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

2.3 Interaction of a multi-level system with non-resonant light
Although there are many cases where the two-level approximation is justified, we
will now consider the more general case of a multi-level atom by going back to the
Schrödinger-equation (eq.2.1) and using time-dependant perturbation theory to solve
the differential equation.
cm(t),
In the second order expansion we get the following expression for the coefficients
cm(t) − cm(0) = − 1
2 t
dt
0
′
t ′
dt
0
′′
k
µnk E(r, t ′ )µkm E(r, t ′′ )e iω nkt ′
e iω kmt ′′
, (2.21)
where the sum has to be evaluated over all possible states. For further analysis we insert
the complex electric field E(t) = 1
2 E0e iω lt + c.c., where we neglect the spatial dependance
again. Thus we obtain
cm(t) − cm(0) = − 1
42 t
dt
0
′
t ′
dt
0
′′
k
µnkE0(t ′ ) e iωlt ′
+ e −iωlt ′
×µkmE0(t ′′ ) e iωlt ′′
+ e −iωlt ′′ e iωnkt ′
e iωkmt ′′
, (2.22)
where we have avoided the Rotating Wave Approximation.
Assuming the atom to be in state |m〉 we can rewrite the coefficients as complex
phase e iφ(t) . The equation above then reads
1 · e iφ(t) − const. = − 1
42 t
dt
0
′
t ′
dt
0
′′
k
µ nkE0(t ′ ) e iωlt ′
+ e −iωlt ′
×µkmE0(t ′′ ) e iωlt ′′
+ e −iωlt ′′ e iωnkt ′
e iωkmt ′′
. (2.23)
By differentiating and averaging over one period this becomes
〈 ˙φ〉
|E0| 2
= −
42
k
|µkm| 2
1
ωkm + ωl 8
1
− . (2.24)
ωkm − ωl