PHYSICAL RKVIE% A VOLUME 21, NUMBER 5 MAY 198Q

Gravity and inertia in quantum mechanics

J.-L. Staudenmann and S. A. %ernerPhysics Department and Research Reactor Facility, University of Missouri-Columbia, Columbia, Missouri 65211

R. Colella and A. %". OverhauserPhysics Department, Purdue Uniuersity, West Lafayette, Indiana 47907

(Received 11 September 1979)

The experiments described in this paper probe the simultaneous effects of gravity, inertia, and quantummechanics on the motion of the neutron. Using a neutron interferometer of the type developed by Bonseand Hart for x rays, we have observed quantum-mechanical interference phenomena induced by thegravitational field of the Earth and by the Earth's rotation relative to the fixed stars. The importance ofthese experiments with regard to the role of the principle of equivalence in quantum mechanics is discussed.

I. INTRODUCTION

The purpose of this paper is to provide a com-prehensive description of a series of neutron-interferometry experiments we have carried outover the last few years, in which the effects ofgravity and inertia on the quantum-mechanicalphase of the neutron have been studied. Separatepreliminary reports on various aspects ot thiswork have already been published. ' We will re-view in this paper all of our previous work andgive many experimental and theoretical detailsthat have not been discussed in our previous re-ports.

In 1965 it was demonstrated by Bonse and Hartthat interference effects could be obtained betweenwell-separated coherent beams of x-ray photons ofabout 1-A wavelength. In their experiment anx-ray beam was split into two spatially separatedcoherent beams, a few centimeters apart, and re-combined in such a way that intensity oscillationscould be observed as the optical path along one ofthe beams was changed. This remarkable achieve-ment opened up the field of interferometry in theangstrom region. The question was immediatelyasked whether or not the same principles couldbe used to get interference effects between co-herent beams of thermal neutrons, which can bediffracted in crystals in the same way as x rays.The feasibility of neutron interferometry was, infact, finally demonstrated in 1974 by Bauch, Trei-mer, and Bonse, ' using essentially the samescheme, apart from dimensions, adopted for xrays. The principles upon which x-ray and neu-tron interferometry are based are obviously verydifferent from those applied in optical interfero-metry. Two new features are at the basis of in-terferometry in the angstrom region: (a) Braggdiffraction, and (b) highly perfect crystals, freeof lattice defects. The usual requirements in

polishing and aligning optical surfaces to frac-tions of wavelength would obviously not be satis-fiable in x-ray and neutron interferometry. Wewill see, however, in the next section how thevery same two new features (a) and (b) above en-able us to circumvent this apparently insurmount-able difficulty. When a neutron beam is coherent-ly split and recombined, a new property of theneutron becomes available for investigation. thephase. In the language of quantum mechanics wecan say that it is the wave function, along with itsown phase, that becomes measurable, whereasbefore the advent of neutron interferometry onlythe probability density ~P~ could be measured.A number of experiments have been performedsince 1974 in which the neutron phase has beenprobed in one way or another. We will concen-trate in this paper on those experiments in whichthe neutron phase is affected by the Earth's gra-vity and by its rotation. Among the various inter-actions in nature, gravity is by far the weakestone. In the hydrogen atom, for example, thegravitational attraction between the proton and theelectron is only 10 times the electrostatic(Coulomb) attraction. The fact that neutrons aresubject to a gravitational pull toward the centerof the Earth has been demonstrated by verifyingthat a neutron beam follows a parabolic trajec-tory. '

The gravitational constant involved in interpre-ting the observed parabolic paths has been veri-fied to coincide, with reasonable accuracy, withthe accepted value for the gravitational accelera-tions as measured with macroscopic bodies. Thisresult has been obtained in a recent improvedversion of this experiment. The important resulthere is that the gravitational mass of the neutronis found to coincide with its intertial mass, asrequired by the principle of equivalence. Thisprinciple has been verified with great accuracy,

21 1419 1980The American Physical Society

1420 STAUDKNMANN, %KRNKR, COLKLLA, AND OVERHAUSKR 21

better than 1 part in 10, for macroscopic bod-ies. There is no guarantee, however, that thesame principle holds for the quantum-mechanicalbehavior of isolated elementary particles. Wewant to emphasize that the parabolic fall of aneutron is a classical experiment, in the sensethat no quantum features of the neutron are beingobserved. This would merely be a requirementimposed by the correspondence principle.

The experiment which we have carried out isone in which a neutron interferometer is orientedin such a way that the two coherent beams, intowhich the primary beam is split, propagate inregions of space with different average gravita-tional potential. A gravity-induced change ofphase provides a manifestation of gravity effectson a quantum-mechanical feature of the neutron.If the change of phase agrees with a gravitationalpotential of the form m~g

' r, where m~ is thegravitational mass of the neutron, g is the gravi-tational acceleration, and r is the position vectorof the neutron, then we can conclude that we haveverified the principle of equivalence for neutronsin the quantgm Bmit. It will be shown, in fact,that the fringe shift can be expressed by a formulain which the gravitational constant and Planck'sconstant are inseparably linked. This is, to ourknowledge, the only experiment in physics inwhich the outcome depends on a simultaneous di-rect combination of gravitational and quantum-mechanical properties of an elementary particle.

A complete test of the principle of equivalencein the quzntum limit would involve repeating theexperiment in an accelerated frame of reference,traveling in a gravitation-free space. We havenot directly done this experiment. However, wesurmise that this experiment does not need to bedone, if we believe that the Schrodinger equationholds in an accelerated frame. In such a case,it is possible to derive the same expression forthe fringe shift as that obtained for the gravita-tional case. '

Since the coordinate frame in which our experi-ments are carried out is not an intertial frame,the Hamiltonian governing the neutron's motionwill involve a third term in addition to the kineticenergy (relative to the Earth) and the gravitationalpotential energy. Our neutron experiment design-ed to detect this effect is the quantum-mechanicalanalog of the optical interferometry experimentof Michelson, Gale, and Pearson" carried out in1925. An experiment which demonstrated theprinciples of detection of rotation by optical inter-ferometry was carried out earlier in 1913bySagnac. ' The physical principle involved for msthe basis for the ring-laser Sagnac gyroscope.

In the latest version of our experiments, a geo-

metry has been chosen in which the effect of gra-vity is suppressed. In this way only effects as-sociated with the relative motion of the Earth withrespect to the stars is detected. If our resultsagree with the insertion in the neutron Hamiltonianof a term of the form cu

' L (where &u is the Earth' sangular rotational velocity, and L is the neutron'sangular momentum with respect to the center ofthe Earth) we can conclude that the principle ofequivalence for neutrons has been verified in thequantum limit, in an accelerated frame free ofgravity.

In the next section we will discuss the basicprinciples of neutron interferometry. In Sec. IIIwe will give some experimental details relevantto the neutron source, the neutron monochroma-tor, the detection system, and the constructionof the interferometer. In Sec. IV we give thetheoretical background necessary to understandthe experimental results. Section V is devotedprimarily to a discussion of gravitationally-in-duced quantum interf erence. The experimentsdiscussed in this section were carried out with anincident beam directed horizontally, that is,parallel to the local surface of the Earth. Theeffect of the rotation of the Earth on the neutronphase has been accurately detected with a verti-cally directed incident beam. These experimentsare discussed in Sec. VI. In Sec. VII we makesome concluding remarks.

H. PRINCIPLES OF NEUTRON INTERFEROMETRY

Various schemes have been proposed, and inpart realized, for obtaining interference effectsbetween spatially separated coherent neutronbeams having a wavelength in the angstrom range. 'We will limit ourselves to consider the schemedepicted in Fig. 1, consisting of three identicalperfect silicon crystal slabs cut perpendicular toa set of strongly reflecting lattice planes, typi-cally (220). The distances dq and d2 between theslabs are usually a few centimeters and are equalto within about 1 p m. A nominally collimated,monochromatic beam is directed along the line SAand is coherently split by the first silicon crystalslab by Bragg reflection. These two coherentbeams are again split in the second crystal in theregions near points B and C. Two of these beamsoverlap and interfere near point D in the thirdSi Slab. We always assume that by "beam" wemean a plane wave of limited, but not infinitesi-mal, lateral extent (a few mm or so). If the beamtraversing the path I is phase shifted by increas-ing the "optical" path length (via some interactionpotential) relative to the beam traversing the pathD, the intensities in the detectors C2 and C3 will

21 GRAVITY AND INERTIA IN QUANTUM MECHANICS 1421

AI h

Lattice

A-=

hah

I I

2 hah

I I

phasehf

which the surfaces of the optical components needto be polished to fractions of wavelength. Thephase shift resulting from a step of thickness t onthe surface is given by the formula

where & is the neutron wavelength and n is theindex of refraction, which differs from 1 by

A.n-1= —Nb.2' (4)

FIG. l. A schematic diagram of the LLL interfero-meter. The incident beam is coherently split in the re-gion of the first slab between A and A'. The two coher-ent beams I and II a.re again coherently split in the sec-ond crystal slab and recombined in the third crystal inthe region of points D and D'. The interfering beamsare detected by counters C2 and C3, a noninterferingbeam is detected by counter C&.

change. We will show here that the expected in-tensities in these detectors, as a function of thephase shift P are of the form

o.' cosp

and

I, = n(l + cosp), (2)

The constants & and y depend upon the incidentflux, the crystal structure and the neutron-nuclearscattering length of Si.

The basic principle of this interferometer seemssimple enough; however, there are certain sub-tleties hidden in the apparent simplicity. Thefirst one concerns the alignment of the three crys-tals. Clearly, in order to preserve the Braggreflecting condition for a given wavelength neu-tron, we must align the three crystals to withinthe "Darwin width, " which is typically 0.1" of arcfor neutrons. Bonse and Hart devised a simpleand ingenious way to achieve this result. Theycut out the three slabs from a large monolithicsilicon single crystal of very high quality, free oflattice defects of any kind. As a consequence ofgreat advances in crystal growth techniques,prompted by the needs of the solid-state-elec-tronics industry, it is possible today to purchasefrom commercial manufacturers silicon crystalsof the required perfection with typical dimensionsof order 5 to 10 cm. Spatial coherence in atomicpositions is preserved to a billionth of a centi-meter over these distances.

The second point concerns the accuracy withwhich the surfaces of the slabs need to be polish-ed. It would clearly be impossible to satisfy theusual requirements of optical interferometry, in

Here & is the atom density and b is the neutron-nuclear scattering length. For Si at &= 1..4.4,(n —1)=0.67&&10 . Thus, a step t=2 p, m willcause a phase shift corresponding to ~th of afringe.

The third consideration is the question of theextent to which the incident beam is required to bemonochromatic. In our neutron-interferometryexperiments, the incident beam is only nominallymonochromatic with &&/& = 0.01. The importantfeature of this type of interferometer is that itutilizes Bragg reflection from perfect crystals.This requires the wavelength, along a given tra-jectory (ray line) to be defined to within about 1part in 10 . But this definition of wavelength isaccomplished by the interferometer itself and notthrough the preparation of the incident beam.

The final point we want to mention is a peculiarfeature of angstrom-wavelength interferometry.We can understand. that the two beams BD and CDin Fig. 1 are coherent and produce interferencefringes localized in space, with spatial separa-

0tions of order 1 A. Strictly speaking, the inter-ferometer could consist of the first two slabsonly. The problem is that no film or detector ofany. kind is able to resolve fringes so closely sep-arated in the region of the overlap. The schemeadopted by Bonse and Hart is to use the thirdcrystal as a receiver, or as a mixer. The crys-tal lattice potential Vg corresponding to the re-ciprocal lattice vector G mixes the two wavestraveling along the rays BD and CD, so that theoutgoing beams depend upon the wave amplitudesof each of the incident beams.

We will now look in detail at the diffractionmechanism of the three crystal LLL interfero-meter. (LLL stands for Laue-Laue-Laue trans-mission geometry. ) The theory of diffraction ofneutron waves by the periodic potential of a per-fect crystal lattice is similar in many ways to thetheory of electron motion in solids. A neutronwave of wave vector Kp oriented on or near aBrillouin-zone boundary will be Bragg reflectedforming a coherent state described by the wavefunction

1422 STAUDENMANN, %KRNER, GOLELLA, AND OVERHAUSER 21

g(r) = gp exp(iKp ' r) + g exp[i(Kp+ G) ' r] . (5) X(r) =Xp exp(ikp r)+Xo exp(i& r). (6)

In electron band theory, there will be two suchwave functions, one for states above the energygap corresponding to the reciprocal lattice vector6, and one below the energy gap. In the dynami-cal theory of neutron diffraction, the energy of theneutron is fixed by the preparation of the incidentbeam, and the periodic potential causes a splittingof the allowed internal wave vectors Kp. Thus, theindex y takes on two possible values. The waveamplitudes go~ are determined by the orientationof the external incident wave vector kp and therequirements of continuity of the neutron wavefunction across the entrant boundary. Continuityof the wave function across the exit boundary de-termines the amplitudes of the diffracted beam(yo) and the forward scattered beam (Xp), suchthat the wave function of the neutron leaving thecrystal is of the form

For the symmetric I aue geometry, the solutionto this problem is given in the Appendix. The re-sults are given by Eqs. (A21) and (A22); they areof the form

X, =D(~8)C, (8)

where 4 is the amplitude of the incident planewave, and the coefficients T and D depend uponthe angular deviation &~ of the incident wave vec-tor kp from the exact Bragg condition for the re-ciprocal lattice point G.

Using these results, it is an easy matter to de-rive expressions for the waves emerging from thethird slab of the interferometer. The wave func-tion in the region of detector C3 ls

U, (r) = 4$D(&8) exp(ikp ' ro)D(-&8) exp[ikp ' (r~ —ro)]T(&8)

+ T(48) exp(ikp ' re)D(&8) exp[iko ' (r~ —re )]D(-&8)e P}exp[ikp ' (r —r~)],and the wave function in the region of detector Cq is

U&(r) = QD(68) exp(iko ' r, )D( &8) exp[ik—p ' (r~ —ro)]D(&8)

+ T(&8) exP(ikp ' re)D(&8) exP[i& ' (rD —rz)]T(- n8) }exP[i& (r —rD)] .

(9)

(10)

I Up I=

Ic'

I (1 + cosP) 2A (x, &)

IU21'= I4 I'[&(~, ~) »(~, ~)—cosP] (12)

The functions A and B are given by

The origin of coordinates is taken to be point A inFig. 1. The phase shift P is assumed to be intro-duced into path I of the interferometer. The ex-ternal diffracted, wave vector Q is given by Eq.(A29). After some algebra, we find for squaresof the wave amplitudes

pa 2m~le )a2 cos0g Vcel 1~p coseB

(i6)

~=IP P .2

A(x &)dxam)Eg)

p Sln g ceps

Here ~& is the Bragg angle, V„» is the volumeof a unit cell, &G is the structure factor, and a isthe slab thickness.

Since the beam incident on the interferometeris divergent, we must integrate these expressionsover the angle &~. This is equivalent to inte-grating over the scaled angular variable x. We,therefore, see that the expressions for the param-eters o.' and r in Eqs. (1) and (2) are

and

A. (x, ~) =, , ), sin'$(x'+ cos'&)(x +1 (is) and

4m/E~ /

Y =IP ~P . 28 y B(xq 'E)dÃq

p sln2 B Vce»(18)

a(x, E)=, , sin'&[(x'+ cos'&)'+ sin ]].x +1The definition of the symbol $ is given by Eq.(A23). These functions are dependent upon thetwo dimensionless parameters

kp sin20~ V„»P 4

and

(i4) where Ip is the incident beam intensity. The inte-grals appearing in Eqs. (17) and (18) are shown inFig. 2. For our experiments a=0.246 cm, atX=1.4 A, the value of p:=40, which gives y/&=2, 6.Thus the predicted contrast in the "0-beam" isconsiderably higher than in the "G-beam", asobserved experimentally.

The analysis presented here assumes that eachplane wave Fourier component in the incident di-

1424 STAUDENMANN, WKRNER, COLE LLA, AND OVERHAUSER

A. The neutron wavelength )o

FIG. 4. Schematic diagram of the neutron interfero-meter and the He detectors used in these experiments.

is about 2X10 neutrons/cm /sec. Three smallHe detectors are mounted behind the interfero-

meter, inside an aluminum box (25 X 25&&25 cm')which is attached to a rotator assembly. The Hedetectors have a diameter of & in. and are filledto a pressure of 40 atm, yielding a counting effi-ciency of order 90/0 for thermal neutrons. Fig-ure 4 shows the arrangements of the three count-ers, with Cg mounted on a noninterfering beamfor the purpose of aiding in orienting the inter-ferometer. Occasionally the three He countersare replaced by a single Ar-filled proportionalcounter when x rays are used instead of neutrons.The rotator assembly, consisting of a large steeltube supported on ball bearings, is rigidly attach-ed to the inside of a large dense masonite neutronshield which is mounted on a vibration isolationpad.

The interpretation of all our experiments re-quires an accurate knowledge of the incident neu-tron wavelength &0 as measured in the laboratoryframe of reference. %e have employed a techni-que to measure &0 which is schematically illus-trated in Fig. 5. A pyrolytic graphite crystal isplaced in the beam which passes directly throughthe interferometer. The beam transmitted throughthis crystal is counted with a fission chamber.By rotating the pyrolytic graphite crystal throughthe same (004) reflection used for monochroma-tion of the incident neutrons; a dip in the trans-mission is observed. A similar dip is observedby reflecting the beam to the right instead of tothe left. The difference in the crystal rotationangles for minimum transmission, i. e. , 02 —eq,determines the neutron wavelength in terms ofthe lattice parameter of pyrolytic graphite (c=8.V08 A). The analysis of the data presentedhere is based on this method of determining &0.

B. The interferometer

The interferometer used in the experimentsdescribed in this paper is a considerably improvedversion of the same kind of device used in ourearly work. ' It has been cut with a 600-grit,4-in. -diameter, diamond blade from a 5-cm-dia-meter high-purity silicon perf ect- single-crystalingot. The room-temperature resistivity is p

20000-counts54 sec

l5000-

8= l9.76:= Xo- I.I54 A

IOOOO-

Ilel-

ationpa

configu

5000-

0- doublemonochromator

Interfero

box M counter

rotatinggraphitecrystal

l8 l9 20 I 59I I

l60 IGI

8 (deg )

FIG. 5. An example of the method used to determine the neutron wavelength by measuring the angular spacing be-tween the transmission dips observed by rotating the pyrolitic graphite crystal in the beam transmitted through theinterferometer.

IT Y AND INKR TIA ININ QUANTUM MECHANICS 1425

C2

c3

monitor

FIG. 6. Geometry of the Si-slab exin Fig. 7. Th

e i-slab experiment showne same geometr was

experiments dy s used for the x-ray

imen s esigned to measure Scase the Si slab is r

qb, ~d ( ec. V}, in whichs a is replaced by a plastic slab.

= 3600 ohm cm. The slab= 3 . a th1ckness is a=2.464

d=mm, and the distance between the slabs is

=34.518+0.002 mm. It has been fn s or silicon do n

n most cases, a surface originally flat corn

he departure from flatness du t thenough in some cases to

e o is effect is

p ase shift across theo produce an appreciabl

no silicon es the beam dimensions. Alth

tchant 1s immune from this undesirabough

in a mixture of 3 parts HÃQ (70%), 2 arts l

re or min. The quality of our interfero-

meter can be seen fromrom a measurement of thephase shift induced by thh y e mean neutron-nuclea1nteraction potential. Placin a

earg y, p

a o homogeneous material in the intmeter as shown in F '

i in e interfero-n in ig. 6 and rotating it about

axis perpendicular to thgi a outan

ar o e interferometer scatter-1ng plane results in an osc'liat' t'

n . We show data obtained inzs way for a Si slab (thickness 7' —0.2ig. 7. The slit size on the neutron beam was

6 'y show that the phase.6 mm . It is eas to

corresponding to a given rotat 1on angle ~ is

sin5 singecos'8& —sin'6 '

where N is the atom density, b the nucleae interf erometer Bra

(220 planes), and A. thragg angle

n e neutron wavelen th. Tperiod of these oscillat

g . heci a 1ons therefore provides a

precision measurement of thb.

en o e scattering length

The exexperiments described in this pa erpresented unusual deafa 1culties with regard tomounting the interferometer. The ex e

ol ot t' thing e interferometer about a hzontal axis and also ab

a or1-

hso a out a vertical axis. W

ave therefore beenen forced to find a method forholding the interferomete ' i .We ha

me er without straining it.g ct1on of waxes, gle ave tried a lar e sele

an epoxies. In all cases th'

ded the performance f th

e 1n uced strain spo1l-

considerable eff orte o e interferome eter. After

e e ort, we have discovered that

counts239 sec

4000-

5000-

2000-

IOOO-

0-I I

-0.26I8 -0.)745 -0.0873I

0

X.= I.229 A

neutronsI

0.0875I'

O.J745 0.26I8

sin 8 sin 8(radians)

FIG. 7. Example of data obtained bshown.

e y rotating an Si slab in the ine interferometer (Fig. 6}. The counts in detector C3 are

1426 STAUDKN MAN N, WKRNKR, COLE LLA, AND OVKRHAUSKR 21

"double-sticky-back" plastic tape works. It pro-vides the necessary adhesive character, yet it issufficiently pliable and resilient so as to mini-mize the transmitted strain.

IV. THEORY GF THE EXPERIMENTS

Classically, the Hamiltonian governing a neu-tron's motion in the gravitational field of the ro-tating Earth is

2P Q g ~ ~ J

2m;(2o)

Here the angular momentum of the neutron's mo-tion about the center of the Earth (r =0),

L=rxp (21)

=p 2

+mego' r —v L+ Vo,2m)(22)

where Vo is the gravitational potential energy atsome reference height above the Earth (say, thecenter of the interferometer), and gp is the ac-celeration due to gravity. The classical equa-tions of motion are Hamilton's equations

ea . aar= =,' and p=- -- —.Bp Br

Here, the dot implies a time derivative, so thatr is the neutron velocity in the coordinate frameof the rotating Earth. The first of these equa-tion gives the canonical momentum

p =m~r + m~Q) X r ~ (24)

p is the canonical momentum of the neutron, co isthe angular rotation velocity of the Earth, M themass of the Earth, m& the inertial mass of theneutron, and m~ the gravitational mass of theneutron. From an epistomological point of view,it is not possible to be confident that this Hamil-tonian correctly describes quantum- mechanicalphenomena, especially. those involving interfer-ence. However, for lack of evidence to the con-trary, we will assume it is also the correct quan-tum-mechanical Hamiltonian, and then see if thepredictions based on it agree with the experiment.The principle of equivalence would require thatthe inertial mass m& and the gravitation mass m~in Eq. (20) are equal.

Since the distances involved within the neutroninterferometer are very small compared to theradius R of the Earth, we can write (20) as

r = rp+ vpt+ —,'(gt') + p((ut'~&& g) . (26)

The transit times for thermal neutrons throughthe interferometer in our experiments are of or-der 5x10' sec. .Based on a sidereal day of 23 h,56 min, one has co=V.29x10 ' sec . Thus, wesee that the term in (28) involving a& is smallerthan ,'gt by a fa—ctor of about 10 . Therefore,the Coriolis force has a negligible effect on thetrajectory over these small distances; however,its effect on the neutron phase is not negligibleas we shall see.

The discussion so far has been based on classi-cal mechanics. In order to calculate the phaseshift P in a neutron interferometer experiment,we assume that we can associate a de Brogliewave of wave vector k with the neutron havingcanonical momentum p:

p=@k q (29)

where @ is Planck's constant divided by 2r. Thephase difference for the neutron wave traversingthe path ACD in Fig. 4 relative to the neutronwave traversing the path ABD is then given by

P= — t p dr- — t p'dr.@ ~ACD @ ~ABD

1p dr, (so)

The momentum appearing in this line integral onthe path ACDBA around the interferometer isgiven by Eq. (24}. The phase shift thus involvestwo terms,

in a rotating frame,~ ~

m;r =m gp —m&&u& ((exr) —am, (ox r. (26)

Thus, the term -~ ' L in the Hamiltonian givesrise to both the centrifugal acceleration and theCoriolis acceleration. Since we will only be in-terested in the neutron's motion over distancescorresponding to the dimensions of the interfero-meter, which are very small compared to theEarth's radius, we can define an effective gravi-tational acceleration in the usual way

g =gp+ (m, /m, )(o && ((u &&%), (2&)

which we take to be independent of position r. Un-der this assumption, we can solve (26} for the lo-cal motion of the neutron in the frame of the ro-tating Earth. To leading order in (o, the solutionls

and the second gives its time derivative

p =mego —co Xp (25) p =™tp r ' d r +™(&u x r) ' d r . (sl)

Combining Eqs. (24) and (25} we obtain the well-known equation of motion for a classical particle

The velocity r is obtained by differentiating Eq.(28). To a very high order of approximation we

GRAVITY AND INERTIA IN QUANTUM MECHANICS 1427

The area A' is given by

A' = (2d + 2ad cos8e) tanee . (33)

The angle P is defined to be zero when the planeABDC of the interferometer is horizontal. Thelaboratory neutron wavelength i.s related to itsvelocity by

~, =h/m, .)rf, (34)

and 8~ is the Bragg angle. The result for a verti-cally directed beam is given in Sec. VI.

The second term in Eq. (31), which we callP~~, is due to the rotation of the Earth. Usingvector calculus, this integral is easy to evaluate,

(35)

The normal area A enclosed by the beam paths is

A = (2d + 2 ad) tan8e . (36)

The formula (35) was obtained by Page usingwave-optical arguments, and by Anandan and20

Stodolsky within the framework of general rela-tivity. Recently, an interesting derivation hasbeen given by Dresden and Yang in which thephase shift for either a rotating neutron or opticalinterferometer is derived from the point of viewof a Doppler shift due to a moving source andmoving reflecting crystals.

The actual path of any given neutron within acrystal slab is more complicated than the linedrawn straight across the crystal as shown inFig. 1. There is a current j carried by the -branch part of the wave function and a current j~carried by the P-branch part of the wave func-

can regard the trajectories between the interfero-meter slabs as straight rather than paraboliccurves. The angular deviation, from a straightline over these distances (for &=1.4 A) is oforder 0.01" of arc. It is fortunate that this angu-lar deviation is about 10 times smaller than the"Darwin acceptance width" for Bragg reflectionin the silicon crystals. If this were not so, aneutron on the trajectory AB, say, would not beBragg reflected by the middle crystal slab of theinterferometer. The first term in (31), which wewill call P„„, is the phase shift due to the gravi-tational field of the Earth. To work out the inte-gral for P~„requires us to specify the directionof the incident beam and the orientation angle Pof the interferometer with respect to this direc-tion (see Fig. 4). For a horizontally directedincident beam the result is

P~„=-2mm, m~(g/h )&OA' sing -=-q~ sing .(32)

V. GRAVITATIONALLY INDUCED QUANTUMINTERFERENCE

In this section we will confine our attention toexperiments in which a horizontally directed inci-dent beam is utilized as shown in Fig. 4. Theexperimental procedure involves turning theinterferometer, including the entrance slit and thethree detectors Cq, C2, and C3, about the incidentbeam line AB. At each angular setting P, neu-trons are counted for a preset length of time(actually based on the incident beam monitor).This procedure allows the neutron on the beamtrajectory CD to be somewhat higher above thesurface of the Earth than for the beam path AB.The difference in the Earth's gravitational poten-tial. between these two levels causes a quantum-mechanical phase shift of the neutron on the tra-jectory ACD relative to the trajectory ABD. Thephase shift on the rising path AC is exactly equalto the phase shift on the opposite rising path BD,as can be shavn by applying Huygen's principle.

The phase shift P~„depends on the product ofthe inertial and gravitational masses m, m~ of theneutron. Thus, measuring this phase shift in-duced by the Earth's gravity can be r egarded asa test of the princple of equivalence in the quan-tum limit if we compare the mass

m„=(m, m, )'" (37)

with the neutron mass obtained from mass spec-troscopy results on the proton and the deuteronaccording to the formula

m„= m~ —m~ + E„/c = 1.6747 x 10 4g . (36)

The deuteron binding energy is obtained from theradiative capture 'gamma-ray energy &„(=2.23MeV). The fact that this experiment is a test inthe quantum limit is apparent since quantum-mechanical interference is involved, and Planck'sconstant appears explicitly in the formula (32).

tion. In addition, there is a current j ~ due tothe interference of the o.'- and P-branch wave func-tions which leads to a sinusoidal trajectory forthe neutron. This is the phenomenon which leadsto Pendellosung interf er ence fringes. Considera-tion of these effects has recently been treated inextensive detail by one of us. The conclusionreached is that to very high order of approxima-tion the trajectory can be regarded as a straightline across the crystal, and that the microscopicdetails of the trajectory do not play a role in cal-culating the nest phase shift due to gravity withinthe crystal medium.

STAIJOENMANN, %KRNKR, COLKLLA, AND OVKRHAUSF R

A. The total phase shift

Unfortunately, as we have pointed out previous-ly, there is an additional effect on the measuredphase shift P resulting from bending (or warping)of the interferometer under its own weight. Thiseffect is dependent upon the rotation angle P, sincethe experiment involves turning the interferometerabout an axis which is not an axis of elastic sym-metry. We eall this effect

Pbeag =-qbeng Sm4 ~

In Subsec. B below we will justify writing P„„„inthis form.

For a horizontally directed incident beam we caneasily work out the dot product involved in evalua-ting Ps~ in Eq. (35); the result is

4mm;Ps~ ——' ~&( cos@cos&~ + sing sini' sin8~ },8 age ac

where ~L, is the colatitude angle, I" is the angle ofthe incident neutron beam west of due south.Beam port B is oriented nearly exactly along anorth-south line, such that the monochromaticbeam incident on the interferometer is directeddue south. Thus, the angle I' in Eq. (40) is zerofor our experiments, and we can write (40) in theform

=0, while the gravity and bending effects aremaximum for g =90'. We can rewrite Eq. (43)as

P = q sin(p —p,),where

2 2 2q' =(q +qb g) +qg

(44)

(45)

qg aan ae

~grav + ~bend(48)

The fact that the phase shift due to the Earth's ro-tation (Sagnac effect) depends upon cosg and notsing comes about as a result of our selection ofdue south as the incident beam direction [I'=0in Eq. (40)]. This leads to an important experi-mental circumstance. Although qs~~ is of or.-der 2.5% of q~„, its contribution to the total fre-quency of oscillation q of the interference patternis very small (of order 3 parts in 10 ). However,it leads directly to a shift P, in the center of theinterferenee pattern. Table I gives the calculatedwavelength dependence of q~„, qs~, and Pg.We have used the colatitude angle 81, ——51.37 andthe acceleration due to gravity q = 980.0 cm/secat Columbia, Missouri, along with the dimen-sions of our interferometer given in Sec. III tocompile this table.

PS agnac qgagnae COSg (41) B. Bending

where

qs~ ——(4~~,.(uAlh) cos8~ . (42)

Consequently, the total phase shift in these experi-ments involves three contributions

P =Pgeae+Psagnae+ Pbeng

=-q~„sing + qs ~ cosQ —qb„g sing . (43)

We see that the Sagnac effect is maximum for P

The values for qb„d given in Table I have beenobtained from a series of experiments using xrays. The procedure involves using molybdenumKn x rays (&=0.71 A). We direct a beam of xrays along the same incident line AB (Fig. 4) andobserve the interfering x-ray beams with an x-ray sensitive porportional gas detector as a func-tion of rotation angle P. The effect of gravity(gravitational red shift) on the x rays over the

TABLE I. Calculated frequencies of oscillation, q.

0 Og

(deg)A.

(cm2) (cm2)&gray

(radians)~Sggnac

(radians)&bend

(radians)6}0

(deg)

0.60.81.01.21.41.61.82.02.22.4

8.988512.023615.093418.208321.380024.622027.950531.385134.950338.6781

4.038 565.437 916.885 828.398 469.99543

11.701 113.547 115.575 417.844 520.438 7

4.035 255.429 976.869 998.370 459.949 61

11.630 313.441 815.423 617.630 120.140 0

9.525517.090427.028539.518154.802473.210995.1913

121.362152.596190.168

0.582 3960.784 1960.992 9971.211131.441 431.687 411.953 622.246122.573 332.947 44

1,406 411.875 212,344 012.812 823.281 623.750 424.219234.688 035.156 835.625 63

3.297 622.512 072.029 541.703 261.468 071.290 621.152 081.041 030.9501090.874 415

21

~ ~ ~ I ~ ~ ~ I ~

IOK

cooqts68 sec, IOK-

5K-

IOK-

5K-

IOK

5K

IOK-

5K

IOK-

5+IOK-

5K-

IOK-

5K-

IOK-

5K-

X RAyS

(220)20

l5

Ip

0

-lp

coUnts&42 sec.

4Qpp. .

40OO-

45oo. -

5000-

55oo.

«pp—

50oo-

550p-

550p

«oo-500p.

X RAYS (44p)

&g& = 2po

@= l5O

loo

5o ~

0' .

5o

@--IP---

C'=- I5

-20

I i ~ I ~ e ~ ~

-0.) 0p5 0 pp5

sin 5 sin 8

setto 'entation angle Q

ction of the

ings 4, of the '(see Fig. 6

0in erferometer

~ ) for varloQs

reflection in Sr. This data '

p o this datae rmine q bend

distances 's involved in the int

bl 8 c s th x-e e x-ray wave field m

y uniform (cohie may not rem

erentp over the cr

th e'e Ulterferometo warping of th

P will not necnecessarily giveensity

g abend y

uso e reme car

~ n

and we do not observe

e circumvent theesuring the shift

ese problems bin the phase

oscillation patt s ltse of an interf

fit lb f 1

Fi. 6p astic in the ' te

fo me ry) as asee

P. Data obtained ifunction of thee angle

il tof 1 t fo th

shift

pI.n ig. 10 whe

erl-w ere we plot th

rn& vs amer W e observe ach justifies the fo

lmear relatione orm of Eq. (89

se plots s Q&end' It is reason bla y cer-

FIG. 9. X-ra d

(radians)

. 9. X-ray data for the (4400) reflection in Si.

@*-20..

4oooI

-Q.I745 -0087'sins sin g

cos g

tain that ththe functional form o p

I I I' I I I I

300-PHASE

(deg}200-

IOO-

0-'I

-0.4 -0.2I I

0 0.2I I

0.4

FIG. 10.sin 4'

A plot of the hi

pions re ' ro ating a plastic

erring x ra b

l i 8 Th- -. "l. ~ "-ese plots gives q

Pb„~—krM= (2m/X )6d0

where &d is th dife' ference in g

the inrelative to ACD d

or theng o

the fact that tha xs very nearl fy our times the

STAUDENMANN, %ERNER, COLKLLA, AND OVKRHAUSER 21

4000

3000

C3

2000,

+1.060 A

100

80

FOURIER TRANSFORM+1.060 A

1000

0-40 -32 -24 -16 -8 0 8 16 24 32 40g [deg ]

FIG. 11. Gravitationally induced quantum interferenceexperiment at X 0=1.060 A. The counting time wasabout 5 min per point.

60

40

slope of the (220) data, we conclude that

~do- sin'8& .Therefore putting the arguments together thathave led to (39), (47), and (48) we find

Pb„d =—q„„~sin@ =-(C/%0) sin es sing;

(48)

(49)

20

0 0l I i i

10 20 30 40 50 60

the numerical value of the constant from thedata is

C =34.5V rad A. (50)

The reason why the bending effect seems to de-pend quadratically on sineB is not yet understood.

C. Experimental results

We show in Figs. 11 and 12 representative dataobtained at two wavelengths: ~0 ——1.060 and 1.419A, respectively. The neutron counting rate indetector C& is plotted versus the interferometerrotation angle P. The contrast (maximum/mini-mum) of these data is seen to be about 3 to l.Contrasts as high as 8 to 1 have been observedin some runs. To obtain the frequency of oscilla-tion q we Fourier transform the data numericallyaccording to

N

sin eQQ si5 (8i)g=1

FIG. 13. Fourier transform of the data of Fig. 11.

tensity, and the index j runs over all & datumpoints. The Fourier transforms of the data ofFigs. 11 and 12 are shown in Figs. 13 and 14.

There is loss of contrast at larger rotationangles g, which we believe to be due to warpingof the interferometer under its own weight as theinterferometer is rotated. This explanation ofthe effect is in accord with various experimentswe have carried out with reduced slit sizes inwhich the loss of contrast is reduced. We havefound that as the neutron wavelength becomeslarger the loss of the contrast occurs at smallerrotation angles P. This observation is also inagreement with the above explanation, since thebending effect measured with x rays is porportion-al to sin'es(= &OG'/4).

In any interferometry experiment, the long-term

8000

6000-g 1&19 A

where I is the oscillatory part of the neutron in-120-

100

Ill 80

FOURI FR TRANSFORM

X; 1.419 A

C34000 60

2000 40

0 I I I I I I

-40 -32 -24 -16 -.8 0 8 16 24 32 40|t)[deg ]

FIG. 12. Gravitationally induced quantum interfer-ence experiment at X 0= 1.419 A. The counting time wasabout 7 min per point.

20

I

0 1 0 20 30 40 50 60 70 80 90

FIG. 14. Fourier transform of the data of Fig. 12.

GRAVITY AND INERTIA IN QUANTUM MECHANICS

5000-

counts360 sec.

2000-

0o= l.05 4

data taken from58.4 to -4I.6 deg

~ M " -58 426 " -52.4

a 26 ' -26 4~ 20 -20.4s mean value

~%'s'

L

'N

I

00

*0

~ 0+

4

*0s

~ gpO

)000-

0s

~ ~

4e

0

0

*D

30 20 IO 0 -CO -20 -30 -404 {deg)

FIG. 15. Gravitationally induced quantum interference experiment at & p=1.050 A. These are data from 6 runs takenover a period of about 75 h. These data show the long-term phase stability of the interferometer.

Frequencyof

oscillation

q

loo-

50-

Gravitationally induced

quantum interference //

//

l4+~2

/

l5$/

//

I /3l. l~O. I / l2.2g2 4/26/

178/30

25'7/34/34. I

/

qb d( experimentaobend

I

0.50 I l5 2

&. (A)FIG. 16. A plot of the frequency of oscillation of q of

a large series of scans of the type shown in Figs. 11and 12 as a function of wavelength Ap. The dashed.curve is the least-squares fit to the data using Eqs. (32}and (49). The labels next to the data points are runnumbers.

phase stability is an important consideration. Weshow in Fig. 15 data obtained at &p ——1.050 A inwhich we have tested the phase stability of ourinterferometer. This figure shows the gravita-

tionally induced quantum interference fringes ob-tained on 5 separate runs. The total data collec-tion time was about 75 h. We see that the phasestability over this period of time is extremelygood. The fact that the interferometer is mountedinside a heavy aluminum box, which in turn ismounted inside a large heavy masonite neutronshield, creates isothermal conditions sufficientto obtain data of this quality.

We have now taken data of the type shown inFigs. 11 and 12 at a wide selection of neutronwavelengths. The frequency of oscillation q foreach run has been obtained both by the Fourier-transform method discussed above and by a least-squares-fitting procedure (which is discussed inthe next section). These two data analysis techni-ques agree to within the statistical uncertainty ofthe raw data. The results of this extensive set ofmeasurements is summarized in Fig. 16 in whichwe have plotted the observed frequency gf oscilla-tion q versus the incident neutron wavelength &p.

The dashed line is the prediction of theory, basedon Eq. (32) and the measured frequency of oscil-lation due to bending. We have fitted the theoreti-cal curve to the data leaving (m, m~)" as an adjust-able parameter. We find

(m, m~)' = (1.675 + 0.003) && 10 g (52)

which agrees with the rest mass of the neutronobtained from mass spectroscopy to within thelimits of error.

STAUDKXMA WX, WKRNER, COLKLLA, AND OVKRHAUSKR

VI. NEUTRON SAGNAC EFFECT

A. Horizontal-incident-beam experiments

According to Eqs. (2) and (44) the counting ratein detector C3 should vary with the rotation angle

Q according to the formula

I,(p) = o'f&+ cos[q sin(g —gp) +tIO]], (53)

We pointed out in the previous section that thecontribution of qs ~ to the total frequency of os-cillation of the interference pattern is very smallas a result of our selection of the direction of theincident neutron beam. However, the Sagnac ef-fect leads to an angular shift $0 in the center ofthe interference pattern given by Eq. (46). lnSubsec. A we will discuss our results on the mea-surement of $0. We have also pursued an alterna-tive approach, utilizing a vertically directed inci-dent beam, to observe the neutron Sagnac effect.These experiments are described in Subsec. B.

(deg)

0

gIe

g l3

theory

34 ~2e

~12.2~ I I X g24Q I6le g

3OQ

+253I.I~ Sm ~+

+i5

~32

Shift C, of the center of

the interference pattern

where P, is the "zero-phase" of the interfero-meter resulting from the fact that the two legsABD and ACD are not precisely equal. It is ap-parent that in order to separately measure $0 and

Pp data must be accumulated over an angularrange of (@—$0} where the sine function departsfrom linearity.

This presents special difficulties since the warp-ing of the interferometer at large P tends to washout the contrast. Because of this loss of contrastthe analytical form of the intensity I,(g) is morecomplicated than Eq. (53). We have pursued thisproblem by a least-squares fitting procedure inwhich we use a parametrized form of the inten-sity profile. Since the experimental results of

.the type shown in Figs, 11 and 12 exhibit an enve-lope and a weakly sloping mean level, we havechosen the form

I~(g) =A + BP + CP

+ D cos[@sing + &q] cos[q sin(g —g p) + P p]

(54)

as a phenomenological generalization of Eq. (53).There are nine parameters in this equation. '

B C D Q &g Pp q, and P,. We are, of course,only interested in q and Po. The results of an ex-tensive series of measurements e,nd data analysesare shown in Fig. 17. The dashed line is the re-sult of a numerical calculation based on Eq. (46)relating the angle $0 to qs, „. Since qa, is in-dependent of wavelength (aside from its depen-dence on the area A), and q~,„ is proportional tothe wavelength (aside from its dependence on thearea A'), we see that except for the small correc-

0 05

).(A)

l.5

FIG. 17. Experimental results for the shift ft)0 of thecenter of the interference patterns as a function ofwavelength Xo. These results were obtained by least-squares fitting of the functional form of Eq. (54) to alarge series of data of the type shavn in Figs. 11 and12. The angle f3t 0 is related to the frequency of oscilla-tion due to the neutron Sagnac effect as given by Eq.(46). The dashed curve libeled "theory" is the resultof using Eq. (46) and the geometric parameters of theinterferometer given in the text. The labels next to thedata points are run numbers.

B. Vertical-incident-beam experiments

We show in Fig. 18 a schematic diagram of ourmost recent experiments designed to detect theneutron Sagnac effect. The initially horizontalbeam from the monochromator is reflected by aberyllium crystal through 90, such that the beamincident on the interferometer is vertical (along a

'plumb line). . The experimental procedure involvesturning the interferometer about the vertical lineAB through various angular settings Q. For abeam which is precisely vertical, the phase shiftdue to gravity, P,„, is independent of the angle@, as can be seen from symmetry. However, theangle between the rotation axis ~ of Earth and theinterferometer normal area vector A is P depen-dent. Thus, this incident beam orientation allows

tion due to q„.„d in Eq. (46}, Qp should be inverselyproportional to ~0. From this data it is clear thatwe are observing the neutron Sagnac effect. How-

ever, the scatter in the data is rather severe.

GRAVITY AND INERTIA IN QUANTUM MECHANICS 1433

phasshifter

vert ical

Cp COUA)S

500 sec.

l000-

500

A) I(C) )l

nterferometerat Bragg pos.

coll ima to r&

tOO"50"

(B)~

. I(Cgb«~ I i r, }

lriterferometer off Bragg pos.

800"(C)

l(C)) = l(4))-l(4))b

beam from & ............ ......,lmonochromato r

P/Ff PFf F I pp

li

p Se crystai500--

FIG. 18. Schematic diagram of the configuration ofthe apparatus for the vertical-beam experiments de-signed to measure the neutron Sagnac effect. The draw-ing is not to scale. The collimator is approximately1 m in length, and the interferometer is approximately8 cm long from point A to point D. The angle 6 of thephase shifter is defined to be zero when it is parallelto the three interferometer slabs.

400

0

200--

loo-0"

400 (H) ~(C)

(P') g (61

0 s ~ ' r

4~ ~ a l

0 Ct

0

~ 4 — ~ 4 1

= g@,~„sing . (55)

The superscript "v" indicates that this expressionis for a vertically directed incident beam. Forthese experiments the incident wavelength is fixedat &o —1.262 A, which in turn determines the valueof the normal area, whicI '.s&=8.864 cm . Theangle P is defined to be zero, when the normalarea vector A is directed due west. The predictedfrequency of oscillation is

„=91.92 deg = 1.604 rad. (theory). (56)

As we turn the interferometer through variousangles P, the counting rate is expected to varyaccording to

f(g) =A+Bcos(qe, „sing+ po).

By allowing A and B to be numerically different,we have taken into account the facf that perfectcontrast is never actually realized in practice.The results of such an experiment are shown in

us to experimentally suppress the effects of gra-vity, leaving only the effect of the rotation of theEarth on the phase shift. The bending effect isalso independent of P since the orientation of theinterferometer with respect to g is fixed. Thesuggestion that the Sagnac effect could be obser-ved with this geometry was first made by Anan-dan."

Using Eq. (35) it is an easy matter to work outthe formula for P8. „as a function of @; we get

Pa = (4vm, /h)&uA sing~ sing

200

(i) f(

lN E S W Nl . l . l . l . l

90 I80 270 0 904(deg )

FIG.. 19. Data taken in a "direct" measurement of theeffect of the Earth's rotation on the neutron phase snift.The various parts of this figure are explained in detailin Sec. VIB.

Fig. 19. There is a difficulty in directly inter-preting the counting rate in detector C, to be f(P).There is a natural variation of the "effective"incident beam intensity which is the angular ac-ceptance range for Bragg scattering by the siliconinterferometer. This is due to the energy-anglecorrelations in the incident beam resujting fromthe monochromation process using single crystals.However, one can measure this variation sepa-rately by blocking off the beam in one leg of theinterferometer (and then the other) and measuringthe effective beam strengths under noninterferingconditions. The series of curves in Fig. 19 arethe results of utilizing this idea. Part (A) of thisfigure is the raw data for the counting rate in de-

1434 STAUDENMANN, WKRNKR, COLKLLA, AND OVKRHAUSER 21

tector Cz versus g. Part (B) is the backgroundcounting rate with the interferometer rotated offthe Bragg reflecting condition. Part (C) is theraw data minus the background, which we calli(P}. The sets of data (D) and (E) are the countingrates in detector C3 when beam I is blocked offand then beam II is blocked off with a cadmiumabsorber. Parts (F) and (G} are the backgroundcounting rates under these conditions. Part (H)gives the average of the data in scans (D) and (E)minus the average background of scans (E) and(G). The data of part (H), called o,'(@), is directlyproportional to the effective incident beam inten-sity. The final graph, part (I), is obtained bydividing f(g) in part (C) by o.'(g) in part (H), thatlsq

f(p) =f(p)/o'(&) . (58)

This procedure allows us to divide out the angle-energy correlation effects, leaving orily variationswith P due to interference. The solid line in part(H) is a least-stiuares fit of the data to the func-tional form (57). We find

qs„„——104.4+ 0.4 deg

= 1.822 + 0.007 rad (expt. ). (59)

This value is to be compared with 91.92 deg in Eq.(56). There are several difficulties with this tech-nique. The most serious one is that it involvesa number of independent steps in the sequence inarriving at the corrected plot in part (I). Thus,the experimental systematic and statistical errorsaccumulate.

In order to circumvent this difficulty we havedeveloped a technique in which we directly mea-sure the phase shift. We insert a slab-shapedphase shifter into the interferometer as shown in

Fig. 18. This slab is, in fact, another Si singlecrystal of thickness T =0.2931 cm, although itcould be made of any material. Rotating this slabthrough an angle & about an axis normal to theparallelogram ABDC results in a phase shift aris-ing from the mean neutron-nuclear potential. Theformula for this phase shift was given in Sec. III,,Eq. (19). As we rotate this phase shifter throughvarious angles &, the counting ratios in detectorsC2 and C3 are observed to oscillate. Repeatingthis procedure at another setting Q of the inter-ferometer results in another oscillating pattern,of the same period, but shifted in phase with re-spect to the first pattern. We show in Fig. 20(a)data taken at @=0 and in Fig. 20(b) data takenat p =-90'. The phase shift between these twopatterns is due to the rotation of the Earth. Theresults of an extensive series of measurementsare shown in Fig. 21. Each datum point in this

200

I I I

.1000counts

. .2pp (e) /=0

I I I I I I

-8 -6 -4 -2 0 2 4 6 8Sfdeg)

FIG. 20. Typical oscillating counting rates observedin detector C3 at two orientation settings fII} of the inter-ferometer. The counting time for each datum point wasapproximately 600 sec.

figure was obtained by least-squares fitting of asine wave of unknown phase to data of the typeshown in Fig. 20. Because of long-term driftsof the interferometer phase, measurements at areference angle (usually A pointing east or west}were repeated after each new setting P.

The labeling of north, south, east, and west onthis diagram was achieved through an astronomi-cal sighting of the star Polaris. This line of sightwas carried inside the reactor hall (which is be-low ground level) by precision surveying techni-ques and transferred onto the interferometer witha laser mounted on a rotary table.

The solid curve in Fig. 21 is the result of aleast-squares fitting of the data to a sine wave.

P (deg)

FIG. 21. A plot of the phase shift P due to the Earth' srotation as a function of orientation fII} of the normalarea A of the interferometer about a vertical axis. Thesymbols N, W, S, and E indicate north, west, south,and east. These data were taken in six sections as dis-cussed in the text. The different symbols are for theinterferometer box facing various directions.

GRA VITY AND INERTIA IN QUANTUM MECHANICS

It gives

q~, ——96.8+0.2 deg

=1.689+ 0.003 rad (expt). (60)

P (deg)

This result is in closer agreement with theorythan the result (59). However, it is obvious fromthe data that there is still a substantial problem.These data were taken in six steps. Because ofgeometrical limitations of the rotator inside theheavy masonite shield, we were only able to accu-mulate data over an angular range of Q of about130'. To obtain data through other ranges of P itwas-necessary to turn the entire masonite shieldbox through large angles (typically 90 ). Thisrequired releveling and orienting of the rotatorassembly. One notes that the data from each ofthe six sequences do not fit perfectly together.This fact points up the need for extreme care andprecision in this experiment. Since the magni-tude of the frequency of oscillation due to gravityis about 50 times the frequency due to the Earth' srotation, a small misalignment of the beam axisoff verticality results in a contribution from P~„If the incident beam is off the axis of a plumb lineby an angle z, the P-dependent part of the phaseshift due to gravity is

P~, =q~„sing siny, (61)

q~„=94.6 + 0.3 deg

=1.651+ 0.005 rad (expt). (63)

This result is within 3/z of the theoretical predic-tion (56). We believe that to improve upon thisresult would require new techniques, of which weare not aware.

where q„„is given in Sec. IV. To give an ideaof the size of the effect, suppose y'=0. 1 . At ~()

=1.262 A, q „=44 deg rad =2521 deg, givingP~„=4.4 deg. Thus, a misalignment of the axisof the beam (and also the axis of rotation) of 0.1'will result in an error of about 5% in the mea-surement of the neutron Sagnac effect. On thebasis of these considerations, we now feel thatthe unusual agreement of experiment with theoryreported in our preliminary paper was somewhatfortuitous.

We have now modified the heavy masonite shieldso that the rotator assembly can be turned througha full 360 deg without realigning the beam axis.We have exercised extreme care in aligning thebeam axis using precision levels. Our most re-cent data taken under these conditions are shownin Fig. 22. The solid curve is again a least-squares fitting of this data to a sine curve. Wefind

0t

50 too

N

l50

-50--E

4 (deg)

-l00--

FIG. 22. A plot of the phase shift' P due to the Earth' srotation as a function of orientation f as in Fig. 21.These data were taken after modification of the rotatorassembly to allow a complete 360' sequence of scansto be performed without realignment of the verticalityof the incident beam.

VII. CONCLUSIONS

Our observation of quantum- mechanical inter-ference phenomena in these experiments confirmsthat the Newtonian potential mg ' r must be includedin Schrodinger's equation, and that this potentialinfluences the Phase of the neutron wave functionin a manner expected for any other potential. Webelieve that this result has a fairly deep signifi-cance which concerns the principle of equivalence.

The equality of inertial and gravitational massis one statement of this principle. The classicalexperiments ' of Eotvos and Dicke have verifiedthis equality to very high precision. An alterna-tive and stronger statement of this princple re-quires that the results of an experiment carriedout in a uniform gravitational field cannot be dis-tinguished from the results of an experiment car-ried out in a gravity-free laboratory experiencinga constant acceleration.

All verifications of the equivalence principle,prior to our experiment, have been in the classi-cal domain. The experimental results did notdepend on Planck's constant. For the experimentsdescribed in this paper, the number of interfer-ence fringes observed for a given rotation of theinterferometer depends on the numerical value ofPlanck's constant, and therefore represent a testof the principle of equivalence in the quantumlimit.

Since the phase shifts observed in our gravita-tionally induced quantum-interference experimentdepend upon the product m;m~, and the phase shiftsin the Sagnac experiment depend only on m&, wecan certainly claim that the combination of theseexperiments demonstrates the equivalence ofinertial and gravitational mass in a quantum-

1486 STAUDKNMANN, %KRNKR, COLKLLA, AND OVERHAUSKR

mechanical phenomenon. We would like to pro-pose that a stronger and deeper conclusion can bereached.In order to truly verify the principle of equiva-

lence one must carry out tmo experiments —one onthe surface of the Earth in a laboratory at rest,and another in a laboratory far out in space havingan acceleration g. We have not done this experi-ment. However, we suggest that the second ex-periment need not be done if one accepts thevalidity of Schrodinger's equation in a gravity-free, inertial frame. If this is granted, then theoutcome of an interferometer experiment in anaccelerated laboratory can be calculated withcertainty. This has been done, and we find thatthe observed phase shift in our Earth-bound ex-periment agrees with this prediction if we replacethe laboratory acceleration a with g in the finalformula.

To summarize then, one must either questionthe validity of Schrodinger's equation under zero-gravity conditions or assert that we have verifiedthe stronger statement of the equivalence principlein the quantum limit. The first alternative seemsunacceptable.

It would be very exciting to carry out these ex-periments at both very low neutron energies andat very high neutron energies. In the ultra-coldneutron energy region where the change in gravi-tation potential energy is comparable to the neu-tron kinetic energy, the WEB approximation forcalculating phase shifts fails and the neutron tra-jectory is not well defined. In the very high neu-tron energy region where terms of order (v/c)cannot be neglected one will begin to see generalrelativistic effects as discussed by Anandan andStodolsky. We believe that experiments in bothregions of neutron energy are possible, and weare currently pur. suing certain new ideas alongthese lines.

ACKNOWLEDGMENTS

The role played by C. Holmes in the successfulcompletion of this experiment was extraordinarilysignificant. We would also bke to acknowledgethe support of the staff of the University of Mis-souri Research Reactor, in particular, %. B.Yelon, R. Berliner, C. Edwards, and R. Brugger.We are grateful for the excellent single crystalsof silicon provided us by B. Stone and J.. Burd ofthe. Monsanto Corporation. We thank our col-leagues G. W. Ford, B. DeFacio, and D. Green-berger for their interest and many helpful discus-sions over several years. And finally, we arevery indebted to S. Paiva and K. I eu for the as-tronomical sighting and surveying. This workwas supported by the National Science Foundation

Atomic, Molecular, and Plasma Physics Programthrough Grant No. '?6 08960.

APPENDIX A: DYNAMICAL THEORY OF NEUTRONDIFFRACTION

We review here the essential aspects of the dy-namical theory of diffraction for the symmetricalLaue-transmission geometry (Fig. 23). For fur-ther details we refer the reader to various reviewarticles. ' We assume that the absorption iszero, which is a very good approximation forsilicon.

Let the incident-neutron wave function be givenby the plane wave

y(r)=C exp(ik, r).To find the wave function g(r) inside the crystal,we must solve the Schrodinger equation

(Ai)

[ (k '/2-~) v'+ V(r)]y =~,y,

where V(r) is the periodic interaction potential ofthe neutron with the lattice. Defining

v(r) =- (2m/k') V(r)

(A2)

(AS)

k20=- (2m/0') &„Eq. (A2) can be written as

(V +ko)g=v(.

We now write P(r) as a Bloch function

g(r) =g go exp(iko'r+iG r),

(A4)

(A5)

(A6)

and expand v(r) in the Fourier series

v(r) =Q v-e@' . (Av)

y,-[k, -(K, +G') ]=~ v; --q (Aa)

+(P):- X

FIG. 23. Schematic diagram of the symmetric Lauegeometry. p is the incident plane wave. P is the wavefunction inside the crystal and x is the wave functionemerging from the back face of the crystal.

The vectors G are the reciprocal lattice vectorsof the crystal. Putting (A6) and (A7) into the waveequation (A5) and equating coefficients of e'o ', wefind

21 GRAVITY AND INERTIA IN QUANTUM MECHANICS

We now make an approximation. We assume thatko is oriented very close to the Bragg conditionfor a particular reciprocal lattice vector G, andassume that the internal incident wave vector +p=kp, as we can easily verify. Thus, the "reso-nance factor" [ko —(K, +G') ] is small only forG' = G and for G' =0. That is, only $0 and go willbe large. Under these conditions, the above in-finite set of equations reduces to two equations;in matrix form they are given by

and

Kp ——kp —N

K~o = ko —N

(A13)

(A 14)

matching, this requires that the internal wave vec-tor Kp can differ from ko by only a component nor-mal to the surface. There are two possible valuesfor Kp for each incident wave vector kp, one onthe o.' branch and one on the P branch. Thus

-VG

where we have defined

2= 2& =&p-vp

(K -Ko) -u 6 =0,(K'- K', ).

(A10)

The relation between these vectors is shown inFig. 24. Thus, the internal wave function is com-posed of four plane waves:

q(r) =rP() exp(iKO r)+P() exp(iKO r}

+ P~ exp(i'~ r}+P~ exp(iK~ r) . (A15)

and

K~ =K, +G. (A 11)

For a nontrivial solution of (A9) to exist, the de-terminant of the matrix of coefficient must bezero, thus

g-+/=0 (A 16)

The diffracted part of this wave function must bezero along the entrant boundary, and the incidentpart must match the external incident wave (Al).Thus'

(K- Ko)(K —Ko) =goy g/4ko. (A 12)

In this equation we have made the approximationsK+K0=2ko and K+Ko =2ko. Equation (A12) de-fines the dispersion surface, that is, the locus ofallowed internal incident wave vectors Kp in kspace. The dispersion surface has two sheets,which are hyperbolas as shown in Fig. 24. Wecall the one branch of this surface + and the otherbranch P

The neutron wave function must be continuousacross the entrant surface. For perfect phase

4o+ Wo——4.

The ratio of the diffracted wave amplitude Po tothe incident wave amplitude g, for each branch isdetermined by (A9). If we define C" to be go/P~(where 'Y= o' or P), then

2k 0(K —K"0) v 6v o 2ko(K-IPg) ' (A 18)

which are known for each incident wave vectorko. Thus, we can use (A16) and (A 17) to expressall of the internal wave amplitudes in terms of theamplitude of the incident wave 4. The resultsare

G g =[c'/(c' c )]c,y,'=-[c /(c~- c )Jc,g&=[c c'/(c~-c )Jc,y~= [c c~/(c~-c )Jc.

(A19)

The wave function X(r) emerging from the backface of the crystal is the sum of two coherentplane waves

y(r) = y., exp(ik, . r) + yo exp(ik . r) . (A20)

'0FIG. 24. Diagram showing the two hyperbolas of the

dispersion surface giving the locus of the allowed inter-nal wave vectors for the symmetric Laue case. Theasymptotes (dashed lines) are circles of radii E [Pq.(A10)] drawn about the point 0 and the point G. Thereciprocal lattice vector is 0 and ko is the external in-cident wave vector.

The boundary condition for continuity of the neu-tron wave function across the back face of thecrystal requires us to match the incident part ofX(r) with the incident part of g(r), and also thediffracted part of y(r) with the diffracted part ofg(r). The algebra is rather tedious, but straight-forward. The results are

1488 STAUDKNMANN, WERNER, COLELLA, AND OVERHAUSER 21

t'. q sin)Xp= Iz ~ z z zgz + cosh I~+& )

(A21) and

g =ko&e sin20~ . (A27)and

2

Xo= z z za sing 8 4'q. k() P

vo ( +P') (A22)

The Fourier components of the neutron-crystalinteraction potential are related to the structurefactor ++ by

where vo =4mFo/1 „» (A26)

g =a(qz+ pz)'~/2 cose, ,

'5p = a[(vp/kp) + z)]/2 cos&z,

5G = 50+ 2koa&e sining .

(A23)

(A24)

(A26)

where the volume of a unit cell is V„». The wavevector of the diffracted wave can be seen from thegeometry of Fig. 24 to be

ko =G+ k(, +2k,«sine, n, (A29)

p =I va I/&p (A26)

The crystal thickness is a, ~& is the nominalBragg angle, and &~ is the angular deviation of theincident wave vector ko from the exact Bragg con-dition. The quantities P and g are given by

where n is a unit vector normal to the surface.Note that

I xo I'+I x, I' =

Ic I', (A3O)

as it must, for the zero-absorption case we areconsidering;

'A. W. Overhauser and B.Colella, Phys. Rev. Lett.33, 1237 (1974).

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U. .Bonse and M. Hart, Appl. Phys. Lett. 6, 155 (1965).~H. Bauch, W. Treimer, and U. Bonse, Phys. Lett. 47A,

425 (1974).For recent reviews of neutron and x-ray interferometrysee U. Bonse and W. Graeff in X-ray Optics, Topicsin Applied Physics, edited by H- J. Queisser (Sprin-ger, Berlin, 1977), Vol. 22; H. Bauch and D. Petros-.chek, in Neutron Diffraction, edited by H. Dachs(Springer, Berlin, 1978), pp. 303-351.

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Dicke, Ann. Phys. (N.Y.) 26, 442 (1967).2For a discussion of this point see D. M. Greenbergerand A. W. Overhauser, Bev. Mod. Phys. 51, 43 (1979).

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~~An early attempt to achieve neutron interferometrywas made by H. Maier-Leibnitz and T. Springer, Z.Phys. 167, 386 (1962), using a prism as a beam split-ter.W. Bauspiess, U. Bonse, and W. Graeff, J. Appl.Crystallogr. 9, 68 (1976).

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