  We examine the role of interactions for a Bose gas trapped in a double-well
potential ("Bose-Josephson junction") when external noise is applied and the
system is initially delocalized with equal probability amplitudes in both
sites. The noise may have two kinds of effects: loss of atoms from the trap,
and random shifts in the relative phase or number difference between the two
wells. The effects of phase noise are mitigated by atom-atom interactions and
tunneling, such that the dephasing rate may be reduced to half its single-atom
value. Decoherence due to number noise (which induces fluctuations in the
relative atom number between the wells) is considerably enhanced by the
interactions. A similar scenario is predicted for the case of atom loss, even
if the loss rates from the two sites are equal. In fact, interactions convert
the increased uncertainty in atom number (difference) into (relative) phase
diffusion and reduce the coherence across the junction. We examine the
parameters relevant for these effects using a simple model of the trapping
potential based on an atom chip device. These results provide a framework for
mapping the dynamical range of barriers engineered for specific applications
and sets the stage for more complex circuits ("atomtronics").
