  We present a cluster Gutzwiller mean-field study for ground states and
time-evolution dynamics in the Bose-Hubbard ladder (BHL), which can be realized
by loading Bose atoms in double-well optical lattices. In our cluster
mean-field approach, we treat each double-well unit of two lattice sites as a
coherent whole for composing the cluster Gutzwiller ansatz, which may remain
some residual correlations in each two-site unit. For a unbiased BHL, in
addition to conventional superfluid phase and integer Mott insulator phases, we
find that there are exotic fractional insulator phases if the inter-chain
tunneling is much stronger than the intra-chain one. The fractional insulator
phases can not be found by using a conventional mean-field treatment based upon
the single-site Gutzwiller ansatz. For a biased BHL, we find there appear
single-atom tunneling and interaction blockade if the system is dominated by
the interplay between the on-site interaction and the inter-chain bias. In the
many-body Landau-Zener process, in which the inter-chain bias is linearly swept
from negative to positive or vice versa, our numerical results are
qualitatively consistent with the experimental observation [Nat. Phys.
\textbf{7}, 61 (2011)]. Our cluster bosonic Gutzwiller treatment is of
promising perspectives in exploring exotic quantum phases and time-evolution
dynamics of bosonic particles in superlattices.
