  We propose a tunable optical setup to engineer topologically nontrivial flat
bands in twisted bilayer graphene under circularly polarized light. Using both
analytical and numerical calculations, we demonstrate that nearly flat bands
can be engineered at small twist angles near the magic angles of the static
system. The flatness and the gaps between these bands can be tuned optically by
varying laser frequency and amplitude. We study the effects of interlayer
hopping variations on Floquet flat bands and find that lattice relaxation
favors their formation. Furthermore, we find that, once formed, the flat bands
carry nonzero Chern numbers. We show that at currently known values of
parameters, such topological flat bands can be realized using circularly
polarized UV laser light. Thus, our work opens the way to creating optically
tunable, strongly correlated topological phases of electrons in moir\'e
superlattices.
