  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.
