  Triplet down conversion, the process of converting one high-energy photon
into three low-energy photons, may soon be experimentally feasible due to
advances in optical resonator technology. We use quantum phase-space techniques
to analyse the process of degenerate intracavity triplet down conversion by
solving stochastic differential equations within the truncated positive-P
representation. The time evolution of both intracavity mode populations are
simulated, and the resulting steady-states are examined as a function of the
pump intensity. Quantum effects are most pronounced in the region immediately
above the semi-classical pumping threshold, where our numerical results differ
significantly from semi-classical predictions. Regimes of measurable squeezing
and bipartite entanglement are identified from steady-state spectra of the
cavity output fields. We validate the truncated positive-P description against
Monte Carlo wave function simulations, finding good agreement for low mode
populations.
