Anomalous exceptional point and non-Markovian Purcell effect at threshold in 1-D open quantum systems*

We show that when a quantum emitter is coupled near threshold to a generic 1-D continuum with a van Hove singularity in the density of states, a characteristic spectral configuration appears involving a bound state, a resonance state and an anti-resonance state, as well as several exceptional points (EPs). At one EP appearing below the threshold, the resonance and anti-resonance states coalesce while the bound state instead experiences an avoided crossing. Meanwhile, if one considers the limit in which the coupling g vanishes, all three states converge on the continuum threshold itself. For small g values the eigenvalue and norm of each of these states can be expanded in a Puiseux expansion in terms of powers of g^2/3, which suggests a third order EP occurs at the threshold. However, in the actual g ––> 0 limit, only two discrete states in fact coalesce as the system can be reduced to a 2x2 Jordan block; the third state instead merges with the continuum. We further demonstrate the influence of the EP on non-Markovian dynamics characterizing the relaxation process of the quantum emitter in the vicinity of the threshold.