Temporal modulation of second harmonic generation in ferroelectrics by a pulsed electric field

We revisit the relationship between electric polarization modulation $\mathrm{\Delta }P$ in ferroelectrics induced by a low-frequency pulsed electric field and the corresponding second harmonic intensity modulation $\mathrm{\Delta }{I}_{\mathrm{S}\mathrm{H}}$. Using nonlinear response theory, we derive their time-domain expressions linear in the pulse amplitude, revealing that not only the electric field but also its time derivative contributes to $\mathrm{\Delta }{I}_{\mathrm{S}\mathrm{H}}$. Furthermore, even when the time-derivative component is negligible, $\mathrm{\Delta }{I}_{\mathrm{S}\mathrm{H}}$ can be in antiphase with the pulsed field and thus with $\mathrm{\Delta }P$. These theoretical predictions are further supported by real-time simulations of a model for electronic ferroelectrics. Our results demonstrate that the commonly assumed relation $\mathrm{\Delta }P\propto \mathrm{\Delta }{I}_{\mathrm{S}\mathrm{H}}$ can break down under certain conditions, reflecting the complex-valued and frequency-dependent nature of nonlinear dynamical susceptibilities.