  Ultracold bosons in optical lattices are one of the few systems where bosonic
matter is known to exhibit strong correlations. Here we push the frontier of
our understanding of interacting bosons in optical lattices by adding synthetic
spin-orbit coupling, and show that new kinds of density- and chiral-orders
develop. The competition between the optical lattice period and the spin-orbit
coupling length -- which can be made comparable in experiments -- along with
the spin hybridization induced by a transverse field (i.e., Rabi coupling) and
interparticle interactions create a rich variety of quantum phases including
uniform, non-uniform and phase-separated superfluids, as well as Mott
insulators. The spontaneous symmetry breaking phenomena at the transitions
between them are explained by a two-order-parameter Ginzburg-Landau model with
multiparticle umklapp processes. Finally, in order to characterize each phase,
we calculated their experimentally measurable crystal momentum distributions.
