PHYSICAL REVIEW A VOLUME 42, NUMBER 7 1 OCTOBER 1990

Low-contrast and low-counting-rate measurements in neutron interferometry

H. Rauch, J. Summhammer, and M. ZawiskyAtominstitut der Osterreichischen Uniuersitaten, A-1020 Wien, Austria

E. JerichaInstitut fiir Kernphysik, Technische Uniuersitat, A 1040-Wien, Austria

(Received 16 April 1990)

Here we discuss the limits of the recognizability of neutron interference patterns either observedin low-contrast measurements or when collecting very few neutrons only. Low contrast can becaused by a strong beam attenuation or by a large phase shift applied to one beam path. Stochasticand deterministic cases have different influences on the interference pattern, which can be interpret-ed as a different wavelike or particlelike behavior of the system. Measurements of interference pat-terns with very few neutrons only are related to the phase —particle-number uncertainty relation,which is discussed on the basis of stochastic and quantum theory arguments. Analogies betweencoherent-state behavior known in atomic physics and the behavior of neutrons in an interferometerare discussed also.

I. INTRODUCTION

Neutron interferometry has been developed forthermal, cold, and ultracold neutrons. ' Since its inven-tion it has been used for many fundamental investigationsconcerning the wave-particle duality and the gravitation-al, electromagnetic, and strong interaction of the neutronwith its surroundings. Neutrons are massive particleswith many well-defined particle properties including aninternal quark structure, but they behave in neutron in-terferometry like waves according to the complementari-ty principle of quantum mechanics. All the performedexperiments belong to the regime of self-interference be-cause the phase-space density of any neutron beam is ex-tremely low (10 '

) and in nearly every case when a neu-tron passes through the interferometer the next neutronis still in a uranium nucleus of the reactor fuel. There-fore, in general, one observes, first-order interferencefringes that are caused by a variable phase difference P oftwo overlapping coherent beams. The degree of coher-ence is defined as the absolute value of the autocorrela-tion function of the overlapping beams, which can bedetermined experimentally from the visibility of the in-terference fringes '

1+ (~r(a)

~I

~r(O)

~ &cosP

with

r(a) =(1(*(o)y(a)&,

where 6 is the optical path difference and P the phasedifference of the interfering beams (P =k 6).

A reduction of the visibility can be caused by a nonuni-form phase shift across the beam, by a spatial shift of onewave train relative to the other in the order of the coher-ence lengths of the beam, " ' or by the attenuation ofone of the interfering beams. ' '" Most interference ex-periments aim high visibility at high counting rates in or-

der to determine the phase di6'erence as accurately aspossible. Contrary to those experiments we are here deal-ing in the first part with experiments that show a lowcontrast and in the second part with such ones of verylow counting rates. Both kinds are related to the ques-tion of the statistical and physical significance of an in-terference pattern and treat the limiting cases where in-terference phenomena can still be detected.

II. CONTRAST REDUCTIONBY BEAM ATTENUATION

The simplest way for beam attenuation is by insertingan absorbing material into one of the coherent beams ofan interferometer. The absorption process can be de-scribed by an imaginary part of the index of refraction,yielding a complex phase shift

p=p'+i/", (2)

with P'= Nb, XD and —P"=Na, D/2, where D is thethickness of the sample along the beam path, X is theparticle density, b, is the coherent scattering length ofthe sample, k is the wavelength of the neutrons, ando, =0., +0., is the total cross section including absorp-tion and scattering e8'ects in order to fulfill the opticaltheorem of general di8'raction theory. ' In the absorptionprocess of a neutron a compound nucleus is formed withan excitation energy of about 7 MeV, which decays by o.,P, or y radiation, which can be registered by variousdetectors. Absorption is an irreversible process causedby the statistical formation of the compound nucleus andcan be seen as the essential step for a measuring processand indeed any absorbed neutron causes a signal identify-ing the beam path the neutron has chosen. The residualinterference pattern reads as' '

I'I =I "(aE, +e2)+ —[(a + 1) '2+/ caosP'],

42 3726 1990 The American Physical Society

42 E MEASUREMENTD LO~-COUNTING RATE MLO~-CONTRAST A

x10

3727

P

—[( +1)+2a cosP'] .I =I"(aE]+E2)+2

a (4)

' the in-I" denotes the no 'robabjlitiesterfering p '

temming fromI) is given

ering part s em'

(in beamnoninterfer'

attenuation~

The beambe measure d sepa-

respectively.~ D) and canIrI =exp( '

ed completelby a

d beam is close.

hieved

s section or y aio the measureen-to-closed ratio

fd i (ti ):theo enh en and close reg'

MCQC

1.8

2-

2%'I

014v 6v

0. .3 0.4 0.'9 (mm)

0.2 0.

of the interference pattern ind(d ii i)d

'h

are ab-( oc astic) a

litude of thse roportion-

sord' h oda and &n

attenua-rst case as a, d

strong beamspecially for sd differences keep' g

ich ives, eein tei

the stochastic case p

mar ke i eem visible in t e s

ig ta er-eriments have beenRelated experi

b' S '

ff"t' Th' 'ntd to coherence eb measured w'

fading due oith and wi oAl han auxiliary potating

1pFi . 1. erobability o( )], i p

'thout the absor e. This demonstrat

ticle in t ees that an in

red that is considera yf th b b hi d

lfld if the statisttcarference

ven be obtaine ier than the inter et become larger r e

+")+ulated par er

ncerning'tude, i.e.,

new results concef th

elitude attenu ation o

shown in Fig. 2.b thto absorptio slow transmissio p

hih bhenomenperimen a

of fluctuations o n robe the presence oents are in progre

hd inboth beams, t eare inserte inbe w

testphoseshifter

Q 1.7C absorbe

detecto

4eI

0.2

8e-6e

0.3 0.4 0.5D (mm)

2%I

0. 0. 1

ithoutnce attern in forward (0) direction within one beamb b

'd

'n Gda sor

in time od with a strong

1 d to a measuringlues r re t(below). e'

. The intensity va u40 s.

ritten

+—[(a ++— +b)+2&ah cosP'],I =I"(ae]+b62)+2

+ = Q exp[ —(b, 5k, ) /2

1 to a.e an also be atttenuated

't rferom t c

Relatedearns in the in e

rocesses.a non-

or inco erenal lace in a n

thdispersive po

This additionaal coherent beam sp '-h deviated

e si nal.mentast e e

'measura

s not constitu ete a measureminto the origina 1 one.1 b' f'd b"kn in rincip e

iffraction romhffects occur inbecome la e e

Similar effe

n the widths othe autocorre a i

th t f1 nd therefore e'on Eq. 1 an,i h order is ch

b i'

d1a

d tib tiomentum iries as a u

'ilid h 6k the visibi ityturn widt s i it

f the phase shift P' asoft ep

~, (~krk) r2]

of scattering leng thp o

( =b) h i ib1 d

*

d d*

changed and the in er

I

t a ntum resolu-that the momentucr stali t rf o et

si nificant di erenff hthmetric, causing signi

3728 H. RAUCH, J. SUMMHAMMER, M. ZAWISKY, AND E. JERICHA 42

0.05

Gd

Q2-1.0

O.S—i&a

a 06—

Q.

&OC—

STOCAB

OE TEHMINIS TIC

ABSORP TION

I 0 005

02'

0 Q2 04 06 08 10TRANSMIS SION PROBABIL I TV o

FIG. 2. Normalized interference amplitudes for stochastic and deterministic absorption with special attention for the lowtransmission case (insets).

of the sample is placed parallel or perpendicular to thereflecting lattice planes of the interferometer crystal. ' '

Many related experiments have been reported in theliterature. "' The reduction of the contrast in the neigh-borhood of the 256th interference order caused by a35.006(17)-mm-thick Bi phase shifter imposed on aGaussian beam [A, =1.9225(24) A] is shown in Fig. 3.'

The obvious reduction of the contrast is caused partly bythe high-order coherence effect [0.12 from Eq. (6)], thebeam attenuation [0.675 from Eq. (3)], and due to theroughness of the surface (0.79). Thus it is partly deter-ministic and can be undone in the course of the experi-ment, and partly stochastic, which cannot be undone.

Similar results have been reported for photon experi-ments where the transmission of the reflecting mirrorswas varied. ' These measurements were discussed interms of an unsharp wave-particle behavior ''- or interms of unsharp measuring procedures, where the term"unsharp" has nothing in common with the term "inac-curate. " Various models mainly based on information-theory arguments claim different conservation laws forthe particle and wave knowledge. ' ' None of themcan account for all the phenomena discussed before.

Therefore an alternative formulation will be given in Sec.IV.

III. LOW-COUNTING-RATE PHENOMENA

In this case only few neutrons have been counted andthe phase difference P between the coherent beams can beobtained with rather large error bars only, which is relat-ed to the phase-particle-number uncertainty principle.In a realistic experiment the intensity behind the inter-ferometer is measured at K different positions j of thephase shifter for a certain period of time or monitorcounts. Interferometer theory predicts

N, =N[1+ Vcos(P~)] (7)

(b, , )'=N ( V —1)+2NN N—

and statistical theory yields for an estimate of pj, if themean counting rate N and the visibility V[V=( I&(&)l/I&(0)l ) of Eq. (1)] are known from otherexperiments,

11000—

5000—

/

I

JI

/

I

!'/

/

'glI

-02-2'8

I

-0.1

/Ii/

II

/

II

.I

35mmi BISMUTH

0I

00

4

I

/

I

/

'm ~Xf

OS 02

AI PHASESHIFTER

DISPERSIyESAMPLE BI

AOAI (mm)

FIG. 3. Interference pattern if a phase shift in the order of the coherence length is applied. The AD values belong to the opticalpath differences caused by the Al auxiliary phase shifter (Ref. 13). The intensity values belong to a measured time of 113 s.

42 LOW-CONTRAST AND LOW-COUNTING-RATE MEASUREMENTS. . . 3729

From the E different phase shift settings one obtains anestimate for the phase difference P by y optimization

K—1

(&p)'= g (&p, )

Approximating the sum by an integral over a definitenumber of interference fringes one obtains

(10)

where AN denotes the standard deviation of the totalcounting rate N registered at this detector, which obeysPoissonian statistics (hN=&X) as a basic feature of thesource emission process.

If one includes the counting rates observed by the oth-er detector, or if already in the measurements one usesthe constraint X"+X, =X (binomial scan), one obtains

V2

1 f + V—(1—2f)cosa —fV cos a~(1+&1 2f) ' —~ 1,

V —-1 1

2

P(No ~N=IVo+ NH =2 N=No+ IVH -2

5000-

2500-.

rg MnNo

e

r

re NMox

Mn0

~' NMoz

rr~

0--012 0 & 2 01234 0 &234 5

No

I)

P(NO J. N=NO+NH N =No+ NH -50

500-Nhfin

0 NM01o

NMH0

NMOzo

250-

0- r

2 8a20 10 A5 22 28 IO 20 8 &4 202632

BINOMIA L POISSON

FIG. 4. Measured counting-rate histograms for a binomial (left) and a Poissonian measuring method (right) for very low countingrates (+=2, above) and a medium counting rate (X=50, below). Each part is divided into a case where there is minimum and m»-imum counting rate in the 0 detector. Calculated values are shown as dashed lines.

3730 H. RAUCH, J. SUMMHAMMER, M. ZAWISKY, AND E. JERICHA 42

(bP) N=2 =2 (12)

where f is the average fraction of counts the detector in

forward direction (0) receives, and N, =EN is the totalnumber of neutrons counted. A comparison of Eq. (10)and (11) shows that using the counts of both detectors in-

creases the accuracy of the phase measurement ideally bya factor of (N/N, )' . A similar result has been estab-lished by a maximum likelihood analysis. A coupling ofthe interference counting rate to the monitor countingenlarges the uncertainty product up to 1% (for a monitorefficiency of 10 ). Similar results have also be obtainedby Yurke for an ideal Mach-Zehnder interferometer.

If the visibility is considerably smaller than 1, the term[1—(1—V )'~ ] ' can be approximated by 2/V, whichyields for monitor scans

and for binomial scans

N 2(bP) N, =

N" V-'(13)

The difference with the exact formula is of the order of5%%uo for V=0.5.

Related experiments have been performed at the inter-ferometer setup at the 250-kW TRIGA reactor in Vien-na, which certainly is an appropriate facility for low-counting-rate phenomena. In the first part of the experi-ment, histograms were taken of how many counts reachthe 0 detector for a fixed monitor counting rate. Manyrepetitive measurements were done to obtain the distribu-tion function P(N, ), where two positions j were chosen,one at the maximum and one at the minimum intensity inthis detector. For both cases a Poisson distribution is ex-pected

—X

P(N )= e (14)

Analogous measurements with a constant total countingrate yielded binomial statistics (N =N, +N, )

No-- 59V= 43+16%

N t x'P(N )= ' f '(1 f)—

No!(N~ —No)!J' J(15)

The experimental results show for both cases good agree-ment with the theoretical predictions (Fig. 4). Whereasin the binomial scans the total number of counts in bothdetectors is constant, in the monitor scan cases the meancounting rate N was chosen to be comparable to that inthe related binomial scans.

~ 04Zl

o l—

No=V= 31+23%

08-

04 —(iod

02-

0~0~ S02 So~ ~04 ~05

N =llV=50+39%

0.2-

8 D (mm)

t

0.10

02 ~04

+o

&05

FIG. 5. Characteristic interference pattern taken with rathersmall numbers of neutrons.

FIG. 6. Measured phase —particle-number uncertainty rela-tion compared to the statistic theory estimate for a binomial(above) and a Poissonian (below) measuring method.

42 LOW-CONTRAST AND LOW-COUNTING-RATE MEASUREMENTS. . . 3731

The same setup was used to extract phase informationout of fringes measured with only very few neutrons (Fig.5). These measurements were also repeated many times(up to 10000 times) and the data points were fitted bymeans of a least-squares-fit procedure according to Eq.(7). The results obtained are compared to the expectedbehavior in Fig. 6. The agreement with the predictions ofEqs. (10)—(13) is again fairly good. At the same timethese experiments demonstrate the buildup of the in-terference pattern from single-neutron events as it hasbeen demonstrated in a similar fashion for electron in-terference patterns.

~~ T=0.5

IV. DISCUSSION

I'( T&)=1—TI =1—& I

1('q"I &,

which yields the conservation law

P+ 8'=1,

(17)

and which describes all the phenomena discussed in Sec.II (Fig. 7). T denotes the general transparency of the sys-tem and represents losses which, in principle, areequivalent to having detected a neutron at a certain posi-tion in the interferometer, and Vis the visibility of the in-terference pattern. The averages have to be taken overthe beam cross section. The different cases are discussedin Ref. 31.

The formulation of the phase —particle-number uncer-tainty relation in Sec. III is based on standard probabilitytheory and has to be completed in the case of a morerigorous treatment including quantum-mechanicaleffects. The fermion character of neutrons does notappear directly because the low intensities involved in

any kind of neutron interference experiment makeparticle-particle interaction negligible. In this caseFermi-Dirac statistics approaches Bose statistics becausesuccessive neutrons are generally separated in time,space, and momentum. Such systems are generally de-

As mentioned in Sec. II the measured reduction of theinterference contrast due to absorbing or scattering phaseshifters can be visualized as unsharp wave-particle behav-ior of the system or alternatively to an unsharp measur-ing procedure. In this respect the behavior of thesystem changes with increasing absorption from a purelywavelike interference one to a particlelike one with dis-tinct beam path detection.

A measure for the wavelike nature can be found fromthe correlation function of the interfering beams [Eqs. (1)and (7) with N =Io T, which defines the transparency ofthe system]

~(TI ) = TV =& lq'q"

I &,

and for the particlelike nature

FIG. 7. Synopsis of wavelike and particlelike character ofneutrons in different experimental situations.

scribed by coherent states whose associated numberstates are Poisson distributed and whose phase-difference —particle-number uncertainty relation can bewritten as

oo N )1

(gy)z 1—N N2 —4N

o n!v'n+1 (19)

ACKNOW LEDGEMENTS

The financial support from the Fonds zur Forderungder Wissenschaftlichen Forschung (Project No. S 4201) isgratefully acknowledged.

which reduces for N &) 1 to the statistical limit discussedbefore [Eq. (8)], but which deviates from that value forN~O due to projection states onto the vacuum state.Similar behavior has also been verified in the analysis oflow-counting-photon experiments.

The analogy between coherent states and free, butcoherently coupled, particle motion inside an interferom-eter addressed here needs further justification. By weak-ening the harmonic potential, which is generally used todefine coherent states, the characteristic level structuredisappears and reaches the limiting case of a momentumdistribution function which characterizes the free-movingparticle beam. The consistency of the coherence lengths,which are determined by the momentum distributions,can be seen as an analog to the uniform wave-packet phe-nomena in coherent atomic states. Additionally, a timeaverage has to be taken instead of an ensemble averagewhich is equivalent in any ergodic interpretation of quan-tum mechanics. Thus the coherently split k states (k andk+G, where G is a reciprocal-lattice vector) with aphase difference P exhibit properties similar to coherent-state phenomena.

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