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

37
to consider the photon shot-noise [Eq. (31)]. 33 Depending
on the details of a particular measurement, either the
spin noise (33) or the photon noise (31) may dominate.
If a measurement is optimized for statistical sensitivity,
the two contributions to the noise are found to be comparable
(Budker et al., 2000b).
Fleischhauer et al. (2000) considered an additional
source of noise in polarimetric spin measurements. When
the input light is off-resonant, independent quantum fluctuations
in intensity of the two oppositely circularly polarized
components of the light couple to the atomic Zeeman
sublevels via ac Stark shifts and cause excess noise
in the direction of the atomic spins. Fleischhauer et al.
(2000) concluded that this source of noise, while negligible
at low light power, actually dominates over the
photon shot noise above a certain critical power. Estimates
based on the formulae derived by Fleischhauer
et al. show that for the optimal conditions found by Budker
et al. (2000b) this effect could contribute to the overall
noise at a level comparable to the photon shot noise.
However, the effect of the ac Stark shifts can, in certain
cases, be minimized by tuning the light frequency so that
the shifts due to different off-resonance levels compensate
each other (Novikova et al., 2001a). Some other possibilities
for minimizing the additional noise may include
compensation of the effect by different atomic isotopes
(Novikova et al., 2001a), or the use, instead of a polarization
rotation measurement, of another combination of
the Stokes parameters (Appendix A) chosen to maximize
the signal-to-noise ratio.
2. Experiments
One approach to using NMOR for precision magnetometry
is to take advantage of the ultra-narrow (width ∼1
µG) resonance widths obtainable in paraffin-coated cells
(Sec. VIII.E). Using the experimental setup described in
Sec. XI.A, Budker et al. (2000b) optimized the sensitivity
of an NMOR-based magnetometer to sub-microgauss
magnetic fields with respect to atomic density, light intensity,
and light frequency near the D1 and D2 lines
of 85 Rb. They found that a shot-noise-limited magnetometric
sensitivity of ∼3 × 10 −12 G Hz −1/2 was achievable
in this system. This sensitivity was close to the shotnoise
limit for an ideal measurement [Eq. (33)] with the
given number of atoms in the vapor cell (∼10 12 at 20 ◦
C and rate of ground state relaxation (∼1 Hz), indicating
that, in principle, NMOR is a nearly optimal technique
for measuring the precession of polarized atoms in
external fields. If limitations due to technical sources
of noise can be overcome, the sensitivity of an NMORbased
magnetometer may surpass that of current optical
pumping (Alexandrov et al., 1996) and SQUID (superconducting
quantum interference device) magnetometers
(Clarke, 1996), both of which operate near their shotnoise-limit,
by an order of magnitude. Even higher sensitivities,
up to 2 × 10 −14 G Hz −1/2 , may be achievable
with an ingenious magnetometric setup of Allred et al.
(2002) in which K vapor at a density of 10 14 cm −3 is used
and the effect of spin-exchange relaxation is reduced by
“locking” the precession of the two ground-state hyperfine
components together via the spin-exchange collisions
themselves (Happer and Tang, 1973).
The magnetic-field dependence of NMOR is strongly
affected by the magnitude and direction of transverse
magnetic fields (Budker et al., 1998a,b). At low light
powers, the transverse-field dependence can be quantitatively
understood using a straightforward extension of
the Kanorsky-Weis model discussed in Sec. VII.A, which
opens the possibility of sensitive three-dimensional magnetic
field measurements.
As discussed in detail in Sec. XI.C, if frequencymodulated
light is used to induce and detect NMOR, the
dynamic range of an NMOR-based magnetometer may be
increased beyond the microgauss range to > 1 G without
appreciable loss of sensitivity.
Novikova et al. (2001a); Novikova and Welch (2002),
using buffer-gas-free uncoated cells, investigated the application
of NMOR in optically thick media to magnetometry
(Sec. VIII.A.3). Although the ultimate sensitivity
that may be obtained with this method appears to
be some two orders of magnitude inferior to that obtained
with paraffin-coated cells, 34 this method does provide a
broad dynamic range, so that Earth-field (∼0.4 G) values
could be measured with a standard polarimeter.
Rochester and Budker (2002) theoretically analyzed
magnetometric sensitivity of thick medium NMOR measurements
optimized with respect to light intensity in the
case of negligible Doppler broadening, and in the case of
large Doppler broadening. In the former case, the sensitivity
improves as the square root of optical density,
while in the latter, it improves linearly—a result which
can be obtained from standard-quantum-limit considerations
(Sec. XII.A.1).
Cold atoms prepared by laser trapping and cooling
were also recently used for NMOR-based magnetometry.
Isayama et al. (1999) employed a pump/probe geometry
with ∼10 8 cold 85 Rb atoms that were trapped in a MOT
and then released (Sec. VIII.H) for measurement of an
applied magnetic field of ∼2 mG. The observation time
for a single measurement was limited to about 10 ms due
to the free fall of the atoms in the Earth’s gravitational
33 We do not consider the use of the so-called spin-squeezed quantum
states (discussed by, for example, Ulam-Orgikh and Kitagawa,
2001, and references therein) or squeezed states of light
(Grangier et al., 1987), which can allow sensitivity beyond the
standard quantum limit.
34 The limitation in practical realizations of thick-medium magnetometry
may come from the effects of radiation trapping (Matsko
et al., 2001b, 2002a; Novikova and Welch, 2002).