  We perform the finite-temperature determinant quantum Monte Carlo simulation
for the attractive Hubbard model on the half-filled bilayer square lattice.
Recent progress on optical lattice experiments lead us to investigate various
single-particle properties such as momentum distribution and double occupancies
which should be easily measured in cold-atom experiments. The pair-pair and the
density-density correlations have been studied in detail, and through
finite-size scaling, we show that there is no competing charge density wave
order in the bilayer band-insulator model and that the superfluid phase is the
stable phase for the interaction range $\abs{U}/t = 5-10$. We show the
existence of two energy scales in the system as we increase the attractive
interaction, one governing the phase coherence and the other one corresponding
to the molecule formation. In the end, we map out the full $T-U$ phase diagram
and compare the $T_c$ obtained through the mean-field analysis. We observe that
the maximum $T_c/t(= 0.27)$ occurs for $\abs{U}/t = 6$, which is roughly twice
the reported $T_c$ of the single-layer attractive Hubbard model.
