Memory effects in spontaneous emission processes

Arne L. Grimsmo, Asle H. Vaskinn, Per K. Rekdal, and Bo-Sture K. Skagerstam

Phys. Rev. A 87, 022101 – Published 4 February 2013

Abstract

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 dimensionless lifetime $τ = t Γ_{0}$ 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 $O (1 / τ^{2})$ or $O (1 / τ \ln^{2} τ)$ 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 cutoff $ω_{c}$ of the electromagnetic fluctuations, i.e., $ω_{c} ≫ ω_{A}$ , one obtains either a $O (1 / τ)$ or a $O (1 / τ^{2})$ 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.

Received 18 September 2012

DOI:https://doi.org/10.1103/PhysRevA.87.022101

Authors & Affiliations

Arne L. Grimsmo^1,2,*, Asle H. Vaskinn^1,†, Per K. Rekdal^3,‡, and Bo-Sture K. Skagerstam^1,4,§

¹Department of Physics, The Norwegian University of Science and Technology, N-7491 Trondheim, Norway
²Department of Physics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
³Molde University College, P.O. Box 2110, N-6402 Molde, Norway
⁴Centre for Advanced Study (CAS), Drammensveien 78, N-0271 Oslo, Norway

^*arne.grimsmo@ntnu.no
^†asle.vaskinn@ntnu.no
^‡per.k.rekdal@himolde.no
^§bo-sture.skagerstam@ntnu.no

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Vol. 87, Iss. 2 — February 2013

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Images

Figure 1
Probability $| c_{e} {(τ) |}^{2} = {| {\tilde{c}}_{e} (τ) |}^{2}$ as a function of $τ \equiv Γ_{0} t$ . The solid curve corresponds to the exact numerical solution, i.e., solution of Eq. (33) with the kernel ${\tilde{κ}}_{0}^{R} (t)$ as given by Eq. (32) in the main text for a two-level system in vacuum. The upper (dash-dotted) curve corresponds to the small-time, ${\tilde{b}}_{A} τ ≪ 1$ , expansion Eq. (39). The lower (dotted) curve corresponds to the exponential decay $\exp (- τ)$ with expected deviations at small $τ$ . The values of the relevant parameters are ${\tilde{b}}_{A} \equiv {\tilde{ω}}_{A} / Γ_{0} = 10$ and $Λ = 1000$ .Reuse & Permissions
Figure 2
Probability $| c_{e} {(τ) |}^{2} = {| {\tilde{c}}_{e} (τ) |}^{2}$ as a function of $τ \equiv Γ_{0} t$ as in Fig. 1, but for a smaller $τ$ interval.Reuse & Permissions
Figure 3
Probability $| c_{e} {(τ) |}^{2} = {| {\tilde{c}}_{e} (τ) |}^{2}$ as a function of $τ \equiv Γ_{0} t$ . The solid curve corresponds to the exact numerical solution, i.e., solution of Eq. (33) with the kernel ${\tilde{κ}}_{0}^{R} (t)$ as in Eq. (32) for a two-level system in vacuum. The almost overlapping dotted curve corresponds to Eq. (41). The values of the relevant parameters are ${\tilde{b}}_{A} \equiv {\tilde{ω}}_{A} / Γ_{0} = 10$ and $Λ = 1000$ . The inset shows a limited part of the larger figure and illustrates the accuracy of the asymptotic form Eq. (41).Reuse & Permissions
Figure 4
Probability $| c_{e} {(τ) |}^{2} = {| {\tilde{c}}_{e} (τ) |}^{2}$ as a function of $τ \equiv Γ_{0} t$ . The upper curve corresponds to the exact numerical solution, i.e., solution of Eq. (33) with the kernel ${\tilde{κ}}_{0}^{R} (t)$ as in Eq. (32) for a two-level system in vacuum which, within the numerical accuracy of the figure, overlaps with the asymptotic formula Eq. (41). The lower curve corresponds to the main result in Ref. [41], i.e., Eq. (73). The asymptotic form according to Eq. (75) is in excellent agreement with the numerical evaluation of Eq. (73). The values of the relevant parameters are ${\tilde{b}}_{A} = 10$ and $Λ = 1000$ . As a reference, the straight line in the figure corresponds to a pure exponential decay, i.e., $| c_{e} {(τ) |}^{2} = e^{- τ}$ .Reuse & Permissions
Figure 5
Probability $| c_{e} {(τ) |}^{2} = {| {\tilde{c}}_{e} (τ) |}^{2}$ as a function of $τ \equiv Γ_{0} t$ . The upper curve corresponds to the exact numerical solution of the Volterra equation as considered in Refs. [43, 44], i.e., of Eq. (33) with the kernel ${\tilde{κ}}_{1}^{R} (τ)$ of Eq. (59) for a two-level system in vacuum. For large values of $τ$ the solution is dominated by the $1 / τ$ asymptotic behavior according to Eq. (72). The lower curve corresponds to Eqs. (73) and (75), which overlaps within the numerical accuracy of the figure, with a characteristic $1 / τ^{2}$ asymptotic behavior for $c_{e} (τ)$ . The values of the relevant parameters are ${\tilde{b}}_{A} = 1000$ and $Λ = 1000$ .Reuse & Permissions

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