achievements in neutron interferometry

Acta Physica Hungarica 75 (1-4), pp. 203-219 (199~)

ACHIEVEMENTS IN NEUTRON INTERFEROMETRY

H. RAUCH

Atominstilut dcr Ostcrreichischen Univcrsit•

Sch~ttelntrasse I15, A-1020 Wien, Austria

Within a perfect crystal interferometer widely separated coherent beams can be produced which can be influenced by nuclear, magnetic and gravitational interaction. The veri¡ of the 4r-symmetry of spinor wave functions and of the spin superposition law at a macroscopic scale and the observation of gravitational effects including the Sagnac effect have been widely debated in literature. The coupling of the neutron magnetic moment to resonator coils permi t ted the coherent energy exchange between the neutron system and the macroscopic resonator. This phenomenon provided the basis for the observation of the magnetic Josephson effect with an energy sensitivity of 10 -19 eV. Partial beam path de- tection experiments are in close connection with the development of quantum mechan]cM measurement theory. A striking spectral modulation effect has been observed by means of a proper post-selection procedure under conditions where the spatial shift of the wave trains greatly exceeds the coherence length of the neutron beams traversing ah interferometer. It is shown that SchrSdinger-cat-like states are created by the superposition of two coherent states generated in the interferometer. These entangled states exhibit under certain circumstances characteristic squeezing phenomena indicating a highly non-classical behaviour.

I n t r o d u c t i o n

Different kinds of neutron interferometers have been tested in the past. The slit interferometer is based on wavefront division and provides long beam paths but only a very small beam separation [1, 2]. The perfect crystal interferometer [3, 4] is based on amplitude division. The interferometer based on grating diffraction is a more recent development and has its main application for very slow neutrons [5]. A schematical comparison is shown in Fig. 1. The perfect crystal interferometer provides highest intensity and highest flexibility for beam handling and is now most frequently used due its wide beam separation and its universal availability for fundamental, nuclear and solid-state physics research.

The perfect crystal interferometer represents a macroscopic quantum device with characteristic dimensions of several centimeters. The basis for this kind of neutron interferometry is provided by the undisturbed arrangement of atoms in a monolithic perfect silicon crystal [3, 6]. Ah incident beam is split coherently at the first crystal plate, re¡ at the middle plate and coherently superposed at the third plate (Fig. 1, middle).

From general symmetry considerations follows immediately that the wave functions in both beam paths, which compose the beam in the forward direction behind the interferometer, are equal (�90243 = r243 because they ate transmitted-reflected-reflected (TRR) and reflected-reflected-transmitted (RRT), respectively. The

Acta Physica Hungarica 75, 1994 Akad› Kiad£ Budapest

2 0 4 H. RAUCH

entrance slit Bi-prism detector

Ÿ L�91

r reflecting planes

ating

grating mirror

Fig. 1. Scheme of a slit, a perfect crys ta l a n d a g ra t ing in te r fe rometer

de Broglie wavelength of the neutrons diffracted from such crystals is about 1.8 and their energy is about 0.025 eV. The theoretical treatment of the diffraction process from the perfect crystal is described by the dynamical diffraction theory, which can also be found in the literature for the neutron case [7, 8, 9, 10]. In- side the perfect crystal two wave fields are excited when the incident beato fulfi]s the Bragg condition, one of them having its nodes at the position of the atoms and the other in between them. Therefore, their wave vectors are slightly different (kl - k2 = 10-Sk0) and due to mutual interference processes, a rather complicated interference pattern is built up, which changes substantially over a characteristic length A0, the so-called Pendell6sung length, which is of the order of 50/~m for an ordinary silicon reflection. To preserve the interference properties over the length of the interferometer, the dimensions of the monolithic system have to be accurate on a scale comparable to this quantity. The whole interferometer crystal has to be placed on a stable goniometer table under conditions avoiding temperature gradi- ents and vibrations. A phase shift between the two coherent beams can be produced by nuclear, magnetic or gravitational interactions. In the first case, the phase shift is most easily calculated using the index of refraction [11, 12]:

A2N ~ o'~ i~r~NA ki,~ = 1 _ (1) n = k0 ~ ~-~ -t- 4Ÿ '

which simpli¡ for weakly absorbing materials (~r ~ 0) to

n = l - A 2 N b T c , (2 )

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NEUTRON INTERFEROMETRY 205

where be is the coherent scattering length and N is the particle density of the phase shifting material. The different k-vector inside the phase shifter causes a spatial shift A of the wave packet which depends on the orientation of the sample surface

k - k0 (1 - n)kw A - - - D 0 with k - k 0 = (3)

As in ordinary light optics the change of the wave functions is obtained as follows:

r --* r i a k = r -lvb~ = r i• (4)

Therefore, the intensity behind the interferometer becomes

I0 ~ Ir243 +r243 ce ( l + c o s x ) . (5)

The intensity of the beam in the deviated direction follows from particle conserva- tion:

Io + IH = const. (6)

Thus, the intensities behind the interferometer vary a s a function of the thickness D of the phase shifter, the particle density N or the neutron wavelength ,X. A wave- packet description has to be used to define coherente properties in real experiments.

r ce /a(k)e'krd3k. (7)

Standard quantum mechanics defines the momentum distribution of the beam by

g(k) = Ir 2 = ]a(k)l z (8)

and, therefore, one gets the real part of the coherence function as the Fourier-trans- forro of the momentum distribution

I r ( a ) l ~ I fg(k)e~kad3kl, (9)

which simplifies for Gaussian momentum distributions

g(k) ce exp [ - (k - ko)2/26k2] , (10)

with characteristic widths 6ki to

Ir(~X0)l = H exp [-(Ai6ki)2/2]. (11)

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206 H. RAUCH

The mean square distante related to IF(A)I defines the coherenee length A which is for Gaussian distribution functions directly related to the minimum uncertainty relation (A~ = 1/(26ki)).

Any experimental deviee deviates forro the ideatized assumptions made by the theory: the perfect crystal can have slight deviations from its perfectness, and its dimensions may vary slightly; the phase shifter contributes to imperfections by variations in its thickness and inhomogeneities; and even the neutron beato itself contributes to a devation form the idealized situation beeause of its momentum spread �91 Therefore, the experimental interference patterns have to be described by a generalized relation

I ce A + B cos(x + (I)0), (12)

where A, B and (I) are characteristic parameters of a certain set-up. It should be mentioned, however, that the idealized behaviour described by Eq. (5) can nearly be approached by a well balanced set-up [13]. The reduction of the contrast at high order results from the longitudinal coherente length which is determined by the momentum spread of the neutron beam (A~ = (26k~) -a. This causes a change in the amplitude factor of Eq. (12) as (B ~ B exp[-(Ai6ki)~/2]). The wavelength depen- dente of X in Eq. (4) disappears in a special sample position where the surface of the sample is oriented parallel to the reflecting planes and the path length through the interferometer becomes Do/s in 0B and, therefore, the phase shift X = -2dhkzNbcDo becomes independent of the wavelength and the damping at high interference orders is strongly reduced as compared to the standard position. Related results of a recent experiment where the interference pattern in the 256th interference order has been measured in the dispersive and the nondispersive sample position are shown in Fig. 2 [14]. The much higher visibility of the interferences in the nondispersive sample arrangement is visible and is eaused by the mueh smaller momentum spread perpendicular to the reflecting planes of the perfect crystal.

Various post-selection measurements in neutron interferometry have shown that interference fringes can be restored even in tases when the overall beam does not exhibit any interference fringes due to spatial phase shifts larger than the coherente length of the interfering beams [15, 16, 17, 18]. This indicates that the simple picture whieh prediets interference only when wave packets spatially overlap is untrue. Interference actually occurs no matter how large the optical path difference may be. From classical optics it has been known for many years that the coherence properties manifest themselves in a spatial intensity variation for phase shifts smaller than the coherence length and in a spectral intensity variation for large phase shifts [19, 20, 21, 22, 23]. This phenomenon becomes more apparent for less monochromatic beams and can cause overall spectral shifts [24, 25] and even squeezing phenomena [26, 27]. The related phenomena for matter waves have been discussed recently [28, 29] and will be elucidated in more detail in this paper.

All the results of interferometric measurements, obtained until now can be explained well in terms of the wave picture of quantum mechanics and the complementarity principle of standard quantum mechanies. Nevertheless, one should bear in mind that the neutron also carries well-de¡ particle properties, which have to be transferred through the interferometer. These properties ate summarized in

Acta Physica Hungarica 75, 199~


~~ position B ,~ ~', (standard) ^ /

, #,4Ÿ ~ / \ - - t~ !1

position A ~ position C interferometer (non dispersive)

crystal i

5 c m

~ ilOOll

~ m !

~ l l i l l

~ i lol ,

t

t q. j ~ ~ t, I 't l ~ �9

I i I I I �9 t I I, I I t t i I i I 1 I

~,' 12 V i i i i i

�9 ,, Ÿ Ÿ

Ÿ ~ I I I P

v \z -~, ,io ~, i2

- - ~ 4 0 ~ t t m ~

a/

b/

Fig. 2. Interference pa t t e rn observed at high order (m = 256) with a dispersively (above) a n d a nondispersively arranged sample [14]. (Dashed lines correspond to measurement at low order.)

Table I together with a formulation in the wave picture. Both particle and wave properties are well established and, therefore, neutrons seem to be a proper tool for testing quantum mechanics with massive particles, where the wave-particle dualism becomes very obvious.

All neutron interferometric experiments pertain to the case of self-interference, where during a certain t ime interval, only one neutron is inside the interferometer, if at all. Usually, at that time the next neutron has not yet been born and is still contained in the uranium nuclei of the reactor fuel. Although there is no interaction between different neutrons, they have a certain common history within predetermined limits which ate defined, e.g., by the neutron moderation process, by their movement along the neutron guide tubes, by the monochromator crystal and by the special interferometer set-up. Therefore, any real interferometer pattern contains single particle and ensemble properties together.

P a r t i a l b e a m p a t h d e t e c t i o n e x p e r i m e n t s

A certain attenuation can be achieved either by a semi-transparent material o r a proper chopper system. The transmission probability in the first case is defined by the absorption cross-section ~a of the material [a = I/Io = exp(-aaND)] and the change of the wave function is obtained directly form the complex index of refraction (Eq. 1):

r ""* COCŸ kD = r 2 --- e i x ~ / • 1 6 2 (13)


208 H. RAUCH

"rabie I Properties of the neutrons

Particle properties:

mass spin magnetic moment half life electric charge electric dipol moment confinement radius quark structure

m = 1.6749543(86). 10 -24 g s = h/2 m = -1.91304308(54)~zh- 711/2 = 641(8) s q < 2.2 �9 lO-2~ d < 4.8 �9 10-25e �9 cm R = 0.7fin n = u - d - d

Wave properties:

Compton wavelength de Broglie wavelength coherence length packet length decay length phase difference

= h/mc = 1.32 �9 10 -13 cm )~B ~- h /mv _ 1 �9 10 -8 cm* A c = A2/A)~ = 1 �9 10 -6 cm* •p = v . • t = 1.10cm* A d = v . T l l 2 = 2 . 1 0 s cm* O_<x_<2r

vMues belong to thermal neutrons (A B = 1.8/~, v = 2200 m/s)

Therefore , the b e a m m o d u l a t i o n behind the in te r fe romete r is ob ta ined in the fol-

lowing form lo ce 1r243 + r243 ce [(1 + a) + 2x / • XI. (14)

On the o ther hand , the t r ansmis s ion p robab i l i t y of a chopper wheel or ano the r shu t te r sys tem is given by the open to closed rat io , a = toven / ( top . . + tdos .d) , and one ob t a in s af ter s t r a igh t fo rward ca lcula t ions

I c e [ ( 1 - a ) l r 2 4 3 1 6 2 1 9 1 1 6 2 2 4 3 2] ce [ ( l + a ) + 2 a c o s x ] , (15)

i.e. the cont ras t of the interference pa t t e rn is p ropo r t i ona l to V• in the first case, and p r o p o r t i o n a l to a in the second case, a l though the same number of neu t rons are abso rbed in bo th cases. T h e absorp t ion represents a measur ing process in bo th cases because a c o m p o u n d nucleus is p roduced wi th an exc i ta t ion energy of several MeV, which is usual ly de-exc i ted by g a m m a rays. These can easi ly be de tec ted by different means .

F igure 3 shows the dependence of the normal i zed cont ras t of the measu red interference p a t t e r n on the t r ansmiss ion p robab i [ i ty [30, 31, 32]. The different con t r a s t becomes especia l ly obvious for low t r ansmis s ion p robab i l i t i e s where the in ter fer ing pa r t of the interference pa t t e rn is d i s t inc t ly larger than the t r ansmis s ion p r o b a b i l i t y t h rough the s e m i - t r a n s p a r e n t absorber sheet . The d iscrepancy diverges for a ---+ 0 bu t it has been shown tha t in this regime the var ia t ions of the t r ansmis s ion due to var ia t ions of the th ickness or of the densi ty of the absorber p la te have to be taken into account which shif ts the poin ts below the x / • [33]. This can mos t easi ly be unde r s tood if the va r i a t ion of the beam a t t e n u a t i o n due to va r i a t ions of the th ickness or densi ty f luc tua t ions are ineluded a = ~ + Aa , which yields af ter

Acta Physica Hungarica 75, 1993


incident ~~~~:o~~;,:n~s ~ ~ (22oo) ~ absorbing-

,oil \ ~ b e L ~ abS~Ÿ

p h a s e ~ e h o p p e r \ ~

interferometer/~~~~"~ ~ /'~ "~ ,~ f crystal ~ ...... ~ ].~ ~ ].- t~/\ H-detector L I " k_

Odetector a) b) c)

00s r-------7-.--- ~ 0 ~ =~ I s t o c h a s t i c ~ [ / ~ I [ + / I =o, I- absorp t ion / /

t/~-I t / !.4 2<../ o ' mo~:o% . . . . ,s ~

t/'-" L ' l i d / / I E L , * .// deterministic V - . . . I~='~~.= ~ ; 1 / I'~,~.. ~ro~'1 / absorption

�9 ooos o k Ÿ :~--I - - I I/ I / t_0J o. - ~ ~ ~ ~ ~ ~ _ ~ ~ ~ _ a ~ 7 "

~ransrnission probability a

Fi$. 3. Sketch of the experimental arrangement for absorber measurements (above). (a) stochastic absorption, (b) deterministic absorption, (c) attenuation by a transmission grating. Reduction of the contrastas a function of beam attenuation for different attenuation methods (below) [31, 32]

averaging v~ < v~ (16)

indicating that the points fall below the , /• The region between the linear and the square root behaviour can be reached

by very narrow chopper slits of by narrow transmission lattices, where one starts to lose information of through which individual slit the neutron went. This is exactly the region which shows the transition between a deterministic a n d a stochastic view and, therefore, it can be formulated by a Bell-like [34, 35] inequality

V• > x > a. (17)

The stochastic limit corresponds to the quantum limit when one does not know any more through which individual slit the neutron went. Which situation exists depends on how the slit widths compare to the coherence lengths (Ai ~ (26ki) -1) in the related direction. In case that the slit widths become smaller than the


210 H. RAUCH

"'V/ /~ a = O ~

~ ~ 0 . ~ O

g lattice ~-~.~~~ ~

@.0@ 6.oo o.o~ O.lO o.15 o.zo interferometer crystal

transmission a

Fig. 4. Lattice absorber in the interferometer approaching the classical limit when the slits are oriented ho¡ and the quantum limit when they are oriented vertically [31, 36]

coherence lengths, the wave function behind the slits shows distinct diffraction peaks which correspond to new quantum states (n :/› 0), which now do not overlap with the undisturbed reference beam. The creation of the new quantum states means that those labelled neutrons carry information about the chosen beam path and, therefore, do not contribute to the interference amplitude [36]. A related experiment has been carried out by rotating an absorption lattice around the beam axis where one changes from I y A~ (vertical slits) to 1 >> Ay (horizontal slits), Fig. 4, because the coherence length parallel to the reflecting lattice vector is much larger than in any other directions. Thus, the attenuation factor a has to be generalized including not only nuclear absorption and scattering processes but also lattice diffraction effects ir they remove neutrons from the original phase space.

A very similar situation exists if a very fast chopper produces beam bursts (packet lengths) shorter than the coherence time Ate = A/v. In this case, diffraction in t ime occurs which also removes neutrons out of the original phase space. This limit is very difficult to reach with a mechanical chopper but it can probably be tackled with a high frequency spin flipper.

N e u t r o n J o s e p h s o n e f f e c t

This phenomenon is based on the dipol coupling of the magnetic moment p of the neutron to a magnetic field B ( H = -q which causes the famous 4zr- symmetry of spinor wave functions, as measured in early neutron interferometer experiments [37, 38]. The change of the wave function reads:

r ---, r -i(mlh) = r -i(l'Bt/h) = r -~~=/2 = r (18)

where = represents a formal description of the Larmor rotation angle around the field B(c~ = (2#/h) f Bdt = (2#/hv) f Bds). This enabled also the realization of the

Acta Physica Hun9arica 75, 199.~

NEUTRON I N T E R F E R O M E T R Y 211

spin superposition experiments where spin-up (I T>) and spin-down states (I ~>) ate superposed producing a final state perpendicular to both initial states [39, 40]. It is interesting to mention that in the case of spin reversal by means of a resonance flipper the spin term is accompanied by an energy exchange equal to the Zeeman energy hw~ = 2#B. This provided the basis for the observation of a new quantum beat effect: the magnetic Josephson analog.

A double coil arrangement can be used for the observation of a new quantum beat effect. If the frequencies of the two coils are chosen to be slightly different, the energy transfer becomes different too ( A E = h(Wrl -w r2 ) ) . The frequency difference can be made very small, if high quality frequency generators ate used for the field generation. The flipping efficiencies for both coils ate always very close to unity (better than 0.99). Now, the wave functions change according to

r ~ �91 I J.> q-eiXei(~-~ I ~> �9 (19)

Therefore, the intensity behind the interferometer exhibits a typical quantum beat effect, given by

I ce 1 -t- cos[x + (w~x - wr2)t]. (20)

Thus, the intensity behind the interferometer oscillates between the forward and deviated beato without any apparent change inside the interferometer [41]. The time constant of this modulation can reach a macroscopic scale which is correlated to ah uncertainty relation A E A t < h/2. Figure 5 shows the result of an experiment, where the periodicity of the intensity modulation, T = 27r/(w~l - wr2), amounts to T = (47.90 5= 0.15) s caused by a frequency difference of about 0.02 Hz. This corresponds to a mean difference of the energy transfer between the two beams, A E = 8.6- 10 -1~ eV, and to an energy sensitivity of 2.7.10 -19 eV, which is better by many orders of magnitude than that of other advanced spectroscopic methods. This high resolution is strongly decoupled from the monochromaticity of the neutron beam, which was A E = 5.5 �9 10 -4 eV around a mean energy of the beam EB = 0.023 eV in this case. It should be mentioned, that the result can also be interpreted as being the effect of a slowly varying phase A(t) between the two flipper fields, but the more physical description is based on the argument of a different energy transfer. The extremely high resolution may be used for fundamental, nuclear and solid-state physics applications.

The quantum beat effect can also be interpreted a s a magnetic Josephson effect analog. In this case, the phase difference is driven by the magnetic energy (ABo = Bo2 - Bre)

6 2#AB0 (21)

and, therefore,

z x ( t ) = ( ~ r ~ - ~ r l ) t = 2#AB0

h . t . ( 2 2 )

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212 H. RAUCH

resonance flipper

IH ~~ .~

m c~

t

interferometer

Ira)

i

Fig. 5. Quantum beat effect observed when the frequencies of the two flipper coils differ by about 0.02 Hz around 71.89979 kHz [41]

This yields the observed modulation (compare Eq. (20))

I ce (1 + cos A(t)). (23)

This is analogous to the well-known Josephson effect in superconducting tunnel junctions [42], where the phase of the Cooper pairs in both superconductors is related to the electrical energies on both sides of the junction according to

5• 1 1 (02 - O1) = ~ ( E 2 - El) = ~2 eV (24)

which is driven by the electrical potential V between both superconductors. This gives

2 eV O(t ) = - y - - . t (25)

a n d a superconducting Josephson current

I s = I , Ma= sin r (26)

P o s t s e l e c t i o n o f s t a t e s

In the course of several neutron interferometer experiments [15, 16, 17, 18] it has been established that smoothed out interference properties at high interference order can be resored even behind the interferometer when a proper spectral filtering is applied. The experimental arrangement with an indication of the wave packets at different parts of the interference experiment is shown in Fig. 6. An additional

Acta Physica Hungarica 75, 199,$


monochromator

~Z.~ A ~/ Bi phase % [ [ S hiffer

~eam from ~ " ~" - - ~ z 3He detector

s,t "' I]1"/1 . . l l L r I~direct 3 I I 1 ~ ' ~o~qV~. q Hedetector I ' ,4 " ~ " , r "~.m./r ~l• d IP£ ~ ~ /~'- C3beam

I I �9 direct"%~L~~ .......... ',= u = --qc b e ~ ~ %

�9 analyser skew-symmetnc a:~lsY:a~r " ~

neutron interferometer "

3 He detector

Fig. 6. Scheme of the experimental arrangement with a skew-symmetrically cut perfect crystal interferometer anda post-selection analyser crystal [18, 28]

monochromatization is applied behind the interferometer by means of various single crystals brought into Bragg position.

Using Eqs (5) and (7), the momentum-dependent intensity reads:

Io ( r ,k ) - - I r243 + r243 + A,k)l 2 ce la(k)1211 § cos(a(k) .k)l ~, (27)

whereas the overall beam reads

lo(Z~o) oc 1 + Ir(ao)l c o s a o k, (28)

where A0 represents the spatial phase shift for the k0-component of the packet. Eq. (28) describes the interference fringes when A0 is varied. The formula also shows that the overall interference fringes disappear for spatial phase shifts larger than the coherence lengths [Ai > A/c = 1/(26k,)] (see Eq. (11)). This behaviour is shown in Fig. 7 and has been verified experimentally by several investigations for Gaussian and non-Gaussian neutron beams [43, 44, 45].

In our experiment we deal with the coherence properties along the interferometer axis (x), where the (tangential) components of the momentum vectors (and coherence length) do not change due to Bragg diffraction. Aceording to basic quantum mechanical laws, the related momentum distribution follows from Eq. (27) and for Gaussian packets it can be rewritten in the form (k = k=, ky, kz)

I0(k) = e x p [ - ( ~ - k0)~/26k ~] 1 + cos x 0 y ,

where the mean phase shift is introduced (Xo = koAo = NbcAoDell). The sur- prising feature is that lo(k) becomes oscillatory for large phase shifts where the

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214 H. RAUCH

interference fringes described by Eq. (28) disappear (see Fig. 7). This indicates that interference in phase space has to be considered [46,47] rather than the simple wave function overlap criterion described by the coherence function (Eq. (10)). The second beam behind the interferometer (H) just shows the complementary modulation I H = I to ta l - - Io .

The amplitude function [48] of the packets arising from beam paths I and II determines the spatial shape of the packets behind the interferometer

lo(x) = le(x) + r + A)I=, (30)

which separates for large phase shifts into two peaks (Fig. 7). For Gaussian packets, £ corresponds to the coherence length Ac and fulfils the minimum uncertainty relation �91 = 1/2. For ah appropriately large displacement (A >~ Ac) , the related state can be interpreted a s a superposition state of two macroscopically distinguishable states, that is, a stationary SchrSdinger-cat-like state [49, 59], but here first for massive particles. These states - - separated in ordinary space and oscillating in momentum spaee - - seem to be notoriously fragile and sensitive to dephasing effects [51, 52, 53, 54].

Schroedinger-cat-like

Jr ...... . . . . . . . . . . . . . . C . . . . . . . . . . . . . o ,r "~ -1~ -SOx ,(A)S~ 1.~0 2.$0 k3l~ i 3.60 -150 -50�9 (A T 15,0 2.60 k3"1~ 360

2 d k / k o :=0.02

0 . . . . . . . . . . . . . . . I

0 50 100 t

21 �91 A ~ 1 ~

"ol ...... / ~ ....... g x(a) k(A-)

Fig. 7. Interference pat tern a s a function of the relative phase shift (middle) and related wave packets and momentum spectra behind the interferometer [18]

Measurements of the wavelength spectrum were made with a silicon crystal with a rather narrow mosaic spread which reflects in the parallel position a very

Acta Physica Hungarica 75, 199~Ÿ


narrow band of neutrons only (•k'/ko ,~ 0.0003) causing an enhanced visibility at large phase shifts (Fig. 8). This feature shows that an interference pattern can be restored even behind the interferometer by means of a proper postselection procedure. In this case the overall beam does not show interference fringes anymore and the wave packets originating from the two different beato paths do not overlap.

The momentum distribution has been measured by scanning the analyser crystal through the Bragg position. The related results are shown in Fig. 9 for different phase shifts. These results clearly demonstrate that the predicted spectral modulation (Eq. (29)) appears when the interference fringes of the overall beato disappear. The modulation is somehow smeared out due to averaging proeesses across the beato due to various imperfections, unavoidably existing in any experimental arrangement. The contrast of the empty interferometer was 60 %.

f i l t e r e d u n f i l t e r e d 4500 l ~ ~ , T r

�9 ooo~. / - ~ . , ~ . ,,oooo a,00t\ / " ~ , / \ D = 0 m m / ~ ~ 3000L . .~ ~ ,,0oo0

/ I .I t . . i seoooo �9 ! i

,00ot A , . ~ ~. D = 4 m m 3500 ~ . ~ ~ ~ 1 4 0 0 0 0

~ 3000 I , " , , _.] S,O000 , , ,

, 0 o o ~ . ~ f ~ D = 1 2 m m

3O00 �9 [ l _ t [ ' lzoooo ~ ! t

�9 ~176 Ff~ �9 A D = 2 0 m m ,,o.~ V \ ' - 0 ~ - - - - - - .o0 ~ , , , , . o o . o ' ' o14

- 0 . 8 - 0 . 4 0 . 0 0 . 4 0 .8 - 0 . - 0 . 4 0 . 0 0 . 8

a ( d e g ) a ( d e g )

Fig. 8. Interference pattern of the overall beato (6k/ko = 0.012) and the beato reflected from a nearly perfect crystal analyser in the antipaxalleI position (�91 = 0.003) [18]

Discuss ion

All the results of the neutron interferometric experiments ate well described by the formalism of quantum mechanics. According to the complementarity principle of the Copenhagen interpretation, the wave picture has to be used to describe the observed phenomena. The question how the well-defined particle properties of the

Acta Physica Hunga~'ica 75, 1994

216 H. RAUCH

neutron are transferred through the interferometer, is n o t a meaningful one within this interpretation, but from the physical point of view it should be an allowed one.

The newly discovered persisting phase space coupling in cases of large spatial shifts of the wave packets may bring some attenuation to the action of plane wave components outside the packet, The results clearly demonstrate that a spectral modulation can be observed in neutron interference experiments at high interference order and that interference has to be treated in phase space rather than in ordinary space. It looks like that the plane wave components of the wave packets, i.e. narrow band width components, interact over a much larger distance than the size of the packets. This interaction guides neutrons of certain momentum bands to the O- or H-beam, respectively. These phenomena throw a new light on the discussion on SchrSdinger-cat-like situations in quantum mechanics and, therefore, on the discussion about the EPR-experiments, too [28, 56, 57, 58]. Spatially separated packets remain entangled in phase space and nonlocality appears as a result of this entanglement. The analogy with optical experiments performed in the time- frequency domain is striking [23]. An analogue situation exists in neutron spin- echo systems where multiple spin rotation plays an equivalent tole as high order interferences discussed here [59].

Each peak in the momentum distribution corresponds to a different number of phase shifts experienced by the neutrons of that wavelength band during its pas- sage through the interferometer. In that sense, the minimum quantum unity of the incident wave packet becomes a new quantity representing different quantum states with distinguishable properties. This kind of labelling shows that constructive interference is restricted to that wavelength band only; a situation similar to that where new states have been created due to lattice diffraction inside the interferometer (Fig. 4)[36].

The new quantum states created behind the interferometer can be analysed with regard to their uncertainty properties. Analogies between a coherent state behaviour a n d a free but coherently coupled particle motion inside the interferometer have been addressed previously [32]. In such cases, the dynamical conjugate variables x and p minimize the uncertainty product with identical uncertainties (Ax) 2 = (Ak) 2 = 1/2 (in dimensionless units). Using lo(k) and lo(x) (Eqs (29) and (30)) as distribution function we get in our case

lim <(Ax)2>=<x2>-<x>2=(6x)2+(@) 2 A >> 6x

and (for 6k/tco << 1)

< (Ak) 2 > =< k 2 > - < k >2

(31)

=(6k)~ 1 - ~ [l+e-(a~162176 2 "(32)

These relations are shown in Fig. 10 indicating that for (Ak) 2 values below the coherent state value can be achieved, which in quantum optic terminology means



spectra l d is t r ibut ion in tegra ted e~ ( d , g )

22++ 222 22o 2~s cont ras t 8000 . . . . 5000

++~176176 +~176176176176176176176 , +, ,~ 2000 ~

2,34 2.36 2.38 2.40 - 0 8 0.0 0 8

r x (i) �9 (d.q) $000 . . . .

Ÿ Z ] ~ sooo , , , t c,,, o+ m +ooo ~

' + m'~P. , , 1 . ~ zo<m I-" ' ' t ~ c ~ w

O l l l + I 2.34 2 .36 Z.38 2.60 - 0 . 8 0 .0 o.s

(1) , (d .~) I 0 0 0 5000 , , ,

l_~^ i ,.., V �91 ! [_.�91 ~1

1.34 I . M I . M 1 .40 - 0 . I I 0 .0 0.II

Fig. 9. Measured spectral modulation of the outgoing beam and the residual interference pattern for different bismuth phase shifter thickness. The wavelength resolution of the analyser was

0.002 A [18]

squeezing [35, 60, 61]. One emphasizes that a single coherent state does not exhibit squeezing, but a state created by superposition of two coherent states can exhibit a considerable amount of squeezing. Thus highly nonclassical states ate made by the power of the quantum mechanical superposition principle.

A cknowledgements

Most of the experimental results discussed in detail have been obtained by our Dortmund- Grenoble-Vienna interferometer group working at the high flux reactor in Grenoble, and some recent ones stem from our cooperation with the Columbia-Missouri group working at the MURR reactor. The cooperations within these groups and especially the eooperation with colleagues from our Institute, which axe cited in the references, are gratefully acknowledged.


2 1 8 H. RAUCH

<a kz~/6k ~ 4

/ /

/ /

/

/ /

/ /

/ /

/

! 1 2 5

x=A/Z6x

4 <Axt>/6x z

l

Fig. 10. Spatial and momentum uncertainties of the outgoing beams with the indicat ion of squeezing in the momentum domain [18]

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