follows from (2.23) that the total flux of GA plus that of any electromagnetic field present is quantised. This extends to the time-varying regime what was already ...
Class. Quantum Grav. 6 (1989) 407-418. Printed in the U K
Particle interferometry in weak gravitational fields Y Q Cai and G Papini Department of Physics and Astronomy, University of Regina, Regina, Saskatchewan, Canada S4S OA2
Received 4 January 1988
Abstract. The problem of particle interferometry in weak inertial or gravitational fields is treated from a unified point of view. The effect of the fields can be confined to a phase factor to be determined by quadratures once the solution of the possibly non-linear wave equation describing the particles is known. The procedure is completely Lorentz invariant and gauge invariant, can be extended to higher orders and applies to a wide range of interferometers, from optical ones to those using superfluids. Results already reported in the literature are re-obtained and in some cases improved. Other results known to hold for stationary fields are extended to time-dependent fields. It is also shown that interferometers hold promise as broad-band detectors of gravitational radiation even at high frequencies.
1. Introduction
It has been repeatedly suggested that the interference of particle beams or currents may ultimately lead to the measurement of small but conceptually important gravitational effects [ 13. Very coherent and intense photon beams, in particular, are presently being employed in attempts to detect gravitational radiation from various astrophysical sources [2], while the effect of inertial fields on superconducting? [3] and neutron interferometers [4] has already been observed [4,5]. Neutron interferometers have also been used to study the effect of gravity on quantum systems [6] and the construction of interferometers based on neutral superfluids [7] and molecular beams [8] is underway or has been discussed. The effect that gravitational or inertial fields have on the wavefunctions of the interfering particles can be determined in a number of ways. If the fields are stationary and the particles are non-relativistic, the wavefunctions must only obey the Schrodinger equation [ 3,9]
allr
ih= ( 1/ 2 m )[ p i - (e/ CIA,- mcyoiI2$+ imc2yo& dt
+ eAo$ + V$
i = 1,2,3
( 1.1)
where
+ Though the works in [ 3 ] consider specificall) charged superfluids, their application to neutral superfluids is immediate.
0264-9381/89/030407+ 12SO2.50
1989 IOP Publishing Ltd
407
408
Y Q Cai and G Papini
is the metric deviation, A , represents the electromagnetic field and V is a potential describing any other interactions present in the system. The effect then amounts to a shift in the phase of which, to leading order, is
+
Since under the coordinate transformations x‘fi = x@+ 5”
5, = 77,”t”
(1.4)
where 5’ is smooth enough to keep yPy small, the quantities yWvtransform as Y L Y
= YWY
- t,,”-
SV,,
(1.5)
it immediately follows that (1.1) and (1.3) are covariant under (1.4) to first order in 5, a n d also gauge invariant. The field yoi does transform here as a vector potential in three-dimensional space because it has been assumed stationary. The generalisation of (1.3) to time-dependent fields has had a more tortuous development. The approaches of Linet a n d Tourrenc [ 101 and later of Stodolsky [ 111 are based on a generalisation of the ‘eikonal’ method of classical optics. They are gauge invariant only in the slow-motion limit a n d d o not take into account the coupling of the angular momentum of the particle to the gravitational or inertial fields, thus giving only a partial effect. The generalisation of Anandan [12] is based on two assumptions. (i) The phase difference AS of particles satisfying the Klein-Gordon equation
( v ~ v-, m 2 c 2 / h 2 ) 4 ( x=)O (1.6) and travelling along a closed path r can be represented by means of a non-integrable wavevector
K, = -V,S
AS = fr
such that K,
dx,.
(1.7)
In (1.6) V, represents covariant differentiation. (ii) The effect of gravity is contained in the phase of the wavefunction and in the propagation of the amplitude. While (ii) is correct for weak fields as shown below, ( i ) is unnecessarily restrictive. It may be justified, but requires in general further assumptions. Finally, another generalisation of (1.3) [13] makes use of the relation [I41 yo,=;
~ ~ ~ , ~ X 0(x3) ~ x r +
(1.8)
and thus requires Fermi normal coordinates. This is an undesirable constraint in some problems, particularly when dealing with high-frequency gravitational radiation. A perhaps simpler, more general way to deal with this problem was briefly suggested by Papini long ago [9], but never discussed in detail. It applies to weak gravitational and inertial fields, and is manifestly covariant and gauge invariant. I n reproposing this method here, the application to the detection of high-frequency gravitons has been kept in mind. No resonance condition is of help in this case and no resonance condition is implied by the theory of the interferometer which would work in principle as a wide-band detector. Another possible application is to the study of rotating heavy fermion systems ( H F S ) as suggested in 3 4.
Particle interferometry in weak gravitational jields
409
Additional motivation is also provided by the desire to qualify those claims that the strong equivalence principle can be tested by interferometry. In fact, the equivalence principle requires locality while interferometry is manifestly non-local. It will be shown below that in ordinary laboratory situations, true non-locality appears only through higher-order terms. What follows applies to a variety of particles and quantum systems, whose dynamical behaviour can be described by (1.6) or (2.1) below, and in particular to charged and neutral superfluids [3] which are ideally suited for the measurement of extremely small physical effects [15] and have been extensively studied in the literature [16, 171. In describing the behaviour of superfluids in inertial and gravitational fields the following approximations are usually made: (i) weak fields, (ii) slow particle motion ( U < < c ) , (iii) negligible depairing, and in addition for superconductors (iv) lattice incompressibility. These are all independent assumptions which, with the exception of (iv), can be safely made in earth-bound laboratory situations. In some experiments and for some metals, (iv) can also be assumed without undue difficulties. Different techniques of measurement [7,18] further distinguish the problem of the neutral superfluids from that of superconductors. With these limitations in mind, the basic result, derived in § 2, can be also applied to superfluids and condition (ii) may be effectively removed. Some applications are given in § 3. Section 4 contains a discussion of the results.
2. Weak-field solution of a class of relativistic wave equations In view of the wide variety of interferometers mentioned in the previous section, it is convenient to study systems whose wavefunctions satisfy equations more general than (1.6). One such equation is [17] {[V, - i ( e l 4 4 , l 2 -
m 2 c 2 / h 2 ) 4 ( x=) P 1 4 ( x ) 1 2 4 ( x )
(2.1)
where p is a constant and A , ( x ) represents the total electromagnetic potential of all external and gravity-induced fields present. The quantity 4 ( x ) here represents a scalar. Equation (2.1) is the fully covariant version of the Landau-Ginzburg equation [19]; it reduces to the Gross-Pitaevskii equation when A, vanishes and to (1.6) for p = 0. It is therefore well suited to discuss a number of systems, from superfluids to scalar particles. If, in particular, heavy fermion systems admit minimal coupling [20], (2.1) may be used in this case too, with the added advantage of a much larger effective coupling to gravity. A solution of (2.1) with e = 0 can be obtained by introducing a new field @ ( x ) related to 4 ( x ) above by Q ( x ) = [elX,+ ( x ) ] = eIX4(x)
where the phase operator y, is defined by
with
(2.2)
410
Y Q Cai and G Papini
and a ( x ) is a function to be determined. The quantities rm,Ap
=
t( ~ a A , p+
Y”~,A
- YAP,“ )
represent the Christoffel symbols of the first kind constructed in terms of y,”. (2.2) in the form
+ a , u Q , , + Q , u a , , + Qa,,,. Contraction of (2.10) with vp” gives
(2.6) Rewriting
(2.10)
It is now convenient to isolate the first-order terms in (2.11) by using the expansion
- yl‘; + yFzi+ . . .
yfiu
(2.12)
where 1 > 1 y;,? > 1 y221 . . . . On using (2.7) one obtains
(2.13)
Particle interferometry in weak gravitational Jields
41 1
where I; collects all higher-order terms coming from applying (2.12) to the Christoffel symbols and to the equations
4 , y = 4,g+Lp -~pp~~Y,cLp+~,,yI)~4,~y
(2.14)
4,,*”= [ ( m 2 c 2 / h 2 ) + ~ l ~ 1 2 1 ~ + ~ a ~ a + ~ , Y + , ~ U .
(2.15)
and
Equation (2.13) yields to first order T,’@,,~ - [ m2c2/h 2 +pl@12]@ = 4;,& - [ m 2 c 2 /h 2 + pl+/’]4 = 0
(2.16)
provided
- r ~ ) + , p + ~ ) + y ~ ~ : , , p=o. ~ + + ~ ) (2.17) +,p
Q ~ , , ~ ~ , ~ + ~ ~ , , Q , ~ + ~ ( - ~ ~ ~ + , L ~ ~
When C = 0, (2.17) has a solution a = 1 . By using (2.8) one can also eliminate a,fiQ,’ from (2.17). Since @,, - 4,, -- Q,, to first order, one finds a,,Q,+ = [(@,,
- 4,,)Q,”- Q,,Q,”I/Q
= 0.
(2.18)
Higher-order corrections to a can thus be simply obtained from the equation a(),)‘-
I
d4~’D(X-X’)I;/Q.
(2.19)
Some caution must be exercised in using (2.19). The quantity Z is not in fact a true scalar in Riemannian space. However, over restricted regions of spacetime and for smooth coordinate transformations a ( h ) appears to be sufficiently coordinate independent. In turn, one can prove that if @(x) satisfies the equation T ~ ~ @’ ,( m , ~2 c 2 /h 2 ) @= pl@12@
(2.20)
+ ( x ) = [e-iX,@(x)]= e-’“@(x)
(2.21)
then
satisfies (2.1) with e = 0. Thus the effect of weak gravitational fields can be described by a phase factor once the solution of the gravity-free equation is known, and @ need not be a constant as in [12]. An alternative way of determining the solution to any order may be obtained by expanding 4 ( x ) as N
4=
+(’”
n
=o
(2.22)
The effect of the electromagnetic field can also be incorporated in the phase factor in a straightforward way by adding to ( 2 . 7 ) the term
412
Y Q Cui and G Papini
It is easy to see that the first-order solution (2.21) is gauge invariant and invariant under the transformations (1.4) and (1.5) only for closed spacetime paths. In this case, Stokes’ theorem gives (2.23) where X p is the surface bound by the closed path, FPy= - A , , , + A , , represents the linearised Riemann curvature tensor 1
Rclvap
(1)
= -2(YFO,”p
(1) - Yva,,b
(1)
- Ypp,“a + Y!;,Fa).
and Rlrvap (2.24)
Equation (2.23) may be considered as the equation governing particle interferometry, whenever the gravitational fields are weak. In the absence of electromagnetic fields, the result is that of [9]. An expression of the form (2.23) for a n infinitesimal loop has also been obtained by Anandan [21]. Equation (2.21) with o! = 1 provides justification and limitations for Anandan’s assumption leading to ( 1 . 7 ) [ 121. GA ( Z, 4 ) does in fact behave as a 4-vector only for weak fields a n d gauge invariance is assured only for closed paths. It immediately follows from (2.23) that gravitational fields finite only over a limited spacetime domain and vanishing everywhere else, would still be observable in the field-free region by particle interferometry (the gravitational BohmAharonov effect [22]). Since (1.1) can be obtained in the non-relativistic limit from the Klein-Gordon equation, one may expect (2.21) to give (1.3) in the appropriate approximation. One can in fact show that (2.21) applied to a closed path gives the correct results even for inertial fields, for which the weak-field approximation must be applied with great care. The case of a slowly steadily rotating system satisfying (1.6) is of particular interest. In this case the phase factor (1.3) is generated wholly by the term -$yk:‘(D,” of CA, where 0 now is a solution of the free Klein-Gordon equation. Differentiating this term twice contributes the expression
- yEj0,”’l
= 2 y b : ’ ~ , ~=, 2 m c y , , ( ~ ’ , ~
which multiplied by the factor ( l l 2 m ) gives the correct term in (1.1).
3. Applications In what follows only the first-order solution of (2.1) is used. This corresponds to taking ~(x)]
d z * ( r : ~ , : ~-r;,:,)[Jop(z),
0
. *
-: d z ” ?:A‘ [ 2
.
Pp,@(x)]
c
(3.1)
Though the form (2.23) for ,y looks more appealing because of its manifest gauge invariance, expression (3.1) is more readily applicable. The phase shifts calculated below are those induced by the gravitational field of the earth, rotation and gravitational waves. I n all cases it is assumed for simplicity that @ ( x ) may be represented by plane-wave solutions.
Particle interferometry in weak gravitational jields
413
3.1. The jield of the Earth
If Earth is assumed spherical, then its gravitational field may be described by the Schwarzschild metric [ 2 3 ] d.r2 = -( 1 - 2 M G / c'r) dt2+ ( 1 / c 2 ) (1 - 2 M G / c 2 r ) - ' dr2 + ( r 2 / c 2 )d e 2 + ( r 2 / c 2 )sin2 8 d 4 '
(3.2)
where M represents the mass of the Earth. Equation ( 3 . 1 ) is now applied to a quadrangular interferometer of vertices A, B, C, D, in which a beam of particles is split at A and the resulting beams interfere at A again after travelling along the opposite paths ABCDA and ADCBA. If the particles are neutral, the contribution to the change in phase may be divided into a part Ax, containing .IlLy and the remainder Ax2. Since the particles are assumed to move with constant speed v, the integration over the time portion of the spacetime loop may be reduced to space integrations by choosing the limits of integration appropriately. A detailed calculation shows that Axl vanishes for both paths ABCDA and ADCBA. Assuming that the linear dimension of the interferometer I is such that I / R
