  We provide a complete study of the phase diagram characterising the distinct
dynamical regimes emerging in a three-dimensional Josephson junction in an
ultracold quantum gas. Considering trapped ultracold superfluids separated into
two reservoirs by a barrier of variable height and width, we analyse the
population imbalance dynamics following a variable initial population mismatch.
We demonstrate that as the chemical potential difference is increased, the
system transitions from Josephson plasma oscillations to either a dissipative
(in the limit of low and narrow barriers) or a self-trapped regime (for large
and wider barriers), with a crossover between the dissipative and the
self-trapping regimes which we explore and characterize for the first time.
This work, which extends beyond the validity of the standard two-mode model,
connects the role of the barrier width, vortex rings and associated acoustic
emission with different regimes of the superfluid dynamics across the junction,
establishing a framework for its experimental observation, which is found to be
within current experimental reach.
