  Single-photon sources are at the heart of quantum-optical networks, with
their uniquely quantum emission and phenomenon of two-photon interference
allowing for the generation and transfer of nonclassical states. Although a few
analytical methods have been briefly investigated for describing pulsed
single-photon sources, these methods apply only to either perfectly ideal or at
least extremely idealized sources. Here, we present the first complete picture
of pulsed single-photon sources by elaborating how to numerically and fully
characterize non-ideal single-photon sources operating in a pulsed regime. In
order to achieve this result, we make the connection between quantum
Monte--Carlo simulations, experimental characterizations, and an extended form
of the quantum regression theorem. We elaborate on how an ideal pulsed
single-photon source is connected to its photocount distribution and its
measured degree of second- and first-order optical coherence. By doing so, we
provide a description of the relationship between instantaneous source
correlations and the typical experimental interferometers (Hanbury-Brown and
Twiss, Hong-Ou-Mandel, and Mach-Zehnder) used to characterize such sources.
Then, we use these techniques to explore several prototypical quantum systems
and their non-ideal behaviors. As an example numerical result, we show that for
the most popular single-photon source---a resonantly excited two-level
system---its error probability is directly related to its excitation pulse
length. We believe that the intuition gained from these representative systems
and characters can be used to interpret future results with more complicated
source Hamiltonians and behaviors. Finally, we have thoroughly documented our
simulation methods with contributions to the Quantum Optics Toolbox in Python
(QuTiP) in order to make our work easily accessible to other scientists and
engineers.
