  Two-photon quantum interference at a beam splitter, commonly known as
Hong-Ou-Mandel interference, was recently demonstrated with
\emph{microwave-frequency} photons by Lang \emph{et
al.}\,\cite{lang:microwaveHOM}. This experiment employed circuit QED systems as
sources of microwave photons, and was based on the measurement of second-order
cross-correlation and auto-correlation functions of the microwave fields at the
outputs of the beam splitter. Here we present the calculation of these
correlation functions for the cases of inputs corresponding to: (i) trains of
\emph{pulsed} Gaussian or Lorentzian single microwave photons, and (ii)
resonant fluorescent microwave fields from \emph{continuously-driven} circuit
QED systems. The calculations include the effects of the finite bandwidth of
the detection scheme. In both cases, the signature of two-photon quantum
interference is a suppression of the second-order cross-correlation function
for small delays. The experiment described in Ref.
\onlinecite{lang:microwaveHOM} was performed with trains of \emph{Lorentzian}
single photons, and very good agreement between the calculations and the
experimental data was obtained.
