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HOME | SEARCH | PACS & MSC | JOURNALS | ABOUT | CONTACT US Gravitational, rotational and topological quantum phase shifts in neutron interferometry This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1994 Class. Quantum Grav. 11 A207 (http://iopscience.iop.org/0264-9381/11/6A/016) The Table of Contents and more related content is available Download details: IP Address: 193.136.128.7 The article was downloaded on 01/10/2009 at 12:24 Please note that terms and conditions apply. 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 [email protected](z, t ) = @I& t ) - @I(z. 1 ) . (5) In neutron interferometry, the phase shift [email protected](r, t ) caused by a potential V(d,v , t') is the quantity that is measured, and is of physical intaest, namely [email protected]" = [email protected] - [email protected], (6) where [email protected] 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 [email protected]"(=, 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 [email protected]: Iz = az - bz cos([email protected]) (1W and I3 = a3 +b3 cos([email protected]). (W The constants az,&. a3 and b3 depend upon the transmission and reflection coefficients of the three Bragg reflecting crystals. As the phase shift [email protected] 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 [email protected]" 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 [email protected],=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 [email protected] 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 [email protected] - 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(&) [email protected]). (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 [email protected] 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 [email protected] Thus, since AQAc is such a small phase angle, we have from (47) (48) * = a3 [email protected][ (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 [email protected] (50) Thus, the count rates are linearly proportional to [email protected] 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) [email protected][email protected] (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: [email protected]= (AC)/Zb = 2.22 f0.49 mad Negative slope: [email protected]*= = (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 [email protected],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 [email protected] 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 [email protected]= -- &(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([email protected]) 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 [email protected] = 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 [email protected] = 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 [email protected]$ = + @ ~ / = 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. References [I] D i m P A M 1945 Rev. Mod. Phys. 17 195 [21 Feynman R P 1948 Rev. Mod. Phys. 20 367 [3] Ahamnov Y and Casher A 1984 Phys. Rev. Lett. 53 319 [4] Cimmino A, Opat G I, Klein A G, Kaiser H, Wemer S A, Arif M and Clothier R 1989 Phys. Rev. Len. 63 380 [SI Kaiser H, Werner S A, Clothier R, Arif M, Klein A G, Opat G I, and Cimmino A 1991 Pmc. Inr. Con$ on Atomic Physics-ICAP-12 AIP Conz Pmc. no 233 p 241 (New York AIP) [6] Sagnnc M G 1913 C. R Acud Sei. 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