  Observation of strong correlations and superconductivity in
twisted-bilayer-graphene have stimulated tremendous interest in fundamental and
applied physics. In this system, the superposition of two twisted honeycomb
lattices, generating a Moir$\acute{\mathrm{e}}$ pattern, is the key to the
observed flat electronic bands, slow electron velocity and large density of
states. Despite these observations, a full understanding of the emerging
superconductivity from the coupled insulating layers and the appearance of a
small magic angle remain a hot topic of research. Here, we demonstrate a
quantum simulation platform to study superfluids in twisted bilayer lattices
based on Bose-Einstein condensates loaded into spin-dependent optical lattices.
The lattices are made of two sets of laser beams that independently address
atoms in different spin states, which form the synthetic dimension of the two
layers. The twisted angle of the two lattices is controlled by the relative
angle of the laser beams. We show that atoms in each spin state only feel one
set of the lattice and the interlayer coupling can be controlled by microwave
coupling between the spin states. Our system allows for flexible control of
both the inter- and intralayer couplings. Furthermore we directly observe the
spatial Moir$\acute{\mathrm{e}}$ pattern and the momentum diffraction, which
confirm the presence of atomic superfluid in the bilayer lattices. Our system
constitutes a powerful platform to investigate the physics underlying the
superconductivity in twisted-bilayer-graphene and to explore other novel
quantum phenomena difficult to realize in materials.
