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

Analytical formulation of subcycle 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 quasifree particles.

Figure 1

Time evolution of ${\rho }_C\left(t\right)$ (red line), and the electric field (blue line). The pump laser has an intensity of (a) $1\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, (c) $1\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and (e) $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (b), (d) and (f) Energy-gap dependence of the excited electron density after pump excitation at $k=|\vec{k}|$.

Figure 2

SCM under a pump intensity of $1\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. The polarization of the pump and probe lights is parallel. (a) Full calculation of the time evolution of $\text{Re}\delta \sigma (\omega ,T_p)$. The ordinate represents the energy from $B_g$. (b) Fourier components of (a) in a logarithmic scale. (c) $\text{Re}{\sigma }_{\text{DFKE}}(\omega ,T_p)$. (d) Fourier components of (c) in a logarithmic scale. (e) and (f) $\text{Re}{\sigma }_{\text{ex}}(\omega ,T_p)$ and its Fourier components, respectively. (g) Applied electric field.

Figure 3

SCM under a pump intensity of $1\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. The description of the each panel corresponds to that in Fig. 2.

Figure 4

SCM under a pump intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The description of each panel corresponds to that in Fig. 2.

Figure 5

SCM under a pump intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The polarization of the pump and probe beam is orthogonal, and the description of each panel corresponds to that in Fig. 2.

Figure 6

SCM under a pump intensity of $1\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ analyzed using the cosine band model. (a) SCM with full component. (b) Time evolution of ${\rho }_c\left(t\right)$ (red) and pump field (blue). (c) Fourier transformation of (a).

Figure 7

SCM under a pump intensity of $2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ analyzed using the cosine band model. (a) Full calculation of the time evolution of $\text{Re}\delta \sigma (\omega ,T_p)$. The ordinate represents the energy from the $B_g$. (b) Fourier components of (a) in a logarithmic scale. (c) $\text{Re}{\sigma }_{\text{DFKE}}(\omega ,T_p)$. (d) Fourier components of (c) in a logarithmic scale. (e) and (f) $\text{Re}{\sigma }_{\text{ex}}(\omega ,T_p)$ and its Fourier components, respectively. (g) Applied electric field (blue) and ${\rho }_C\left(t\right)$ (red).

Figure 8

SCM under a pump intensity of $2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ analyzed using a one-dimensional parabolic band. (a) SCM with full component. (b) Time evolution of ${\rho }_c\left(t\right)$ (red) and pump field (blue). (c) Fourier transformation of (a).

Figure 9

Comparison of excitation rates from Keldysh theory (blue dotted line) and the calculation using Eq. (C1) with Kane's model (red solid line).