Floquet engineering nearly flat bands through quantum-geometric light-matter coupling with surface polaritons

We investigate Floquet engineering in a sawtooth chain -- a minimal model hosting a nearly flat band endowed with nontrivial quantum geometry -- coupled to driven surface polaritons. In this paradigmatic flat band model, light-matter coupling to a flat band is enabled by quantum geometry despite the vanishing band velocity and band curvature. We show that light polarization and finite momentum transfer in polaritonic settings provide sufficient tunability to flatten or unflatten bands, with sometimes drastic band structure modifications beyond what is attainable with laser pulses in free space. Possible implications for light-driven phenomena in prototypical flat-band moir\'e or kagome materials are discussed.