  We investigate the behavior of a one-dimensional Bose-Hubbard gas in both a
ring and a hard-wall box, whose kinetic energy is made to oscillate with zero
time-average, which suppresses first-order particle hopping. For intermediate
and large driving amplitudes the system in the ring has similarities to the
Richardson model, but with a peculiar type of pairing and an attractive
interaction in momentum space. This analogy permits an understanding of some
key features of the interacting boson problem. The ground state is a
macroscopic quantum superposition, or cat state, of two many-body states
collectively occupying opposite momentum eigenstates. Interactions give rise to
a reduction (or modified depletion) cloud that is common to both
macroscopically distinct states. Symmetry arguments permit a precise
identification of the two orthonormal macroscopic many-body branches which
combine to yield the ground state. In the ring, the system is sensitive to
variations of the effective flux but in such a way that the macroscopic
superposition is preserved. We discuss other physical aspects that contribute
to protect the cat-like nature of the ground state.
