  Bosons in periodic potentials with very strong local interactions, known as
the constrained bosons often exhibit interesting physical behavior. We
investigate the ground state properties of a two-leg Bose-Hubbard ladder by
imposing three-body constraint in one leg and hardcore constraint in the other.
By using the cluster-mean-field theory approximation and the density matrix
renormalization group method, we show that at unit filling, for strong two-body
attraction among the three-body constrained bosons, the system becomes a gapped
pair-Mott insulator where all the bosons form strong bound pairs and occupy the
leg with three-body constraint. With increase in hopping strength this
pair-Mott insulator phase undergoes a phase transition to the gapless
superfluid phase for equal leg and rung hopping strengths. However, when the
rung hopping is stronger compared to the leg hopping, we obtain a crossover to
another gapped phase which is called the rung-Mott insulator phase where the
bosons prefer to delocalize on the rungs than the legs. By moving away from
unit filling, the system remains in the superfluid phase except for a small
region below the gapped phase where a pair superfluid phase is stabilized in
the regime of strong attractive interaction. We further extend our studies by
considering three-body constraint on both the legs and find that the crossover
from the gapped to gapped phase does not occur rather the system undergoes a
transition from a pair-rung-Mott insulator phase to the superfluid phase at
unit filling. Moreover, in this case we find the signature of the pair
superfluid phase on either sides of this gapped phase.
