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Modern Physics Letters B

DEVIATIONS FROM SPONTANEOUS LYMAN-α EXPONENTIAL DECAY REVISITED: A RIGOROUS TREATMENT WITHIN A TWO-LEVEL NONRELATIVISTIC MODEL

JOSIP SEKE, A. V. SOLDATOV, Nikolay N. Bogolubov

Published Jul 10, 1999• 4 citations
Publisher

Abstract

A recent paper by Facchi and Pascazio, in which the advantage of numerical evaluations of deviations from exponential decay was pointed out, makes a revisitation of the subject necessary. All the more, since the above-mentioned authors have not estimated the error bounds (being absolutely necessary in any calculation) for their results. On the contrary, the accuracy of all results presented in this paper is proved by error estimates. Moreover, for the first time, the power of the simple iteration method in calculating both the exponential and non-exponential time evolution of an unstable atomic state is demonstrated.

DOI
10.1142/s0217984999000683

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A recent paper by Facchi and Pascazio, in which the advantage of numerical evaluations of deviations from exponential decay was pointed out, makes a revisitation of the subject necessary. All the more, since the above-mentioned authors have not estimated the error bounds (being absolutely necessary in any calculation) for their results. On the contrary, the accuracy of all results presented in this paper is proved by error estimates. Moreover, for the first time, the power of the simple iteration method in calculating both the exponential and non-exponential time evolution of an unstable atomic state is demonstrated.
For the first time to the authors' knowledge a mathematically rigorous method is used for treating finite-time deviations from the exponential decay in the case of spontaneous Lyman-\ensuremath{\alpha} transitions in a two-level hydrogenic atom. First, in the so-called Weisskopf-Wigner model (where the rotating-wave approximation is implied) finite-time deviations with a rigorous validity range, based on accurate error estimations, are derived. Second, in order to obtain the frequency shift (Lamb shift), counter-rotating terms are taken into account by using a projection-operator method developed a decade ago by one of the present authors [J. Seke, Phys. Rev. A 21, 2156 (1980)]. By means of this method, equations of motion for dipole-moment expectation values are derived without employing the usual Born approximation. These equations are solved by using the method of the Laplace transform and its inverse. Again finite-time deviations from the exponential decay of the dipole-moment expectation values with a definite validity range are obtained. Finally, it should be emphasized that all the results presented in this paper contain accurate error estimates which are absolutely necessary in a rigorous mathematical method.
A mathematically rigorous treatment of the Weisskopf-Wigner model of atomic spontaneous emission is presented. For the first time to the authors' knowledge, it is shown that in the correct asymptotic treatment with a cutoff frequency at which the dipole approximation breaks down, the main contribution to the long-time deviation from the exponential decay is of the order of 1/(t ln2{\mathrm{ln}}^{2}ln2t). This contradicts the results of previous authors who have obtained a long-time behavior of the order of 1/t2{t}^{2}t2 by nonrigorous mathematical treatment of the same model in the dipole approximation. However, we will show that the result 1/t2{t}^{2}t2 can still be obtained if the retardation effects are taken into account, i.e., if no dipole approximation is made.
It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes possible a complete and gauge-invariant treatment of spontaneously decaying atomic state. All the shortcomings of the Weisskopf-Wigner gauge-dependent method (Markov approximation and the neglect of virtual transition, retardation and electron-spin effects) are removed. After deriving a gauge-invariant equation of motion for the decaying state, a new physical interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) is introduced. In this new interaction picture, apart from a natural solution of the renormalization problem, explicit finite analytic results for the whole non-Markovian Lyman- alpha spontaneous decay are derived. Furthermore, a quasi-Lorentzian expression for the radiative line shape containing the correct (complete) finite non-relativistic frequency shift, stemming from the Lamb shifts of both levels 1S and 2P, is calculated.