  We analyse the ground-state properties of three-body constrained bosons in a
one dimensional optical lattice with staggered hoppings analogous to the double
well optical lattice. By considering attractive and repulsive on-site
interactions between the bosons, we obtain the phase diagram which exhibits
various quantum phases. Due to the double-well geometry and three-body
constraint several gapped phases such as the Mott insulators and
dimer/bond-order phases emerge at commensurate densities in the repulsive
interaction regime. Attractive interaction leads to the pair formation which
leads to the pair bond order phase at unit filling which resembles the
valence-bond solid phase of composite bosonic pairs. At incommensurate
densities we see the signatures of the gapless pair superfluid phase.
