  We have developed a theory of a Josephson junction formed by two
tunnel-coupled Bose-Einstein condensates in a double-well potential in the
regime of strong atom-atom interaction for an arbitrary total number $N$ of
bosons in the condensates. The tunnel resonances in the junction are shown to
be periodically spaced by the interaction energy, forming a single-atom
staircase sensitive to the parity of $N$ even for large $N$. One of the
manifestations of the staircase structure is the periodic modulation with the
bias energy of the visibility of the interference pattern in lattices of
junctions.
