Anomalous-order exceptional point and non-Markovian Purcell effect at threshold in one-dimensional continuum systems

For a system consisting of a quantum emitter coupled near threshold (band edge) to a one-dimensional continuum with a van Hove singularity in the density of states, we demonstrate general conditions such that a characteristic triple-level convergence occurs directly on the threshold as the coupling g is shut off. 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 the influence of a third-order exceptional point. However, in the actual g -> 0 limit, only two discrete states in fact coalesce as the system can be reduced to a 2 x 2 Jordan block; the third state instead merges with the continuum. Moreover, the decay width of the resonance state involved in this convergence is significantly enhanced compared to the usual Fermi golden rule, which is consistent with the Purcell effect. However, non-Markovian dynamics due to the branch-point effect are also enhanced near the threshold. Applying a perturbative analysis in terms of the Puiseux expansion that takes into account the threshold influence, we show that the combination of these effects results in quantum emitter decay of the unusual form 1 - Ct(3/2) on the key timescale during which most of the decay occurs. We then present two conditions that must be satisfied at the threshold for the anomalous exceptional point to occur: the density of states must contain an inverse square-root divergence and the potential must be nonsingular. We further show that when the energy of the quantum emitter is detuned from threshold, the anomalous exceptional point splits into three ordinary exceptional points, two of which appear in the complex-extended parameter space. These results provide deeper insight into a well-known problem in spontaneous decay at a photonic band edge.