  We study ground-state properties and dynamics of a dilute ultracold atomic
gas in a double well potential. The Gaussian barrier separating the two wells
derives from the interaction between the atoms and a quantized field of a
driven Fabry-Perot cavity. Due to intrinsic atom-field nonlinearity, several
novel phenomena arise being the focus of this work. For the ground state, there
is a critical pumping amplitude in which the atoms self-organize and the intra
cavity field amplitude drastically increases. In the dynamical analysis, we
show that the Josephson oscillations depend strongly on the atomic density and
may be greatly suppressed within certain regimes, reminiscent of self-trapping
of Bose-Einstein condensates in double-well setups. This pseudo self-trapping
effect is studied within a mean-field treatment valid for large atom numbers.
For small numbers of atoms, we consider the analogous many-body problem and
demonstrate a collapse-revival structure in the Josephson oscillations.
