  We discuss the possibility of exponential quantum localization in systems of
ultracold bosonic atoms with repulsive interactions in open optical lattices
without disorder. We show that exponential localization occurs in the maximally
excited state of the lowest energy band. We establish the conditions under
which the presence of the upper energy bands can be neglected, determine the
successive stages and the quantum phase boundaries at which localization
occurs, and discuss schemes to detect it experimentally by visibility
measurements. The discussed mechanism is a particular type of quantum
localization that is intuitively understood in terms of the interplay between
nonlinearity and a bounded energy spectrum.
