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We propose a tunable optical setup to engineer topologically nontrivial flatbands in twisted bilayer graphene under circularly polarized light. Using bothanalytical and numerical calculations, we demonstrate that nearly flat bandscan be engineered at small twist angles near the magic angles of the staticsystem. The flatness and the gaps between these bands can be tuned optically byvarying laser frequency and amplitude. We study the effects of interlayerhopping variations on Floquet flat bands and find that lattice relaxationfavors their formation. Furthermore, we find that, once formed, the flat bandscarry nonzero Chern numbers. We show that at currently known values ofparameters, such topological flat bands can be realized using circularlypolarized UV laser light. Thus, our work opens the way to creating opticallytunable, strongly correlated topological phases of electrons in moir\'esuperlattices.

Floquet-Engineered Topological Flat Bands in Irradiated Twisted Bilayer Graphene