Experiments to Control Atom Number and Phase-Space Density in ...

of the two angular momenta can be expressed by the total electron angular momentum
J = L+ S. Noting that
the fine structure Hamiltonian (2.4) can be rewritten as
J 2 = S 2 +L 2 +2 L· S (2.5)
Hfs = A
2 (J2 −L 2 −S 2 ). (2.6)
The energy-splitting due to the fine structure can be treated as a perturbation to
the central potential approximation. The difference in the energy levels is hence given
by
∆E = A
[J(J +1)−S(S +1)−L(L+1)]. (2.7)
2
The resulting energy level structure and configurations for the cases of 87Rb and 6Li are
shown in figure 2.3. The coupling of the angular momenta leads to J = 1/2 for the
ground state and J = 1/2 or J = 3/2 for the excited state.
Central potential
approximation
2 P (L=1)
2 S (L=0)
2 P3/2 (J=3/2)
2 P1/2 (J=1/2)
2 S1/2 (J=1/2)
Fine Structure
7.123 THz
D line D 1 line D 2 line
87 Rubidium
Central potential
approximation
2 P (L=1)
2 S (L=0)
2 P3/2 (J=3/2)
2 P1/2 (J=1/2)
2 S1/2 (J=1/2)
Fine Structure
10.053 GHz
D line D 1 line D 2 line
794.978 nm 780.941 nm 670.992nm 670.977 nm
6 Lithium
Figure 2.3: Ground and first excited states of 87 Rb and 6 Li in the central potential and
fine structure approximations. Energy splittings are not to scale.
The D-line transition is split into two separate lines, called the D1 and D2line.
The D2-line transition is the transition relevant to the experiments described in this
dissertation.
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