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

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Index of refraction Number of atoms
Pump
B⩵0
Probe
B≠0
0
Atomic velocity
Ren 1, Ren 1
0
Probe laser detuning
a
12, 12
b
φ ∝ Ren Ren
FIG. 11 The Bennett-structure effect on a 1/2 → 1/2 transition
in which the upper state decays to levels other than
the lower state; B pump = 0, B probe ≠ 0. (a) In the pump
region, monochromatic laser light produces Bennett holes in
the velocity distributions of atoms in the lower state |+1/2〉,
|−1/2〉 sublevels. Since there is no magnetic field, the holes
occur in the same velocity group (indicated by the dashed
line) for each sublevel. (b) In the probe region, a magnetic
field is applied, shifting n + and n − relative to each other
(upper trace). The real part of the indices of refraction are
shown; so that the features in plot (a) correspond to dispersive
shapes in this plot. The shifted central detunings of the
BSR features are indicated by the dashed lines. Polarization
rotation of the probe laser light is proportional to the difference
Re n + − n −(lower trace). Features due to the Doppler
distribution and the Bennett holes can be seen. Since the Bennett
related feature is caused by the removal of atoms from
the Doppler distribution, the sign of rotation due to this effect
is opposite to that of the linear rotation.
whether a magnetic field is present in the pump region,
and whether the excited state decays to the ground state
or other levels, BSR NMOR can be due to holes, (and
have one sign), be due to “peaks” generated by spontaneous
decay to the ground state (and have the opposite
sign), or not be present at all.
Note also that mechanical action of light on atoms
could, under certain conditions, lead to redistribution of
atoms among velocity groups, thus deforming the Bennett
structures and modifying the nonlinear optical properties
of the medium (Kazantsev et al., 1986).
B. Coherence effects
The coherence effects can produce even narrower
widths than the Bennett-structure-related NMOE, thus
leading to significantly higher small-field Faraday rotation
(Arimondo, 1996; Gawlik, 1994). At low light power,
the effect of the pump light can be conceptually separated
from that of the magnetic field, so that the coherence effect
can be thought of as occurring in three stages. First,
atoms are optically pumped into an aligned state, causing
the atomic vapor to acquire linear dichroism; second, the
atomic alignment precesses in the magnetic field, rotating
the axis of dichroism; third, the light polarization is
rotated by interaction with the dichroic atomic medium,
since the alignment is no longer along the initial light
polarization. The third, “probing,” step does not require
high light intensity, and can be performed either with a
weak probe beam or with the same pump light as used
in the first step.
Consider atoms with total angular momentum F = 1
that are not aligned initially subject to linearly polarized
laser light with frequency corresponding to a transition
to a F ′ = 0 state. One can view the atoms as being in
an incoherent mixture of the following states: |M = 0〉,
(|M = 1〉 ± |M = −1〉)/ √ 2. The first of these states can
be excited to the F ′ = 0 state only by z-polarized radiation;
it is decoupled from x- and y-polarized light. Similarly,
the other two states (which are coherent superpositions
of the Zeeman sublevels) are y- and x-absorbing
states, respectively. Suppose the laser light is polarized
along the x-axis. Optical pumping by this light causes
depletion of the x-absorbing state, leaving atoms in the
“dark” y- and z-absorbing states. 21 The medium becomes
transparent for the x-polarized radiation; however,
it can still absorb and refract light of an orthogonal polarization.
The medium is aligned 22 (Appendix B.1) and
possesses linear dichroism and birefringence. 23
In the presence of a magnetic field, the atomic alignment
axis precesses around the direction of the field at
the Larmor frequency [precession of atomic alignment in
a magnetic field is explained in detail in, for example,
Corney (1988, Ch. 15)]. One can understand the effect
of this precession on the light polarization by thinking of
the atomic medium as a layer of polarizing material like
a polaroid film (Budker et al., 1999b; Kanorsky et al.,
1993). It is easy to show that the rotation of the “polarizer”
around the z-axis causes the linear output po-
21 This process is known as coherent population trapping (Arimondo,
1996) because, as a result of optical pumping by linearly
polarized light, atoms in the |M = ±1〉 substates are not completely
pumped out as would seem to be the case at first glance
(see Fig. II.A), but largely remain in a dark coherent superposition.
22 In general, alignment can exist for an F ≥ 1 state even when
there is no dark state [i.e., for F → F + 1 transitions (Kazantsev
et al., 1984)].
23 The linear birefringence turns out not to be important for the coherence
effect because the refractive index is unity on resonance.
Note however, that at high light powers, circular birefringence
emerges as a dominant effect responsible for nonlinear Faraday
rotation (see Sec. V.C).