  The intention of this work is twofold, first to present a most simple system
capable of simulating the intrinsic bosonic Josephson effect with photons, and
second to study various outcomes deriving from inherent or external
decoherence. A qubit induces an effective coupling between two externally
pumped cavity modes. Without cavity losses and in the dispersive regime,
intrinsic Josephson oscillations of photons between the two modes occurs. In
this case, contrary to regular Markovian decoherence, the qubit purity shows a
Gaussian decay and recurrence of its coherence. Due to intrinsic
non-linearities, both the Josephson oscillations as well as the qubit
properties display a rich collapse-revival structure, where, however, the
complexity of the qubit evolution is in some sense stronger. The qubit as a
meter of the photon dynamics is considered, and it is shown that qubit
dephasing, originating for example from non-demolition measurements, results in
an exponential destruction of the oscillations which manifests the
collectiveness of the Josephson effect. Non-selective qubit measurements, on
the other hand, render a Zeno effect seen in a slowing down of the Josephson
oscillations. Contrary to dephasing, cavity dissipation results in a Gaussian
decay of the scaled Josephson oscillations. Finally, following Ponomarev et al.
[Phys. Rev. Lett. 106, 010405 (2011)] we analyze aspects of thermalization. In
particular, despite similarities with the generic model studied by Ponomarev
{\it et al.}, our system does not seem to thermalize.
