  When a mesoscopic conductor is coupled to a high-quality electromagnetic
cavity the flow of charges and the flux of photons leaking out of the cavity
can both depend strongly on the coupled quantum dynamics of the system. Using a
voltage-biased Josephson junction as a model system, we demonstrate that there
is a simple connection between the full counting statistics of the charges and
the photons in the long time limit. We then reveal the intimate relation
between the counting statistics and the nonlinear dynamics of the system,
uncovering novel regimes of coherent charge and photon transport associated
with bifurcations in the classical dynamics of the system.
