  The dynamics of coupled condensates is a wide-encompassing problem with
relevance to superconductors, BECs in traps, superfluids, etc. Here, we provide
a unified picture of this fundamental problem that includes i) detuning of the
free energies, ii) different self-interaction strengths and iii) finite
lifetime of the modes. At such, this is particularly relevant for the dynamics
of polaritons, both for their internal dynamics between their light and matter
constituents, as well as for the more conventional dynamics of two spatially
separated condensates. Polaritons are short-lived, interact only through their
material fraction and are easily detuned. At such, they bring several
variations to their atomic counterpart. We show that the combination of these
parameters results in important twists to the phenomenology of the Josephson
effect, such as the behaviour of the relative phase (running or oscillating) or
the occurrence of self-trapping. We undertake a comprehensive stability
analysis of the fixed points on a normalized Bloch sphere, that allows us to
provide a generalized criterion to identify the Rabi and Josephson regimes in
presence of detuning and decay.
