Survival probability of a local excitation in a non-Markovian environment: Survival collapse, Zeno and anti-Zeno effects

The decay dynamics of a local excitation interacting with a non-Markovian environment, modeled by a semi-infinite tight-binding chain, is exactly evaluated. We identify distinctive regimes for the dynamics. Sequentially: (i) early quadratic decay of the initial-state survival probability, up to a spreading time tS, (ii) exponential decay described by a self-consistent Fermi Golden Rule, and (iii) asymptotic behavior governed by quantum diffusion through the return processes, leading to an inverse power law decay. At this last cross-over time tR a survival collapse becomes possible. This could reduce the survival probability by several orders of magnitude. The cross-over times tS and tR allow to assess the range of applicability of the Fermi Golden Rule and give the conditions for the observation of the Zeno and anti-Zeno effect.