Single-Photon Atomic Cooling - Raizen Lab - The University of ...

trap center through a Taylor expansion yielding
where the Zeeman shift at a displacement z is
FMOT ≈ −2 ∂F ∂F
kv + 2 βz (2.85)
∂ω ∂ω0
βz = gµB
dB
z. (2.86)
dz
As before, the scattering force (Eq. 2.77) depends on the frequency detuning
∆ = ω − ω0, so ∂F/∂ω0 = −∂F/∂ω so that these two terms can be combined
and written as
FMOT = −2 ∂F
(kv + βz). (2.87)
∂ω
This can be brought into a particularly simple form by introducing the variable
α originally defined in the discussion of optical molasses
FMOT = −αv − αβ
z. (2.88)
k
This form emphasizes that the imbalance in the scattering rates caused
by the Zeeman shift in energy levels leads to a restoring force with a spring
constant αβ/k. In typical experimental situations the atom undergoes strongly
over-damped motion.
2.6 Branching Ratios
During the course of the single-photon cooling process 87 Rb atoms are
excited from the |F = 2,mF = 2〉 state into the |F ′ = 1,mF = 1〉 state
from where they spontaneously decay. Electric dipole selection rules, ∆F =
68