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

eigenfunctions |n〉 with the corresponding energy eigenvalues En = ωn,
H0|n〉 = En|n〉. We will write the time-dependant wavefunction |Ψ(r, t)〉 as superposition
of eigenstates |n(r)〉,
|Ψ(r, t)〉 = cn(t)e −iωnt
|n(r)〉 (2.2)
n
with the time-dependant coefficients cn(t). The Schrödinger equation thus becomes
H(t)|Ψ(r, t)〉 = (H0 + H ′
(t)) cn(t)e −iωnt
|n(r)〉
= i ∂
∂t
n
cn(t)e −iωnt
|n(r)〉 (2.3)
n
Eq. 2.3 can be further evaluated by multiplication from the left with 〈m| and integrating
over spatial coordinates. This leads to a set of differential equations for the coefficients
cn(t),
i dcm(t)
dt =
where H ′ mn = 〈m|H ′ (t)|n〉 and ωmn = ωm − ωn.
cn(t)H ′ mn(t)e iωmnt
, (2.4)
n
2.2 Interaction of a two-level atom with non-resonant light
Let us consider the case of a periodic perturbation with frequency ω, more precisely
an oscillating electric field described by E(r, t) = 1
2 E0e i( k·r−ω lt) + c.c.. In most cases the
spatial dependance of the electric field is negligible when considering the interaction
of atoms with light, since the extend of the electric field is on the order of λ (a few
hundred nanometers) while the atoms are several orders of magnitude smaller (a few
Ångström). The formalism presented here is for non-resonant light; the resonant case
has to be treated seperately.
We will apply the common dipole approximation for radiative transitions of the
atoms [7]. Thus the Hamiltonian becomes
H(t) = H0 − µ · E(t), (2.5)
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