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

35
tically pump the atomic medium into a polarized state
which precessed with the Larmor frequency.
D. Magnetic shielding
Nonlinear magneto-optical effects are generally very
sensitive to magnetic-field magnitude, direction, and gradients.
For ground-state spin-relaxation times of < ∼ 1 s,
NMOE are maximum at sub-microgauss magnetic fields
[Eq. (2)], necessitating their control at this level. 32 This
control can be achieved with ferromagnetic shielding in
spite of the fact that residual fields due to imperfections
of the shield material are usually ∼10–50 µG (Kozlov
et al., 1982). With a carefully designed four-layer magnetic
shield and three-dimensional coil system inside the
shield (see Fig. XI.A), it is possible to achieve long-term
magnetic-field stability inside the innermost shield at the
level of 0.1 µG (Budker et al., 1998a; Yashchuk et al.,
1999a, 2002a).
Closed ferromagnetic shells shield different frequency
external fields via different physical mechanisms. For
static and very low-frequency external fields, the most
important mechanism is flux shunting due to the high
permeability of the material. At high frequencies, the
skin effect becomes the most important mechanism. A
review of the early work on the shielding problem, a theoretical
treatment, and a survey of practical realizations
was given by Sumner et al. (1987); see also books by
Khriplovich and Lamoreaux (1997); Rikitake (1987).
For static fields, the dependence of the shielding ratio
on the shape and size of n shielding layers is given by an
approximate formula:
S tot ≡ B in
B 0
n−1
∏
≃ S n
i=1
( ) ] k −1
Xi+1
S i
[1 −
, (32)
X i
where S i ≃ X i
µ i t i
.
Here B 0 is the magnetic field applied to the shield, B in
is the field inside the shield, S i is the shielding factor
of the i-th layer, X i is the layer’s radius or length (depending
on the relative orientation of the magnetic field
and the layer), and t i and µ i are the thickness and magnetic
permeability of each layer, respectively. We assume
X i > X i+1 and µ i ≫ X i /t i . The power k depends on
the geometry of the shield: k ≃ 3 for a spherical shield;
k ≃ 2 for the transverse and k ≃ 1 for the axial shielding
factor of a cylindrical shield with flat lids. Thus, for
shields of comparable dimensions, spherical shells provide
the best shielding. In the design of the three outer
32 There are specific requirements to magnetic shielding for the
work with atoms in a cryogenic buffer gas environment discussed
in Sec. VIII.D.2, in which relaxation times of up to minutes are
expected.
layers of the four-layer shield shown in Fig. XI.A, an approximation
to a spherical shape, simpler to manufacture
than a true sphere, is used. The overall shielding ratio
is measured to be ∼10 −6 , roughly the same in all directions.
The innermost shield is in the shape of a cube with
rounded edges. This allows compensation of the residual
magnetic field and its gradients as well as application
of relatively homogeneous fields with a simple system of
nested 3-D coils of cubic shape (Yashchuk et al., 2002a).
The field homogeneity is increased by image currents, due
to the boundary conditions at the interface of the highpermeability
material, which effectively make the short
coils into essentially infinite solenoidal windings.
An important characteristic of a magnetic shield, in
addition to the shielding factor and the residual fields
and gradients within the shielded volume, is the residual
magnetic noise. One source of such noise is the Johnson
thermal currents (Allred et al., 2002; Lamoreaux, 1999;
Nenonen et al., 1996) in the shield itself.
Superconducting shielding (Cabrera, 1988; Cabrera
and Halmilton, 1973, and references therein) yields the
highest field stability. The shielding properties of superconductors
originate in the Meissner effect, i.e., the exclusion,
due to persistent currents, of magnetic flux from
the bulk of a superconductor. Hamilton (1970); Taber
et al. (1993) developed a technique for reducing the magnetic
field inside a superconducting shield by expanding
the shield. When the dimensions of the superconducting
enclosure increase, the enclosed magnetic field decreases
due to conservation of magnetic flux “frozen” into the superconductor.
In one experiment, expansion of a folded
lead-foil (100 µm-thick) balloon was used; an ultra-low
residual magnetic field (∼ 0.06 µG) was achieved over a
liter-sized volume (Cabrera and Halmilton, 1973). With
expanded lead bags as superconducting shields and a surrounding
conventional ferromagnetic shield, a magnetic
field of < 0.1 µG in the flight dewar of the Gravity Probe
B Relativity Mission was maintained in a 1.3 m-by-0.25
m-diameter volume enclosing a gyroscope (Mester et al.,
2000), a shielding ratio of about 5 × 10 −9 . Some limitations
of superconducting shields were discussed by Dietzfelbinger
et al. (1990).
An essential difference between superconducting and
ferromagnetic shields is their opposite boundary conditions.
Since a superconductor is an ideal diamagnetic
material, magnetic field lines cannot penetrate into it,
and so image currents are of opposite sign as those in ferromagnetic
material. This is important for determining
the shield geometry that provides the best field homogeneity.
E. Laser-frequency stabilization using magneto-optical
effects
The development of frequency-stabilized diode laser
systems has led to recent progress in experimental atomic
and molecular physics in general, and the study of NMOE