Hyperpolarizabilities for the one-dimensional infinite single-electron periodic systems: I. Analytical solutions under dipole-dipole correlations

TL;DR

This paper derives analytical expressions for third-order hyperpolarizabilities in one-dimensional infinite chains, revealing how intraband contributions and dipole-dipole correlations break permutation symmetry and Kleinman symmetry, with implications for nonlinear optical experiments.

Contribution

It provides the first analytical solutions for hyperpolarizabilities in infinite chains under specific models, highlighting the role of intraband effects and symmetry breaking in nonlinear optics.

Findings

Intraband contributions dominate hyperpolarizabilities when included.

Inclusion of bla_k terms breaks permutation and Kleinman symmetry.

Predicted trends of  in low and resonant frequency regions.

Abstract

The analytical solutions for the general-four-wave-mixing hyperpolarizabilities $\chi^{(3)}(-(w_1+w_2+w_3);w_1,w_2,w_3)$ on infinite chains under both Su-Shrieffer-Heeger and Takayama-Lin-Liu-Maki models of trans-polyacetylene are obtained through the scheme of dipole-dipole correlation. Analytical expressions of DC Kerr effect $\chi^{(3)}(-w;0,0,w)$, DC-induced second harmonic generation $\chi^{(3)}(-2w;0,w,w)$, optical Kerr effect $\chi^{(3)}(-w;w,-w,w)$ and DC-electric-field-induced optical rectification $\chi^{(3)}(0;w,-w,0)$ are derived. By including or excluding ${\bf \nabla_k}$ terms in the calculations, comparisons show that the intraband contributions dominate the hyperpolarizabilities if they are included. $\nabla_k$ term or intraband transition leads to the break of the overall permutation symmetry in $\chi^{(3)}$ even for the low frequency and non-resonant regions. Hence it breaks the Kleinman symmetry that is directly based on the overall permutation symmetry. Our calculations provide a clear understanding of the Kleinman symmetry breaks that are widely observed in many experiments. We also suggest a feasible experiment on $\chi^{(3)}$ to test the validity of overall permutation symmetry and our theoretical prediction. Finally, our calculations show the following trends for the various third-order nonlinear optical processes in the low frequency and non-resonant region: $\chi^{(3)}(-3w;w,w,w)> \chi^{(3)}(-2w;0,w,w)> \chi^{(3)}(-w;w,-w,w)>\chi^{(3)}(-w; 0,0,w)>= \chi^{(3)}(0;w,-w,0)$, and in the resonant region: $\chi^{(3)}(-w;0,0,w)> \chi^{(3)}(-w;w,-w,w)> \chi^{(3)}(-2w;0,w,w)>\chi^{(3)}(0;w,-w,0)>\chi^{(3)}(-3w;w,w,w)$. (w=\omega)