Experiments with Supersonic Beams as a Source of Cold Atoms

For intermediate fields, the full perturbing Hamiltonians from the magnetic
field and the spin-orbit coupling must be accounted for by diagonalizing the sum of
the individual Hamiltonians. However, for the special case where the state of interest
has maximal J, mJ resulting from the maximal possible projections of S and L,the
state transitions smoothly between low and high field with the same slope of the shift
as a function of B. This is convenient since it is desirable to target such a state for
slowing by the coilgun as this state will have the greatest possible energy shift in
the field. Thus, for sub-levels targeted by the coilgun, it is frequently unnecessary to
diagonalize a Hamiltonian to calculate the energy shifts.
4.1.3 Adiabatic Following and Spin Flips
The Hamiltonian described in the previous sections, depends on the projection
of the spin onto the axis of the magnetic field. An important consideration is what
will happen when the apparent direction of the magnetic field changes. There are
two possible outcomes of a change in the direction of the field: the projection of
the spin may adiabatically follow the field for slow enough changes in the field, or
the original quantization axis may be lost causing expectation value of the spin will
change. This last event is referred to as a spin flip, as a particle can change from
being low-field-seeking to high-field-seeking, or vice versa.
It is important to determine the limits in the rate of change for which the
projection of the spin will adiabatically follow the field. As noted above, the angular
momentum vector J in the Zeeman regime, or S and L, in the Paschen-Back regime
precess around the magnetic field. This is known as Larmor precession, and the rate
of this precession is
ωL = μatom| B|
, (4.15)
which is referred to as the Larmor frequency. The projection of the spin onto the
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