This interaction Hamiltonian contains operators to create and annihilate photons
with transitions between atomic states.
previous study of time dependent perturbation theory,
we know that transitions between initial and final states are proportional to the
matrix element of the perturbing Hamiltonian between the states,
The initial state
should include a direct product of the atomic state and the photon state.
Lets concentrate on one type of photon for now.
We then could write
We will first consider the absorption of one photon from the field. Assume there are photons of this type in the initial state and that one photon is absorbed. We therefore will need a term in the interaction Hamiltonian that contains on annihilation operator (only). This will just come from the linear term in A.
Jim Branson 2013-04-22