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

Eq. 2.4 yields
J 2 = S 2 + L 2 + 2 L · S, (2.6)
allowing us to rewrite the fine structure Hamiltonian (Eq. 2.3) as
Hfs = A
2 (J2 − S 2 − L 2 ). (2.7)
Because the energy shift of this Hamiltonian is small compared to the
Bohr splittings, it can be introduced as a perturbation to those energy levels.
Time independent perturbation theory allows us to write the shift in an energy
level to first order due to this perturbation as
∆E = 〈Ψ|Hfs|Ψ〉. (2.8)
This notation introduces the question of what basis should be taken to make
the evaluation of this matrix element the simplest. The coupling of L and
S has caused Lz and Sz to be unconserved quantities so the uncoupled basis
|L,Lz,S,Sz〉 is clearly not an eigenvector of Eq. 2.7. However, from inspection
it is clear that elements of the coupled basis |S,L,J,Jz〉 are eigenvectors of the
fine structure Hamiltonian and can be used to evaluate Eq. 2.7, immediately
resulting in
∆E = A2
[J(J + 1) − S(S + 1) − L(L + 1)]. (2.9)
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The ground state configuration of 87 Rb is [Kr]5s. This means that
there is one unpaired electron and that this electron has no orbital angular
momentum, therefore S = 1/2 and L = 0. This means that the only value J
can take is J = 1/2. Evidently, the ground state is not split by Hfs because
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