Dynamical nonlinear optical response in time-periodic quantum systems

We present a comprehensive theoretical framework for calculating the linear and nonlinear optical responses of time-periodic quantum systems. Using density matrix evolution in the Floquet basis and adopting the length gauge, our approach incorporates both interband and intraband contributions of the position operator, enabling detailed insights into photon-assisted transitions and their associated optical phenomena. Notably, we identify a divergent ac response to dc fields in Floquet systems, reminiscent of a Drude peak at finite frequencies. This framework generalizes to optical tensor conductivity calculations at arbitrary perturbation orders and captures various dc photocurrents, including shift current, injection current, and Berry dipole contributions, under specific limits. To demonstrate the versatility of our method, we compute linear and nonlinear optical conductivities for one- and two-dimensional systems, revealing phenomena such as band inversions, $j$-photon-assisted transitions, and high harmonic generation. These results highlight the interplay between periodic driving and nonlinear optical effects, offering different avenues for exploring dynamical topological properties and their applications in ultrafast spintronics, optoelectronics, and strongly correlated systems. Our findings provide a robust platform for analyzing the complex optical behavior of driven quantum systems and guiding experimental investigations in this rapidly evolving field.