标题： 非厄米系统在特殊点的广义彼得曼因子 显示英文标题

标题： Generalized Petermann factor of non-Hermitian systems at exceptional points

摘要： 开放系统中模式的非正交性显著修改了它们的共振响应，导致与布雷特-维格纳共振关系在定量和定性上出现偏差。 对于具有洛伦兹线型的孤立共振，这种偏差表现为共振线宽由于彼得曼因子（PF）而增强，该因子由有效非厄米哈密顿量的左、右本征模式的重叠给出。 彼得曼因子在例外点（EPs）处发散，此时共振频率退化，并且右、左本征模式彼此正交。 这种发散标志着从洛伦兹线型的定性偏离，这一现象最近引起了广泛关注。 在这项工作中，我们将这一概念扩展到例外点，并描述该EP PF如何在各种物理场景中表现出来。 首先，我们从物理角度识别这个PF为系统对外部或参数扰动响应的增强。 利用基于右、左特征向量的两个自然正交投影参考系，我们表明每种选择都带有精确的几何解释，这自然地将孤立共振的PF概念推广到EPs。 这两种选择可以组合成一个总体的EP PF，它再次可以用纯粹的几何术语来表达。 其次，我们通过考虑参与退化的模式以及那些仍然光谱分离的模式的作用，阐明决定EP PF大小的几何机制。 第三，我们设计了一个系统来研究特定物理设置中的EP PF，包括两个耦合到带有嵌入半透明镜的波导的微环。 此示例显示我们的方法比传统的截断提供了更准确的光谱响应强度。 这些结果以与原始PF对孤立共振相同的方式完成了对EPs处系统的描述。 显示英文摘要

摘要： The nonorthogonality of modes in open systems significantly modifies their resonant response, resulting in quantitative and qualitative deviations from Breit-Wigner resonance relations. For isolated resonances with a Lorentzian lineshape, the deviations amount to an enhancement of the resonance linewidth by the Petermann factor (PF), given by the overlap of left and right eigenmodes of the underlying effectively non-Hermitian Hamiltonian. The PF diverges at exceptional points (EPs), where resonance frequencies degenerate, and right and left eigenmodes are orthogonal to each other. This divergence signifies a qualitative departure from a Lorentzian lineshape, which has gained recent attention. In this work, we extend this concept to EPs, and describe how this EP PF manifests in a variety of physical scenarios. Firstly, we identify this PF in physical terms as an enhancement of the response of a system to external or parametric perturbations. Utilizing two natural orthogonally projected reference systems based on the right and left eigenvectors, we show that each choice carries a precise geometric interpretation that naturally extends the notion of the PF for isolated resonances to EPs. The two choices can be combined into an overall EP PF, which again can be expressed in purely geometric terms. Secondly, we illuminate the geometric mechanisms that determine the size of the EP PF, by considering the role of modes participating in the degeneracy and those that remain spectrally separated. Thirdly, we design a system to study the EP PF in a specific physical setup, consisting of two microrings coupled to a waveguide with embedded semitransparent mirrors. This example shows our approach yields a more accurate spectral response strength than conventional truncation. These results complete the description of systems at EPs in the same way as the original PF does for isolated resonances.