Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

in brackets describes a Wigner-3j-symbol, which includes the selection rules for optical
transitions. It is related to the Clebsch-Gordan-coefficients via the following equation
Jk 1 Jj
−Mk q Mj
= (−1) J k−1−Mj
(2Jj + 1) 〈Jk − Mk, 1q|Jj − Mj〉. (2.31)
Further evaluation of the reduced matrix element is necessary in order to cal-
culate the dipole matrix element. The reduced matrix element in the F-basis can be
transferred into the J-basis by factoring out the F-dependance,
〈IkJkFk||r||IjJjFj〉 = (−1) 1+J k+Ij+Fj
2Fj + 1 2Fk + 1
The term in curled brackets represents a Wigner-6j-symbol.
Jk Ij Fk
Fj 1 Jj
〈Jk||r||Jj〉. (2.32)
By inserting eq. 2.30 and 2.32 into 2.28 we express the matrix element as
µ jk = e〈J kF kM k|rq|JjFjMj〉
Jk 1 Jj
= e · (−1) 1+Jk+I+Fj+J k−Mk −Mk q Mj
× 2Fj + 1
Jk I Fk 2Fk + 1 〈Jk||r||Jj〉. (2.33)
Fj 1 Jj
We can connect the reduced matrix element 〈J ′ ||r||J〉 with the Einstein-coefficients Ajk
for spontaneous emission, respectively the inverse of the partial lifetime [10],
A kj = 1
τ kj
=
ω 3
kj
3πɛ0c 3
1
2J k + 1 |〈J k||er||Jj〉| 2 . (2.34)
Hence the dynamics of the system are included in the Einstein-coefficients, while the
geometrical properties are represented by the Wigner-symbols. By applying eq. 2.34
and eq. 2.33 we get the following result for the energy shift [13],
11