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

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effects (e.g, quantum beats), excited state level crossings
and the Hanle effect (Sec. II.D.1). Generally, even-order
terms of ρ are stationary or slowly oscillating, while oddorder
terms oscillate with the frequencies in the optical
range. In particular, the next order ρ (3)
ab
is the lowest
order in which a nonlinear polarization at the optical frequency
appears. It is responsible for the onset of the
nonlinear (saturated) absorption and dispersion effects,
such as the nonlinear Faraday and Voigt effects. The appearance
of ρ (4) marks the onset of nonlinear (saturated)
fluorescence and the nonlinear Hanle effect. In terms of
quantum interference (Sec. II.D.1), these effects can be
associated with opening of additional interference channels
(in analogy to a multi-slit Young’s experiment).
The Liouville equation shows how subsequent orders
of ρ are coupled (see discussion by Boyd, 1992):
˙ρ (n)
ab
= −(iω ab + γ ab )ρ (n)
ab
− ī h
[
−d · E, ρ (n−1)] ab, (9)
where ¯hω ab is the energy difference between levels a and
b. The density matrix ρ to a given order ρ (n) can thus
be expressed in terms of the elements ρ (n−1) of the next
lowest order. Odd orders of ρ describe optical coherences
and even orders describe level populations and Zeeman
coherences [according to Eq. (9) the latter are the source
terms for the former and vice-versa]. The magnitude of
the linear absorption coefficient, governed by ρ (1)
ab
, can be
expressed in terms of the populations ρ (0)
aa . Similarly, ρ (3)
ab ,
which determines the lowest-order saturated absorption
and the lowest-order nonlinear Faraday effect, is driven
by ρ (2)
aa and ρ (2)
bb
(excited- and ground-state populations),
ρ (2)
bb
(excited-state coherences) and ρ (2)
′ aa ′ (ground-state coherences).
In the language of nonlinear optics, the induced polarization
can be expanded into powers of E:
∞∑
∞∑
P = P (n) = χ (n)· E n , (10)
n=1
n=1
where the χ (n) are the n-th order electric susceptibilities.
Expression (10) is a particular case of the general expansion
used in nonlinear optics in which n optical fields E i
oscillating at frequencies ω i generate n-th order polarizations
oscillating at all possible combinations of ω i . In the
present article, we focus on the case of a single optical
field oscillating at frequency ω. Here the only polarizations
oscillating at the same frequency are of odd orders
in P . Coherent forward scattering is governed by P and is
thus directly sensitive only to odd orders of ρ or χ. Even
orders can be directly probed by observing (incoherently
emitted) fluorescence light. Linear dispersion and absorption
effects such as the linear Faraday effect are χ (1)
processes, while the nonlinear Faraday effect arises first
as a χ (3) process.
In forward-scattering experiments, the absorbed power
is proportional to 〈Ṗ · E〉, which is given in (odd) order
l by (χ (l) · E l ) · E ∝ χ (l) I (l+1)/2 , where I is the light
intensity. 13 Forward-scattering in lowest order (l = 1)
involves χ (1) and is a linear process in the sense that the
experimental signal is proportional to I. In light-induced
fluorescence experiments, the scattered power is proportional
to the excited-state population, which, to lowest
order [ρ (2)
bb
] is also proportional to I. We adopt the following
classification (which is consistent with the commonly
used notions): Processes which involve ρ (1) and ρ (2) will
be called linear processes, while all higher-order processes
will be referred to as nonlinear processes. 14
This perturbative approach is useful for understanding
the onset of a given effect with increasing light power.
It works well in most cases involving nonresonant light
fields. In the case of resonant laser excitation, however,
caution must be used, since there is no way to
make meaningful distinctions between different perturbative
orders once the transition is saturated [the series
(9) and (10) are non-converging and all orders of these
expansions are coupled.]
This discussion is generalized in a straightforward
manner to the case of pump-probe experiments (i.e., in
which separate light beams are used for pumping and
probing the medium). Here the signals that depend on
both beams are, to lowest order, proportional to the
product of the intensities of these beams and thus correspond
to a χ (3) nonlinear process.
B. Saturation parameters
Nonlinear optical processes are usually associated with
high light intensities. However, nonlinear effects can
show up even with weak illumination—conventional spectral
lamps were used in early optical pumping experiments
with resonant vapors to demonstrate nonlinear
effects (Sec. IV.A). It is important to realize that the
degree of nonlinearity strongly depends on the specific
mechanism of atomic saturation by the light field as well
as the relaxation rate of atomic polarization.
A standard discussion of optical nonlinearity and saturation
for a two-level atom is given by Allen and Eberly
(1987). The degree of saturation of a transition is fre-
13 This follows directly from the Maxwell’s equations in a medium
(see discussion by, for example, Shen, 1984).
14 The formalism of nonlinear wave mixing (see discussion by Boyd,
1992; Shen, 1984) allows one to describe linear electro- and
magneto-optical effects as three- or four-wave-mixing processes,
in which at least one of the mixing waves is at zero frequency.
For example, linear Faraday rotation is seen in this picture as
mixing of the incident linearly polarized optical field with a zerofrequency
magnetic field to produce an optical field of orthogonal
polarization. While this is a perfectly consistent view, we prefer
to adhere to a more common practice of calling these processes
linear. In fact, for the purpose of the present review, the term
nonlinear will only refer to the dependence of the process on
optical fields. Thus, we also consider as linear a broad class of
optical-rf and optical-microwave double-resonance phenomena.