Anomalous decay of an atom in structured band gap reservoirs

TL;DR

This paper analytically studies the spontaneous emission of a two-level atom in structured reservoirs with band gaps, revealing inverse power law decay behaviors and their dependence on the number of atoms and reservoir properties.

Contribution

It provides an exact analytical solution for atom decay in structured reservoirs with band gaps, including the effects of multiple atoms and identifying conditions where power law decay disappears.

Findings

Inverse power law decay with powers decreasing to 2 over time

No trapping of excited state population occurs

Power law decay vanishes for large number of atoms N

Abstract

We analyze the spontaneous emission of a two-level atom interacting with a special class of structured reservoirs of field modes with band gap edge coinciding with the atomic transition frequency. The exact time evolution of the population of the excited level is evaluated analytically through series of Fox-$H$ functions. Over estimated long time scales, inverse power law relaxations emerge, with powers decreasing continuously to 2 according to the choice of the special reservoir. No trapping of the population of the excited level emerges. The same results are recovered in presence of $N-1$ atoms, each one in the ground state, described by the Dicke model. The power of the inverse power law decay results to be independent of $N$. A critical number $N_{\alpha}^{(\star)}$ is evaluated, such that, for $N \gg N_{\alpha}^{(\star)}$, the inverse power law decay vanishes.