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

orem can be used to find all of the others which differ only in the values of
m,m ′ and q.
We can use the Wigner-Eckhart theorem to factor the electric dipole
transition matrix element between two hyperfine states into two terms.
〈F,mF |µq|F ′ ,m ′ F 〉 = 〈F ||µ||F ′ 〉〈F,mF |F ′ , 1,m ′ F,q〉 (2.91)
This can be rewritten using the more symmetrical Wigner 3-j symbols which
are related to the Clebsch-Gordan coefficients in the following way [75–77]
〈F,mF |F ′ , 1,m ′ F,q〉 = (−1) F ′
√ ′
−1+mF F
2F + 1
1 F
(2.92)
as
m ′ F q −mF
〈F,mF |µq|F ′ ,m ′ F 〉 = 〈F ||µ||F ′ 〉(−1) F ′
√ ′
−1+mF F 1 F
2F + 1
m ′ F q −mF
.
(2.93)
This reduced matrix element can be further reduced by using the Wigner-
Eckart theorem once more to factor out the F and F ′ dependence into a Wigner
6−j symbol.
〈F ||µ||F ′ 〉 = 〈J||µ||J ′ 〉(−1) F ′ +J+1+I (2F ′ + 1)(2J + 1)
J
F
′ J 1
′ F
.
I
(2.94)
The values of the 3-j and 6−j symbols are tabulated in [78] and can also be
found from the built in functions ThreeJSymbol[] and SixJSymbol[] in the
program Mathematica.
These formulas can be used to find the spontaneous decay branching
ratios from the excited state |F ′ = 1,mF = 1〉 which are shown in Fig. 2.12. In
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