Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

5.4.1 Multiple initial and final states
Each hyperfine level splits into (2F+1) magnetic sublevels. We therefore have to deal
with three inital levels (F=1) and five final levels (F=2). In an external magnetic field
the levels are split as shown in fig. 5.5, providing the possibility for a spatially resolved
measurement. If this splitting is larger than the Raman transition linewidth, only one
transition will be resonant. It is then possible to treat each transition as a separate
two-level Raman process. The shift due to an external magnetic field is described in
more detail in appendix B.2.
In the result for the final state probability we have to take into account the
fact that the atoms are in different magnetic sublevels, where we can assume an equal
distribution in each of these if there is no condition favorable for one above another. In
addition to the detuning δ due to the difference frequency of the two laser beams another
detuning δB(r) arises due to the magnetic field splitting of the magnetic hyperfine states.
i
This detuning is therefore dependant on the magnetic state of an atom. Including the
detuning due to the magnetic field splitting the final state probability develops to
P f (t → ∞, δ) = 1
3
sin
where we have to sum over all initial states.
5.4.2 Multiple intermediate states
i
2 |βi(r)|τ
2
(δ − δB
2
sinc
i (r))τ
, (5.31)
2
In addition to multiple initial and final states we also have to take multiple intermediate
levels into account. We have argued that we can treat multiple initial and final states
separately. However, things are not quite as easy if one considers different intermediate
levels. A treatment similar to the one in section 5.3 can be done by including additional
intermediate levels. Adiabatic elimination of these levels results in an effective Raman-
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