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

3
@Ω
Σ Σ
n
n
n
gΜB
@
M = −1 M = 0 M = +1
gΜB
FIG. 2 An F = 1 → F ′ = 0 atomic transition. In the presence
of a longitudinal magnetic field, the Zeeman sublevels
of the ground state are shifted in energy by gµBM. This
leads to a difference in resonance frequencies for left- (σ + )
and right- (σ − ) circularly polarized light.
Refractive index
1
0
n n
10 20 greater rotation “per atom” than heavy flint.
In this paper, we briefly review the physics and applications
of resonant linear magneto-optical effects, and then
turn to our main focus—the nonlinear effects (i.e., effects
in which optical properties of the medium are modified
by interaction with light). We will also discuss various
applications of nonlinear magneto-optics in atomic vapors,
and the relation between nonlinear magneto-optics
and a variety of other phenomena and techniques, such
as coherent population trapping, electromagnetically induced
transparency, nonlinear electro-optics effects, and
self-rotation.
II. LINEAR MAGNETO-OPTICS
In order to provide essential background for understanding
nonlinear magneto-optical effects (NMOE), we
first review linear magneto-optics of atoms and molecules
near resonance. (See Sec. III for a discussion of the difference
between the linear and nonlinear magneto-optical
effects.)
A. Mechanisms of the linear magneto-optical effects
At the conclusion of the 19th century, Macaluso and
Corbino (1898a, 1899, 1898b), studying absorption spectra
of the alkali atoms in the presence of magnetic fields,
discovered that the Faraday effect (magneto-optical activity)
in the vicinity of resonance absorption lines also
has a distinct resonant character (see also work by Fork
and Bradley III, 1964).
The principal mechanism of the linear Macaluso-
Corbino effect can be illustrated by the case of an F =
1 → F ′ = 0 transition (Fig. II.A), where F, F ′ are the
total angular momenta. 2 Linearly polarized light inci-
8 4 0 4 8
Light detuning (
FIG. 3 The dependence of the refractive index on light frequency
detuning ∆ in the absence (n) and in the presence
(n ± ) of a magnetic field. Shown is the case of 2gµB = ¯hΓ
and a Lorentzian model for line broadening. The lower curve
shows the difference of refractive indices for the two circular
polarization components. This is the characteristic spectral
profile of linear optical rotation.
dent on the sample can be decomposed into two counterrotating
circular components σ ± . In the absence of a
magnetic field, the M = ±1 sublevels are degenerate and
the optical resonance frequencies for σ + and σ − coincide.
The real part of the refractive index n associated with the
atomic medium is shown in Fig. II.A as a function of the
light frequency detuning ∆ (the dispersion curve). The
refractive index is the same for the two circular components.
When a magnetic field is applied, however, the Zeeman
shifts 3 lead to a difference between the resonance frequencies
for the two circular polarizations. This displaces
the dispersion curves for the two polarizations as shown
in Fig. II.A. A characteristic width of these dispersion
curves, Γ, corresponds to the spectral width (FWHM)
of an absorption line. Under typical experimental conditions
in a vapor cell this width is dominated by the
Doppler width and is on the order of 1 GHz for optical
transitions. The difference between n + and n − (Fig.
II.A) signifies a difference in phase velocities of the two
circular components of light and, as a result, the plane
of polarization rotates through an angle
ϕ = π(n + − n − ) l λ . (1)
Here l is the length of the sample, and λ is the wavelength
of light. In addition to the difference in refraction for the
two circular polarizations (circular birefringence), there
2 Throughout this article, we designate as F, F ′ total angular momenta
of the lower and the upper states of the transition, respectively.
For atoms with zero nuclear spin, F, F ′ coincide with the
total electronic angular momenta J, J ′ .
3 The connection between the Faraday and the Zeeman effects was
first established by Voigt (1898b), who also explained the observations
of Macaluso and Corbino (Voigt, 1898a).