  Spontaneous parametric down conversion (SPDC) has been one of the foremost
tools in quantum optics for over five decades. Over that time it has been used
to demonstrate some of the curious features that arise from quantum mechanics.
Despite the success of SPDC, its higher-order analogues have never been
observed, even though it has been suggested that they generate far more unique
and exotic states than SPDC. An example of this is the emergence of
non-Gaussian states without the need for post-selection. Here we calculate the
expected rate of emission for nth-order SPDC with and without external
stimulation (seeding). Focusing primarily on third-order parametric
down-conversion (TOPDC), we estimate the photon detection rates in a rutile
crystal, for both the unseeded and seeded regimes.
