The onset time of Fermi's golden rule

TL;DR

This paper revises the understanding of when Fermi's golden rule applies, showing that for broadband reservoirs with certain spectral properties, the onset time can be much later than previously thought, affecting quantum decay predictions.

Contribution

It introduces a new criterion for the onset time of the Fermi regime in broadband reservoirs, extending the applicability of Fermi's golden rule to more general conditions.

Findings

For ta>1, the onset time scales with the reservoir cutoff frequency.

Standard criterion is valid for ta1 and narrowband reservoirs.

Fermi regime onset can be significantly delayed in broadband reservoirs.

Abstract

Fermi's golden rule describes the decay dynamics of unstable quantum systems coupled to a reservoir, and predicts a linear decay in time. Although it arises at relatively short times, the Fermi regime does not take hold in the earliest stages of the quantum dynamics. The standard criterion in the literature for the onset time of the Fermi regime is $t_F\sim1/\Delta\omega$, with $\Delta\omega$ the frequency interval around the resonant transition frequency $\omega_0$ of the system, over which the coupling to the reservoir does not vary appreciably. In this work, this criterion is shown to be inappropriate in general for broadband reservoirs, where the reservoir coupling spectrum takes the form $R\left(\omega\right)\propto\omega^\eta$, and for which it is found that for $\eta>1$, the onset time of the Fermi regime is given by $t_F\propto\left(\omega_{\mathrm{X}}/\omega_0\right)^{\eta-1}\times1/\omega_0$ where $\omega_{\mathrm{X}}$ is the high-frequency cutoff of the reservoir. Therefore, the onset of the Fermi regime can take place at times orders of magnitude larger than those predicted by the standard criterion. This phenomenon is shown to be related to the excitation of the off-resonant frequencies of the reservoir at short times. For broadband reservoirs with $\eta\leq1$, and for narrowband reservoirs, it is shown that the standard criterion is correct. Our findings revisit the conditions of applicability of Fermi's golden rule and improve our understanding of the dynamics of unstable quantum systems.