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

the latter of which only applies to the excited manifold of the D2 transition
and not to levels with J = 1/2.
This Hamiltonian can be solved in a manner analogous to that used in
the case of fine structure splitting. We note that by squaring both sides of
Eq. 2.10 one can write,
J · I = 1
2 (F 2 − J 2 − I 2 ). (2.13)
This can be used in Eq. 2.12 to solve for the shifts in energy levels due to this
interaction. Again we note that the coupling of I and J causes the uncoupled
basis |I,Iz,J,Jz〉 to no longer be an eigenvector of the Hamiltonian describing
the atom. But as before, simple inspection revels that |I,J,F,mF 〉 is an eigen-
vector of Eq. 2.12. Taking this as the basis in evaluating the perturbation due
to the hyperfine Hamiltonian results in energy splittings given by
where
∆Ehfs = 1
2 AhfsK + Bhfs
3
2
K(K + 1) − 2I(I + 1)J(J + 1)
, (2.14)
4I(2I − 1)J(2J − 1)
K = F(F + 1) − I(I + 1) − J(J + 1) (2.15)
is introduced for notational convenience.
The nuclear moment constants for the 87 Rb D2 line are given in Ta-
ble 2.2. The ground state value was taken from a precise atomic fountain
measurement [48], while the excited state values were measured using a het-
erodyne technique between two ultra stable lasers referenced to atomic 87 Rb
[49].
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