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

The energy shift for the level m is then seen to be
and for the ground state
∆Em = 〈 ˙φ〉
|E0| 2
= −
|E0| 2
∆Eg = −
4
k
k
|µkg| 2
|µkm| 2
1
ωkg + ωl
1
ωkm + ωl
−
−
1
ωkg − ωl
1
ωkm − ωl
(2.25)
. (2.26)
A comparison with the classical result ∆Eg = − α(ω)
2 |E0| 2 gives the quantum-mechanical
description for the polarizability α(ω) [11]
α(ω) = − 1
2
k
|µ kg| 2
1
ωkg + ωl
1
− . (2.27)
ωkg − ωl In fig. 2.2 we show the AC-Stark shift for the Sodium ground state and the
3 2 P3/2(F = 0) excited state as an example. The poles appear due to resonant transitions.
2.4 Dipole matrix elements, Einstein-coefficients and the AC-
Stark shift
In order to calculate the AC-Stark shift it is necessary to evaluate the dipole matrix
elements
µ jk = 〈j|µ|k〉 = 〈j|erq|k〉. (2.28)
Here e denotes the charge of an electron, r the electron coordinate and q the polarization
of the incident electric field.
Since in alkali atoms hyperfine splittings play an important role, the eigenstates
of the atoms have to be expressed in the F-basis. In the following section we will label
the states |k〉 by the quantum numbers J k and F k for the angular momenta and the
9