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Physics Letters A 311 (2003) 6–12

www.elsevier.com/locate/pla

Phase modulation of a neutron wave and diffraction of ultraconeutrons on a moving grating

A.I. Franka,∗, S.N. Balashovb, I.V. Bondarenkoa, P. Geltenbortc, P. Høghøjc,S.V. Masalovichb, V.G. Nosovb

a Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Russiab Institute of General and Nuclear Physics, RNC “Kurchatov Institute”, Moscow, Russia

c Institut Laue-Langevin, Grenoble, France

Received 29 April 2002; received in revised form 8 March 2003; accepted 10 March 2003

Communicated by J.P. Vigier

Abstract

We report the result of the experiment of UCN diffraction on a moving grating. The resulting spectrum is found to bein good agreement with theory. This purely quantum effect may be interpret as a result of phase modulation of the neuor as diffraction in time. Also, this experiment demonstrates the validity of the Galilean transformation of the neutrofunction in a new and very clear way. 2003 Elsevier Science B.V. All rights reserved.

PACS: 03.75.Be; 03.75.-b

Keywords: Neutron optics; Quantum mechanics

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1. Introduction

In general, quantum experiments are performebetter understand quantum mechanics and to demstrate the validity of its conclusions. But their impotance is not limited to undoubtedly great pedagogrole. Equally crucial is that such experiments woustimulate the development of new methods andvices based on quantum principles.

* Corresponding author.E-mail address: frank@nf.jinr.ru (A.I. Frank).

0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rigdoi:10.1016/S0375-9601(03)00460-2

-

It is well known that slow neutrons are an exclent candidate to examine nonstationary interactionmatter waves due to their low velocities and quantfrequencies. In contrast to the devices most commoused in optics, nonstationary devices change theergy of the neutron. The methods of time-dependor, which is the same, dynamic neutron optics baon nonstationary quantum phenomena allow conof the variance of the direction, intensity and energythe neutron beam making it possible to perform ntron focusing not only in space but also in time.

A number of nonstationary phenomena in neutoptics have already been analyzed theoreticallyverified experimentally. It appears to be first discus

hts reserved.

A.I. Frank et al. / Physics Letters A 311 (2003) 6–12 7

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by Moshinsky [1] in 1952. He considered the prolem of neutron wave evolution following an instataneous removal of an ideal absorber from a beof monochromatic neutrons. In 1976 Gerasimov aKazarnovsky [2] analyzed a number of nonstationquantum phenomena which, in principle, could beserved in experiments with ultracold neutrons (UCAfter that a number of theoretical papers devotedthis subject were published [3–14].

During the last decade some nonstationary qutum effects in neutron optics were observed expmentally. In Ref. [15] the results of the experimeon neutron reflection from a vibrating surface wasported (see also the earlier work [16] on neutron dfraction on surface acoustics waves). In the recentper [17] inelastic neutron scattering on high-frequeultrasonic waves in the crystal was observed. All thexperiments may be interpreted in terms of neutenergy exchange with phonons by analogy withexperiment on neutron interaction with an oscilling electromagnetic field [8,18,19] assumed to refrom multi-photon exchange. More recently, thesults of the experiment on neutron energy specttransformation due to transmission through fast,called quantum, chopper were reported [20]. Theperiment was proposed in Refs. [3–5]. The obtainresults are in good agreement with theory althouin fact, they were just the observation of spectrwidening in accordance with the uncertainty relatEt ∼= h. A discrete spectrum resulting from thphase modulation of the atomic de Broglie waves wobserved in Ref. [21]. The same group demonstlater atomic wave diffraction in time using temporslits [22].

In the present Letter we report the resultsthe quantum experiment to observe ultracold neu(UCN) diffraction on a moving grating proposedRef. [10].

2. Periodic action on a neutron wave and neutrondiffraction on a moving grating

To present the simplest explanation of the physnature of the investigated phenomenon, we will sfrom an analysis of a general case of neutron wmodulation. Assume that the plane waveΨ (x, t) =ei(kx−ωt) propagates in the positivex-direction. A

device, which we will call a modulator, is locatedthe origin of the reference frame. The operationthe modulator performs the periodic variation of tamplitude or phase of the primary wave. At smdistances from the modulator,k−1 < x vT , thewave function will have the form

(1)Ψ (x, t)∼= f (t)ei(kx−ωt),

where f (t), is the function of the periodT , k isthe wave number andv is the neutron velocityRepresentingf (t) as a Fourier expansion,f (t) =∑∞n=−∞ ane−inΩt , Ω = 2π/T , one concludes tha

the wave function of the distance from the modulax > 0 is [9,10]

(2)Ψ (x, t)=∞∑

n=−∞an exp

[i(knx −ωnt)

],

with

ωn = ω+ nΩ,

(3)kn = k

(1+ n

Ω

ω

)1/2

forΩ/ω 1.

Eqs. (2) and (3) show that the modulator’s actresults in that the transmitted state is a superposof coherent waves with a discrete frequency spectrThe resulting frequenciesωn correspond to the particlenergiesEn = h(ω + nΩ), where h is the Planckconstant. The intensities of partial waves areIn =|an|2 while a0 is the amplitude of the wave with thinitial energy.

Assume now that a periodical structure or gratwith a space periodL is moving at the velocityVacross the neutron beam in the positive direction ofY -axis. Assume further that the different elementsthe grating have different properties and as a resthe waves that are transmitted through its differparts differ in their intensities or phases. In tsimplest case, one half of each grating element ftransmits neutrons while the other half absorbs thcompletely. It is then easy to see that movinggrating we modulate the transmitted wave in eapoint of the beam cross section with the frequeΩ = 2πV/L. This means that the resulting spectruof the transmitted waves will therefore be discrete.

In Refs. [9,10] this result was obtained usianother approach. Solving the stationary problemdiffraction in the moving reference frame where t

8 A.I. Frank et al. / Physics Letters A 311 (2003) 6–12

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grating is at rest it is found under the assumptionkL 1, T ω−1 that

Ψ0(x, y, t)

= exp[−i(kV y +ωV t)

]

(4)

×∑n

cn exp[i(√k2 + 2kV qn − q2x

+ qny −ωt)],

with

kV = mV

h, ωV = mV 2

2h,

(5)qn = 2π

Ln, cn = 1

L

L∫0

Φ(y)eiqny dy,

whereΦ(y) is the amplitude of the transmitted wawhich may be complex. (It was assumed here thatinitial state was the plane wave exp[i(kx − ωt)] andthe grating was located inx = 0.)

Next, we go to the laboratory frame of referencby applying the Galilean transformation to the wafunction. To do that, in accordance with Section 17Ref. [23], we must replace the variabley in Eq. (4) byy −V t and then multiply the whole expression for twave function by exp[i(kV y −ωV t)]. An addition, wetake into account that

(6)qnV = 2πV

Ln= nΩ,

and arrive at the final formula

Ψ (x, y, t)

(7)

=∑n

cn exp[i(√k2 + 2kV qn − q2

n x

+ qny − (ω+ nΩ)t)].

It is easy to see from this equation that insteadthe energyE = hω the quasi-energy [24]

(8)En = hωn, ωn = ω+ nΩ,

arises as a result of diffraction on a moving gratiProvided the value for the grating space frequeL−1 is small

(9)L−1 kV , T h

mv2,

one can neglect the third term under the square rin Eq. (7) and taking into account (6) arrive at tfollowing expression for thex-components of thek-vectors of partial waves

√k2 + 2kV qn − q2

n∼=

√k2 + 2mV

hqn

(10)= k

(1+ nΩ

ω

)1/2

= kn.

So, at small diffraction angles

(11)Ψ (x, y, t)=∑n

cn exp[i(knx + qny −ωnt)

].

Eq. (11) differs from Eq. (2) by the diffraction termqny. Besides, Fourier coefficientscn in (11) were ob-tained by thek ↔ x transformation of the spatial dependence of the transmitted wave amplitude whileanin (2) results fromω ↔ t transformation of time dependent amplitude. One can verify thatcn, coincidentwith an if the last are obtained with functionf (y/V ).Eq. (11) transforms into Eq. (2) if the grating vlocity V and grating space periodL are simultane-ously increased without changing the modulation tiT = L/V .

3. Experiment

The experiment was performed at the UCN sou[25] of the Institute Laue-Langevin (Grenoble, Frace). As proposed earlier [10,26], both primary neutmonochromatization and analysis of the diffracspectrum were done using neutron interference fil[27–29] (NIF), a quantum device analogous toFabry–Perot interferometer for light. In the simplecase, a NIF is made of a three-layered structurethin films coated on a substrate, which is transpato UCN. It is well known that the interaction of lonwavelength neutrons with matter can be describedthe introduction of an effective potential associawith the medium,U = (2πh2/m)ρb, wherem isthe neutron mass,ρ is the number of nuclei peunit volume andb is the coherent scattering lengtWhen the outer layers have a greater value ofρb

than the inner one, the potential structure of the fihas a double-hump barrier with a well in betweas shown in Fig. 1. For a sufficiently thick midd

A.I. Frank et al. / Physics Letters A 311 (2003) 6–12 9

otential

Fig. 1. Neutron interference filter. (a) Three thin films (around 200 Å) composed of two different materials on a silicon wafer. (b) Pstructure. (c) Transmission probability as a function of energy.

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layer, the width of the gap is large enough for tformation of quasi-bound states. As a result, the filtetransmission function is essentially resonant. Tquantum resonant tunnelling through two (or evmore) potential barriers was successfully used inspectroscopy of ultracold neutrons [30,31].

The principle of the experiment is illustratedFig. 2. Instead of a linearly moving grating, wused a rotating one, which was manufactured onsurface of a silicon disk, 150 mm in diameter a0.6 mm thick. In its peripheral region radial groovwere made by the lithographic technique. Its anguperiod was 3.325× 10−4 rad−1 corresponding to aspace period of 20 µ at a diameter of 120 mm. Tgrooves covered about half of the period. Usingmotor, the grating could be spin about the vertical ato 6000 rpm. Ultracold neutrons reached the gratthrough an annular corridor with an inner and an oudiameter of 110 mm and 130 mm, respectively.

The NIF-monochromator was then placed betwthe exit of the corridor and the grating. Only UCNwith a narrow spectrum of vertical velocities, i.ea maximum of 4.52 m/s and FWHM of about 0.08m/s, passed through the monochromator and reac

Fig. 2. Principle of the experiment. A phase grating is preparedthe surface of the silicon disc. When the disc is spinning, each pof the monochromatic UCN beam is crossed by the moving gra

the grating. The noted UCN velocity correspondedan energy of 107 neV. Due to refraction in silico

10 A.I. Frank et al. / Physics Letters A 311 (2003) 6–12

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neutron waves, which had passed through the diffeelements of the grating, possessed the different phaThis phase difference could be expressed as

(12)ϕ = k(1− n)d,

wherek is the neutron wave number,n is the refractiveindex andd is the depth of the grooves. The depthdwas chosen to be about 0.14 µ, a measurement wcorresponds exactly to the phase shiftϕ = π forthe 107 neV neutrons refracted in silicon. The Foucoefficientscn corresponding to this kind of phaseπ -modulation are

(13)cn = 2

iπn, n= 2s − 1,

wheres is integer. It must be emphasized that the lwith the initial energyhω (n= 0) is absent in the finastate.

The major component of the setup is an UCgravity spectrometer with interference filters [30,3(see Fig. 3). Ultracold neutrons were fed to tspectrometer using an electropolished stainlessneutron guide. After some reflections, the UCNsdown to the annular corridor, NIF-monochromatoragrating. The UCNs transmitted through the gratwent further down to the glass mirror neutron gu50 cm high. Falling down this guide, UCNs aaccelerated in the Earth’s gravitational field and thenergy increases approximately to 1 neV/cm of thefree fall. A second NIF was placed inside this glaguide to act as an analyzer. Changing the posiof this second analyzing filter allowed the energythe transmitted neutrons to be scanned. The resoenergy of the second filter was 127 neV to compena change in the energy due to gravity. The neutrtransmitted through the analyzer filter were countea 3He gaseous detector.

Strictly speaking, this spectrometer is not sensito the total neutron energy but just the vertical coponent of thek-vector. It is this very component thais modified by fast grating moving in the horizontplane. Because there is no any collimation of themary beam, the initial wave does not only havevertical component of thek-vector,k0z, but alsok0xandk0y . This means that to compare our experimenresults with theoretical predictions we have to modslightly the equations presented in Section 2. An exform of verticalk-vector components after transm

.

l

t

Fig. 3. Gravity UCN spectrometer with neutron interference filt(NIF). 1 entrance chamber, 2 vacuum chamber, 3 motor (to rotathe grating), 4 mirror neutron guide, 5 rotating grating, 6 analyzfilter, 7 filter-monochromator, 8 detector, 9 step motor.

sion through the grating is described by the equatio

(14)knz =√k2

0z + 2qn(kV − k0y)− q2n.

We choose a reference frame where the local gravelocity V is directed along theY -axis. Eq. (14)has the same form as Eq. (10) when the condikV k0y satisfies together with conditions (9). In thdescribed experiment the value ofv0y was limited bythe boundary velocity of glass (vl ≈ 4.5 m/s) becauseneutrons with a higher horizontal velocity were nreflected by the wall of the guide and could not reathe detector.

A.I. Frank et al. / Physics Letters A 311 (2003) 6–12 11

red with

Fig. 4. Measured energy splitting after the diffraction of UCN in a moving grating due to phase modulation of a neutron wave compatheory.

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The neutron count rate as a function of the analyposition was measured for two very different gratirotation frequencies: 180 and 5400 rpm (Fig. 4).the case of slow spinning, the effective modulatfrequencyΩ = πV/a was small and the corresponing energy splittingE = hΩ was much less thathe width of the initial energy spectrum of UCN. Thscanning curve was simply a convolution of the mochromator and analyzer spectral functionsf (E) andg(E),

(15)F1(z)=∫f (E)g(E − ξ)dE,

whereξ = mgH , H is the distance between thfilters andE = (h/2m)k2

0z.The results of the corresponding measurements

presented in Fig. 4 with open dots. The dash linethe result of fitting of these data by the Gaussian wthe dispersionσ ∼= 6.2 neV that corresponds to thexperimental resolution of the spectrometer FWHM∼=7.3 neV.

The fast rotation regime (5400 rpm) correspondto the linear velocity of the gratingV ≈ 36 m/s andthe modulation frequencyΩ around 1.07× 107 rad/s.In this case, the spectrum of the neutrons that pa

through the grating is

(16)Φ(E)=∑n

|cn|2f (E + nhΩ),

if the termsq2n and k0y in brackets in Eq. (14) ar

neglected. The scanning curve is then

(17)F2(ξ)=∑n

|cn|2F1(ξ + nhΩ),

where cn and F1(ξ) are defined by Eqs. (13) an(15), respectively. This type of a “theoretical” scaning curve was calculated for the values ofcn cor-responding to the Gaussian resolution functionF1(z)

obtained in measurements at slow grating rotationis shown in Fig. 4 as a solid curve and is compawith the measured results shown with solid dots. Figclearly shows a discrete resulting spectrum with thetensities|a±1|2 of the first satellites and energy splting in good agreement with theory.

4. Conclusion

An experiment of UCN diffraction on a movingrating was performed. This experiment is notdiffraction one in the true meaning of the worWhen grating is at rest or rotating slowly, the neutr

12 A.I. Frank et al. / Physics Letters A 311 (2003) 6–12

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coherence length is less than the grating period andobservation of the diffraction pattern is impossible.the case of fast rotation, the resulting spectrum wfound to be discrete in good agreement with theoAs already noted in Ref. [10], all interpretationsthese results by means of a real force that may chathe neutron energy have failed. Instead, the conof quasi-energy [24] and the uncertainty principEt ∼= h itself are much more appropriate for thinterpretation of this purely quantum effect. Speakin terms diffraction in time [1] or phase modulatioof the neutron wave is also correct in this case.addition, this experiment demonstrates the validof the Galilean transformation of the neutron wafunction in a new and very clear way. We believe ththis effect, which has been observed for the first timmay be used for controlled changing of the neutenergy in other experiments. The idea of neuttime-focusing, as proposed in [32], now looks morealistic.

Acknowledgements

This work was partly supported by the RussFoundation for Basic Research (project No. 00-17172) and INTAS (project No. 00-00043).

References

[1] M. Moshinsky, Phys. Rev. 88 (1952) 625.[2] A.S. Gerasimov, M.V. Kazarnovsky, Zh. Eksp. Teor. Fiz.

(1976) 1700.[3] R. Gähler, R. Golub, Z. Phys. B 56 (1984) 5.[4] J. Felber, R. Gähler, R. Golub, Physica B 151 (1988) 135.[5] J. Felber, G. Muller, R. Gähler, R. Golub, Physica B 1

(1990) 191.[6] V.G. Nosov, A.I. Frank, J. Moscow Phys. Soc. 1 (1991) 1.[7] R. Golub, S.K. Lamoreaux, Phys. Lett. A 162 (1992) 122.

[8] J. Summhammer, Phys. Rev. A 47 (1993) 556.[9] V.G. Nosov, A.I. Frank, Phys. At. Nucl. 57 (1994) 968.

[10] V.G. Nosov, A.I. Frank, Phys. Lett. A 188 (1994) 120.[11] A.I. Frank, V.G. Nosov, Ann. New York Acad. Sci. 755 (199

293.[12] A.I. Frank, D.B. Amandzholova, Ann. New York Acad

Sci. 755 (1995) 858.[13] A.I. Frank, V.G. Nosov, J. Phys. Soc. Jpn. (Suppl. A) 68 (19

13.[14] V.G. Nosov, A.I. Frank, Phys. At. Nucl. 62 (1999) 754.[15] J. Felber, R. Gähler, C. Raush, R. Golub, Phys. Rev. A

(1996) 319.[16] W.A. Hamilton, A.G. Klein, G.I. Opat, P.A. Timmins, Phy

Rev. Lett. 58 (1987) 2770.[17] E. Iolin, B. Farago, F. Mezei, E. Raitman, L. Rusevich, Phys

B 241–243 (1998) 1213.[18] J. Summhammer, K.A. Hamacher, H. Kaiser, et al., Phys. R

Lett. 75 (1995) 3206.[19] J. Summhammer, Phys. Rev. A 54 (1996) 3155.[20] Th. Hils, J. Felber, R. Gähler, et al., Phys. Rev. A 58 (19

4784.[21] A. Stean, P. Szriftgiser, P. Desbiolles, J. Dalibard, Phys. R

Lett. 74 (1995) 4972.[22] P. Szriftgiser, D. Guéry-Odelin, M. Arndt, J. Dalibard, Phy

Rev. Lett. 77 (1996) 4.[23] L.D. Landau, Ye.M. Lifshitz, Quantum Mechanics, Nauk

Moscow, 1974.[24] Ya.B. Zeldovich, Zh. Eksp. Teor. Fiz. 51 (1966) 1492.[25] A. Steyerl, H. Nagel, F. Schriber, et al., Phys. Lett. A 1

(1986) 347.[26] I.V. Bondarenko, A.I. Frank, S.N. Balashov, et al., J. Phys. S

Jpn. (Suppl. A) 68 (1996).[27] A.A. Seregin, Sov. Phys. JETP 73 (1977) 1634.[28] K.-A. Steinhauser, A. Steyerl, H. Scheckenhofer, S.S. Ma

Phys. Rev. Lett. 44 (1980) 1306.[29] A. Steyerl, W. Drexel, S.S. Malik, E. Gutsmiedle, Phys

B 151 (1988) 36.[30] I.V. Bondarenko, V.I. Bodnarchuk, S.N. Balashov, P. G

tenbort, et al., Phys. At. Nucl. 62 (1999) 721.[31] I.V. Bondarenko, A.I. Frank, S.N. Balashov, et al., Nu

Instrum. Methods A 440 (2000) 591.[32] A. Frank, R. Gähler, in: Proceedings of IV Internation

Seminar on Interaction of Neutrons with Nuclei ISINN-JINR E3-96-336, Dubna, 1996, pp. 308–312;A.I. Frank, R. Gähler, Phys. At. Nucl. 63 (2000) 545.