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

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