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

to vary linearly with position along ˆz. The counter-propagating beams are
tuned below the atomic resonance by an amount ∆ and have opposite circular
polarizations as shown in the figure. To see how this leads to a trap based
on imbalanced scattering rates consider an atom displaced from the center
of the trap to a position z = z ′ . At this location the magnetic field causes
the Zeeman sublevel mJ = −1 to be brought closer into resonance with the
atomic transition while bringing the mJ = 1 sublevel further from resonance.
The detuning of each of these states from resonance is indicated by ∆− and
∆+ respectively, with ∆+ > ∆−. Transition selection rules then indicate that
the atom will scatter at a greater rate from the σ − beam, pushing it back
towards the center. If the atom is displaced in the opposite direction then
the Zeeman shift in the sublevels will cause it to scatter preferentially out of
the σ + beam, again pushing back towards the center. Of course with the 3-D
geometry shown in Fig. 2.10 any displacement from the trap center will lead
to a restoring force. Additionally, the red detuning of the beams causes the
atoms to cool according to the discussion of optical molasses, therefore MOTs
both confine and cool atomic ensembles.
We can describe the simple 1-D situation more quantitatively by in-
cluding the Zeeman shift into Eq. 2.78, which was used to describe optical
molasses
FMOT = F σ+
scatt[ω − kv − (ω0 + βz)] + F σ−
scatt[ω + kv − (ω0 + βz)], (2.84)
where β is the magnetic field gradient. This can be approximated near the
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