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

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Rotation angle (mrad)
I 0.038 mWcm 2
87 Rb F ⩵ 2
F ′ ⩵ 1,2,3
I 1.3 mWcm 2
85 Rb F ⩵ 3
F ′ ⩵ 2,3,4
(a)
(b)
Laser detuning (GHz)
FIG. 14 Comparison of experimental NMOR spectra obtained
by Budker et al. (2000a) to density-matrix calculations
performed in the same work for a natural mixture of
Rb isotopes contained in an uncoated buffer-gas-free vapor
cell. Dots – data points, solid curves – theory (without free
parameters). Offset vertical bars indicate the central frequencies
and calculated relative contributions of different hyperfine
components (F → F ′ ) to the overall rotation. Laser detuning
is relative to the D2 line center. Magnetic field is ∼0.1 G (at
which NMOR is relatively large and coherence effects dominate),
laser beam diameter is ∼3.5 mm, and Rb density is
∼10 10 cm −3 . Residual discrepancies between data and theory
may be due to nonuniform spatial distribution of light intensity.
Note that at low light power (a), the sign of rotation for
F → F + 1 transitions is opposite to that for F → F − 1, F
transitions, whereas at high light power (b) the sign of rotation
is the same for all hyperfine transitions.
teracts with the light field more strongly (Kazantsev
et al., 1984). Thus when the magnetic field causes the
ground-state alignment to precess, the transmission of
light polarized along the initial axis of light polarization
decreases for F → F − 1, F transitions, and increases
for F → F + 1 transitions. By considering the
“rotating polarizer” model (Fig. V.B), one can see that
the sign of rotation is opposite in the two cases (for an
F → F + 1 transition, the initial “polarizer” is crossed
with the light polarization). At higher light powers at
which alignment-to-orientation becomes the dominant
mechanism for NMOR (Sec. V.C), the sign of optical
rotation is the same for all types of transitions.
2. Peculiarities in the magnetic field dependence
A peculiar property of the NMOR transit effect signal,
observed by Chen (1989), remained unexplained for
many years. All theories described in the literature predicted
the shape of the ϕ(B) dependence to be a dispersive
Lorentzian [Eq. (17)], linear near the zero crossing at
B = 0. However, experimental measurement of ϕ(B) for
small B revealed a slight deviation from this anticipated
behavior: the signal had higher than expected derivative
at B = 0.
This observation was later explained as follows. The
shape of ϕ(B) was expected to be Lorentzian as a consequence
of an assumed exponential relaxation of atomic
coherences in time. In room-temperature buffer-gas-free
vapor cells, however, the width of the NMOR resonance
was determined by the effective relaxation due to the
finite interaction time of the pumped atoms with the
light beam. The nonexponential character of this effective
relaxation led to the observed discrepancy. The
simplicity of the theoretical model of the NMOR transit
effect developed by Kanorsky et al. allowed them to
treat these time-of-flight effects quantitatively by averaging
over the Gaussian-distributed light beam intensity
and the Maxwellian velocity distribution. Their
approach produced quasianalytical expressions for ϕ(B)
which were beautifully confirmed in an experiment by
Pfleghaar et al. (1993).
3. NMOE in optically thick vapors
For a given medium used in studies of near-resonant
NMOE, the optimum optical thickness depends on the
light power employed; if most of the light is absorbed
upon traversal of the medium, there is a loss in statistical
sensitivity. At low light power, when the saturation
parameter κ (Sec. III.B) is not much greater than unity,
it is generally useful to use samples of no more than a
few absorption lengths. However, at high light powers,
the medium may bleach due to optical pumping, and
greater optical thicknesses may be used to enhance the
NMOE without unduly compromising statistical sensitivity.
Nonlinear Faraday rotation produced by a optically
thick vapor of rubidium was studied by Matsko et al.
(2001b); Novikova et al. (2001b); Novikova and Welch
(2002); Sautenkov et al. (2000). Novikova et al. (2001b)
used a 5-cm-long vapor cell containing 87 Rb and a 2.5-
mW, 2-mm-diameter laser beam tuned to the maximum
of the NMOR D1-line spectrum. They measured maximum
polarization rotation ϕ max (with respect to laser
frequency and magnetic field) as a function of atomic
density. It was found that ϕ max increases essentially linearly
up to n ≃ 3.5 × 10 12 cm −3 . At higher densities,
dϕ max /dn decreases and eventually becomes negative.
The maximum observed rotation, ∼10 rad, was obtained
with magnetic field ∼0.6 G. (This shows the effect of
power broadening on the magnetic-field dependence; at
low light power, the maximum rotation would occur at a
magnetic field about an order of magnitude smaller.)
A model theory of NMOR in optically thick media,
accounting for the change in light power and polariza-