Experiments to Control Atom Number and Phase-Space Density in ...

z -y’
k
y z’
σ polarized
x x’
Figure 7.44: Coordinate systems at high magnetic fields. The quantization axis is determined
by the magnetic field and is along the z axis. The black coordinates are in the
lab-frame, the red primed coordinates represent the rotated frame used for calculating
the scattering cross section.
not be specified. The incident polarization is ˆǫinc = 1/ √ 2(ˆx±iˆz) in the lab frame, which
corresponds to ˆǫ ′ inc = 1/ √ 2(ˆx ′ ∓iˆy ′ ) in the rotated frame.
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The effect of the atomic cloud on the field is described by the differential equation
∂E(x ′ ,y ′ ,z ′ )
∂z ′
= −
′ ′ ′ n(x ,y ,z )σ0
2(1+δ 2
(1+iδ)(I− ˆz
)
′ ˆz ′ )·ˆσ − ˆσ −∗ ·E(x ′ ,y ′ ,z ′ ). (7.10)
I is the identity matrix, δ ≡ 2∆ is the detuning from resonance in units of half linewidth,
Γ
and ˆσ − is the left-circular unit vector. Taking a closer look at this differential equation
one important physical conclusion can be drawn immediately: only the projection of the
field onto the excitation vector ˆσ −∗ ·E(x ′ ,y ′ ,z ′ ) can excite the atom.
This differential equation can be solved by looking for eigenvector solutions for
which the incoming polarization is preserved. This leads to the following ansatz
E(x ′ ,y ′ ,z ′
−nξσ0
) = E0 exp
2(1+δ 2
(1+iδ) ˆǫ. (7.11)
)
σ0 = 3λ 2 /(2π) is the on-resonance scattering cross section of a two-level atom and
n = n(x ′ ,y ′ ) = ∞
−∞ n(x′ ,y ′ ,z ′ )dz ′ is the atomic column density. ξ has been added as
a factor to account for the difference to the effective scattering cross section σ0. ˆǫ is
the polarization unit vector of the field. Using this ansatz in equation 7.10 leads to the
following eigenvalue equation
ξˆǫ = (I− ˆz ′ ˆz ′ )·ˆσ− ˆσ ∗ − ·ˆǫ. (7.12)
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