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 \(\Gamma_0\) 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.