  Recent experiments have demonstrated an open system realization of the Dicke
quantum phase transition in the motional degrees of freedom of an optically
driven Bose-Einstein condensate in a cavity. Relevant collective excitations of
this light-matter system are polaritonic in nature, allowing access to the
quantum critical behavior of the Dicke model through light leaking out of the
cavity. This opens the path to using photodetection based quantum optical
techniques to study the dynamics and excitations of this elementary quantum
critical system. We first discuss the photon flux observed at the cavity face
and find that it displays a different scaling law near criticality than that
obtained from the mean field theory for the equivalent closed system. Next, we
study the second order correlation measurements of photons leaking out of the
cavity. Finally, we discuss a modulation technique that directly captures the
softening of polaritonic excitations. Our analysis takes into account the
effect of the finite size of the system which may result in an effective
symmetry breaking term.
