Experiments with Supersonic Beams as a Source of Cold Atoms

iments are described in detail, as are the methods used to produce and characterize
the magnetic fields produced in the coils. The concepts and experiments presented
in this chapter are also described in [68–73].
4.1 Atoms in External Magnetic Fields
In the presence of an external magnetic field, the internal energy levels of a
particle can undergo a shift, which is referred to as the Zeeman effect in low field,
and the Paschen-Back effect in high field. The description of these effects presented
here is limited to the fine structure level, though the mechanics described can easily
be extended to the hyperfine structure.
At the fine structure level, each energy level is described by the total angular
momentum J = L + S where L is the orbital angular momentum and S is the spin
angular momentum of the state. The perturbing Hamiltonian which leads to the
fine structure splitting couples L and S, making J a good quantum number. Each
state with total angular momentum J has 2J + 1 sublevels, labeled by mJ, giving
states |J, mJ〉 which are eigenvectors of the operators J 2 and Jz (assuming z is the
quantization axis) with eigenvalues
and
J 2 |J, mJ〉 = J (J +1)|J, mJ〉 , (4.1)
Jz |J, mJ〉 = mJ |J, mJ〉 . (4.2)
The sublevels of the state J are degenerate, but an external magnetic field
breaks this degeneracy due to the additional perturbing Hamiltonian
HB = −μatom · B, (4.3)
which shifts the sublevels. The magnetic moment of the atom, μatom, mustbede-
termined in order to calculate the shifts in the energy levels caused by the magnetic
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