  We explore photon coincidence counting statistics in the ultrastrong-coupling
regime where the atom-cavity coupling rate becomes comparable to the cavity
resonance frequency. In this regime usual normal order correlation functions
fail to describe the output photon statistics. By expressing the electric-field
operator in the cavity-emitter dressed basis we are able to propose correlation
functions that are valid for arbitrary degrees of light-matter interaction. Our
results show that the standard photon blockade scenario is significantly
modified for ultrastrong coupling. We observe parametric processes even for
two-level emitters and temporal oscillations of intensity correlation functions
at a frequency given by the ultrastrong photon emitter coupling. These effects
can be traced back to the presence of two-photon cascade decays induced by
counter-rotating interaction terms.
