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

The Wigner-Eckhart theorem can be used to determine the relative strength of
each of the allowed transition by reducing the transition matrix element [26],
〈F,mF|µq|F ′ ,mF ′〉 = 〈F||µ||F′ 〉(−1) F′
√ ′
−1+mF F
2F +1
1 F
, (2.10)
mF ′ q −mF ′
where |F,mF〉 (|F ′ ,mF ′〉) are the ground (excited) state hyperfine quantum numbers,
µq is the component of the spherical electric dipole operator for polarization q and the
quantity in parentheses is known as the Wigner 3-j symbol. Applying the Wigner-
Eckhart theorem again, the transition matrix element reduces even further to
〈F||µ||F ′ 〉 = 〈J||µ||J ′ 〉(−1) F′ +J+1+I (2F ′ +1)(2J +1)
with the use of the Wigner 6-j symbol.
J J ′ 1
F ′ F I
, (2.11)
The matrix element 〈J||µ||J ′ 〉 can be determined experimentally by measuring
the transition frequency and the lifetime of a given transition, where
1
τ
= ω3
3πǫ0c 3
Values of 〈J||µ||J ′ 〉 for 87 Rb and 6 Li can be found in [17, 20].
5 2 P 3/2
5 2 S 1/2
2J +1
2J ′ +1 |〈J||µ||J′ 〉| 2 . (2.12)
m -1 0 1
F F’=1
m F
m F
.02
.42
.05
.42
.10
-2 -1 0 1 2
Figure 2.6: Branching ratios for the 87 Rb transitions for the implementation of singlephoton
cooling.
Figure 2.6 shows the relevant branching ratios for the transition for the imple-
mentation of single-photon cooling in rubidium.
10
F=2
F=1