Memory Effects in Spontaneous Emission Processes

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We consider a quantum-mechanical analysis of spontaneous emission in terms of an effective two-level system with a vacuum decay rate Γ₀ and transition angular frequency ωₐ. 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 τ= tΓ₀ of spontaneous emission, we obtain deviations from exponential decay of the form O (1/τ) for the decay amplitude as well as the previously obtained asymptotic behaviors of the form O (1/τ²) or O (1/τ²τ) for τ≫ 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 ωc of the electro-magnetic fluctuations, i.e. ωc ≫ ωₐ, one obtains either a O (1/τ) or a O (1/τ²) 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.