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

with the perturbation H ′ (t) = −µ·E(t). The set of differential equations, eq. 2.4, specifies
to
∂cm(t)
∂t
= i
where µmn describes the dipole matrix element 〈m|µ|n〉.
µmnE(t)cn(t)e iωmnt
, (2.6)
n
Let us first consider the analytically solvable case of a two-level atom. The more
general treatment of a multi-level system will be presented in chapter 2.3. In a two-level
atom there are two coupled differential equations
and
∂cg(t)
∂t
∂ce(t)
∂t
i
= µgeE(t)ce(t)e iωat
i
= µeg E(t)cg(t)e
iωat
(2.7)
(2.8)
with the resonant absorption frequency ωa = ωe − ωg. The ground and excited states
are labeled by the indices g and e respectively. By introducing the Rabi frequency for a
two-level atom
these differential equations can be simplified to
and
Ω = E
〈e|µ|g〉, (2.9)
∂cg(t)
∂t = iΩ∗ ce(t)e −iωat
(2.10)
∂ce(t)
∂t = iΩcg(t)e iωat , (2.11)
where we have ignored the spatial dependance of the oscillating electric field.
Generally it is possible to solve these equations by transforming them into a
rotating frame. Here we will follow another ansatz [8] and substitute the coefficients
cg(t) and ce(t) with
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