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Memory Effects in Spontaneous Emission Processes

We consider a quantum-mechanical analysis of spontaneous emission in terms of an effective two-level system with a vacuum decay rate $Γ_0$ and transition angular frequency $ω_A$. 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 $τ= tΓ_0$ of spontaneous emission, we obtain deviations from exponential decay of the form ${\cal O} (1/τ)$ for the decay amplitude as well as the previously obtained asymptotic behaviors of the form ${\cal O} (1/τ^2)$ or ${\cal O} (1/τ\ln^2τ)$ for $τ\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 $τ$ and for a sufficiently large value of the required angular frequency cut-off $ω_c$ of the electro-magnetic fluctuations, i.e. $ω_c \gg ω_A$, one obtains either a ${\cal O} (1/τ)$ or a ${\cal O} (1/τ^2)$ 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.