Memory Effects in Spontaneous Emission Processes

TL;DR

This paper provides an exact quantum-mechanical analysis of spontaneous emission, revealing deviations from exponential decay influenced by memory effects, regularization procedures, and physical boundaries, with results applicable across various physical parameters.

Contribution

It offers a numerically exact study of memory effects in spontaneous emission, clarifying asymptotic decay behaviors and their dependence on regularization and boundary conditions.

Findings

Deviations from exponential decay are of order 1/τ or 1/τ^2 for large τ.

Asymptotic behaviors depend on regularization procedures and physical parameters.

Physical boundaries can significantly alter decay rates without changing the fundamental decay behavior.

Abstract

We consider a quantum-mechanical analysis of spontaneous emission in terms of an effective two-level system with a vacuum decay rate $\Gamma_0$ and transition angular frequency $\omega_A$. Our analysis is in principle exact, even though presented as a numerical solution of the time-evolution including memory effects. The results so obtained are confronted with previous discussions in the literature. In terms of the {\it dimensionless} lifetime $\tau = t\Gamma_0$ of spontaneous emission, we obtain deviations from exponential decay of the form ${\cal O} (1/\tau)$ for the decay amplitude as well as the previously obtained asymptotic behaviors of the form ${\cal O} (1/\tau^2)$ or ${\cal O} (1/\tau \ln^2\tau)$ for $\tau \gg 1 $. The actual asymptotic behavior depends on the adopted regularization procedure as well as on the physical parameters at hand. We show that for any reasonable range of $\tau$ and for a sufficiently large value of the required angular frequency cut-off $\omega_c$ of the electro-magnetic fluctuations, i.e. $\omega_c \gg \omega_A$, one obtains either a ${\cal O} (1/\tau)$ or a ${\cal O} (1/\tau^2)$ dependence. In the presence of physical boundaries, which can change the decay rate with many orders of magnitude, the conclusions remains the same after a suitable rescaling of parameters.