Resonant nonlinear magneto-optical effects in atoms∗ - The Budker ...

39
Photodiode signal (V)
10
8
6
4
2
0
7
6
5
4
33.0 33.2 33.4 33.6 33.8 34.0
0 20 40 60 80 100 120 140
Time (s)
FIG. 20 Experimental optical rotation signal obtained by Romalis
et al. (2001b) produced by Hg atomic polarization in a
magnetic field of 17 mG. The inset shows a one-second segment
of the data.
from resonance by 20 GHz, is detected (Fig. XII.B). 35
The spin-precession frequency is measured in two identical
cells with opposite directions of the electric field
with respect to the magnetic field. This device has a
sensitivity to energy level shifts of ∼0.3 µHz/Hz 1/2 over
a measurement time of a hundred seconds, better than
any other existing device (see Sec. XII.A). With several
months of data taking, it has set a limit on the atomic
EDM of ∣ ∣d ( 199 Hg )∣ ∣ < 2.1×10 −28 e cm (95% confidence).
C. The Aharonov-Casher phase shift
The Aharonov-Casher effect was studied interferometrically
with neutrons (Cimmino et al., 1989) and
with the fluorine nucleus in the TlF molecule (Sangster
et al., 1993, 1995) using conventional Ramsey molecular
beam spectroscopy. Faraday-Ramsey spectroscopy
(Sec. VIII.C) is a convenient method for measuring the
Aharonov-Casher effect with atoms. Gorlitz et al. (1995)
performed such an experiment with Rb atoms and found
agreement with the theoretical prediction at the level
of 1.4%. They verified the linear dependence of the
phase shift on the strength of the electric field as well
as its nondispersive nature (independence of velocity).
An atomic Aharonov-Casher effect was also measured by
Zeiske et al. (1995) using an atomic interferometer.
In Faraday-Ramsey geometry, the phase shift accumulated
in a homogeneous static electric field between
∆M = 2 sublevels of an aligned atom is given by
φ AC = 2gµ
¯hc
∫
EdL, (37)
where the field integral extends over the flight region between
the two optical interaction regions. This phase
shift enters directly in the Faraday-Ramsey line-shape
function given by Eq. (26). When using atoms with a
well known g-factor, the measurement of φ AC can thus
be used to experimentally determine the electric field integral
(Rasbach et al., 2001).
D. Measurement of tensor electric polarizabilities
The quadratic Stark effect in a Zeeman manifold
|F, M〉 in a homogeneous static electric field E can be
parameterized by
An atom moving in an electric field E with velocity v
experiences a magnetic field B = E × v/c. This “motional”
magnetic field induces phase shifts of the atomic
magnetic sublevels and thus affects nonlinear magnetooptical
signals. The E × v effect is a linear Stark effect
that represents a severe systematic problem in many
EDM experiments. The induced phase shift is given by
the Aharonov-Casher (1984) phase
φ AC = 1
¯hc
∫
(c)
µ × E(s) · ds (35)
acquired by a magnetic moment µ carried on a path (c). 36
The phase (35) is independent of the shape of the trajectory
(c) and is thus referred to as a “topological phase”.
∆ s (F, M) = − 1 2 αE2 , (38)
where the static electric polarizability α can be decomposed
into scalar (α 0 ) and tensor (α 2 ) parts according to
(Angel and Sandars, 1968):
α = α 0 + 3M 2 − F (F + 1)
F (2F − 1)
α 2 . (39)
States with J = 1/2, such as the alkali ground states,
can only have tensor properties due to nonzero nuclear
spin. Tensor polarizabilities of such states are suppressed
compared to the scalar polarizabilities by the ratio of the
hyperfine splitting to the energy separation between the
35 A previous (several times less sensitive) version of this experiment
(Jacobs et al., 1995) used the same light beam for both
pumping and probing.
36 Equation (35) can be easily derived by considering a particle with
magnetic moment moving along a trajectory whose element is
given by ds = vdt. The particle acquires a differential phase shift
dφ = · B
¯h
dt = · E × v
¯hc
which gives (35) upon integration.
dt = · E × ds , (36)
¯hc