  We study the phase coherence and visibility of trapped atomic condensates on
one-dimensional optical lattices, by means of quantum Monte-Carlo simulations.
We obtain structures in the visibility similar to the kinks recently observed
experimentally by Gerbier et.al.[Phy. Rev. Lett. 95, 050404 (2005); Phys. Rev.
A 72, 053606 (2005)]. We examine these features in detail and offer a
connection to the evolution of the density profiles as the depth of the lattice
is increased. Our simulations reveal that as the interaction strength, U, is
increased, the evolution of superfluid and Mott-insulating domains stall for
finite intervals of U. The density profiles do not change with increasing U. We
show here that in one dimension the visibility provides unequivocal signatures
of the melting of Mott domains with densities larger than one.
