  We address the quantum dynamics of second harmonic generation with a
perturbative approach. By inspecting the Taylor expansion of the unitary
evolution, we identify the subsequent application of annihilation and creation
operators as elementary processes and find out how the expansion of the
second-harmonic photon-number probability distribution can be expressed in
terms of the interplay of these processes. We show that overlaps between the
output states of different elementary processes contribute to the expansion of
the probability distribution and provide a diagrammatic technique to
analytically retrieve terms of the distribution expansion at any order.
