Memory effects in spontaneous emission processes

Abstract

We consider a quantum-mechanical analysis of spontaneous emission in terms of an effective two-level system with a vacuum decay rate ${\ensuremath{\Gamma}}_{0}$ and transition angular frequency ${\ensuremath{\omega}}_{A}$. 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 dimensionless lifetime $\ensuremath{\tau}=t{\ensuremath{\Gamma}}_{0}$ of spontaneous emission, we obtain deviations from exponential decay of the form $O(1/\ensuremath{\tau})$ for the decay amplitude as well as the previously obtained asymptotic behaviors of the form $O(1/{\ensuremath{\tau}}^{2})$ or $O(1/\ensuremath{\tau}{\mathrm{ln}}^{2}\ensuremath{\tau})$ for $\ensuremath{\tau}\ensuremath{\gg}1$. 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 $\ensuremath{\tau}$ and for a sufficiently large value of the required angular frequency cutoff ${\ensuremath{\omega}}_{c}$ of the electromagnetic fluctuations, i.e., ${\ensuremath{\omega}}_{c}\ensuremath{\gg}{\ensuremath{\omega}}_{A}$, one obtains either a $O(1/\ensuremath{\tau})$ or a $O(1/{\ensuremath{\tau}}^{2})$ 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.