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Neutron interferometry
Samuel A. Werner
Citation: Physics Today 33, 12, 24 (1980); doi: 10.1063/1.2913855 View online: https://doi.org/10.1063/1.2913855 View Table of Contents: https://physicstoday.scitation.org/toc/pto/33/12 Published by the American Institute of Physics
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Neutron interferometry

Coherent beams of neutrons, split and recombined by Bragg diffraction in a perfect single crystal of silicon, demonstrate effects on the phase of the wavefunction due to gravity and other phenomena.

Samuel A. Werner

Diffraction effects at wavelengths on
the order of angstroms have been known since Max von Laue's demonstrations of x-ray diffraction in 1912. Interference between well-separated, coherent beams, however. is much more difficult to arrange.
In 1965 Ulrich Bonse and Michael Hart (then at Cornell University) were able to obtain interference effects between beams of x rays with a wave-
length of about 1 A and spatially sepa-
rated by about 1 cm. They used a perfect single crystal of pure silicon to split an x-ray beam into two coherent parts by Bragg reflection, and then used further Bragg reflections to re-
combine the beams. When they varied the optical path of the x rays in one of the beams they found oscillations in the intensity of the recombined beam. This remarkable achievement opened up the field ofinterferometry to the region of angstrom wavelengths and raised the question of whether one could use
the same principles to obtain interference effects between coherent beams of thermal neutrons, which are diffracted
by crystals in the same way as x rays. Helmut Rauch, Wolfgang Treimer
and Bonse finally demonstrated neutron interferometry in 1974 in a series ofexperiments at a small reactor at the
Austrian Atomic Insti tute in Vie nna.' An earlier attempt, in 1968, by Heinz Maier-Leibnitz and Tasso Springer to
construct a neutron interferometer based on diffraction by a s lit and subsequent deflection by a biprism proved only partially successful.
Obvious ly, the principles upon which neutron interferometry is based are very different from those applied in
optical Interferometry, in part because we are dealing with much smaJler wavelengths and in part because we are dealing with neutrons, For these shorl wavelengths one generally uses Bragg reflection from crystal planes to spliL and recombine the beam; lo ensure the coherence of the recombmed beams the crystals must be large and alm,1s1 pt,r-

fectly free from lattice imperfections. Obviously it is not possible to polish and align the optical surfaces to fractions of a wavelength (in this case fractions of an angstrom) as one does for visible light. However, the use of Bragg reflection from sections of large, perfect crystals circumvents this difficulty and permits ready observation of interference bet ween the two beams. With neutron interferometry the phase of neutron's wavefunction, r/,, becomes directly accessible to measurement, whereas earlier only the amplitude-or, rather the probability density, !1/Jl2-was directly measurable.
Since 1974 a number of experiments have been carried out that use the Banse-Hart type of interferometer to probe tb.e phase of the neutron wave-function. Figure 1 shows the interferometer and a typical experimental arrangement. Among these experiments have been: ► demonstrations of the effects of the Earth's rotation and gravitational field on the neutron phase, as predicted by the Scbrodinger equation ► measurements of lhe neutron-nucleus interaction potential ► demonstration that a fermion wavefunction reverses its sign after a rotation by 211" and is only restored to its initial phase by a rotation through. 4r,. ► searches for non.linear variants of a Schrodinger equation.
In this a rticle I will give an overall review of these experiments in an attempt to convey the beauty and s im• plicity of this technique in probing certain fundamental aspects of quan-
tum physics. r will also speculate on
fut ure experiments to give some flavor of the scope of new applications.
A neutron interferometer
Va rious schemes have been proposed, and to some extent realized, for obtaining interference effects between spatially separated coherent thermal neutron beams havi ng a wavelength in l he angstrom range. l will limit myself

here to considering the sort o( arrange-
ment shown in figure 1 and schematically depicted i:n figure 2. The interferometer consists of three identical, perfect slabs of silicon, cut perpendicula r to a set of strongly reflecting lattice planes, typically the (220) planes; the slabs are machined from a single silicon crystal to ensure the perfect alignment of the crystal planes from slab t-0 slab. The distances between slabs are usually a few centimeters and must be equal to within abou t a micron, A nominally collimated, monochromatic beam is directed from the source to the first slab of the interferometer (point A in figure 2), where it is coherently split
by Bragg reflection. The two resulting beams are split by the second silicon slab in the regions near points Band C. The central two of these four beams overlap, in the region of point Don the third silicon slab. The two beams are each partly reflected. so that the "G" and "0" beams leaving the third slabs are coherent superpositions of tbe beams, I and II, that have passed through the interferometer.
lf the beam traversing path l is shifted in phase by an amount /J with respect to the beam along path 11-by some interaction that increases the
optical path length-the intensities measured by the detectors Cl and C
3
will change. The expected intensities at tbese two detectors are of the form

l 2=r- acos{J

( 1)

T3 = a (l + cos /J)

(2l

whe re f3 is the difference in the relative phase of the wave functions between paths I and LI, and a and 1' are constants t hat depend upon the incident

flux, the crystal structure and the neut ron- nuclear scattering length of silicon. Equations1 and 2 predfot that tb.e neutron current is "swapped" back end

forth between C2 and Ca as the phase shift fJ is varied. Note that the con•

Sarnuel Werner Is professor of physics at the University of Missouri, Columbia campus.

24 PHYSICS TODAY / DECEMBER 1980

I 0 0 3 1 - ~

t B O

I 1 2 0 0 2~ 7

500 50

CJ

1 $ B O A m 8 ' > c a n I O S 1 1 1 u1 e o f P l w > a

strast of the interference observed by counter C9 is, in principle, 100%, while the contrast in 12 can never be 100% (a is always less than y). One generally records the output from both coU'nters to enhance the rate at wMch signifi-
cant data are collected. (Our graphs, however, will only show the experimental points from one counter.} The detector C, counts the neutrons in the non-interfering beam directed along
the Line AC: it serves as a reference. Although the basic principle of this
interferometer seems simple enough, we must remember that there are perhaps 109 oscillations of the neutron wave on each of the paths. Thus, there
areclearly verystringentrequirements on the thermal and microphonic stabil-
ity of the apparatus. It is necessary to isolate the interferometer from the vibrations in the reactor hall, and to maintain an isothermal enclosure around it. The effect of vibrations is much more important for neutrons than for x rays because the transit time of neutrons across the interferometer (typically on the order of50 microsecJ is much longer than that ofx rays, which travel with thespeed oflight. Thus, for a neutron interferometer. if the length
of path I varies (because of vibrations, say) relative to path 11 by as much as a
fra.ction of an angstrom during the 50 mforosec transit time, the interference fringes are wiped out. To preserve the Bragg reflection condition for a given wavelength, the three crystals must be aligned to within the "Darwin width" of the beams. (This width is the angular width of the incident beam over which strong Bragg reflection occurs; for neutrons it is on the order of0.1 sec of arc.) Bonse and Elart achieved this alignment in a simple and ingenious way: they cut all three slabs from a large,
monolithic single crystal. As a conse-
quence of great advances in crystal growth techniques, prompted by the
needs of the solid-state electronics industry, it is possible today to purchase (at a modest cost) silicon crystals of the required perfection with typical dimen-
sions of 5 to 10 cm from commercial manufacturers. The accuracy with which the surfaces of the slabs need to be polished is not as severe as might be
anticipated. The reason for this is that the index of refraction of silicon (or most any other material) for thermal neutrons di.ffers from 1 by only a few parts in 106• Calculation shows that a step of two microns on the surface of one of the slabs causes a phase shift of only 1/1110 of a fringe for 1.4-A neutrons. Thus, the requirements on polishing are very similar to the requirements in ordinary optical i_nterferometry. Finally, there is the question ofthe ext~nt to which the incident beam must. be monochromalic. Because the interferometer is based on Bragg reflection, the

The neutron Interferometer used in the author's laboratory at the University of Missouri. The

upper photo shows the s1l1con slabs machined from a single crystal of high-purity s,hcon to

maintain alignment ol lhe crystal planes from slab to slab. The lower photo stiows such an

interferometer in place In Its housing at the reactor The entrance silt ,sat the letl and the three

counters C,, C2 and C1 (see hgure2) are on the right

Figure 1

PHYSICS TODAY / DECEMBER 1980 25

wavelength a long a given trajectory (ray line) must be defined to within abouLone pa rt in 10''. However, this
definition is accomplished by the interferometer its elf, and not through the

La11,ce planes

d,

B

Phase SlllllP

Countet c,

preparation of the incident beam. Thus

one can carry out the experiments witb

beams forwh1ch !M /J.. is on the order of

l %, which can be provided by nuclear reactors with standard techniques for

Gbeam

producing monochromatic beams. For

a complete understanding of the quan-

titative performance of the interferom-

Source

eters, one must analyze the diffraction

of neutrons within slabs ofl)erfect sin-

gle crystals using the dynamical theor y

of diffraction?

Coherent scattering The scattering of thermaJ neutrons

Aelerence beam
Counler C,

by nuclei is s-wave scattering because the neutron wavelength is much larger (by a factor of about 10") than the raclius of a typical nucleus. Another

Schematic diagram of the neutron Interferometer shown In figure 1. The lattice planes, usually

the (220) planes, are continuous from slab to slab, and the dimensions a, d, and d2 are machined

10 optical precision. The phase shift In path Ican be produced by Inserting a gascell, a slab of

solid material or a magnetic field Into the marked region.

Figure 2

way of seeing this is as follows: Tbe

angular momentum of an incident neu-

tron (mass rn. speed u) is on the order of collaborators from the Austrian Atom- larger the scattering length, the more

muR0, where R0 is the radius of the ic l.nstitute in Vienna and the Universinucleus. Because u is on the order of ty of Dortmund in Germany have car2200 m/ sec and R0 is around 10- 12 or ried out a series of precise experiments 10- •3 cm, the angulai: momentum is at the Institut Laue-Langevin in Gre-
much smaller than fiv l(l + 1) for l = l noble, France, to measure scattering
or larger. Thus only the s-wave Cl = 0) lengths of nuclei, using neutron inter-

rapid the interference oscillations. For example, for heliu.rn-4 the scattering lengthis 3.26 X10- 13 cm (or 3.26 fenni)
with a precision of 1% . For deuterium
the scattering length is 6.55 fm, and the frequency of oscillation with atom den-

component of the incident beam inter- ferometry. The principle of the tech- sity is clearly more rapid. The scatter-

acts w:ith the nu.clear potential. The nique is straightforward: ff one of the ing length for hydrogen is negative.

outgoing wave is therefore a spherical coherent beams traverses a sample of This is known from other experiments,

wave, of the form

thickness t and refractive index n in butthesign ofthescattering length can

be 1••1r

one leg of the interferometer, then the two beams will differ in phase by

The parameter bis called the coherent scattering length; it is proportional to

/J = 217{ l - n )t / A

(3)

the strength of tbe neutron-nucleus when they recombine. Here A is the

interaction. The total scattering cross neutron wavelength. The index of resection is 41rb2 , so that (except for tbe fraction n, is given by

factor of 4, which is a quantum-mechanical effect) bis an effective radius

71=1 - NbA2/ 21r

(4-)

of the scattering nucleus; for repulsive where Nis the density of atoms, so the

potentials b is positive while for attrac- phase shift is proportional to t he scat-

tive potentials it is negative.

tering length:

Traditionally, one has determined. tbe scattering lengths of nuclei from

/3 = Nt.J.b

diffTaction data from single crystals or For solids one can insert a slab of

from polycrystals; some experimenters materiaJ in one of the arms and rotate

have also used mirror-reflection or re- it about an axis perpendicular to the

fraction of neutron beams to determine plane of the interferometer, thus

scatterrng lengths. These experiments changing the effective thickness I and

typicalLy have a precision of few per- inducing interference osciJJations in

cent. Tbe neutron interferometer pro- the counting rates observed in detec-

vides us with a new and very precise tors Czand C3 . For gases one inserts a technique for determining scattering gas cell into one of the arms and ob-

lengths (sometimes to within one part serves the interference oscillations as in 1041, free from the usual uncertain- one varies the gas pressure.

ties of the Debye-WaJJerand extinction Figure 3 shows a few of the beautiful

effects always present rn crys tal dif- results« that the Vienna- Dortmund

fract10n experiments. (The Debye- group have recently obtained at tbe

Waller effect is the change in the inten- l nstitut Laue-Langevin. These experi-

s ity of Bragg reflections due to the ments were performed with a gas cell in

t hermal moLions of t he nuclei: t,he ex- one ofthe interferometer arms; one can

ti nction effect lS t he change in the compute the density of atoms from the

Bragg reflections due to a bsorption and (measured) pressure, with corrections

multiple scattering).

from the known temperatu re depen-

Over the past five years a group of det1ce of the virial coefficients. The

also be determined in an interferometer experiment by using a quarter-wave plate, made, say, of aluminum (0.05
mm thick), for which the sc.attering length is known to be positive a:nd observing whether theintensity oscillations shift to the left or to theright bya phase of 90° in graphs like the ones shown in figure 3.
To calculate the scattering length from .first principles one must solve the problem of the interaction of the neutron and all the nucleons in the scattering nucleus. This can be solved exactly only for two bodies, that is for scattering from hydrogen. There are some very useful results for tbe three-body problem (neutron-4euterium scattering), and recently developed methods for calculations of the few-body problem to allow a more fundamental treat;. ment of the four-body problem. V. F. Katchenko and V. P. Levasbere in Russia have carried out a detailed analysis
using the Faddeev-Yacubovsky equations with a charge-independent separable central potential. Neutron scattering from heliu.m-3 and from tri-
tium are experimental realizations of the four-body problem. In the case of He3, a strong effective attraction ex.ists in the state where the compound nucleus has total spin of zero; it depends
markedly on the details of the nuclear force. Because of the large cross sec-

26 PHYSICS TODAY I DECEMBER 1980

tion for neutron absorption by helium3, measurement of the scattering length is quite difficult. However, the
group at Grenoble has carried out a careful neutron-interferometry experi-
ment and has obtained reasonable agreement with the theory. They are currently involved in measuring the scattering length for tritium, with the
idea of gaining insight into the charge dependence of nuclear forces.
The neutron interferometer mea-
sures the average neutron-nuclear potentialofthesamplethrough which one ofthebeams passes. Because the interferometer is very sensitive lo small changes in scattering length, it is also sensitive to smallchanges in the composition ofthe scatteringmaterial. Thus, for example, by comparing the interfer-
ence pattern obtained from a metal sample containing hydrogen with that from a pure metal sample, the Grenoble group has been able to measure the hydrogen content in samples of various transition metals to a precision ofabout 0.05 atomic percent-asensitivity com• petitive with the best analytical-chemical techniques.

Hydrogen
'\IV

Deuterium

I '\1\1\/\J

[

.ii

t$ . . - - - - - - - - - -- - --

~"'
~

~"-../~

Helium

Quantum interference due to gravity

ln most phenomena of interest in

physics, gravity and quantum mechan-
ics do not simultaneously play an important role. However. the neutron interferometer is sufficiently sensitive

JWM/\

to detect the tiny changes in the phase of the wave function that arise from

N,trogen

I

I

changes in gravitational potential ener• gy. Here at the University ofMissouri

0

5

10

Research Reactor a group consisting of

DENSITY (10"' pa~tclestcm'I

Jean-Louis Staudenmann (now at Iowa State University), Roberto Colella and Albert Overhauser (both of Purdue University) and myself has recently

Interference oscillations (or various gases. As the diagram at top shows, a gas cell introduces a phase difference between paths I and II. The rate at which the phase changes

carried out• a pTecision experiment w1lh density serves to measure the scattering

whose outcome depends necessarily length of the gas.

Figure 3

upon both the gravitational constant

and Planck's constant. Preliminary

experiments were carried outfive years segment and a sloping segment. The

ago at the UniveTSity of Michigari. average gravitationaJ potential of the

Figure 4 shows the overall setup at s loping segments is the same. but de-

the Missouri reactor. The thermal- pending on the ani,;le </>, the horizontal
neutron beam emergesfrom the reactor segment for path n will be either a bove
through a helium-filled tube. A pair of or below that for path I. The difference

pyrolytic graphite crystals serves as a in the Earth's gravitational potential monochromator for the beam; this dou• between these two levels causes a qua n-

ble crystal monochromator allows one tum-mechanical phase shift of the neu-

to carry out experiments at various t ron wave on path J relative to palh ll.

neulron wavelengths. The direction of To calculate the phase shift, one simply the incident beam is fixed along lhe uses the de Broglie relat.ions b_ip be-

local North-South axis ofthe Earth; we tween the momentum p a nd wa ve-

will see that this is important in these length ,l of the neutrons:

e x:pe_rimen ts.
The ex:perimental pr ocedure in-

p= hli.

(SJ

volves turning the interferometer, in- The momentum de pends on the height,

cluding the entrance slit and the three z, of t,he neutrons because energy is

detectors C1, C2 and C:i, about the incl· conserved:

dent beam, as s hown in Figure 5. At each angular setting ¢,, neutrons a re

E =p~l2m,+ m~gz

(6 )

counted for a preset length of time. where m, is t he inertial mass and m~ 1s

Paths l and JI each bave a horizontal the gravitational mass of Lhe neutron.

g is the acceleration due to gravity.
From these relationships it is straiglltforward to show that the phase shi~ is
/3~ = - 211m , m. \ glh2 )M sint/i (7)
where A is the areaenclosed by the two beam trajectories. Thus, as we turn the interferometer through various angles r/J about the incident beam direc-
tion, always maintaining the Bragg condition, we expect to see oscillations, induced by the Earth's ,gravitational field, in the outputs from detectors 0.,. and C3 .
The results of an experiment carried out with incident neutrons of wave-
length ,l =1.419 A is shown in figure
Sb. The contrast of the interference pattern dies out with increasing rota· tion angle because the interferometer warps and bends slightly under its own weight (on the scale of angstroms) as it is rotated about the axis of the incident beam, which is not an axis of elastic
sy=etry of the device. This bending effect also causes a small change of the period of the main oscillations. These effects have been studied and quantitatively measured with in-situ x-ray experiments, in which x rays are directed alongthesame incidentbeam path and the interfering x-ray beams are observed as a functiori of rotation angled,. The Rlfect of gravity (gravitational red shift) on x rays over the distances involved in the interferometer is negligible; one consequently assumes that the phaseshiftsobserved a.redueonlyto the bending of the interferometer. The frequency of oscillation due to bending as measured by x rays can be subtracted f-rorn the frequency of oscillation measured with neutrons, leaving only the effects of gravitationaJly induced quantum interference. We have carried out an extensive series of measurements at various neutron wavelengths. the agreement of our most recent results with theory is at the 0.] % level.
This interference experiment demonst.rates that a gravitational potential coherently changes the phase of a neutron wavef'unction. Furthermore Daniel Greenbe rger (Cit.y College of .New York) and Overhau.ser have de• rived5 a result like equation 7, but with m,·J instead of m, m ., for the
s ituation when the neutr~n source, beam slits a nd the inte rferometer have a uniform acceleration g. In comparing our experime ntal results witb the theoretical res ults we have used the neutron mass as measured in a mass spectrometer (again, essenlially m, 2 instead of the combination of m. m~ t hat a ppears in equation 71. Thus, the agreement of our experiment with equation 7 provides t he first verification of the equivalence principle in t he quant.um limit.
In 1925, Al bert Michelson, Reary Gale a nd Fred Pe.arson carried out a n

PHYSICS TODAY / DECEMBER 1980 27

Poma,ybeam collima101

lnterferome1er

Fuel elements REACTOR

MONOCHROMATOR

monocnromator c,ys1als

Overall arrangement of the neutron-lnterterometry experiment at the University of Missouri reactor1n Columbia. The small box at rightcontains the equipment shownin figure 1. Figure4

heroic experiment designed to detect the effect of the Earth's rotation on the s peed of light.. They constructed an interferometer in the form of a rectangle 2010 ft '<. 1113 ft a nd were able to detect a retardation of light due to the Earth's rotation corresponding to about 'I• of a fringe, in agreement with the theory of relativity. The French scientist Maurice Sagnac had demonstrated in 1913 that rotational motion (unlike rectilinear motion) can be detected with optical interferometry. The physical principle involved in t he Sagnac effect is, in fact. t he basis for the modern ringlaser gyroscope.
Neutron Sagnac effect
In view of t he differences in the way light waves and matter waves behave under coordinate transformations, it is
not obvious that an analogous quantum-mechanical effect should exist for neutrons. Because the gravitationally induced quantum interference experiment is carried out on the surface of our rotating Earth, a noninertial frame, the Hamiltonian governing the neutron's motion contains a term dependent on the a ngula r velocity of the Earth, w. and the angular momentum L, of the neutron's motion a bout the center of the Earth. This term, - w-L, gives rise to the classical Coriolis

force. Although this force has an ex-
ceedingly small effect of changing the neutron's trajectory in the interferom-
eter, its effect on the neutron phase is not small. Calculation shows that t he
phase shift in the interferometer due to the Earth's rotation is expected to be6

8, = 2m ,w-A l fi

(8 )

Here A is the normal vector for the a rea enclosed by the beam trajectories in the interferometer. The magnitude
of 8, is about 2% of 83 , which should
have been easily observable in the precision experiments described in the last section. The effects of gravity and rotation can readily be separated because the two phase shifts change in very different ways as one changes the interferometer's axis (the incidentbeam direction) from the local northsouth direction. In fact, as one can see
from equation 8, /3, vanishes if the area
vector is normal to the Earth's axis of rotation. On the other hand, if the
interferometer axis is vertical. the gravitational shift remafos constant as the interferometer rotates, because of the symmetry of the situation.
Staudenman n, Collella and l performed a n experiment with a vertical incident beam at Missou ri, measuring the phase shift as a function of the interfe rometer's or ientation angle
about the vertical direction. Because

{3~ remains constant, tbis experiment is

a dlrect test of equation 8. The result

is shown in figure 6. When the normal

area vector, A, points east or west, t he

phase shift vanishes, while if it points
north or south the phase shift is +95'

or - 95', respectively. T his result is in

reasonable agreement with theory
which predicts it should be ± 92".

When we began this experiment we

did not know the location of the north-

south axis of the Earth relative to the

reactor hall. Because the buildingcon-

tains a great deal of steel and magne-

tite concrete, a compass is useless in

determining true north. We finally

solved this problem by using a telescope

to sjght on the star Polaris outside the

reactor building, and then using preci-

sion surveying t-echniqu.es to carry this

line ofsightinto the reactor hall (which

is below ground level J.

Because the results of this experi-

ment depend only on t.he inertial mass

ofthe neutron m ,, and the resu.ltsofthe

gravity experiment depend on theprod-

uncattimon1mo1t

, one can interpret the combithe two experiments as inde-

pendent measurements of the inertial

and gravitational neutron masses in a

quantum-mechanical experiment.

Recently Max Dresden and Chen-

Ning Yang (SUNY at Stony Brook)

have given7 an interestingderivation of

equation 8 by considering the phase

shift of the rotating interferometer as

ar ising from the Dopp!er shift due to a

moving source and moving reflecting

crystals. According to the general the-

ory of relativity there are phase shifts

of uery small magnitude in addition to

/3~ and /3, which become larger as the

neutron velocity increases. In particu-

lar, there is a phase shift due to the

coupling of the neutron spin to the

curvature of space-time. These effects

have been studied t heoreticallyll by

Jeeva Ana ndan (University of Mary-

land) a nd, independently, by Leo Sto-

dolsky (Max Planck Institut, Munich).

Whether or not experiments can be

carried out in this velocity regime is an

open question; its answer will probably

require new technology.

Magnetic effects
The operator for rotation through 2,r radians causes a reversal of sign of the

¥ 8 6

Gravity-Induced interference pattern, found by rotating the Interferometer about a honzontal axis fDala obtained by Werner. Staudenmann. Colella and Overhauser. Figure 5
28 PHYSICS TODAY I DECEMBER 19B0

40

32

24

16

8

0

8

ANGLE ~ 1deg1ees)

16

24

32 40

wavefunction for a fermion. Although

this principle is well known and is

deeply imbedded in quantum theory, it

had not been directly tested before the

development of neutron interferome-

try. An experiment to observe this

effect with the neutron interferometer

was suggested in 1967 by Herbert Bern-

-50

stein9 (then at the Institute for Advanced Study, Princeton). ln 1976, nearly simultaneously, both the Grenoble interferometry group in France and our group in the US performed the

i
~
r t

experiment.10 Anthony Klein and Geoffrey Opat (University of Mel-

[/)
~

bourne) later demonstrated 11 the same

ft

principle very nicely using a novel

50

Fresnel-diffraction interferometer.

The basic idea of the experiment is

that after a neutron (or other fermion)

rotates through an angle of2ir, the sign

100

of the wave function changes, or, in

other words, the quantum phase shifts by 1r. We use a magnetic field to change the orientation of the neutrons

0 From reactor
Jf Beryll1um crystal

90

180

270

360

DIRECTION ,l(degrees)

in one arm of our interferometer and

observe the resulting phase shift. The

geometry of the experiment is shown in fignre 7. If the neutron travels for a distance / through a magnetic field B,

Neutron Sagnac effect: phase shift due to the Earth's rotation. The graph shows the phase ditference between paths I and II as a function of the orfentation of the normal vec'lor for the area
enclosed by the paths. The data were obtained by Staudenmann, Werner and Colella. Figure 6

one expects its phase to shift by an

amount

Pm= ±2trgnµnm)..Bl/h2

r9)

usual interpretations of quantum theory, in particular, the Born interpreta-

Here the ± signs are for spin-up and tion of the wavefunction, Galilean in-

that point, the integrated effect of the
intensity-dependent potenitial on the neutron phase is

spin-down neutrons; Cn is the neutron variance, and the conservation of promagnetic moment in nuclear magne- bability. Furthermore, they concluded

Pn1 = b(l/uh) lnla l2

(11)

tons ( - 1.91), µn is the nuclear magne- that the function F should be logarith- Here u is the neutron velocity, and a2 is

ton and m is the neutron mass. Put- mic to obtain physically attractive cor- the i ntensity attenuation ofthe absor b-

ting in the numerical values, we expect relations between non-interacting par- ing plate.

that for a precession of 41T (or a phase ticles. Thus, in the case or a single- Clifford G. Shull and his col'labora-

shi~ of 2-ir)

particle Schrodinger equation they pro- tors at MIT have recently carried out

BM = 272 gauss cm A

posed thal a term of the form

We show in figure9 the data from the

- bl/I In (a3l1/!l2)

(10)

two beautiful experiments to test these ideas, using a two-crystal interferometer and absorbing plates of lithium flu-

original Grenoble experiment, in which should be added to the usual Linear oride and cadmium. They detected no
the neutron wavelength was 1.82 A. Schrodinger equation. The length a phase shift. This experiment places an

The graph clearly shows that a phase need not be a universal constant, be- upper Limit on the value of the funda•

shift of 21r corresponds to a precession cause changing its value is equivalent mental constant b of 3.4 '< 10- •a eV. A

of 41r radians. The neutron wavefunc- to adding an unobservable constant Fresnel-diffraction experiment carried

tion thus is indeed a s pinor. In classi- potential to the Hamiltonian The en- oul by R. Giihler, A. G. Klein and A.

cal physics, of course, no such behavior ergy constant b, however, must be a Zeilinger at the lnstitut Laue-Langevin

is possible: a rotation through 21r ex- universal constant, the same for all

actly restores the origina I state_

systems. They estimated from experi-

mental data on the Lamb shift in hy-

Nonlinear SchrOdinger equations

drogen that b must be less than

placed an even lower limit of 3.3 >< 10 16 eVonthevalueofb. Thisis more than Jive orders of magnitude
lower than the limit previously in-

Last year, Abner Shimony (Boston 4 x 10- 111 eV.

ferred from Lamb shift data!12

University) suggested thal the neutron Shimooy pointed out that merely interferometer offers sufficient sensi- repositioning an intensity-attenuating Speculations on future applications

tivity to test the physical reality of plate downstream in a particle beam In a field as new as neutron interfero-

certain nonlinear variants ofthe Schro- should cause a change of phase not metry 1t is. ofcourse. difficult to predict

dingerequation, in which an additional predicted by a linear Schrodinger equa- the future. Over the past several years

term of the form F <i1bl2) appears in the lion: The factor - b ln(oJltbltl in equaHamiltonian. In earlier theoretical tion 10 can be simply viewed as a small studies, Iwo Bialynicki-Birula and J intensity-dependent potenltal. and beMyscielski had concluded that, unlike cause all potentials cause phase s hifts the linear Schrodinger equation, some in an interferometer experiment, this of these nonlinear equations can have one should also. A straightforward solutions in which traveling wave pack- calculation shows that if one positions ets do not s pread out in space with an aLtenual ing plate first at a point in

a large number of "back-of-the-envelope'' ideas for experiments have been suggested. An international workshop on neutron interferometry13 was held in Grenoble two years ago, at which time some of these ideas were put forward. 1n the way of a conclusion. I will bnefly discuss some of these s ug-

time. These nonlinear equations. how- one of the beams of an interferometer gestions.

ever, allow us to retain many of our and then downstream a distance / from Quantum theory rests on the princi-

PHYSICS TODAY I DECEMBER 1980 29

pie of superposition, which states that

if ,f,1 and ,/,2 a re two possible stat.es of a system, and c, and c2 are arbitrary
numbers, then c1tb, + c2,t,2 is also a pos-
sible state of the system. It is usually

taken for granted that the coefficients

c1 and c2 are complex numbers. However, it is possible to imagine a quan-

tum theory based onquaternions (num-
bers of the form a + I b + Ic -t k d),

Asher Peres (Technion, Israel) has pro-

posed a simple test (at least in princi-

ple) to see if the neutron-nuclear scat-

tering lengths are strictly complex

numbers or whether they have an addi-

tional "dimension" allowed by a qua-

ternion quantum theory. The idea is to

use two flat plates of two different

materials, say A and B. Placing materi-

al A in one leg of the interferometer

will ca.use a pbase shift, say AA, while

placing material Bin the interferomet-

er will cause a phase shift t. 8 . Placing both slabs in one of the interferometer

beams together will cause a phase shift
AAB = 6.,,., + t.8 , ifthe neu.tron scatter-
ing lengths are complex numbers. How-

ever, ifthey have a small "quaternion"

compon
A11. + A8

ent, A,._ 8 will not ; furthermore, because

equal quater-

nion multiplications do not commute,

we expect 6.,.,,8 :jA8 ;0. so we should expect a difference if we interchange the

order of A and B with respect to the

direction of the beam. It is obvious

that the effects to be searched for are

very small; otherwise they would al-

ready have been detected in neutron

ditfracti<1n crystallography.

In 1853 Fizeau carried out an expeci-

ment designed to measure the velocity

of light in a moving fluid. The result

was found to be in agreement with

Fresnel's calculation, based on elastic

vibrations of a stationary ether. We

now know t hat tliis result can be de-

rived from conventional Maxwell elec-

tromagoet.ic theory, and. that it .is

therefore consistent with the special

theory of relativity. Several people,

among them Michael Horne {Stonehill

College), Anton Zeilinger (Vienna}and

Anthony Klein (Melbourne), have sug-

gested carrying out the analogous

quantum-mechanical "Fizeau experi-

ment" using the neutron interferomet-

er to measure the phase shift in a

moving medium. 1n fact, experiments

along these lines are under way in

Grenoble. Of course, we think we

know how to predict the result, but

maybe we are over-confident.

Richard Deslattes (National Bureau

of Standards) has proposed construct-

ing a Michelson-type interferometer

and repeating the Michelson-Morley

experiment us ing neutrons.

Greenberger and I have s uggested

that canyi.og out an experiment to

measure gravitationally induced qua n-

tum interference with ult ra-cold ne u-

trons could give sur prising results. be-

0

20

40

60

BO

CURRENT lmAI

Demonstration of splnor rotation. The

magnetic field Induces e rotation of the neu-

tron sprn and changes the phase of the

neutron wave function: a rotation through 4,r

produces phase shift of 21r.

Figure 7

cause the neutron trajectory would no longer be uniquely defined, and the usual WKB techniques for calculating phase shifts fails in the limit of zero velocity. U one could construct an interferometer th.at works with these very slowly moving neutrons, it is conceivable that it would also be usefulin a precision search for an electric dipole moment on the neutron.
There bas been considerable interest in recent years on coherent parity violations. Because the neutron partici• pates in weak interactions, which do not conserve parity, one might expect that in the forward scattering of neut rons through matter ther e is a weak pacity violatingspin dependence. F . C.
Michel and independently, Stodolsky have estimated the rotary power of matter composed of heavy atoms a.t normal densities to be about 10- 8 radians/cm. This small rotation appears to be beyond the limits of sensitivity of the interferometers that we are currently using. However, Gabciel Karl has suggested that the effect can be
enhanced if the phase-shifting medium is composed of twisted molecuJes with a given sense of helicity.
With pulsed neutron-spa) lation sources on the horizon, it is clear that energy dependence of neutron scattering lengths through the epithermal
Breit-Wigner resonances of many isotopes will need to be measu red. It is also clear lhat in the fu.ture one will be able to carry out experiments using both polarized neutrons and polarized
targets (probably dynamically polarized), so that one can directly measure the scattering lengths for both spin states. Neutron interferometry will be

useful for both kinds of experiments. Finally, there is the possibility of
using the two coherent beams available in a neutron interferometer to help solve the "phase problem" in certain crystallographic studies. Although this possibility has been discuBSed for some time, it still appears to require levels of perfection in the sample crystals and stability in the sample position
that are as yet difficult to achieve. I have not attempted to be all-inclu-
sive in mentioning the wide variety of new ideas wh.icb have come to my attention, but only to give the reader a glimpse of the exciting possibilities for future experiments.

The preparation of this manW1cript waa
supported by the Naticna/ &knee FoundJJ.tion's program in Atomic, Mol«ular and Plasma Physia through grant no. 7920979. I would also like to thank the
neutron scattering group al the Oak Ridge National Labororory for the hosp,tality afforded. me during the month of June, 1980 when the writing of this manuscript was initiated. The t«hnical information pro-
uided by many o/my friends and oo//eagues
engaged in neutron interferometry world, wide has been. i'.n.ualuable- in preparing this article.

References

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2. See, for example, U . Boose, W. Graeff', in X -Ray Optics, Topicsin Applied Pb.ysics Vol. 22, H ..J. Queisser, ed., Springer,
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4. J .-L. Staudenmann, S. A. Werner, R.
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12. C. G. Shull. D. K. Atwood, J . Arthur, M. A. Home, Phys. Rev. Lett.44, 765(1980). R. Gabler, A. 0 . Klein, A. Zeilinger, submitted ta Phys. Rev. A (1980).

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an International Workshop al Institut

Laue-Langevin, Grenoble, France, U.

Bonse, H. Rauch, eds., Oxford U. P,

(1979),

0

30 PHYSICS TODAY I DECEMBER 1980