  We study the full field and frequency filtered output photon statistics of a
resonator in thermal equilibrium with a bath and containing an arbitrarily
large quartic nonlinearity. According to the general theory of photodetection,
we derive general input-output relations valid for the ultra-anharmonic regime,
where the nonlinearity becomes comparable to the energy of the resonator, and
show how the emission properties are modified as compared to the generally
assumed simple anharmonic regime. We analyse the impact of the nonlinearity on
the full statistics of the emission and its spectral properties. In particular
we derive a semi-analytical expression for the frequency resolved two-photon
correlations or two-photon spectrum of the system in terms of the master
equation coefficients and density matrix. This provides a very clear insight
into the level structure and emission possibilities of the system.
