Kiefer C. Quantum gravity

QUANTUM THEORY AND THE GRAVITATIONAL FIELD 9
ω describes the influence of the terrestrial rotation on the interference pattern
(‘neutron Sagnac effect’). It yields
∆β Sagnac = m ∮
i
(ω × r)dr = 2m i
ωA , (1.14)
where A denotes the normal area vector of the loop ABDC.
Of main interest here is the gravitational part of the phase shift. Since the
contributions of the sides AC and DB cancel, one has
∆β g = m i
∮
vdr ≈ m i(v 0 − v 1 )
AB , (1.15)
where v 0 and v 1 denote the absolute values of the velocities along AB and CD,
respectively. From energy conservation one gets
√
v 1 = v 0
1 − 2∆V
m i v 2 0
≈ v 0 − m ggh 0 sin θ
m i v 0
,
where ∆V = m g gh 0 sin θ is the potential difference, h 0 denotes the perpendicular
distance between AB and CD, and the limit 2∆V/m i v0
2 ≪ 1 (about 10−8 in
the experiment) has been used. The neutrons are prepared with a de Broglie
wavelength λ =2π/p ≈ 2π/m i v 0 (neglecting the ω part, since the Sagnac
effect contributes only 2% of the effect), attaining a value of about 1.4 Åinthe
experiment. One then gets for the gravitational phase shift the final result
∆β g ≈ m im g gλA sin θ
2π 2 , (1.16)
where A denotes the area of the parallelogram ABDC. This result has been
confirmed by ‘COW’ with 1% accuracy. The phase shift (1.16) can be rewritten
in an alternative form such that only those quantities appear that are directly
observable in the experiment (Lämmerzahl 1996). It then reads
∆β g ≈ m g
m i
gGTT ′ , (1.17)
where T (T ′ ) denotes the flight time of the neutron from A to B (from A to
C), and G is the reciprocal lattice vector of the crystal layers (from which the
neutrons are scattered in the beam splitter). Now m g and m i appear as in the
classical theory as a ratio, not as a product. The ‘COW’ experiment has also
confirmed the validity of the (weak) equivalence principle in the quantum domain.
Modern tests prefer to use atom interferometry because atoms are easier
to handle and the experiments allow tests of higher precision (Lämmerzahl 1996,
1998). There the flight time is just the time between laser pulses, that is, the
interaction time with the gravitational field; T is chosen by the experimentalist.
For example, Peters et al. (2001) have used atom interferometry to measure g
with a resolution of ∆g/g ∼ 10 −10 .