Interferometric method to estimate the eigenvalues of a Non-Hermitian two-level optical system

Non-Hermitian physics has found a fertile ground in optics. Recently, the study of mode coalescence, i.e., exceptional points, has led to the discovery of intriguing and counterintuitive phenomena. Degeneracies are typically modeled through the coupled mode theory to determine the behavior of eigenstates and eigenvalues. However, the complex nature of the eigenvalues makes their characterization from the response spectrum difficult. Here, we demonstrate that a coherent interferometric excitation allows estimation of both the real and imaginary parts of the eigenvalues. We study the clockwise and counter-clockwise modes in optical microresonators both in the case of Hermitian and non-Hermitian intermodal coupling. We show the conditions by which a resonant doublet, due to the dissipative coupling of counter-propagating modes caused by surface roughness backscattering, merges to a single Lorentzian. This permits us to estimate the optimal quality factor of the microresonator in the absence of modal coupling caused by backscattering. Furthermore, we demonstrate that a taiji microresonator working at an exceptional point shows a degeneracy splitting only in one propagation direction and not in the other. This follows from the strongly non-Hermitian intermodal coupling caused by the inner S-shaped waveguide.