  We investigate the ground state phases of an extended Bose-Hubbard ladder of
unit filling via the density-matrix-renormalization-group method and, in
particular, the effect of rung-hoppings. In contrast to a single-chain, a
commensurate supersolid emerges, and based on the Luttinger parameter, we
classify them into two types. The latter leads to a reentrant gapless behavior
as the onsite interaction is increased while keeping all other parameters
intact. A reentrant gapped transition is also found as a function of
nearest-neighbor interactions. Further, we show that the string order
characterizing the Haldane phase vanishes for a finite inter-chain hopping
amplitude, however small it is. Finally, we propose two experimental platforms
to observe our findings, using either dipolar atoms or polar molecules and
Rydberg admixed atoms.
