Analytical formulation for modulation of time-resolved dynamical Franz-Keldysh effect by electron excitation in dielectrics

Abstract

Analytical formulation of sub-cycle modulation (SCM) of dielectrics including electron excitation is presented. The SCM is sensitive to not only the time-resolved dynamical Franz-Keldysh effect (Tr-DFKE) [T. Otobe, et al., Phys. Rev. B 93, 045124 (2016)], which is the nonlinear response without the electron excitation, but also the excited electrons. The excited electrons enhance the modulation with even-harmonics of pump laser frequency, and generate the odd-harmonics components. The new aspect of SCM is a consequence of i) the interference between the electrons excited by the pump laser and those excited by the probe pulse laser and ii) oscillation of the generated wave packed by the pump laser. When the probe- and pump-pulse polarizations are parallel, the enhancement of the even harmonics and the generation of the odd harmonics modulation appear. However, if the polarizations are orthogonal, the effect arising from the electron excitations becomes weak. By comparing the parabolic and cosine band models, I found that the electrons under the intense laser field move as quasi-free particles.