Emergence of Fermi's Golden Rule

TL;DR

This paper explores how Fermi's Golden Rule transitions from its traditional continuum limit to a discrete level setting, revealing universal decay behaviors and the role of spectral form factors in many-body quantum systems.

Contribution

It provides analytical and numerical analysis of the crossover from discrete to continuum regimes in quantum decay, highlighting universal behaviors and corrections to FGR.

Findings

Universal decay crossover identified

Spectral form factor influences long-time decay

Golden Rule emerges in the continuum limit

Abstract

Fermi's Golden Rule (FGR) applies in the limit where an initial quantum state is weakly coupled to a {\it continuum} of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the Golden Rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by FGR is the appearance of the spectral form factor in the long-time regime for small but nonzero level spacing.