Femtosecond time-resolved dynamical Franz-Keldysh effect

We theoretically investigate the dynamical Franz-Keldysh effect in femtosecond time resolution, that is, the time-dependent modulation of a dielectric function at around the band gap under an irradiation of an intense laser field. We develop a pump-probe formalism in two distinct approaches: first-principles simulation based on real-time time-dependent density functional theory and analytic consideration of a simple two-band model. We find that, while time-average modulation can be reasonably described by the static Franz-Keldysh theory, a remarkable phase shift is found to appear between the dielectric response and the applied electric field.