We present a quasinormal-mode (QNM) theory for coupled loss and gain resonators working in the vicinity of an exceptional point. Assuming linear media, which can be fully quantified using the complex pole properties of the QNMs, we show how the QNMs yield a quantitatively accurate model to a full classical dipole spontaneous-emission response in Maxwell’s equations at a variety of spatial positions and frequencies (under linear response). We also develop an intuitive QNM coupled-mode theory, which can be used to accurately model such systems using only the QNMs of the bare resonators, where the hybrid QNMs of the complete system are automatically obtained. Near a lossy exceptional point, whose general properties are broadened and corrected through use of QNM theory, we analytically show how the QNMs yield a Lorentzian-like and a Lorentzian-squared-like response for the spontaneous-emission line shape consistent with other works. However, using rigorous analytical and numerical solutions for microdisk resonators, we demonstrate that the general line shapes are far richer than what has been previously predicted. Indeed, the classical picture of spontaneous emission can take on a wide range of positive and negative Purcell factors from the hybrid modes of the coupled loss-gain system. The negative Purcell factors are unphysical and signal a clear breakdown of the classical dipole picture of spontaneous emission in such media, though the concept of a negative local density of states is correct. This finding has enabled a quantum fix to the decay of a two-level-system dipole emitter in amplifying and lossy media [Franke et al., Phys. Rev. Lett. 127, 013602 (2021)], and we further show and discuss the impact of this fix using the QNMs of the microdisk resonators. We also show the rich spectral features of the Green’s function propagators, which can be used to model various physical observables, such as photon detection.

中文翻译：

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耦合损耗-增益谐振器的准正规模式、局部状态密度和经典赛尔因子

我们提出了一种用于在异常点附近工作的耦合损耗和增益谐振器的准正态模式 (QNM) 理论。假设线性介质可以使用 QNM 的复杂极点属性进行完全量化，我们展示了 QNM 如何在各种空间位置和频率（下线性响应）。我们还开发了一种直观的 QNM 耦合模式理论，该理论可用于仅使用裸谐振器的 QNM 对此类系统进行精确建模，其中自动获得完整系统的混合 QNM。接近有损异常点，通过使用 QNM 理论扩展和校正其一般属性，我们分析地展示了 QNM 如何产生与其他作品一致的自发发射线形状的类洛伦兹和类洛伦兹平方响应。然而，使用微盘谐振器的严格解析和数值解决方案，我们证明了一般的线形状比以前预测的要丰富得多。事实上，自发辐射的经典图景可以从耦合损耗-增益系统的混合模式中采用广泛的正负珀塞尔因子。尽管负局部态密度的概念是正确的，但负的 Purcell 因子是非物理的，并且表明这种介质中自发辐射的经典偶极子图的明显崩溃。等。，物理。牧师莱特。 127 , 013602 (2021)]，我们使用微盘谐振器的 QNM 进一步展示和讨论此修复的影响。我们还展示了格林函数传播器的丰富光谱特征，可用于模拟各种物理可观察量，例如光子检测。