Title: Exploring the equivalence of causality-based and quantum mechanics-based sum rules for harmonic generation in nonlinear optical materials Show Chinese title

Title: 探索非线性光学材料中基于因果关系和基于量子力学的谐波生成求和规则的等价性

Abstract: The Kramers-Kronig relations and various oscillator strength sum rules represent strong constraints on the physical response of materials. In this work, taking inspiration from the well-established equivalence between $f-$sum rules and Thomas--Reiche--Kuhn sum rules in linear optics, we explore the connection between causality-based and quantum-mechanics-based sum rules in the context of nonlinear optical processes. Specifically, by considering the sum-over-states expression for the second harmonic generation susceptibility, we deduce a new representation basis for the imaginary part of this susceptibility and we use it to derive, from causality-based integral sum rules, a new set of discrete sum rules that the transition dipole moments must satisfy. As in the case of the Thomas--Reiche--Kuhn sum rules, we also show that these results can alternatively be derived through an independent quantum mechanical analysis. Finally, we consider the implications of the derived sum rules for the second-harmonic-generation susceptibility of two- and three-level systems and, more broadly, we discuss the possible significance and challenges of using these results for the goal of identifying fundamental limits to the response of nonlinear optical materials. Show Chinese abstract

Abstract: 克朗尼格-克勒默关系和各种振子强度求和规则对材料的物理响应施加了严格的约束。 在本工作中，受线性光学中$f-$求和规则与托马斯-赖希-库恩求和规则之间已建立的等价性的启发，我们探讨了非线性光学过程中基于因果性和基于量子力学的求和规则之间的联系。 具体而言，通过考虑二次谐波产生极化率的态求和表达式，我们推导出该极化率虚部的新表示基，并利用它从基于因果性的积分求和规则中推导出一组新的离散求和规则，这些规则必须由跃迁偶极矩满足。 与托马斯-赖希-库恩求和规则的情况类似，我们还表明这些结果可以通过独立的量子力学分析得出。 最后，我们考虑了所推导的求和规则对两能级和三能级系统二次谐波产生极化率的影响，并更广泛地讨论了这些结果在确定非线性光学材料响应基本极限目标中的可能意义和挑战。