Dawn and Twilight Time in Quantum Tunneling

Abstract

Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert $W$ function, making its parametric dependence manifest without fitting. For square, modified square, and P\"oschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation $\Gamma T = T_{\text{trans}}$ between the decay rate $\Gamma$, oscillation period $T$, and transmission probability $T_{\text{trans}}$, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.