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Gravitational, rotational and topological quantum phase shifts in neutron interferometry
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1994 Class. Quantum Grav. 11 A207
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Class. Quantum Grav. 11 (1994) MOl-AZ26. Printed in the UK
Gravitational, rotational and topological quantum phase
shifts in neutron interferometry
Samuel A Werner
D e m e n t of Physics and Astmnomy, University of MissourKslnmbia Columbia, MO 65211.
USA
-
Abstract. Experiments involving quantum interference of neutron de Broglie waves. having
a wavelength A
2
k extending over macroscopic distances of order 10 cm am described.
In partiwlnr I discuss interferenre induced by me Eanh's gravitational field and by its rotation
(Sagnac effect), the topologicd phase shifts resulting from the diffraction of the neutron, carrying
its magnetic moment p, around a charged elechode (AhamnovXasher effect) and then f i d v
during its passage thmugh a pulsed magnetic field (scalar Ahamnov-Bohm effea).
1. Introduction to quantum phase shifts of matter waves
A thermal neutron having a de Broglie wavelength of 2 A, has a kinetic energy of about
20 meV and moves with a velocity of 2000 ms-'. This curious dual nature of neutrons,
sometimes a particle, sometimes a wave, is wonderfully manifested in the highly non-local
effects of neutron interferometry. The seemingly incompatible point-by-point motion of
particles in spacetime as described by relativity considerations and these non-local quantummechanical interference phonomena are brought into close juxtaposition by interference
experiments induced by gravity, rotation and topology.
We begin with a general discussion of the quantum phase Q(z,t ) of a matter wave
evolving in space and time according to the Feynman-Dirac path integral [1,2] along the
trajectories defined by classical mechanics. The perfect Si-crystal neutron interferometer is
geometrically identical to the classical optics Mach-Zehnder interferometer. Topologically
it is equivalent to a ring as shown in figure 1. At some point A on the ring an incident wave
is brought in and split coherently into two parts, one propagates clockwise on path I and the
other counterclockwise on path 11 around the ring. At some point B on the opposite si& of
the ring, these two coherent waves are recombined and allowed to leak out to be detected by
an observer, who interprets the results in terms of constructive and destructive interference
of waves. But, of course, only particles are detected one by one, and the interference pattern
is noted only after many neutrons have been detected.
The phase @(z,I) of the wavefunction Y(z, t ) should be regarded as a scalar field.
The source emits a train of waves at position zo at time to. This train of waves evolves
in space and time along the two spatially separated paths in an interferometer as shown in
figure 1. The phase accumulated on either path is a line integral over the Lagan*
12 in
spacetime given by
I , .
The Iagrangian
L is related to the Hamiltonian via a Legendre transformation
L=p.v-li
0264-9381/94/6A0207+20$19.50
(2)
@ 1994 IOP Publishing Ltd
A207
A208
S A Werner
Figuc 1. Generic interfemmter.
where p is the canonical momentum of the neutron. and v is its classical p u p velocity, i.e.
U = d8/dr. Thus, equation (1) gives the phase at the detector at position r as a function
of time t, namely
i6
:L
@(*,t)=-
p*d8--
'Hdt'
t
=[k-ds-L
d'
(3)
where k = 2 n / A is the wavevector, and w is the frequency of the wave at any point (d.
t')
along the trajectory. For each of the two paths in figure 1 we must separately evaluate (3).
namely
f
h(r,t)=f~h.d8-f~?(ldr'
(44
fl
(46)
and
@n(z,t)= i/,"m * d e-
I
7fndt'.
Thus, the phave difference between the two paths is
A@(z, t ) = @I&
t ) - @I(z. 1 ) .
(5)
In neutron interferometry, the phase shift A@v(r, t ) caused by a potential V(d,v , t') is
the quantity that is measured, and is of physical intaest, namely
A@" = A@ - A@,
(6)
where A@o is the 'empty' interferometer phase difference, i.e. when V(d.
U, t ) = 0. The
phase shill due to the potential is the line integral along the classical trajectories of the
neutron around the interferometer.
/* -
1 z
A@"(=, 1 ) = i;
Apn d8 -
- !h. s,'AHndt'+
i
z Am * d8
h/,
i6
AXldt'.
0
The canonical momentum p must be used in evaluating this p o t e n t i a l d c p c ~phasc
t
shin.
It consists of two p m , namely
P ="
i c
+phiddcn
(8)
w h m the kinetic momentum is given by the neutron's velocity U,
mhc= m u .
(9)
Quantum phose ships in mu" interferometry
A209
The hidden momentum comes from the electromagnetic field in the Aharonov-Casher (AC)
effect [3-51, namely
Pbldde"
1
=-p x
C
E
(10)
where 1.1is the neutron's magnetic moment, and E is the electric field from a line of charge.
In the Sagnac effect [&SI the hidden momentum comes from the state of rotation of the
frame of the interferometer. As we shall see in section 5, it is
phdden = mw
x r.
(11)
There are three general situations to consider:
(1) The potential V depends only upon position E' and is independent of time t' and
velocity v, that is
v = V(z) .
(12)
In this case the Hamiltonian X is timeindependent and the neutron's energy E = p2/2m+V
is a constant of the motion. There is a deceleration when the neutron enters the region R
of the potential, and an acceleration when it leaves it. The phase shift comes entirely from
the change of the kinetic momentum, namely
This is the situation for a neutron passing through a slab of material and interacting with the
(unpolarized) nuclei, for gravitationally induced interference [S-101 and for spin precession
in a magnetic field B which is constant in time [ l l , 121.
(2) No force. situations when X is independent of time. The phase shii depends
explicitly upon geometry and topology. In the AC geometry, the acceleration due to the
electric field E is
I
a =--(p.
me
V ) ( ux E ) .
This acceleration is zero, if p = /.LO? and E = E ( x , y), as is the case for a tine charge
along the z-axis. Thus, the phase shift is
ft f A p , i a m . d s .
AOV=A
(15)
The energy E, and the kinetic momentum are constants of the motion.
(3) No force situations where the potential depends only upon time t' and is independent
'. That is
of position z
AV = AV(t') = AX(t')
(16)
and the phase. shift
will depend explicitly upon the time r of detection relative to the time smcture of the
'pulsed' potential. This is the situation appropriate to the scalar AB effect experiment
[13,14]. The neulmn feels no force and the kinetic momentum p is therefore a constant of
the motion.
A210
S A Wemer
F w 2.
Schemltic diagram ofour LLL Bonse-Han Si-crystll interfemmeteraf Missouri. The
dimensions a~ d = 34.518 I O . 0 0 2 m a o = 2.464 i 0.002 mm. for the inrerfemmeter used in
thc gravity experiments.
2. Perfeet silicon crystal neutron interferometers
A schematic diagram of a three-crystal LLL interferometer, of the type deveIoped by Bonse
and Hart [I51 for x-rays, is shown in figure 2. It consists of three perfect crystal slabs cut
perpendicular to a set of strongly reflecting Bragg planes, typically (220). A collimated,
nominally monochromatic beam of thermal neutrons is directed along the line SA and is
coherently split by Bragg reflection in the first crystal slab. These two coherent beams
are split again in the second crystal slab near points B and C. l b o of these beams are
directed toward point D in the third crystal slab where they overlap and interfere. The label
LLL means that this interferometer involves three Laue transmission-geometry crystals. The
neutrons are detected in three 3He gas-filled proportional detectors CI,Cz and C3.
If the beam traversing path II is phase shifted relative to the beam traversing path I,
by introducing a slab of material in path II,causing a change in the ‘optical‘ path length,
the counting rates in detectors Cz and C3 will change. On very general grounds, it can be
shown that the intensities 12 and I3 vary sinusoidally with the phase shift A@:
Iz = az - bz cos(A@)
(1W
and
I3
= a3
+b3 cos(A@).
(W
The constants az,&. a3 and b3 depend upon the transmission and reflection coefficients of
the three Bragg reflecting crystals. As the phase shift A@ is varied the neutron intensity is
swapped back and forth between detectors CZand C3, such that
is constant, requiring
bz = b3.
For an interferometer of the size shown in figure 2, there are of order lo9 oscillations of
the thermal neutron de Broglie wave on each path. Upon recombination in the third crystal
slab, stable interference fringes are formed. For this ‘miracle’ to occur, very stringent
requirements on microphonic and thermal stability must be met. In our two interferometry
set-ups at the University of Missouri Research Reactor (MURR), the interferometers are
positioned inside of metallic, isothermal enclosures, which are mounted upon a vibration
isolation pad.
The incident neutron beam is not precisely monochromatic. Typically the wavelength’
dispersion is 8A/A PZ 0.01. The important feature of this interferometer is that it uses Bragg
reflection in perfect crystals, where the Darwin width is about 0.1 arc sec. This means that
Quantum phase shi&
in neutmn interfcmmetry
plyrr 3. A n e u m ofwcrgy th parsing rbmugh a+
A211
ofoplid pohmtw .,V
the Bragg reflection p m s itself defines the wavelength along a given trajectory (ray) to
within about 1 part in Id.
3. Phase shift due to the interaction of neutrons with noelei
Only the s-wave component of the scattering amplitude is significant for thermal neutrons
scattering from nuclei. It is easy to show from this that the neutmn optical potential of a
slab of material having an atom density N and nuclear scattering amplitude. b, is
2zfi2bN
V,=-.
(19)
m
This potential is typically of order lW7eV [16].
Suppose that this slab is inserted into one leg of the interferometer creating a potential
step as shown in figure 3. Upon entering R the neutron is slowed down, and upon leaving
R it is speeded up. Knowing this, it is tempting to calculate the phase shift A@" caused
by V, using (1) with limits on the integral given by the time-of-flight a"the region
R
' . Surprisingly, this gives the wrong answer! It is important to rcalizc that the cm%y is
conwed h a ,
2
& = - -Po
- . +-VP
2
(20)
2 m 2 m '
since the Hamiltonian 7 i is independent of time. Time does not entcr the calculation. Since
V, is assumed to be small compand to the kinetic energy pi/&, the shift in momentum
calculated from (20) is
Ap
1 v,
= p - po u ---
(21)
2&".
Thcn according to (13). the phase shift of the neutron wave traversing R (of length L ) due
to the potential V,. is
L
'
'*POL = - b N b L
(22)
=Eo
where in the last step we have used (19) and the de Broglic relahion m =fib = h/&. For an
A@,=kg
Ap.de=---
-
aluminum slab, 1 mm thick, using 2 A neutrons. this phase shift is 420 rad 150000deg. It
is clear frmn this that neutron interferometry provides a very sensitive method for accurately
measuring neutron nuclear scattering amplitudes.
A212
SA
Werner
Figure 4.
Schematic diagram of h doubkcrystsl "sun assembly nod t
k
Bt beam pon B at hwm
mterferomstry --up
4. Gravitafionally induced quaatom interference
Lilrc all matter. neutrons are subject to the universal gravitational face. This fact has
been demonsmated by verifying that neutruns fall on a parabolic trajectory in the Earth's
gravitational field [17]. This is a consequence of classical mechanics and is expected from
the principle of equivalence. Is there a quantum phase shift f a the neutron Bccompanying
its fall toward the Earth? The answer is yes, and it is measurable. Gravity and quantum
mechanics do not simultanmusly play an important d e in most phenomena accessible to
terreshial experimentation. However, a neutron interfemmetg experiment. for which the
outcome n d l y depends upon both the gravitational constant G and Planck's constant
h was first carried out by Colella et al (cow) [9]. Sukquently a series of increasingly
p i s e experiments have been completed by my group at Missouri 18, IO]. I now describe
briefly the apparatus and geometry of these experiments. Collectively these experiments arc
called COW expiments.
The overall experimental setup is shown in figure 4. A thermal (Maxwellian spectrum)
neutron beam is brought out of the 10 MW MURR thrwgh a helium gas-filled beam tube, and
monochmmated by a doublacrystal monochromator assembly, which uscs eitha pydytic
graphite crystals or copper crystals. The monochromatic beam then p e s through a d e s
of collimating slits onto the interferometer. The doublecrystal monochromator provides a
variable wavelength incident beam directed along the local north-south axis of the Earth, a
fact which we will see is important in these experiments.
The experimental procedure involves tilting the interfaomta. including the entrance
slits and the thrcc 3He detectors CI,Cz and C3 about the incident beam line SAB shown
in figure 2. For each angular setting a,neutrons me counted for a prcsd length of time
(actually determined by a monitor detector in the incident beam). TU- the in"*a
allows the path CD to be somewhat higher above the W
s surface than the beam path
AB. The difference in the Earth's gravitational potential between these two levels rtsults
in a quantum "id phase shii of the neutron wave on trajectory ACD relative to
that on the trajectory ABD. The phase accumulated on the rising path AC is equal to the
plme accumulated on the opposite rising path BD.The curvature of the trajectdcs ova the
distances involved in the interferometer is negligible. If wc assthat the gravitational
Quantum phase shffis in neutron interferometry
A213
potential is Newtonian, then as a function of the tilt angle U, we have
AV(@ = mgH0 sin(a)
(23)
where m is the neutron mass, g is the local acceleration due to gravity, and Ho is the
perpendicular distance between the lines AB and CD. At Columbia, Missouri, mg =
1.023 x 10-9eVcm-*. The differences in the neutron's momentum between these two
levels is
1 mgH0 sin(or)
A p = p - po = -(W
2
%
where the energy EO is a constant of the motion; in terms of the initial wavevector k,, it is
(25)
The phase difference for neutron waves on path II relative to path I, induced by the Eaah's
gravity is then easily evaluated according to (13):
AGP, =
1h { Ap. ds
8
= -2Rm 2 -&A0
sin(.)
h2
The a m A0 enclosed by the beam paths is
qcw sin(or).
(26)
(27)
Ao=(2d2+2ud)tanBg
where d is the distance between the crystals slabs in the interferometer, a is their thickness,
and
is the Bragg angle for the Si(220) reflection for neutmns of wavelength ho. For
AO = 1.4A. this area is about 10 cm2 for the interferometer used in our experiments. The
formula (26) contains both the gravitational constant and Planck's constant.
As the interferometer is tilted through various angles 01, always maintaining the Bragg
condition, (18) and (26) predict that we should observe quantum oscillations in the counting
rates in detectors C,and CB,induced by the gravitational field of the Earth. An interferogram
obtained with 1.419A incident neutrons is shown in figure 5 [SI. The unexpected loss of
contrast was originally interpreted as being due to the fact that the interferometer bends and
warps under its own weight (on the scale of Angstroms) as it is rotated about the incident
beam axis, which is not an axis of elastic symmetry of the Si-crystal interferometer. We have
studied these effects with in situ x-ray experiments. X-rays were directed along the same
incident beam path and the interfering x-ray beams were monitored as a function of rotation
angle. The effect of gravity on x-rays (gravitational red shift) over the distances involved
in the interferometer is negligible. The frequency of oscillation of the interferogram due
ar41
Figore 5. Gmvimtionally induced quantum interfemgram taken with the interferometer of figure
2 with 1.419A neutrons (Stnuden" etal [SI).
A214
Figure 6. The eight beam tiajectories in an LLL interfemmeter (Home [IS]).
to bending was therefore measured directly with x-rays and subtracted from the frequency
of oscillation measured with neutrons, leaving only the effects of gravitationally-induced
quantum interference. Home [18] has reanalysed these cow-type experiments, pointing out
that the threecrystal LLL interferometer is not a simple two-path device, but really an eightpath interferometer. This has the effect that the single interferometer area Ao appearing in
the cow phase-shift fomnla, (26), should be replaced by a dynamical &%action intensity
weighted average over three areas, Ao, Ao+GA and Ao - 6 A , where SAIAo = r a / d . Here
a is the thickness of each of the three slabs, d is the distance between them and r is a
factor (Iess than unity) dependent upon the misset angle A0 of a given incident ray from
the exact B r a g condition. The eight paths are shown in figure 6. Using Home's theory,
we find that a correction of 4.8% to the experimental frequency of the gravity interferogram
should be made before comparison with the cow phaseshift formula The fact that the
experiment actually involves more than one interferometer area also explains the loss of
contrast with increasing tilt angles (Y. Similar conclusions, using a complete spherical wave
dynamical diffraction treatment were reached by Bonse and Wroblewski [19] in analysing
their acceleration-induced interferometry experiment. We have shown that this geometrical
correction to the COW formula for the frequency of oscillation is in fact given by [lo]
For our interferometer a / d = 0.0714, so that
qmv = 1.0476qc,, .
(29)
We now turn to comparing this predicted frequency of the gravitationally induced
quantum interferogram with our most recent experiments. We have followed a somewhat
different procedure than in the original COW experiments. The phase shift AaWV(ar)=
qWvsin(.) is measured directly by first setting (Y = 0, and rotating the phase rotator through
successive angles S. Due to the neutron-nuclear optical potential of the phase rotator, this
results in a sinusoidal interferogram. Then tilting the interferometer through an angle (Y, and
repeating the scan of the phase rotator, gives rise to another sinusoidal interferogram. The
Quantum phase sh@s in neutron interferometry
A215
SIN (a-
Figure 7. Gravity-induced phase shift as a funchon of the sine of the interferometer tilt angle
a (Werner e t d [IO]).
difference in phase between these two interferograms is A @ J ~ ~ ( CThe
Y ) results
.
of a series
of very accurate measurements, using 1.417A neutrons are summarized in figure 7. The
absicissa is sin(cr-cro), wherecro is a correction due to the Sagnac effect (cl0 = 1.41”), which
will be discussed in section 5. The slope of AQW versus sin@ - 010) is the experimental
frequency qexpof the gravity interferogram. To compare this frequency with qWv, we must
make a correction for bending and for the Sagnac effect, namely,
2
2
(30)
(qcrp- qsagmc)”2 - !?tend ’
Theory gives qsaF = 1.45 rad, and with x-rays we measure q k , d = 1.41 rad. Thus, our
current experimental result is
qpv
q,,(observed)
= 58.72 & 0.03 rad.
(31)
Equation (29) predicts
qpv(theory) = 59.19 rad.
(32)
Thus, the observed frequency due to gravity is 0.8% lower than theory predicts.
Layer and Greene suggested that this discrepancy might be due to the fact that x-rays
interrogate a slightly different region of the Si crystal slabs than the neutron beams [ZO].
That is, the correction for bending effects using x-rays may be in error. Recent x-ray
experiments of Arif et QZ [Zl] have shown that the bending effects are considerably more
complicated than originally thought, and corrections using x-rays to the neutron frequency
of oscillation will require additional analysis.
A way to eliminate the effects of bending, and thereby facilitate a more accurate
comparison with theory, is to float the interferometer in a fluid of density equal to that of Si.
This suggestion was fist made by Zeilinger (see comments at end of [IO]). Implementation
of tbis idea presents several experimental challenges.
5.
The neutron Sagnac effect
In 1913the French scientist MG Sagnac demonstratedthat optical interferomehyis sensitive
to angular rotation. This effect is now used in navigation and is the basis for the ring-laser
gyroscope. In 1925, Michelson, Gale and Pearson carried out an hemic experiment They
constructed an interferometer in the form of a rectangle of size 2010 ft x 1113 ft and were
able to detect the retardation of light due to the Earth‘s rotation, corresponding to about
A216
S A Werner
of a fringe, in agreement with theory [22]. Since the coordinate transformation properties
of light waves and neutron matter waves are different, it cannot he taken for granted that
an analogous quantum mechanical effect will exist for neutrons.
The gravitationally induced quantum-interference experiments were carried out on the
surface of our rotating Eaah,a non-inertial frame of reference. The classical Hamiltonian
governing the neutrons motion in the gravitational field and frame of our rotating Earth is
[231
E = - P2
-
.
mg g T - w . L
(33)
2mi
where L is the angular momentum of the neutron's motion about the centre of the Earth
(T = 0), namely
L = T x ~
(34
and p is the canonical momentum, and w is the angular rotation velocity of the Earth. The
third term here gives rise to the Coriolis and centripetal accelerations. Over the distances
involved in the interferometer, the Coriolis force has a negligible effect on the neutron's
trajectory. However, its effect on the neutron's phase is measurable. Using Hamilton's
equation v = aEH/ap,we see that the neutron's canonical momentum is
p = miv +miw x T .
(35)
We have been more careful here to distinguish between the neutron's inertial mass mi and its
gravitational mass mg. The Hamiltonian in (33) is velocity-dependent, but timeindependent;
thus the energy E is conserved. According to our prescription of section 1 for calculating
the phase shiit we get two terms from ( 3 3 , namely
A4
p . d s = A4gw + A4sGoac.
Rl f
(36)
Using elementary vector calculus the Sagnac phase shift is easily found to be
2mi
A4sagnac= - W .
Ao
(37)
R
where & is the normal area vector of the interferometer loop ABDCA. When the incident
beam is horizontal, and directed along the local north-south axis of the Earth, (37) gives
The colatitude angleL'6 = 51.376" at Columbia, Missouri. Since this phase shift depends
upon the cosine of the interferometer tilt angle a!, and
depends upon the sine of a!,
the frequency of oscillation of the interferogram can be written as
Numerically, qSagoac/qgav zz 0.02. Thus, the effect of the F " s rotation adds a small
correction of only 2 parts in lo4 to the interferogram frequency due to the Earth's gravity.
However, in an experiment carried out by Werner eta[ [7] a vertically directed beam was
used as shown in figure 8. The Sagnac phase shift was measured (using the phaserotator
technique described earlier for the gravity experiments) as a function of the interferometer
orientation angle a! about the vertical axis. From symmetry, it is clear that there is no
or-dependent gravity-induced phase shift for this geometry. In terms of the colatitude angle
Quantum phase shifrs in neutron interferometry
A211
I.ltle.1
Figure 8. Schematic diagram of k veltica-beam geomelq used in obwving the neutron
S a m effect Werner er d [71).
Figure 9. Phase shift due to the Euth's rotation. The angle 01 specifiesthe onemalion of the
interfemmeter normal asea veuor with respect to the local N S axes of the Earth Wmer et 01
m,.
OLat the point on the Earth's surface where the experiment was done, and the angle CY,the
Sagnac phase shift (from (37)) is given by
2m,
Aj3agoac= --oAo
sin(&) sin@).
(40)
fi
points
The experimental results are shown in figure 9. When the normal area vector
west or east, the phase shift is zero: while when
is directed north or south it is +95" and
-95", respectively. This experimental result is in reasonable agreement with (40) which preshould be +9Z0 and -92" for the north and south orientations, respectively.
dicts A@sagnaC
It is interesting to note that the results of this experiment depend only upon the inertial
neutron mass m,, while the results of the gravity experiment depend upon the product of the
gravitational mass m, with the inertial mass m,. Thus, one can intexpret the combination
of the two experiments as a quantum-mechanical interference measurement of the inertial
and gravitational masses. We point out that neutrons are much more sensitive to the Sagnac
effect than photons. This can be seen if we write mu = frk, and replace m , p in (37)by
k / u . For photons, U = c, the velocity of light, which is about 16 times larger than for our
thermal neutrons.
A218
S A Wemer
6. The Aharonov-Casher effect
According to Maxwell theory of electrodynamics, potentials are merely convenient
mathematical tools for calculating the electromagnetic fields of force. In quantum
mechanics, potentials enter the Schrodinger equation and produce phase shifts, even in
geometries where the potential gives rise to no electromagnetic field. These are the situations
of interest in the topological interference effects of Aharonov and Bohm [24]. For electrons,
the effects are of two types: the magnetic (or vector) AB effect, and the electric (or scalar)
AB effect. We have observed directly analogous effects with neutrons. We describe the
neulron vector AB effect in this section, called the Aharonov-Casher (AC) effect 131,and
the neutron scalar AB effect in the next section.
In 1984 Aharonov and Casher [3] proposed that a beam of neutral particles with a
magnetic dipole moment passing around opposite sides of a h e charge will undergo a
relative quantum phase shift. This AC effect is considered to be an electdynamic and
quantum mechanical dual of the Aharonov-Bohm effect for charged particles, as can be
understood by the schematic diashown in figure 10. The AB flux tube has been
replaced by a line of electric charge of lineal density A, and the electron beam of chaqe
e- has been replaced by a beam of neutrons with magnetic moment p. If one views the
AB flux tube as a line of magnetic dipoles, one sees that the role of charge and magnetic
dipole have been interchanged between the AB effect and the A c effect.
For a neutron of mass m and magnetic moment p, moving in the region of an electric
field E, the Hamiltonian is [251
P2 - - 1p .
1-1 = 2m
mc
(E x p ) .
(41)
Thus, using Hamilton's equation U = a'lfjap, we see that the canonical momentum is
p = m u + -Px E .
C
(42)
It is a straightforward matter to show that there is no force on the neutron if the electric
field comes from a line charge along the z-axis, and the neutrons are polarized along z.
That is, the velocity w is a constant of the motion. For a neutron diffracting m u n d a line
charge of lineal density A, one obtains the AC phase shjft by evaluating the line integal of
P, namely
where U = kl depending upon whether the neutron spin is up or down with respect to the
plane of the neutron's motion. We have used the fact that the electric field due to a line
charge is E = 2A+/r. As pointed out by Ac, this phase shift depends only upon the lineal
(a)
F
(b)
(C)
i 10. Duality beween the Aharonov-Bohm topology and the Ahamnov-Cashex topology.
Quantum phase shi& in neutron integeromtry
A219
U
Figure 11. Schematic of the ACeffeCtexperiment. An unpolarized neutron beam of wavdmgth
)i = l.477A is used. The nemo" wave on path 11 passes thmugh a region of elechic field
E and then through a vertical magnetic V i field B. The neuhon wave in path I passes on
the opposite side of the centre electrode, whose polatily was reversed periodically as described
in the text. The interferometer, along with the
slit, the vaeuum electrostatic d.
and
the bias magnet could be tilted about the incident beam direction to adjust the spin-independent
gravitational phase shift ADpv (Cimmino el ul[41)
charge density A, enclosed by the beam paths, but not on any details of their geometrical
shape. In this Sense the effect is topological. Thus, instead of a line charge, a prism-shaped
electrode was placed between the splitter (S)and mirror plate (M)of the interferometer as
shown in figure 11, thereby enabling a much larger lineal charge density to be obtained.
An electrode can be regarded as the superposition of many line charges. The experiment
was a collaborative University of Melboume-University of Missouri project carried out in
Columbia, Missouri.
For the electrode assembly of figure 11, Gauss' law allows us to replace A by
2VL/4zD. Here V is the potential difference between the electrodes, D is their separation,
and L is the effective path length as shown in figure 11. In terms of these parameters, for
D = 0.154 cm, L = 2.53 cm and V = 45 kV (= 150 statvolts), we find
A Q A=
~ 1.50~milliradians.
(44)
It was assumed in the derivation of (43) that the neutrons are polarized along an axis parallel
to the line charge the z-axis. However, it is not necessary to use polarized neutrons if an
additional spin-independent phase shift is judiciously introduced and fine-tuned. In this
experiment, gravity was used for this purpose. The introduction of a further spin-dependent
phase shift A ~ Mby
, means of a magnetic bias field, enabled maximum sensitivity to the
AC effect to be accomplished.
An unpolarized beam can be thought of as consisting of two beams, one spin-up and
A220
S A Werner
the other spin-down. For the spin-up component the counting rate in detector C3 is
I$ = ;[U3
+ 6 3 COS(Aa!+ AB)]
(45)
whereas, for spin-down neutrons, it is
I,” = $ [ U 3 4-h COS(A(Y- AB)].
(46)
The spin-independent phase shift is called Acu here, and the spin-dependent phase shift is
called AB. Thus, for an incident beam of unpolarized neutrons we have
13
Z3t + I,”
= ~3
+
b 3 COS(AO~)
COS(AB).
(47)
We adjust Ahor by gravitationally induced quantum interference to be zero (mod Zn), and
we adjust A& to be rr/2 or 3 r / 2 , where
+
AB = ABM A@AC
Thus, since AQAc is such a small phase angle, we have from (47)
(48)
*
= a3 63IA@Ac[
(49)
where the sign is for negative centreelectrode polarity, and the - sign is for positive
centre-electrode polarity. A similar expression applies to the counting rate in detector Cz,
namely
13(%)
+
I z W =Q+~zIA@AcI.
(50)
Thus, the count rates are linearly proportional to A@Ac.
The magnetic bias field was varied by changing the reluctance of a magnetic circuit. To
find the correct operating point for Acu and ABM, the following procedure was followed
with the magnetic field B set equal to zero, the interferometer was tiIted in steps. The
first maximum near zero tilt angle, sets Acu = 0 (mod 27). Leaving the tilt angle fixed at
this maximizing value, operating points of negative slope (ABM= x / 2 ) and positive slope
(ABM = 3 x / 2 ) were selected by scanning the magnetic field as shown in figure 12. The
magnetic field was then changed in small incremene around the value giving ABM % n / Z
or 3xj2. Gravity scans were then carried out for each value of the magnetic field and the
optimum was chosen as that giving minimum oscillation.
After the optimum operating conditions were established, the 45 kV high voltage was
switched was on across the electrode gap, and then periodically reversed in polarity. Each
switching cycle (positive, zero, negative) took about 30 minutes. The total data accumulation
time was about 2 years. A summary of the data is given in table 1. Cz(+) and C3(+) are the
Figure 12. Magnetic scan interferogram (Kaiser et& 151).
Quantum phase shi$s in neutron interferometry
A221
are the total number of counts accumulated in detectors C2 and C3 per cycle for a positive
centre-electrodepolarity, while Cz(-) and C3(-) are for a negative centre-electrode polarity.
Only the difference counts ACZ = CZ(+) - CZ(-) and AC3 = C3(+) - C3(-) per cycle
are given in table 1. The positive slope data applies to when the bias magnetic field is
adjusted to given A& = 3x/2, the negative slope data applies of the case A& = r / 2 .
Notice that the expected symmetry and sign reversals for the difference counts are found in
the data of table 1.
Statistically combining the results for the positive and negative slopes according to the
rule
(AC) =@zACz+@3AC3
(51)
where 02(=0,24) and q(=0.76) are the statistical weighting factors (inversely proportional
to the square of the standard deviations), we obtain the thiid column of table 1. A fit to
the magnetic scan of figure 12 gives the slope of the interferogram at the operating points,
namely
b = bz = 63 = 1234 f 15 counwcycle .
The phase shift due to the
slope as shown in table 1.
AC
(52)
effect is then obtained from the difference counts and this
Table 1. Difference counts/cycle
Experimental condition
CZ(t)- Cz(-)
CS(+)
- Cd-)
Positive slope
3539 cycles
t7.66 2.48
-4.80 1.39
74 days
Negative slope
3654 cycles
-5.43i2.36
f4.66f1.32
76 days
Positive slope: A@AC= (AC)/Zb = 2.22 f0.49 mad
Negative slope: A@*= = (A0126= 1.96 0.46 “d
*
+
(AC)
5.49
+ 121
4.Fdf1.15
*
Combining the positive and negative slope data, this experiment yields the first
measurement of the AC phase shift
AQA~=2.11f0.34mad.
(53)
This result is to he compared to the theoretical prediction of 1.50 milliradians. The accuracy
of the experiment is limited by available neutron intensity and long-term apparatus stability.
Obviously, the much larger interferometers currently being designed and fabricated in ow
laboratory will be important in the next generation AC effect experiments. Changing the
charged electrode geometry is necessary to elucidate the topological aspects of the AC
predictions. An experiment is being developed to pursue this feature of the theory.
The essential necessity of the AB effect for the self-consistency of quantum mechanics
was first clearly elucidated by Furry and Ramsey [26] in 1960. Recently, Ramsey [271
has given a similar argument for the AC effect, and discussed the complementarity of twopath neutron interferences with separated oscillatory field resonances. An experimental
observation of the AC phase shift by the Ramsey technique, which can be viewed as
interferometry in spin space, has recently been achieved by Sangster er al [ZSI in a beautiful
molecular beam experiment using thallium fluoride molecules. They were able to verify
A222
SA Werner
the velocity independence of A@,w and its linear dependence on the elecIic field E by
observing the interference of the two spin states of the fluorine nucleus.
7. The scalar Ahamnov-Bohm effect
The prediction of a scalar AB phase shift has proved to be difficult to confirm experimentally
for electrical systems. However, it was suggested by &ginger [29] and by Anandan [30]
that an analogous effect should be observable for neutral spinning particles. Conceptually
the analogy is displayed in figure 13.
A divided electron wave packet travelling down two conducting cylinders is.shown in
panel (a) of figure 13. These conducting cylinders have a field-free interior independent
of the electrostatic potentials U, and U,, i.e. they act as Faraday cages. The potential of
cylinder 2 alone is pulsed during a time when the wave packet is contained inside it. In
spite of the absence of a force at all times, a relative phase shift A@mis expected,
This is the situation discussed in section 1 where the Hamiltonian ‘H is timedependent.
In our experiment with neutrons [14], the phase shift is due to the scalar potential
V ( t ) = -p * B(t).
Consider a split neutmn wavepacket entering the solenoids of figure 13(b). A current
pulse i&)
is applied while the neutron is in the force-free interior of solenoid 2. The
,
0)
P i p 13. Schematic diagram of (a)the scalar Ahamnov-Bohm exprimnt for electrons and
(b)the scdar Aiuronov-Bohm expriment for neumns. The wave f m of the applied pulses
a.re also shown (AUman et d [141).
Quantum phase shifts in neutron interferometry
A223
Figure 14. Layout of AB experiment using a skew-symeuic SingleSiCrystal neumu
interferometer. Inset: m isometric view of the interferometer crystal (AUman et d [14]).
resulting magnetic field &(t) gives rise to a phase shift,
“J
A@AB= -- &(t’)dt‘
(55)
Rdirectly analogous to (54). Here U = &1 depending on whether the neutron is spin up or
down relative to the magnetic field. In the actual experiment, short duration pulses (8 ps
wide) were applied to a small solenoid (about 4 cm long). For the 2.35 A neutrons used in
the experimenr the time of flight through the uniform-field portion (3 cm)of the solenoid
was about 1 8 ~ s The
.
obvious question is: how does one know when to apply the pulses
so as to catch the neutrons just as they traverse the centre of the solenoid? The answer
is to apply the pulses cyclically and to gate each detected neutron into a separate scaler
according to its arrival time within a given cycle. Thus, in this way the phase shift of
neutrons which traversed the solenoids when the current was zero can be compared with
the phase shift of neutrons which traversed the centre of the solenoids when the current,
and hence the magnetic field was not zero. The experiment was carried out at the beam
port C interferometer set-up at MUM, using a skew symmetric interferometer. A schematic
of the layout is shown in figure 14.
The axis of quantization was aligned with the solenoid axes by means of a rather
uniform field, produced by four permanent magnets, each a soft iron bar with neodymiumiron-boron caps, mounted symmetricallyaround the interferometer. As in the AC experiment,
unpolarized neutrons were used, and the spin-independent phase was adjusted to O(mod2r)
with an aluminum phase plate. The bias magnetic field was adjusted such that the spindependent phase was n/Z. With these adjustments, we have established the situation such
that the counting rates in detectors CZ and C3 are given by
NZ = a2 - bz sin(A@a)
and
(564
A224
S A Wemer
where again h = b3, as required by conservation of neuimns. At the de Broglie wavelength
of this experiment the neutron speed is 1.68 mm ps-', yielding 17.8 ps of flight time in the
uniform region of the pulsed solenoid. The 8 ps-wide pulse (risetime < 1 ps) corresponds
of travel. The pulsed field is uniform over 30 mm, to within about 1%. The
to 13.5"
pulsed field was turned on every 64ps-', but with reversed polarity on alternate cycles,
giving a full period of 128p.s as shown in figure 13. A pulsed field of 21 Gauss, on for
8 ps, yields a phase shift A@- = A rad, according to (55).
Pulse Coil Volts ( 7 )
U00
- '"t
c
w
Nz
-lo
,
-60 4. -.I 0
,
,
.
,
*I
,
.
.I U
.
IO
,
i
Magnetic Field (Gauss)
Delay (psec)
Figure 15. Scan of counts per channel in the two
detectors 122, C3 and their sum, plotted againa delay
time, for the particulaw e of pulsed field amplitudes of
*I9 G,which corresponds to A@u = 1.4 nd (Allman
et d 1141).
Figure 16. Interferometer difference counts as a
fundion of pulse-Ed field strenps. This is obained
f"the average of the central four points in each
plateau region of dnta sets such as the one shown in
f i p 5 (Allman erd [141).
The experimental runs, each lasting 10 h, consisted of 6 x 108 full cycles of 128ps.
The results of such a run for a pulsed field of 19 gauss is shown in figure 15. A large
number of runs were carried out with increasing magnitude of the field pulse from zero up
to = 30 gauss, corresponding to A O m = 0 up to about 3z/4 radians. Since all neutron
counts were accumulated cyclically in a multiscaler, comparison could be made between
neutrons that traversed the solenoid for zero field with those that were within the uniform
field of the solenoid for the entire duration of the current pulse, thus experiencing the AB
phase shift. The difference counts between positive and negative polarity pulses,
N3(-) - N3(+) = 2b3sin(AOm)
(57)
shows the expected sinusoidal profile, as shown in figure 16, in beautiful agreement with
the AB prediction.
It mi&t appear that the most direct, unambiguous demonstration of the scalar AB effect
with neutrons would be to use neutrons polarized initially along the pulsed magnetic field.
In this experiment, an unpolarized incident beam was used. This fact seems to have caused
some confusion [31]. Each neutron in an unpolarized beam is, of course, 100% polarized
along some direction. However, a neutron in a magnetic field E has two eigenstatesU = +1
and U = -1, with respect to the z-axisof quantization, i.e. along B. In this experiment
E is spatially uniform, but time-dependent. Therefore, the pulsed magnetic field exerts no
Quantum phase shifts in neutron interferometry
A225
force on the neutron, so that the momentum PO = mu0 remains a constant of the neutron’s
motion. Nevertheless, the neutron experiences a torque p x B,and therefore precesses
about B with Larmor frequency OL = y B ( y = 2p./h), such that its wavefunction is
1:
y(z,t ) = cos - e - w ~ / ~ l a +
) i sin 8
-ewf/21p)]
evw/fi-ibr/h
,
(58)
2
Here B is the polar angle of the neutron spin angular momentum vector S with respect to
z , la)is the spin-up eigenstate, Ip) is the spin-down eigenstate, and &O is the initial (fixed)
kinetic energy of the neutron. Equation (58) shows us that the phase shift for the spin-up
state for a magnetic field pulse of duration At is A @ += - @ ~ / 2= - m ~ A t / 2 ,and for the
spin-down state the phase shift is A@$ = + @ ~ / =
2 + w ~ A t / 2 .The phase shift is the
precession angle, a fact which led to the demonstration of the sign reversal of a fermion
wavefunction during 21r precession in earlier neutron interferometry experiments [32,33].
The important difference between this scalar AB experiment and those earlier experiments
is that here the magnetic field is spatially uniform, but time-dependent. Consequently, in
this experiment the magnetic field exerts no force on the neutron but still creates a phase
shift. This is in the spirit of all topological AB effect experiments.
In the earlier experiments, neutrons on one leg of the interferometer were allowed to pass
through a region of constant magnetic field. The neutron experiences a force upon entering
and leaving the region of magnetic field. The spatial and spin parts of the wavefunction
Y ( x ,t ) are entangled, since the spin-up p a t carries a momentum p t and the spin-down
part carries a momentum PI. In fact, the wavepackets corresponding to lor) and
separate
spatially in the region containing the magnetic field B,since their group velocities differ.
This is not the case for the scalar AB effect discussed above. This non-dispersive feature of
the AB effect has recently been demonstrated in a clever experiment hy Badurek et al [MI
using a single beam of polarized neutrons, and polarization analysis after the neutrons have
traversed a region of pulsed magnetic field.
4
1s)
8. Concluding remarks
Neutron interferomety began 20 years ago at the small TRIGA reactor in Vienna, Austria
[35]. Over the intervening years, my group has carried out 14 separate experiments which
bring into clear focus various aspects of quantum mechanics and matter-wave optics. At
least an equal number of interesting neutron interferometry experiments have been carried
by Helmut Rauch’s group in Vienna, working primarily in Grenoble, France, and also by
Clifford Shull’s group at MIT during the 1980s. The four experiments discussed here
certainly engender new refinements and extensions for future work as alluded to in the text.
One can envision a number of directions for this field in the coming years. Some
experiments will require only incremental advances in the technique, while others will
require new technology and entirely new ideas. For example, it is feasible to think
of measuring the gravitational phase shift due to the neutron’s interaction with large,
nearby IaboratoIy masses, i.e. a ‘neutron Cavendish’ experiment. This will require
much larger, split-component neutron interferometers with linear dimensions p a t e r than 1
metre. Perfect silicon crystals of this large size can be obtained today. If the laboratory
masses were to be brought up to the interferometer in a time-dependent (rapid!) manner,
symmetrically encompassing one of the beams in the interferometer, a phase shift due to the
gravitational potential of these masses is expected, but under zero-force conditions. This
is the gravitational version of electromagnetic AB effects. Whether an experiment of this
conceptual complexity will be carried out in my lifetime, I hesitate to speculate.
S A Wemer
A226
I believe that as in optical interferometry, matter wave interferometry will find
practical applications. Sagnac gyroscopes, based upon atom-beam interferometers are being
discussed. With neutrons I believe the applications will appear first in materials science.
The Advanced Neutron Source under development at the Oak Ridge National Laboratory
will have neutron beams of at least a factor of 100 greater intensity than those used in the
experiments discussed here. Neutron phase-contrast microscopy, using the Si-interferometer
can then be used to study density fluctuations inside opaque materials on the resolution scale
of 1 p m or better.
Acknowledgment
This work is supported by the Physics Division of the NSF, grant number 9024608.
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