Left A, 47 (1974) 369. [2] BONSE U. and RAUCH H. (Editors), Neutron Interferometry (Clarendon Press, Oxford)

1979. [3] BADUREK G., RAUCH H. and ZEILINGER A. (Editors), Matter Wave Interferometvy

(North-Holland) 1988. [4] KLEIN A. G. and WERNER S. A., Rep. Progr. Phys., 46 (1983) 259. [5] GREENBERGER D., Rev. Mod~ Phys., 55 (1983) 875. [6] RAUCR H., Contemp. Phys., 27 (1986) 345. [7] SEARS V. F., Neutron Optics (Oxford University Press) 1989. [8] RAUCH H., in Lehrbuch der Experimentalphysik, edited by BERGMANN-SC~FER and

H. NIEDRIG (V~. de Gruyter) 1993, Chapt. 11.


[9] B~NE J., Neutrons, Nuclei and Matter (Institute of Physics Publ.) 1994. [10] F E ~ R. P., LEIGHTON R. B, and SANDS M., The Feynman Lectures on Physics,

Vol. III (Addison-Wesley) 1965. [11] RAUCH H., SEtDL E., TUPPINOER D., PETRASCHECK D. and SC~F~I R., Z. Phys. B, 69

(1987) 69. [12] RAUCI4 H., S U M ~ I ~ E R J., ZAWiSKY M. and JERm~L~ E., Phys. Rev. A, 42 (1990)

3726. [13] RAUCH H. and S t r M M ~ E R J., Phys. Rev. A, 46 (1992) 7244. [14] RAUCH H., in Neutron Interferometry, edited by U. BONSE and H. RAUCH (Clarendon

Press, Oxford) 1979. [15] KAISER H., WERNER S. A. and GEORGE E. A., Phys. Rev. Left., 50 (1983) 560. [16] CLOTHIER n., KAISER H., WERNER S. A., RAUCH H. and WOLWITSCH H., Phys. Rev. A, 44,

No. 9 (1990). [17] KAISER H., CLOTHIER R., WERNER S. A., RAUCH H. arid WOLWITSCH H.: Phys. Rev. A, 45

(1992) 31. [18] JACOBSON D. L., WERNER S. A. and RAUCH H., Phys. Rev. A, 49 (1994) 3196. [19] MANDEL L. and WOLF E., Re?). Mod~ Phys., 37 (1965) 231. [20] JAMES D. F. V. and WOLF E., Phys. Le~ A, 157 (1991) 6. [21] SCHLEICtt W., PERNIGO M. and KIEN F. L., Phys. Rev. A, 44 (1991) 2172. [22] ZOU X. Y., GRAYSON T. P. and .MANDEL L., Phys. Rev. LetL, 69 (1992) 3041. [23] RAUCH H. and StrDA M., AppL Phys. B, 60 (1995) 181. [24] MEZEI F. (Editor), Neutron Spin Echo, Lect. Notes Phys., 128 (Springer) 1980. [25] BADUREK G., RAUCH H. and ZEILINGER A., in Neutron Spin Echo, edited by F. MEZEI,

Lect. Notes Phys., 128 (Springer) 1980. [26] HEINRICI~ M., PETRASCHECK D. and RAUCH H., Z. Phys. B, 72 (1988) 357. [27] MERMIN N. D., Phys. Rev. Lett., 65 (1990) 1838. [28] RAUCH H., Phys. Lett. A, 173 (1993) 240. [29] GIAUBER R. J., Fundamental Problems in Statistical Mechanics, edited by E. G. D.

COHE~ (North-Holland) 1968, p. 140. [30] ZAWISKY M., RAUCH H. and HASEGAWA Y., Phys. Rev. A, 50 (1994) 5000. [31] LORENTZ H. A., Theorie der Strahlung (Akad. Verlagsges., Leipzig) 1927. [32] PRIGO(~INE I., Proceedings of the International Workshop on Ecological Physical

Chemistry, Siena (Elsevier, Amsterdam) 1991, p. 8. [33] H ~ F., Commun. Math. Phys., 123 (1990) 245. [34] BLANCHARD P. and JADCZYK /L, Phys. Lett~ A, 175 (1993) 157.

superposition experiments in neutron interferometry

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