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

24
A). By substituting (22) into the wave equation
( ) ω
2
c 2 + d2
dz 2 E = − 4π d 2
c 2 P, (23)
dt2 per unit distance for an optically thin medium can be
found in terms of the density matrix elements:
where P = Tr ρd is the polarization of the medium, the
absorption, rotation, phase shift, and change of ellipticity
dϕ
dz = −2πω E 0 c sec 2ɛ[cos ϕ(P 1 sin ɛ + P 4 cos ɛ) + sin ϕ(−P 2 cos ɛ + P 3 sin ɛ)],
(24a)
dE 0
dz = −2πω c [sin ϕ(−P 1 sin ɛ + P 4 cos ɛ) + cos ϕ(P 2 cos ɛ + P 3 sin ɛ)], (24b)
dφ
dz = −2πω E 0 c sec 2ɛ[cos ϕ(P 1 cos ɛ + P 4 sin ɛ) + sin ϕ(−P 2 sin ɛ + P 3 cos ɛ)],
(24c)
dɛ
dz = 2πω
E 0 c [sin ϕ(P 1 cos ɛ + P 4 sin ɛ) + cos ϕ(P 2 sin ɛ − P 3 cos ɛ)],
(24d)
where the components of polarization are defined by
P = 1 [
]
(P 1 − iP 2 )e i(ωt−kz) + c.c. ˆx
2
+ 1 [
]
(P 3 − iP 4 )e i(ωt−kz) + c.c. ŷ.
2
(25)
Since we neglect collisions between atoms, the solutions
for a Doppler-free medium can be generalized to the case
of Doppler broadening by simply convolving the calculated
spectra with a Gaussian function representing the
velocity distribution. In addition, an integration along
the light path can be performed to generalize the thinmedium
result to media of arbitrary optical thickness.
VIII. NONLINEAR MAGNETO-OPTICAL EFFECTS IN
SPECIFIC SITUATIONS
In preceding sections, we have outlined the basic
NMOE phenomenology and the methods used for theoretical
description of these effects. In this section, we
discuss the specific features of NMOE in a wide variety
of physical situations.
A. Buffer-gas-free uncoated vapor cells
1. Basic features
There have been numerous studies of NMOR in buffergas-free
vapor cells without antirelaxation coating (by,
for example, Barkov et al., 1989a; Budker et al., 2000a;
Kanorsky et al., 1993, and references therein). In alkaliatom
cells at room temperature, the atomic density is low
enough so that the mean free path of the atoms is orders
of magnitude larger than the cell dimensions. In addition,
because collisions with the walls of the cell destroy atomic
polarization, atoms entering the light beam are unpolarized.
In such cells, the coherence NMOR (Sec. V.B) is
due to the “transit” effect—the pumping, precession, and
subsequent probing of atoms during a single pass through
the laser beam [Bennett-structure-related NMOR (Sec.
V.A) can also be observed in these cells]. Only the region
illuminated by the light beam need be considered
in a calculation, and the relaxation rate of atomic coherence
is determined by the transit time of atoms through
the beam. Since velocity-changing collisions rarely occur,
the atoms have a constant velocity as they make a
single pass through the light beam. Thus when calculating
NMOR spectra (Sec. VII), averaging over the atomic
velocity distribution is equivalent to averaging over the
Doppler-free spectral profiles. At sufficiently low light
power, the coherence transit effect is well described by
the Kanorsky-Weis model (Sec. VII.A). At higher light
powers, where saturation effects and the alignment-toorientation
conversion effect (Sec. V.C) are important, a
density-matrix approach (Sec. VII.B) can be used to describe
both the coherence and Bennett-structure effects
in uncoated cells.
In addition to being many orders of magnitude greater
for small fields, NMOR can have a considerably different
spectrum than linear magneto-optical rotation (Sec.
II). One of the signature features of the low-light-power
NMOR transit effect is the dependence of the sign of
optical rotation on the nature of the transition (Fig.
VIII.A.1). The sign of optical rotation for F → F − 1, F
transitions is opposite to that for closed F → F + 1 transitions
(for ground states with the same Landé factors).
This is because linearly polarized light resonant with
F → F − 1, F transitions pumps atoms into an aligned
dark state (see Sec. V.B), while atoms are pumped on
F → F + 1 transitions into a bright state which in-