  Third-order parametric down-conversion (TOPDC) describes a class of nonlinear
interactions in which a pump photon is converted into a photon triplet. This
process can occur spontaneously, or it can be stimulated by seeding fields. In
the former case, one typically has the generation of non-Gaussian states of
light. In the latter, the situation is more variegated, for stimulated TOPDC
(StTOPDC) can be implemented in many ways, depending on the number and
properties of the seeding fields. Here we show that StTOPDC can be exploited
for the generation of quantum correlated photon pairs. We examine the peculiar
features of this approach when compared with second-order spontaneous
parametric down-conversion and spontaneous four-wave mixing. We model StTOPDC
in a microring resonator, predicting observable generation rates in a microring
engineered for third-harmonic generation. We conclude that if the experimental
difficulties associated with implementing StTOPDC can be overcome, it may soon
be possible to demonstrate this process in resonant integrated devices.
