Floquet engineering the band structure of materials with optimal control theory

We demonstrate that the electronic structure of a material can be deformed into Floquet pseudobands with arbi-trarily tailored shapes. We achieve this goal with a combination of quantum optimal control theory and Floquet engineering. The power and versatility of this framework is demonstrated here by utilizing the independent -electron tight-binding description of the pi electronic system of graphene. We show several prototype examples focusing on the region around the K (Dirac) point of the Brillouin zone: creation of a gap with opposing flat valence and conduction bands, creation of a gap with opposing concave symmetric valence and conduction bands (which would correspond to a material with an effective negative electron-hole mass), and closure of the gap when departing from a modified graphene model with a nonzero field-free gap. We employ time-periodic drives with several frequency components and polarizations, in contrast to the usual monochromatic fields, and use control theory to find the amplitudes of each component that optimize the shape of the bands as desired. In addition, we use quantum control methods to find realistic switch-on pulses that bring the material into the predefined stationary Floquet band structure, i.e., into a state in which the desired Floquet modes of the target bands are fully occupied, so that they should remain stroboscopically stationary, with long lifetimes, when the weak periodic drives are started. Finally, we note that although we have focused on solid state materials, the technique that we propose could be equally used for the Floquet engineering of ultracold atoms in optical lattices and for other nonequilibrium dynamical and correlated systems.