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

We revisit the relationship between electric polarization modulation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi mathvariant="normal">Δ</a:mi><a:mi>P</a:mi></a:mrow></a:math> in ferroelectrics induced by a low-frequency pulsed electric field and the corresponding second harmonic intensity modulation <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="normal">Δ</c:mi><c:msub><c:mi>I</c:mi><c:mi>SH</c:mi></c:msub></c:mrow></c:math>. 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi mathvariant="normal">Δ</e:mi><e:msub><e:mi>I</e:mi><e:mi>SH</e:mi></e:msub></e:mrow></e:math>. Furthermore, even when the time-derivative component is negligible, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi mathvariant="normal">Δ</g:mi><g:msub><g:mi>I</g:mi><g:mi>SH</g:mi></g:msub></g:mrow></g:math> can be in antiphase with the pulsed field and thus with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi mathvariant="normal">Δ</i:mi><i:mi>P</i:mi></i:mrow></i:math>. These theoretical predictions are further supported by real-time simulations of a model for electronic ferroelectrics. Our results demonstrate that the commonly assumed relation <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi mathvariant="normal">Δ</k:mi><k:mi>P</k:mi><k:mo>∝</k:mo><k:mi mathvariant="normal">Δ</k:mi><k:msub><k:mi>I</k:mi><k:mi>SH</k:mi></k:msub></k:mrow></k:math> can break down under certain conditions, reflecting the complex-valued and frequency-dependent nature of nonlinear dynamical susceptibilities.