Atomic Bose-Einstein condensate in twisted-bilayer optical lattices.

Observation of strong correlations and superconductivity in twisted-bilayer graphene1-4 has stimulated tremendous interest in fundamental and applied physics5-8. In this system, the superposition of two twisted honeycomb lattices, generating a moiré pattern, is the key to the observed flat electronic bands, slow electron velocity and large density of states9-12. Extension of the twisted-bilayer system to new configurations is highly desired, which can provide exciting prospects to investigate twistronics beyond bilayer graphene. Here we demonstrate a quantum simulation of superfluid to Mott insulator transition in twisted-bilayer square lattices based on atomic 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 accommodating the two layers. The interlayer coupling is highly controllable by a microwave field, which enables the occurrence of a lowest flat band and new correlated phases in the strong coupling limit. We directly observe the spatial moiré pattern and the momentum diffraction, which confirm the presence of two forms of superfluid and a modified superfluid to insulator transition in twisted-bilayer lattices. Our scheme is generic and can be applied to different lattice geometries and for both boson and fermion systems. This opens up a new direction for exploring moiré physics in ultracold atoms with highly controllable optical lattices.