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

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frequency detuning from resonance, and ρ F M are the
ground state Zeeman sublevel populations. The rotation
angle is given by
ϕ = Im ( )ωl
ñ ‖ − ñ ⊥ sin 2θ, (16)
2c
where l is the sample length and θ is the average orientation
of the axis of dichroism with respect to the linear
polarization. The angle θ is calculated by time-averaging
the contributions of individual atoms, each of which precesses
in the magnetic field before relaxing at rate γ t ,
with the result
ϕ = Im ( )ωl x
ñ ‖ − ñ ⊥
2c x 2 + 1 , (17)
where the dimensionless parameter x = 2gµB/(¯hγ t ).
B. Density matrix calculations
This section describes a nonperturbative density matrix
calculation (Budker et al., 2000a, 2002a; Rochester
et al., 2001) of NMOE for a transition ξJ → ξ ′ J ′ in the
presence of nuclear spin I. Here J is total electronic angular
momentum, and ξ represents additional quantum
numbers. The calculation is carried out in the collisionfree
approximation and assumes that atoms outside the
volume of the laser beam are unpolarized. Thus the calculation
is valid for an atomic beam or a cell without antirelaxation
wall coating containing a low-density vapor.
It is also assumed that light is of a uniform intensity over
an effective area, and that the passage of atoms through
the laser beam can be described by one effective relaxation
rate γ t . The time evolution of the atomic density
matrix ρ is given by the Liouville equation (see discussion
by, for example, Stenholm, 1984):
dρ
dt = 1
i¯h [H, ρ] − 1 2 {Γ R, ρ} + Λ, (18)
where the square brackets denote the commutator and
the curly brackets the anticommutator. The total Hamiltonian
H is the sum of the light-atom interaction Hamiltonian
H L = −d · E (where E is the electric field vector,
and d is the electric dipole operator), the magnetic fieldatom
interaction Hamiltonian H B = −µ · B (where B is
the magnetic field and µ is the magnetic moment), and
the unperturbed Hamiltonian H 0 . The Hamiltonian due
to interaction with a static electric field can also be included
if necessary. Γ R is the relaxation matrix (diagonal
in the collision-free approximation)
〈ξJF M|Γ R |ξJF M〉 = γ t ,
〈ξ ′ J ′ F ′ M ′ |Γ R |ξ ′ J ′ F ′ M ′ 〉 = γ t + γ 0 .
(19)
Λ = Λ 0 +Λ repop is the pumping term, where the diagonal
matrix
〈ξJF M|Λ t |ξJF M〉 =
γ t ρ 0
(2I + 1)(2J + 1)
(20)
describes incoherent ground state pumping (ρ 0 is the
atomic density), and
∑
{ } 2
〈ξJF M 1 |Λ repop |ξJF M 2 〉 = γ 0 (2J ′ J F I
+ 1)(2F + 1)
F ′ J ′ 1
F ′
× ∑
(21)
〈J, M 1 , 1, q|J ′ , M 1〉〈J, ′ M 2 , 1, q|J ′ , M 2〉ρ ′ ξ ′ J ′ F ′ M 1 ′ ,ξ′ J ′ F ′ M ′ , 2
M 1 ′ ,M 2 ′ ,q
describes repopulation due to spontaneous relaxation
from the upper level (see, for example, discussion by
Rautian and Shalagin, 1991). Here 〈. . . | . . .〉 are the
Clebsch-Gordan coefficients and the curly brackets represent
the six-J symbol. A solution for the steady-state
density matrix can be found by applying the rotatingwave
approximation and setting dρ/dt = 0. For the
alkali-atom D lines, one can take advantage of the wellresolved
ground-state hyperfine structure by treating
transitions arising from different ground-state hyperfine
sublevels separately and then summing the results.
The electric field vector is written (see discussion by,
for example, Huard, 1997)
E = 1 [
]
E 0 e iφ (cos ϕ cos ɛ − i sin ϕ sin ɛ)e i(ωt−kz) + c.c. ˆx
2
+ 1 [
]
E 0 e iφ (sin ϕ cos ɛ + i cos ϕ sin ɛ)e i(ωt−kz) + c.c. ŷ,
2
(22)
where ω is the light frequency, k = ω/c is the vacuum
wave number, E 0 is the electric field amplitude, ϕ is the
polarization angle, ɛ is the ellipticity (arctangent of the
ratio of the major and minor axes of the polarization
ellipse), and φ is the overall phase (the field can also be
written in terms of the Stokes parameters—see Appendix