Quasinormal Modes, Local Density of States, and Classical Purcell Factors for Coupled Loss-Gain Resonators

Abstract

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.5 MoreReceived 18 January 2021Revised 31 May 2021Accepted 17 August 2021DOI:https://doi.org/10.1103/PhysRevX.11.041020Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCavity quantum electrodynamicsOptical microcavitiesPhotonicsAtomic, Molecular & Optical