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

Figure 2.2: Plot of the AC-Stark shift. Numerical values are for a beam with P=1W,
w0=10µm and π-polarized light; the hyperfine structure belongs to the 3 2 P3/2 state. The
potential is attractive for ∆E < 0 and repulsive for ∆E > 0.
magnetic quantum number M k. The dipole moment therefore reads
µ jk = e〈J kF kM k|rq|JjFjMj〉. (2.29)
In general, the value of the dipole matrix element will depend on the polarization of
the incoming beam; q = ±1 corresponds to circular (σ ± ) polarised light, for linear (π)
polarised light q equals 0.
The dipole matrix elements can be transformed into reduced matrix elements
by applying the Wigner-Eckart Theorem [12]
〈JkFkMk|rq|JjFjMj〉 = (−1) J k−M k
Jk 1 Jj
−M k q Mj
〈JkFk||r||JjFj〉. (2.30)
On the right-hand side of eq. 2.30 we succeeded in separating the geometry and
symmetry from the dynamics of the system using a reduced matrix element. The term
10