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