Analytical study of quadratic and non-quadratic short-time behavior of quantum decay

TL;DR

This paper investigates the short-time decay behavior of unstable quantum states, revealing that it can follow different power laws depending on the initial state, with implications for interpreting experimental results.

Contribution

It provides a detailed analysis showing that quantum decay can exhibit either quadratic or non-quadratic short-time behavior, depending on initial conditions, supported by solvable models.

Findings

Short-time decay can behave as $1 - ext{const} imes t^{3/2}$ or as $1 - ext{const} imes t^{2}$.

The initial state influences the short-time decay law.

Existing experiments may not distinguish between these behaviors.

Abstract

The short-time behavior of quantum decay of an unstable state initially located within an interaction region of finite range is investigated using a resonant expansion of the survival amplitude. It is shown that in general the short-time behavior of the survival probability $S(t)$ has a dependence on the initial state and may behave either as $S(t)=1-\mathcal{O}(t^{3/2})$ or as $S(t)=1-\mathcal{O}(t^{2})$. The above cases are illustrated by solvable models. The experiment reported in Ref. [1] does not distinguish between the above short-time behaviors.