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Comment on “A proposed method for measuring the electric dipole moment of the neutron using acceleration in an electric ?eld gradient and ultracold neutron interferometry ”, II

S. K. Lamoreaux* and R.Golub#

* Los Alamos National Laboratory, Physics Div., M.S. H803, Los Alamos, NM 87545, USA Hahn Meitner Institut, Glienicker Str. 100, D-14109, Berlin, Germany (February 8, 2008)

#

arXiv:nucl-ex/9901007v4 22 Apr 1999

Abstract

We discuss the proposal of Freedman, Ringo and Dombeck [1] to search for the neutron electric dipole moment by use of the acceleration of ultracold neutrons in an inhomogeneous electric ?eld followed by ampli?cation of the resulting displacement by several methods involving spin independent interactions (gravity) or re?ection from curved (spin independent) mirrors. We show that the proposed technique is inferior to the usual methods based on magnetic resonance.

PACS: 21.20.Ky, 14.20.-c, 03.75.Dg

I. INTRODUCTION

Since the appearance of our original paper [2] there has been some additional discussion [3] and [4], so that it now seems reasonable to publish a revised version with a more detailed exposition of our quantum mechanical treatment (sec. II B.) Searches for a neutron (or elementary particle) electric dipole moment (edm) are interesting because an observation of an edm would be a demonstration of the violation of time reversal (T) invariance outside the K 0 system. Freedman et al. [1] have proposed a new method to search for the neutron edm. The method is claimed to o?er the possibility of vastly improved sensitivity due to the ampli?cation of the e?ects of the interaction of an edm with an electric ?eld by means of subsequent motion in a gravitational ?eld or re?ection from a convex mirror. In the present paper we will review this proposal and show that the claimed gain in sensitivity is based on a misunderstanding of a semi-classical model of the processes involved. According to the proposal, Ultra-cold neutrons (UCN) polarized along the x axis enter a chamber (accelerator) where the electric ?eld directed along z has a gradient in the x direction. Since the incident spin state which is an eigenstate of σx can be considered as a coherent superposition of the two eigenstates |± z of σz , we can suppose that neutrons possessing a non-zero edm in one of the eigenstates will su?er an acceleration ax = ± ?e m 1 ?Ez ?x . (1)

After time T1 the two spin states will be separated by a distance δx = ?e m ?Ez ?x

2 T1

(2)

After the period of acceleration, the neutrons are allowed to leave the chamber in a vertical direction (+z ) rising against gravity. After re?ection from a surface inclined at 450 , the horizontal separation δx is converted into a vertical separation δx = δz . The neutrons then follow parabolic trajectories under the in?uence of gravity. Because of the di?erence in initial heights, δz , hence a di?erence in kinetic energy, the two spins will accumulate a phase di?erence along the two trajectories 1 ?φ = ?V dt = mg ?zT2 (3) h ? where T2 is the time of ?ight along the parabolic trajector(ies). This is called a ’gravitational ampli?er’ by the authors [1]. It is then proposed to measure this ampli?ed phase di?erence between the states |± z as a precession of the polarization vector in the x, y plane: ?φ = gT2 ?e

?Ez 2 T (4) ?x 1 Putting in practical values for the parameters the authors expect a sensitivity to the edm of 10?28 e ? cm, which is superior to that expected for other methods [5]. It is seen that the proposed e?ect depends crucially on the description of the spin 1/2 system where the phase di?erence is calculated along trajectories ending at di?erent points while the polarization is calculated by assuming the two states combine coherently at a single point. The authors also propose a second type of ampli?er based on repeated re?ections from a curved surface, given by Z (x). Then the two ‘points’ representing the two spin states separated by δx due to the edm acceleration (1) will re?ect from portions of the surface with slightly di?erent slopes, di?ering by:

? 2 Z (x) α (x) = δx (5) ?x2 and, after re?ection the angle between the trajectories will increase by 2α (x). Again, the increased separation is supposed to result in an ampli?cation of the phase di?erence and hence of the precession in the x, y plane. This has the same feature as the “gravitational ampli?er”: trajectories ending at increasingly distant points are used to calculate a phase di?erence which is then thought to be measured as a precession of the polarization which is calculated by considering that the two states combine coherently at a single point.

II. DISCUSSION OF THE PROPOSAL ACCORDING TO DIFFERENT MODELS

As the present proposal is based on a mixing of models we will present a discussion of the proposal from several view points. It is important that the models be well de?ned and applied in a consistent manner. Mixing di?erent models leads to errors and confusion. A detailed discussion of the relation between classical, semi-classical and quantum mechanical descriptions of similar situations has been given in [6] with a more complete quantum mechanical discussion in [7]. We will review these ideas with emphasis on the application to the present model. 2

A. Classical Model

As an edm interacting with an electric ?eld behaves identically to a magnetic moment interacting with a magnetic ?eld we choose to discuss the problem in terms of magnetic moments in a constant magnetic ?eld as this is probably more familiar. We will see later that the introduction of ?eld gradients in the author’s proposal is neither essential nor desirable. ? → → ? → → A classical magnetic moment ? ? coupled to an angular moment j , ? ? = γ j in an ? → external ?eld B , (of course the same discussion will apply to an edm in an electric ?eld) obeys the equation of motion ? → dj ? → → =? ? ×B (6) dt ? → ? → whose solution is seen to be a precession of j around B with the Larmor frequency ωL = γB ? → ? → independent of the angle between j and B and which is a constant of the motion. If the ?eld exists in a region of length L, then the particle will cross the ?eld in a time t = L/v during which time the spin will precess through an angle ?L = ωL t = ωL L/v = γBL/v (7)

The spin components, σx,y , averaged over the velocity spectrum f (v ) of the beam, will then be given by σx = σy = dvf (v ) cos ωL L v ωL L dvf (v ) sin v (8)

This is the basis of a technique that is often used to measure velocity distributions. Note that as ωL L increases enough the polarization |σ | is expected to decrease. In a classical model the separation of states in space does not appear. As the gravitational interaction is spin-independent there is no gravitational ampli?cation in this model. Moments ? → → entering the ?eld directed perpendicular to it undergo no energy change since ? ? · B = 0. The classical model is expected to give accurate results so long as the separation between trajectories associated with di?erent states is small compared to the correlation (coherence) lengths of the wave function. Thus the classical model breaks down in the Stern-Gerlach e?ect but gives a very good description of Larmor precession. [7]. Therefore one could conclude on this basis that the proposed ampli?cation does not exist but we are aware that this argument would not be seen as compelling.

B. Quantum mechanical treatment

Since the semi-classical model is in some ways the most di?cult, primarily because it is prone to misinterpretation, we will ?rst discuss the quantum mechanical model. Note that 3

to our knowledge no quantum mechanical demonstration of the alleged ampli?cation e?ect has been presented, [1], [3], [4]. We start with some general, elementary remarks. A wave function ψ (x) and its Fourier transform Ψ (k ) are respectively, the probability amplitudes for ?nding the particle in a given region of position or momentum space. Then a displacement of the particle is equivalent to a phase shift (linear in momentum) of the momentum wave function: ? → → ψ ? x + δx = A more general phase shift ? → ? → i? k ? → Ψ k = Ψ k e ? → can be considered as leading to a displacement, δx, such that ? → ? → ? → ? → ? → ? → → → ? k =? ko +? κ · ?k? k = ? k o + ? κ · δx (10) → ? → ? ? →? → ? → i k · δx d3 kei k · x Ψ k e (9)

(11)

? → only under the condition that Ψ k is narrow enough that higher order terms in the ? → ? → expansion of ? k can be neglected. Also ? k o should not play an important role. In the case of neutron spin echo [6] (or an edm accelerator with constant electric ?eld) where ? ∝ 1/k the condition for this is (κ/ko ? 1) where κ is the width of the wave in momentum space centered around ko . In the spin echo case we also have δx ? 1/κ so that the concept of a displacement between the states has physical meaning and, as will be seen below, no polarization can be observed under these conditions. It is necessary to cancel the phase shift (this is called obtaining the echo), and thus eliminate the displacement δ , in order to observe the polarization. The reason for belaboring these rather self-evident points is that in the work in question there is a tendency to make a distinction between the displacement and the precession methods of searching for an edm whereas we have seen that the two concepts are only di?erent ways of looking at the same phenomenon. There is only one phase for each quantum state and attempts to separate this phase into various components can often lead to confusion. This applies to the present case where a term linear in k is singled out as a ‘displacement’ or in attempts at separating the phase into ’geometric’ and ‘dynamic’ parts. This latter separation is strongly dependent on the coordinate system used for the calculation - only the total phase remains unchanged [8]. ? → If we consider a wave function Ψ k for a state where the spins are initially polarized in the x direction ? → |Ψ (t = 0) = Ψ k 1 1 (12)

and, as a result of a spin dependent interaction, the spin states pick up an additional phase ?± so that at some time t we have ? → Ψ k ,t ? → =Ψ k 4 ei?k (t) ? ei?k (t)

+

(13)

→ then the expectation value of ? σ will be σx + iσy = Ψ (t) |σx + iσy | Ψ (t) = ? → Ψ k

2 i ?+ ? ?? 3 ( k k)

e

dk

(14)

In the case where the spin dependent interaction, V± = ±Vo , is small compared to the kinetic energy of the beam particles (this is certainly the case for any interaction involving a particle edm and holds for the spin echo case as well [7]) and slowly varying we can use the WKB approximation to write ?± = dx k 2 ? 2mV± /h ? 2 ≈ kx ? dx mVo (x) h ? 2k (15)

For Vo (x) = const = ?B or ?e E we can write, taking α = ωL Lm/2? h, with ωL = 2?B/h ? or 2?e E/h ? ?± k = ± α = ± (?o ? κδx) k (16)

where we have expanded around ko , (k = ko + κ) for a narrow spectrum centered on ko 2 using δx = α/ko , see (11), (?o = α/ko ). The case where Vo (x) has a constant gradient, ?E Vo = ?e x ?x is seen not to introduce any signi?cant di?erences into (15). Then σx + iσy = ψ ? (x ? δx, t) ψ (x + δx, t) ei2?o

? = ψ+ (x, t) ψ? (x, t) ≡ I (t)

(17) (18)

where ψ (x) is the Fourier transform of Ψ (k ) and equ. (17) is the result of the WienerKhintchin theorem applied to (14). We see that σx + iσy in equ. (14) as a function of ? represents the Fourier transform of the beam velocity spectrum in agreement with the classical calculation, equ. (8). From equ. (17) it follows that this is a decreasing function of ? ?k = ?+ k ? ?k or δx for large enough ?k so that increasing ?k or δx far enough will result in a reduction of the net polarization. As δx increases so much that the correlation function in (17) approaches zero we approach the case of the Stern-Gerlach e?ect where the spin states can be considered as truly separated and one can talk about a displacement without any confusion. Thus we have seen that the discussion can be carried out equally either in terms of ?k or δx, they both represent the same physical situation. The main thrust of the proposal under discussion [1] is that after a period in the electric ?eld region the x velocity of the UCN is changed into the z direction by re?ection from a mirror at 45o . Subsequent motion in the gravitational ?eld is supposed to signi?cantly increase the phase di?erence ?, according to the (incorrect) semi-classical argument presented in [1] and outlined in the introduction. The key point is that no spin-independent interaction can in?uence the phase shift between the two spin states. At the entrance to the gravitational ?eld drift region (called the ”ampli?er” by the authors) the wave function will be of the form given by (13) and (16); taking the Fourier transform → | ψ (? x , t)

o

? → → ei?o ψ o ? x + δx, t ? ?= = ? → → e?i?o ψ o ? x ? δx, t

? ?

o ? ψ+ (→ x , t) o ? ψ? (→ x , t)

(19)

5

for a narrow spectrum. The superscript o refers to the incoming beam at the entrance ori?ce of the ampli?er region. We now calculate the wave function at the exit of this region. It is su?cient to consider only the motion in the z direction and con?ne ourselves to the steady state situation. Then using the WKB method for the case of the slowly varying gravitational potential, which is not necessarily small compared to the particle kinetic energy, we can write the wave function at the output (in the case of zero electric ?eld in the accelerator) ψ (z ) =

k

A (k ) e

i

z 0

√

2 )dz (k 2 +Kz

=

k

A (k ) e

i

h ?2 3m2 g

2 ) (k 2 + K z

3/2

e

?i

h ?2 k3 3m2 g

(20)

2 2 where Kz ≡ 2m2 gz/h ? 2 = m2 vz /h ? 2 . vz is the velocity of a particle that falls a distance z , starting from rest. In order to investigate the relation between phase shifts and displacements we wish to study the behavior of ψ in a small region located around z = Z , so we set z = Z +?, ? ? Z and expand the exponential in (20) around Z .

ψ (z = Z + ?) =

k

h ?2 2 k 2 + KZ A (k ) exp i 2 3m g

3/2

2 + k 2 + KZ

1/2

? e

?i

h ?2 k3 3m2 g

(21)

Since A (k ) represents a fairly narrow wave packet centered around ko we can write k = ko + κ, κ ? ko ? KZ (the latter condition is necessary for the ’ampli?cation’ to be signi?cant, see below) and then we obtain from (21), keeping only the terms in κ and ? ψ (z = Z + ?) ≈

2 ko ko h ? 2 ko A (ko + κ) exp i ( K Z ? k o ) κ + KZ ? + ?+ κ? 2 mg 2 KZ KZ

(22)

κ

The terms in (22) can be interpreted as follows: The ?rst term represents a displacement in position by vo tZ , the distance the particle with the initial velocity, vo , would go in the time tZ that the particle takes to fall to Z and an additional displacement Eo /mg associated with the boundary condition at z = 0. The next two terms represent the change in wavelength due to the acceleration during the fall. Since κ A (ko + κ) eiκ? represents the envelope of the wave function at z = 0, the last term in (22) shows that the envelope is spread by the factor η = KZ /ko ? 1 due to the energy dependence of the index of refraction (dispersion) for the Schr¨ odinger wave. When the electric ?eld is switched on we have (using equ. (16).) ψ± (z = Z + ?) ≈ ≈ A (ko + κ) e

κ

ko iK κ? i?± k Z

e e

(23)

A (ko + κ) e

κ

ko κ? ±i(?o ?κδx) iK Z

displaying only the relevant terms. Now if we calculate the center of the wave packet, ?o , according to ??/?κ|?o = 0, we ?nd the center of the packet is indeed shifted by δ? = ηδx (24)

in agreement with the ideas of ref. [1]. The wave packet spreading by the factor η , e?ects the electric ?eld induced displacement by the same factor during the transit through the 6

ampli?er region. This is the quantum equivalent of the semi-classical argument given in [1]. However we see immediately from (23) that the κδx term in the phase shift is a small part of the total edm induced, spin dependent phase shift and neither term in the phase shift is altered by travel through the ’ampli?er’ region. Thus from equ. (23) we have

? ψ+ ψ? (z =Z +?)

≈

A? (ko + κ′ ) A (ko + κ) e

κ,κ′

ko iK (κ?κ′ )? i(2?o ?(κ?κ′ )δx) Z

e

(25)

We see that the edm induced phase shift at the output is exactly the same as at the input to the ’ampli?er’. Thus any measurement performed subsequently, whether using a polarization analyzer or interferometer with magnetic beam splitters or any other scheme, will yield the phase shift as it was at the input and there is no ampli?cation. Any attempts to measure the increased displacement directly as a displacement are easily seen to be completely impractical as is evidently recognized by the authors of [1]. In the next section we show that a correct semi-classical argument leads to the same conclusion. In response to the circulation of an earlier version of this paper, [2], Peshkin [3] introduces → what he calls the ”No-Go” theorem. Since ψ± (? x , t) satisfy the same Schroedinger equation in the ampli?er region (the Hamiltonian is spin independent) the quantity, I (t), (see equ. (18)) I (t) =

? ? → d3 x ψ+ (→ x , t) ψ? (? x , t)

(26)

is independent of time due to unitarity. However this leaves open the possibility that the integrand over some limited region of space might be time dependent, thus allowing the ampli?er to work without violating the theorem. Thus Peshkin states that measuring the polarization ”in a small range of x ..instead of over all x at one time , to avoid the integral over all space in I (t) ..avoids the No-Go theorem in principle”. He also claims that the No-Go theorem does not address interferometer experiments as ”there the phase shift shows up as an overlap integral between two partial wave packets in one emergent beam only, not as the conserved integral over all space. For the same reason, the theorem also does not speak usefully to versions of the proposal [1] in which the phase shift is measured with a Mach-Zender interferometer instead of a polarimeter”. The same argument is presented in a recent Letter to the Editor of Nuc. Instr. and Meth. in Physics Research, by Dombeck and Ringo, [4]. The idea is that at the output of an interferometer (with the spin states separated by a magnetic mirror and travelling through the di?erent arms of the interferometer) the output (considering that one spin state is ?ipped inside the interferometer) would be given by → → |ψ+ (? x , t) + ψ? (? x , t)|2 the cross terms giving the integral I (t), equ. (26) but with the integral taken only over one → of the emergent beams. However the partial beams ψ± (? x , t) at the output of the interferometer share all the same phase properties of the beams at the input of the interferometer and the theorem will apply equally to both situations. Be that as it may we propose a more stringent version of the ’No-Go’ theorem, the ’Never-Go’ theorem, i.e. the motion of the 7

spin is una?ected by a gravitational ?eld due to the spin independence of the Hamiltonian as we have shown above (equs. 23 and 25) We have shown that while an ampli?cation of the ’displacement’ betwen the two spin states does occur in the ’gravitational amplifer’ due to the wavelength dependence of the index of refraction, this is virtually unobservable; the phase shift between the two spin states remains unchanged on traversing the region as does the direction of the neutron polarization. As we have not integrated over space or time our result shows that attempts to avoid the No-Go theorem by con?ning the measurements to limited regions will not work. The inapplicability of the theorem is not a su?cient condition for the existence of an ampli?cation e?ect.

C. Semi-classical model

This model employs the geometrical optics approach to quantum mechanics, where each spin eigenstate is represented by a di?erent trajectory. In a sense this is the most di?cult model as it is prone to misunderstanding. In [1], at least three errors are made in application of the semi-classical model to the spin ampli?cation process. Let us ?rst consider ampli?cation by vertical displacements, as discussed before equ. (3) above. First, assuming that the concept of a displacement of the two spin eigenstates is correct, we can calculate the gravitational e?ect on the spin precession angle. The ?rst error in [1] occurs when the assumption that the phase between the two wave functions is simply the phase di?erence between the two eigenstates evaluated at the respective maxima of the wave function envelopes. This is a quantum mechanically incorrect procedure because the phase determination does not commute with the determination of the wave function center at a given time; in other words, it makes no more sense to compare the phases between the two eigenfunctions at two distinct spatial points than it does to compare the phases at two di?erent times. The correct procedure is to make a point by point comparison between the two wave functions, then average over the two envelopes. This is the procedure normally used when calculating the interference between two scalar or vector ?elds as is commonly done in electrodynamics (see, for example, [9], Sec. 7.2). That this is the correct procedure can also be seen from the fact that detection occurs at a single point in space-time (for example, the neutron is absorbed on a 3 He nucleus thereby “collapsing” the wave function to a single space-time point; the spin direction is given by the two wave function phases at that point in space-time). We can now properly calculate the phase di?erence between the two eigenstates at a ?xed point in the ?nal polarimeter/detector; it is the phase di?erence between two classical trajectories that meet at the same space-time point, initially separated a distance δx perpendicular to the momentum k . The change in phase is simply the change in action along the two trajectories, and this can be easily calculated to ?rst order by use of a theorem, which is of crucial importance to interferometry (but universally ignored) due to Chiu and Stodolsky [10]. This theorem states that a change in the action when one of the endpoints of a classical trajectory is displaced is given by

D 0 0 δS = P? δxD ? ? P? δx?

(27)

8

with summation notation over spatial coordinates implied (? = x, y, z ), and where S is the action (equal to the quantum mechanical phase up to a factor of h ? ), P? refers to the momentum, x? the path endpoint coordinates, 0 refers to the trajectory beginning, and D the trajectory end (at the polarization analyzer/detector). As discussed already, δxD ? , that is, the relative displacement of the path at the detector, is identically zero because a neutron is detected at a single point (e.g., a polarized 3 He nucleus). The displacement of the path starting point is given by δxx ≡ δx as given by equ. (2) above. However, it is assumed in [1] that δx is perpendicular to the neutron momentum; therefore, the change in action is exactly zero, as given by the Chiu and Stodolsky theorem, and there is no gravitational acceleration e?ect, in essence, by de?nition. Any other e?ects that could change S , particularly those relating to the electric ?eld gradient or gravitational acceleration in the speci?c geometry given in [1], enter only in second or higher order. The above arguments can be immediately applied to the curved mirror ampli?er. We again assume that the trajectory endpoints must meet in order for there to be an interference, and we calculated the change in action as above. Again we ?nd that δS is identically zero to ?rst order in the electric ?eld gradient. The ?nal semiclassical misconception in [1] concerns the use of an electric ?eld gradient over the storage volume. It seems to us that a larger (at least two times) δx (hence phase shift) can be generated by sending the “bipolarized” neutrons from a region of zero electric ?eld to a region of constant high electric ?eld. Each eigenfunction will acquire a change in energy, hence a change in velocity, as it enters the electric ?eld region; although δx only increases linearly in time, for a given storage time T1 , suddenly accelerating the two eigenstates to their ?nal velocities would lead to a larger δx than if the two eigenstates were subjected to a weaker electric ?eld gradient averaged over the storage volume, but giving the same ?nal velocities only after storing for a time T1 . The implication is that the use of an electric ?eld gradient is completely pointless and only leads to a dilution of a possible edm e?ect. However, this is not surprising based on the foregoing considerations: to achieve the maximum sensitivity to an edm e?ect, whether that be interpreted as δx or a change in phase between the two spin eigenfunctions, the interaction energy given by the usual Hamiltonian H = ??e j · E (28)

must be as large as possible over the duration of an experiment, for it is the integral of H over time that gives the relative action between the two eigenstates. We thus see immediately that if the average magnitude of E is compromised by the wasting of electric ?eld strength toward the establishment of gradients in the system, the ?nal net sensitivity to the edm interaction given by H must also be compromised.

III. CONCLUSION

We have analyzed the proposal for a new type of neutron edm experiment from three di?erent perspectives, and in each case, have arrived at the same conclusion: The new technique o?ers no gain in sensitivity as compared to the usual magnetic resonance technique. In 9

fact, a careful analysis reveals that the new method is inferior to the conventional methods. In [1], a semiclassical approach was improperly used to analyze the proposed technique, and a comparison with the atomic interferometer of Kasevich and Chu [11] was used to justify the approach. However, there really is no point of comparison between the atomic interferometer and the system described in [1]. The Kasevich and Chu “interferometer” is based on a superposition of internal quantum states of the Cs atom, speci?cally, the ground state hyper?ne levels; there is no discussion here of the center of mass wave function, but only of the phase di?erence between the internal states. This phase di?erence evolves at the hyper?ne frequency (approximately 10 GHz) and the beauty of the system is that the freely falling atom experiences a Doppler shift relative to an oscillator ?xed in the laboratory; this relative frequency shift in the accelerating system makes possible, for example, a precise measurement of the Earth’s gravitational ?eld. In a certain sense, the Kasevich and Chu system really isn’t an interferometer (this point is addressed in [12]); the evolving internal quantum state phase di?erence serves as a clock which can be compared to the stationary laboratory oscillator, and the system is best described by the classical approach given above. We might invoke the quantum or semiclassical model to calculate the result of some force that would cause the two hyper?ne levels to spatially separate; the result of this calculation would simply show a diminishing of the internal interference e?ect because the two hyper?ne eigenfunctions no longer fully overlap in space-time; in the limit where the separation is greater than the center of mass wave function coherence length, the concept of a superposition of internal quantum states entirely loses its meaning.

IV. ACKNOWLEDGMENT

We would like to thank L. Stodolsky, who independently arrived at the same conclusions, for sharing his thoughts on this matter with us, and for encouraging us to write this manuscript.

10

REFERENCES

[1] Freedman, M.S., Ringo, G.R. and Dombeck, T.W., Nuc. Instr. and Meth. in Physics Research A396, 181 (1997). [2] Lamoreaux, S.K. and Golub, R., Nuc. Instr. and Meth. in Physics Research, to be published, 1999 and Los Alamos archive (xxx) nucl-ex/9901007 v3 [3] Peshkin, M., Los Alamos archive (xxx) nucl-ex/9903012 v2 [4] Dombeck, T.W. and Ringo, G.R., Nuc. Instr. and Meth. in Physics Research, to be published, 1999 [5] Golub, R. and Lamoreaux, S.K., Physics Reports 237,1, (1994). [6] G¨ ahler, R., Golub, R. Habicht, K., Keller, T., Felber, J., Physica B229, 1, (1996). [7] Golub, R., G¨ ahler, R. and Keller, T. Am. J. Phys. 62, 779 (1994). [8] Berry, M., Lecture held at Technical University Berlin, 14 May, 1998. [9] Wolf, E. and Born, M., Principles of Optics, 6th ed.(Pergamon, Oxford 1980). [10] Chiu, C. and Stodolsky, S., Phys. Rev. D 22, 1337 (1980). [11] Kasevich, M. and Chu, S., Phys. Rev. Lett. 67, 181 (1991). [12] Lamoreaux, S.K., Int. Jour. of Mod. Phys. A 7, 6691 (1992).

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