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A treatment of general relativistic effects in quantum interference
Physics Letters A 182 (1993) 330—334 North-Holland
PHYSICS LETTERS A
A treatment of general relativistic effects in quantum interference Josuke Kuroiwa, Masumi Kasai and Toshifumi Futamase Department of Physics, Faculty of Science, Hirosaki University, Hirosaki, Aomori-ken 036, Japan Received 15 February 1993; revised manuscript received 21 September 1993; accepted for publication 21 September 1993 Communicated by J.P. Vigier
The phase shift in a neutron interferometercaused by the gravitational field and the rotation of the earth is derived in a unified way from the standpoint ofgeneral relativity. Starting from the covariant Klein—Gordon equation and using the nonrelativistic approximation, it is shown that the result includes the gravitational potential term and the Sagnac term which have been actually observed, and the ancillary effect due to the dragging of inertia.
1. Introduction The phenomenon of interference is fundamentally different in classical and quantum physics. Quantum interference is of great significance because the observation of the phase shift provides information on the external field which affects the wave function of the particles. The importance of the phase in the presence of electromagnetic potentials was suggested by Aharonov and Bohm [1] for quantum interference and was subsequently confirmed by experiments [2—4].Overhauser and Colella [5] proposed an experiment to test the effect of the gravitational field of the earth on the phase shift, which was suecessfully performed by Colella, Overhauser and Werner [61 using a neutron interferometer. According to their experiment, the observed phase difference due to the gravitational potential was in good agreement with the theoretical prediction within an error of 1% without distinguishing the intertial mass from the gravitational mass of the neutron. Thus this result supports the validity of the equivalence principle on the quantum level, After that, Page [71 pointed out the existence of another effect due to the rotation of the earth (the Coriolis force) and this effect was detected by Werner, Staudenmann and Colella [8]. This effect, which is a quantum mechanical analogue of the Sagnac effeet [9] in optical interferometry, is so remarkable that the theoretical expression has been derived by 330
various authors in several different ways: optical analogy [7], relativistic eikonal approximation [10], the WKB approximation [8], the Doppler effect of moving media [11] and an analogy of the Aharonov—Bohm effect [12]. Furthermore, Anandan and Chiao [13] proposed the gravitational radiation antenna using the Sagnac effect. The detection of grayitational radiation provides not only a verification of the predictions of general relativity, but also opens a new window for astronomical observations. In this respect, it is desirable to understand thoroughly the general relativistic effects on the quantum mechanical interference. It would be of great significance to have a formalism to derive these various effects in a unified way. The aim of this paper is to show such a treatment. We show that the phase shift in a neutron interferometer due to gravitation and the rotation ofthe earth may be derived within the framework of general relativity, which deals with both gravitational and inertial phenomena in a unified way. The paper is organized as follows. In section 2, a basic formulation is presented to derive the general relativistic effect on the quantum mechanical phase shift. Starting from the covariant Klein—Gordon equation in the Kerr spacetime and taking a nonrelativistic approximation, we obtain a Schrodinger type equation with correction terms which cause the phase difference of the wave functions. In section 3, the result is applied to the case of a neutron inter-
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 182, number 4,5,6
PHYSICS LETTERS A
ferometer on the earth and the effects are evaluated. It might not be sufficient to describe the behaviour of neutrons by the Klein—Gordon equation because of the existence of spin. It is of interest to consider the effect of the spin—curvature coupling, or spin— rotation coupling [14—17].In most situations, however, these effects turn out to be sufficiently small [14—17],so we do not consider the neutron spin in this paper. Finally section 4 is devoted to conclusions,
2. Nonrelativistic approximation of the Klein—Gordon equation Here we shall give a general relativistic consideration of the effect of gravity and the rotation of reference frames. Our starting point is the covariant Klein—Gordon equation (2.1) The Klein—Gordon equation is naively regarded as inappropriate because of the failure to interpretate the wave function cI’ in terms of probability without introducing the procedure of second quantization. In the nonrelativistic limit, however, the inconsistency in the probability interpretation can be resolved in the following way. Consider a flat spacetime for the sake of simplicity. To remove the rest mass energy from the Hamiltonian, let us define the nonrelativistic wave function ~Pby 2
/
~=
v’exp( —i
~—
t
~l.
(2.2)
/1 / Then (2.1) becomes
22 November 1993
allows one to have the probability interpretation for the wave function W. Hence (2.3) reduces to the ordinary Schrodinger equation ~ (2.4) ih-~—= ~—V2W. m If we start from the covariant Klein—Gordon equation and follow the above approximation, the effects of gravity and the rotation of reference frames are naturally included in (2.1) through the metric. Thus, as far as the validity of the nonrelativistic approximation is maintained, we can treat the phase shifts due to the gravitational and inertial force in a unified way. We assume that the external gravitational field of the earth is described by the Kerr metric. In Boyer— Lindquist coordinates, it is given by —
A —
(cdt—a sin2Odçb)2
—~
sin20 +
[(1.2+~2)
fdr2 dcb—ac dt]2+p2~—~+d02 (2.5)
where Am r2 —2 (GM/c2)r+ a 2, Mis the mass of the earth, p2 r2 + a2 cos20 and a is the Kerr parameter which corresponds to the angular momentum J of the earth per unit mass, a = J/Mc. In the present situation, both GM/rc2 and a/r are sufficiently small and their higher orderterms can be neglected. Therefore, to the first order in M and a and neglecting the terms of O((v/c)2), the Klein—Gordon equation in the Kerr spacetime becomes 89’ h2 [1 8 / 8 \ L2 1 ih— —I ——I r2— I— 19’ 8t 2mLr28r\ 8rJ r2h2J GMm ~ 2GMaLW (2.6) .
=—
1 329’ mô!t’ +2i—— +V29’=0, c2 8t2 11 ôt
(2.3)
— —~
.
.
where V2 denotes the three-dimensional Laplacian operator. The relative order of the first term to the others in (2.3) is O((v/c)2), where v denotes the characteristic velocity of wave packets, or equivalently O((A 2),where 2~is the Compton wavelength and 0/)~) is the de Broglie wavelength of a partide. For the neutron interferometer used by Colella, Overhauser and Werner [6], a nonrelativistic treatment gives a sufficiently accurate description so the term of 0 ((v/c)2) can be neglected. This corresponds to discarding the negative energy state and
—
r rc where m is the mass of the particle and L2
— —
h 21 ~
(sin 0 + -p-— [sinOäO\. 80) sin2OaØ2J’
—
-~--
-~-‘~
—~——
(2 7)
,~
L~=
—
ih
~—
(2.8)
are the orbital angular momentum operators of the particle with respect to the center ofthe earth. After the coordinate transformation Ø—*Ø— wt, where w is 331
Volume 182, number 4,5,6
PHYSICS LETTERS A
22 November 1993
the angular velocity of the earth, we obtain the “Schroding;r” equation for an observer fixed on the
B
(i,oerfererrce region)
~-)_-~]w
i=_~_[-~-(r
____
___
di
A
—
GMm 2GMa r 9’~wL~W+ r c L~9’.
D
(source)
(2.9)
Fig. 1. Schematic illustration of alternative paths in a neutron interferometer. 3grav =
I
~ (ABD)
—
/31 (ACD)
[V
3. The phase shift in a neutron interferometer
5(R+c4 sin ~,) V8(R)] mXd1 —
The Hamiltonian derived in the last section is divided into the four terms H—H0 +H1 +H2 +H3,
3 1
—
where 2 1 8
H 0=
h —
~
~
1 ~r
2
8\
‘
(3.2)
is the Hamiltonian for a freely propagating particle, H1
=
—
GMm r
vg
(3 3)
is the Newtonian gravitational potential energy, and H2=—wL0, H3=
3c L~. 2GMa r
(3.4)
,
where is the acceleration of gravity, A=d1d2 is the area ofg the parallelogram, and X is the de Brogue
phase difference observed by Colella, Overhauser and Werner [61 in a neutron interferometer experiment. Next, let us calculate the phase shift /32 principally due to the rotation of the interferometer, which is in our case caused by the earth’s rotation. This is a quantum mechanical analogue of the Sagnac effect [9] in optical interferometry. In the same way as for /31, /3~may be written in the form of the integration fl
1
1
(3.5)
where /3, denotes the phase shift caused by H, (z = 1— 3). In the first place, let us calculate the phase shift due to the gravitational potential. For the purpose of the present discussion, the quasi-classical approximation is valid and the phase shift is given by the integration along a classical trajectory/3~= Fl—’ f J/~ di. Therefore, for a beam of thermal neutrons which splits into two alternative paths and then recombines as in fig. 1 the phase difference fl~is
(3.7)
where Q = (0, 0, a), and L= (Lx, L11,, L~)is treated as the classical angular momentum L = mrx r. The phase difference flsag due to the Sagnac effect [7—12] becomes flsag=fl2(ABD)fl2(AcD)=
332
(3.6)
2=_~Ja)Lzdt=_~JQ.Ldt,
If 9’o is a solution for the Hamiltonian H0, then the solution for the total Hamiltonian H may be in the form 9’=!1~exp[—i(fl1+fl2+fl3)]
mgA X sin ~,,
wavelength divided by 27t. The theoretical prediction (3.6) Overhauser Colella [5] was and itfirst wasderived in goodby agreement withand the
L2
~)+
=
mf TP~(?~Xd1’) ‘~ .‘
2mQ’A ,
(3.8)
where A is the area vector of the sector ACDB. The third term of the phase shift is a general relativistic effect due to the dragging ofthe reference frames, the so-called Lense—Thirring effect [18]. A similar calculation leads to
Volume 182, number 4,5,6
PHYSICS LETTERS A
1 r2GMa 2G fJL = j —i— dt, (3.9) = ~ r3c where /= (0, 0, J) is the angular momentum vector of the earth. The phase difference is
i
~
J3drag = P3(ABD) —
=
/33 (ACD)
r .1. (rx dr)
—
[~ (~ r (~ ~~], LA
2Gm Flc2R31
2Gm hc2R3’
=
r (R+r’)xdr’ I R + r’ I
2Gm
_____ —
2Gm —
r’ X dr’ _3 ~
.rc) ~ X dr’]
—
(3.10)
~
where R represents the position vector of the instrument from the center of the earth. If we assume that the earth is a sphere of radius R with uniform density, then 1 ~MR2Q
(3.11)
and fldrag =
=
—
1 rg j~ /3sag
—
2r
——
~~flSag
~
~Ri £1 [A 3(~A)
~
if R is perpendicular to A,
5
if R is parallel to A
,
(3.12)
2 is the Schwarzschild radius of where rg=phase 2GM/c earth. The difference due to the Lense—Thirring effect is r 5/R 1 0~times that due to the Sagnac effect, and a very sensitive interferometer would be required to distinguish fldrag from flSag. As pointed out by Anandan and Chiao [13], however, in principle these effects can be separated by a “figure-eight” interferometer. This separate detectability derives from the fact that the phase shift due to the Sagnac effect depends only on the area and direction of the loop path whereas that due to the Lense—Thirring effect depends on the distance from the center of the earth as well as the size and direction ofthe interferometer (see eq. (3.12)). Another experiment using a new type of interferometer has also been proposed which ideally would be insensitive to both the Sagnac and —~
22 November 1993
the gravity effects [19] and which could be used to detect the general relativistic effect. 4. Conclusion In this paper we investigated the effect of the grayitation and the rotation of the earth on the quantum interferometer. We have derived the expressions for the phase shift due to the gravitational and the Coriolis forces in a unified way from the viewpoint of general relativity. We have also shown that the general relativistic effect due to the dragging of the reference frames could cause a shift in the interference fringes, although its magnitude is l0~times that of the Sagnac effect caused by the rotation of the earth. A new type of quantum interferometer is proposed and now under active development [19] which could be used to measure such relativistic effects as well as the phase shift due to the spin—rotation coupling and/ or the spin—curvature coupling [14—17]. These effects may be treated in our formalism starting with the covariant Dirac equation. Among various possible relativistic corrections, we have concentrated our attention on the dragging effect in this paper. This effect can in principle be separated from others because of the direction-dependence as shown in (3.10). A general and more careful treatment of general relativistic corrections will be given in a separate paper. It is of great significance to explore the possibility detector. of quantum It may interferometers be straightforward as the gravitational to study the wave interference phase shift caused by gravitational waves in our formulation. We would like to examine this problem in the near future.
Acknowledgement T.F. thanks Professor Y. Fujii for discussions at the early stage of this work. We also thank Dr. S. Bildhauer for discussions.
References [1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485.
333
Volume 182, number 4,5,6
PHYSICS LETTERS A
[2] R.G. Chambers, Phys. Rev. Lett. 5 (~i960)3. [3] G. Mdllenstedt and W. Bayh, Naturwissenschaften 49 (1962) 81. [4] A. Tonomura et al., Phys. Rev. Lett. 48 (1982)1443. [5] A.W. Overhauser and R. Colella, Phys. Rev. Lett. 33 (1974) 1237. [6]R. Colella, A.W. Overhauser and S.A. Werner, Phys. Rev. Lett. 34(1975) 1472. [7]L.A. Page, Phys. Rev. Lett. 35(1975)543. [8]S.A. Werner, J.-L. Staudenmann and R. Colella, Phys. Rev. Lett. 42 (1979) 1103. [9]M.G. Sagnac, C.R. Acad. Sci. 157 (1913) 708, 1410. [lO]J. Anandan, Phys. Rev. D 15 (1977) 1448.
334
22 November 1993
Lii] M. Dresden and C.N. Yang, Phys. Rev. D 20 [121J.J. Sakurai, Phys. Rev. D 21(1980) 2993.
(1979) 1846.
[l3]J.AnandanandR.Y. Chiao, Gen.Rel. Gray. 14(1982)515. [14] J. Audretsch and C. Lämmerzahl, J. Phys. A 16 (1983) 2457. [15] V.G. Baryshevskii and S.V. Cherepitsa, Class. Quantum Gray. 3(1986)713. [161 B. Mashhoon, Phys. Rev. Leti. 61(1988) 2639. [17]Y.Q.Cai and G. Papini, Phys. Rev. Lett. 66(1991)1259. [18]J. Lense and H. Thirring, Phys. Z. 19 (1918)156. [19]S.A. Werner and H. Kaiser, in: Quantum mechanics in curved space—time, eds. J. Audretsch and V. DeSabbata (Plenum, NewYork, 1990).
PHYSICS LETTERS A
A treatment of general relativistic effects in quantum interference Josuke Kuroiwa, Masumi Kasai and Toshifumi Futamase Department of Physics, Faculty of Science, Hirosaki University, Hirosaki, Aomori-ken 036, Japan Received 15 February 1993; revised manuscript received 21 September 1993; accepted for publication 21 September 1993 Communicated by J.P. Vigier
The phase shift in a neutron interferometercaused by the gravitational field and the rotation of the earth is derived in a unified way from the standpoint ofgeneral relativity. Starting from the covariant Klein—Gordon equation and using the nonrelativistic approximation, it is shown that the result includes the gravitational potential term and the Sagnac term which have been actually observed, and the ancillary effect due to the dragging of inertia.
1. Introduction The phenomenon of interference is fundamentally different in classical and quantum physics. Quantum interference is of great significance because the observation of the phase shift provides information on the external field which affects the wave function of the particles. The importance of the phase in the presence of electromagnetic potentials was suggested by Aharonov and Bohm [1] for quantum interference and was subsequently confirmed by experiments [2—4].Overhauser and Colella [5] proposed an experiment to test the effect of the gravitational field of the earth on the phase shift, which was suecessfully performed by Colella, Overhauser and Werner [61 using a neutron interferometer. According to their experiment, the observed phase difference due to the gravitational potential was in good agreement with the theoretical prediction within an error of 1% without distinguishing the intertial mass from the gravitational mass of the neutron. Thus this result supports the validity of the equivalence principle on the quantum level, After that, Page [71 pointed out the existence of another effect due to the rotation of the earth (the Coriolis force) and this effect was detected by Werner, Staudenmann and Colella [8]. This effect, which is a quantum mechanical analogue of the Sagnac effeet [9] in optical interferometry, is so remarkable that the theoretical expression has been derived by 330
various authors in several different ways: optical analogy [7], relativistic eikonal approximation [10], the WKB approximation [8], the Doppler effect of moving media [11] and an analogy of the Aharonov—Bohm effect [12]. Furthermore, Anandan and Chiao [13] proposed the gravitational radiation antenna using the Sagnac effect. The detection of grayitational radiation provides not only a verification of the predictions of general relativity, but also opens a new window for astronomical observations. In this respect, it is desirable to understand thoroughly the general relativistic effects on the quantum mechanical interference. It would be of great significance to have a formalism to derive these various effects in a unified way. The aim of this paper is to show such a treatment. We show that the phase shift in a neutron interferometer due to gravitation and the rotation ofthe earth may be derived within the framework of general relativity, which deals with both gravitational and inertial phenomena in a unified way. The paper is organized as follows. In section 2, a basic formulation is presented to derive the general relativistic effect on the quantum mechanical phase shift. Starting from the covariant Klein—Gordon equation in the Kerr spacetime and taking a nonrelativistic approximation, we obtain a Schrodinger type equation with correction terms which cause the phase difference of the wave functions. In section 3, the result is applied to the case of a neutron inter-
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 182, number 4,5,6
PHYSICS LETTERS A
ferometer on the earth and the effects are evaluated. It might not be sufficient to describe the behaviour of neutrons by the Klein—Gordon equation because of the existence of spin. It is of interest to consider the effect of the spin—curvature coupling, or spin— rotation coupling [14—17].In most situations, however, these effects turn out to be sufficiently small [14—17],so we do not consider the neutron spin in this paper. Finally section 4 is devoted to conclusions,
2. Nonrelativistic approximation of the Klein—Gordon equation Here we shall give a general relativistic consideration of the effect of gravity and the rotation of reference frames. Our starting point is the covariant Klein—Gordon equation (2.1) The Klein—Gordon equation is naively regarded as inappropriate because of the failure to interpretate the wave function cI’ in terms of probability without introducing the procedure of second quantization. In the nonrelativistic limit, however, the inconsistency in the probability interpretation can be resolved in the following way. Consider a flat spacetime for the sake of simplicity. To remove the rest mass energy from the Hamiltonian, let us define the nonrelativistic wave function ~Pby 2
/
~=
v’exp( —i
~—
t
~l.
(2.2)
/1 / Then (2.1) becomes
22 November 1993
allows one to have the probability interpretation for the wave function W. Hence (2.3) reduces to the ordinary Schrodinger equation ~ (2.4) ih-~—= ~—V2W. m If we start from the covariant Klein—Gordon equation and follow the above approximation, the effects of gravity and the rotation of reference frames are naturally included in (2.1) through the metric. Thus, as far as the validity of the nonrelativistic approximation is maintained, we can treat the phase shifts due to the gravitational and inertial force in a unified way. We assume that the external gravitational field of the earth is described by the Kerr metric. In Boyer— Lindquist coordinates, it is given by —
A —
(cdt—a sin2Odçb)2
—~
sin20 +
[(1.2+~2)
fdr2 dcb—ac dt]2+p2~—~+d02 (2.5)
where Am r2 —2 (GM/c2)r+ a 2, Mis the mass of the earth, p2 r2 + a2 cos20 and a is the Kerr parameter which corresponds to the angular momentum J of the earth per unit mass, a = J/Mc. In the present situation, both GM/rc2 and a/r are sufficiently small and their higher orderterms can be neglected. Therefore, to the first order in M and a and neglecting the terms of O((v/c)2), the Klein—Gordon equation in the Kerr spacetime becomes 89’ h2 [1 8 / 8 \ L2 1 ih— —I ——I r2— I— 19’ 8t 2mLr28r\ 8rJ r2h2J GMm ~ 2GMaLW (2.6) .
=—
1 329’ mô!t’ +2i—— +V29’=0, c2 8t2 11 ôt
(2.3)
— —~
.
.
where V2 denotes the three-dimensional Laplacian operator. The relative order of the first term to the others in (2.3) is O((v/c)2), where v denotes the characteristic velocity of wave packets, or equivalently O((A 2),where 2~is the Compton wavelength and 0/)~) is the de Broglie wavelength of a partide. For the neutron interferometer used by Colella, Overhauser and Werner [6], a nonrelativistic treatment gives a sufficiently accurate description so the term of 0 ((v/c)2) can be neglected. This corresponds to discarding the negative energy state and
—
r rc where m is the mass of the particle and L2
— —
h 21 ~
(sin 0 + -p-— [sinOäO\. 80) sin2OaØ2J’
—
-~--
-~-‘~
—~——
(2 7)
,~
L~=
—
ih
~—
(2.8)
are the orbital angular momentum operators of the particle with respect to the center ofthe earth. After the coordinate transformation Ø—*Ø— wt, where w is 331
Volume 182, number 4,5,6
PHYSICS LETTERS A
22 November 1993
the angular velocity of the earth, we obtain the “Schroding;r” equation for an observer fixed on the
B
(i,oerfererrce region)
~-)_-~]w
i=_~_[-~-(r
____
___
di
A
—
GMm 2GMa r 9’~wL~W+ r c L~9’.
D
(source)
(2.9)
Fig. 1. Schematic illustration of alternative paths in a neutron interferometer. 3grav =
I
~ (ABD)
—
/31 (ACD)
[V
3. The phase shift in a neutron interferometer
5(R+c4 sin ~,) V8(R)] mXd1 —
The Hamiltonian derived in the last section is divided into the four terms H—H0 +H1 +H2 +H3,
3 1
—
where 2 1 8
H 0=
h —
~
~
1 ~r
2
8\
‘
(3.2)
is the Hamiltonian for a freely propagating particle, H1
=
—
GMm r
vg
(3 3)
is the Newtonian gravitational potential energy, and H2=—wL0, H3=
3c L~. 2GMa r
(3.4)
,
where is the acceleration of gravity, A=d1d2 is the area ofg the parallelogram, and X is the de Brogue
phase difference observed by Colella, Overhauser and Werner [61 in a neutron interferometer experiment. Next, let us calculate the phase shift /32 principally due to the rotation of the interferometer, which is in our case caused by the earth’s rotation. This is a quantum mechanical analogue of the Sagnac effect [9] in optical interferometry. In the same way as for /31, /3~may be written in the form of the integration fl
1
1
(3.5)
where /3, denotes the phase shift caused by H, (z = 1— 3). In the first place, let us calculate the phase shift due to the gravitational potential. For the purpose of the present discussion, the quasi-classical approximation is valid and the phase shift is given by the integration along a classical trajectory/3~= Fl—’ f J/~ di. Therefore, for a beam of thermal neutrons which splits into two alternative paths and then recombines as in fig. 1 the phase difference fl~is
(3.7)
where Q = (0, 0, a), and L= (Lx, L11,, L~)is treated as the classical angular momentum L = mrx r. The phase difference flsag due to the Sagnac effect [7—12] becomes flsag=fl2(ABD)fl2(AcD)=
332
(3.6)
2=_~Ja)Lzdt=_~JQ.Ldt,
If 9’o is a solution for the Hamiltonian H0, then the solution for the total Hamiltonian H may be in the form 9’=!1~exp[—i(fl1+fl2+fl3)]
mgA X sin ~,,
wavelength divided by 27t. The theoretical prediction (3.6) Overhauser Colella [5] was and itfirst wasderived in goodby agreement withand the
L2
~)+
=
mf TP~(?~Xd1’) ‘~ .‘
2mQ’A ,
(3.8)
where A is the area vector of the sector ACDB. The third term of the phase shift is a general relativistic effect due to the dragging ofthe reference frames, the so-called Lense—Thirring effect [18]. A similar calculation leads to
Volume 182, number 4,5,6
PHYSICS LETTERS A
1 r2GMa 2G fJL = j —i— dt, (3.9) = ~ r3c where /= (0, 0, J) is the angular momentum vector of the earth. The phase difference is
i
~
J3drag = P3(ABD) —
=
/33 (ACD)
r .1. (rx dr)
—
[~ (~ r (~ ~~], LA
2Gm Flc2R31
2Gm hc2R3’
=
r (R+r’)xdr’ I R + r’ I
2Gm
_____ —
2Gm —
r’ X dr’ _3 ~
.rc) ~ X dr’]
—
(3.10)
~
where R represents the position vector of the instrument from the center of the earth. If we assume that the earth is a sphere of radius R with uniform density, then 1 ~MR2Q
(3.11)
and fldrag =
=
—
1 rg j~ /3sag
—
2r
——
~~flSag
~
~Ri £1 [A 3(~A)
~
if R is perpendicular to A,
5
if R is parallel to A
,
(3.12)
2 is the Schwarzschild radius of where rg=phase 2GM/c earth. The difference due to the Lense—Thirring effect is r 5/R 1 0~times that due to the Sagnac effect, and a very sensitive interferometer would be required to distinguish fldrag from flSag. As pointed out by Anandan and Chiao [13], however, in principle these effects can be separated by a “figure-eight” interferometer. This separate detectability derives from the fact that the phase shift due to the Sagnac effect depends only on the area and direction of the loop path whereas that due to the Lense—Thirring effect depends on the distance from the center of the earth as well as the size and direction ofthe interferometer (see eq. (3.12)). Another experiment using a new type of interferometer has also been proposed which ideally would be insensitive to both the Sagnac and —~
22 November 1993
the gravity effects [19] and which could be used to detect the general relativistic effect. 4. Conclusion In this paper we investigated the effect of the grayitation and the rotation of the earth on the quantum interferometer. We have derived the expressions for the phase shift due to the gravitational and the Coriolis forces in a unified way from the viewpoint of general relativity. We have also shown that the general relativistic effect due to the dragging of the reference frames could cause a shift in the interference fringes, although its magnitude is l0~times that of the Sagnac effect caused by the rotation of the earth. A new type of quantum interferometer is proposed and now under active development [19] which could be used to measure such relativistic effects as well as the phase shift due to the spin—rotation coupling and/ or the spin—curvature coupling [14—17]. These effects may be treated in our formalism starting with the covariant Dirac equation. Among various possible relativistic corrections, we have concentrated our attention on the dragging effect in this paper. This effect can in principle be separated from others because of the direction-dependence as shown in (3.10). A general and more careful treatment of general relativistic corrections will be given in a separate paper. It is of great significance to explore the possibility detector. of quantum It may interferometers be straightforward as the gravitational to study the wave interference phase shift caused by gravitational waves in our formulation. We would like to examine this problem in the near future.
Acknowledgement T.F. thanks Professor Y. Fujii for discussions at the early stage of this work. We also thank Dr. S. Bildhauer for discussions.
References [1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485.
333
Volume 182, number 4,5,6
PHYSICS LETTERS A
[2] R.G. Chambers, Phys. Rev. Lett. 5 (~i960)3. [3] G. Mdllenstedt and W. Bayh, Naturwissenschaften 49 (1962) 81. [4] A. Tonomura et al., Phys. Rev. Lett. 48 (1982)1443. [5] A.W. Overhauser and R. Colella, Phys. Rev. Lett. 33 (1974) 1237. [6]R. Colella, A.W. Overhauser and S.A. Werner, Phys. Rev. Lett. 34(1975) 1472. [7]L.A. Page, Phys. Rev. Lett. 35(1975)543. [8]S.A. Werner, J.-L. Staudenmann and R. Colella, Phys. Rev. Lett. 42 (1979) 1103. [9]M.G. Sagnac, C.R. Acad. Sci. 157 (1913) 708, 1410. [lO]J. Anandan, Phys. Rev. D 15 (1977) 1448.
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