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Physical Review A

Spectrum in spontaneous emission: Beyond the Weisskopf-Wigner approximation

P. R. Berman, G. W. Ford

Published Aug 27, 2010• 12 citations
Publisher

Abstract

The theory of spontaneous emission presented by Weisskopf and Wigner [V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930)] provides an excellent approximation of the actual decay that atoms undergo on optically allowed transitions. However, the theory cannot be rigorously correct since it leads to a Lorentzian spectrum that extends to negative frequencies. Within the rotating-wave approximation, we derive a closed-form expression for the spectrum that is valid for all frequencies.

DOI
10.1103/physreva.82.023818

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The theory of spontaneous emission presented by Weisskopf and Wigner [V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930)] provides an excellent approximation of the actual decay that atoms undergo on optically allowed transitions. However, the theory cannot be rigorously correct since it leads to a Lorentzian spectrum that extends to negative frequencies. Within the rotating-wave approximation, we derive a closed-form expression for the spectrum that is valid for all frequencies.
The exact determination of the spontaneous emission coefficient for an excited atom is an extremely complex problem so various approximations are typically used. One of the most popular ones is the use of the dipole approximation of a two-level atom followed by rotating wave approximation (RWA). However, such an approach applied to the entire frequency spectrum results in the appearance of divergent integrals in the derivations, which are not treated rigorously in typical student textbooks. It is known from the literature that the introduction of cutoff for frequencies, justified by the finite size of the atom, may solve this problem. For didactic purposes, in this paper, we introduce a mathematically simple cutoff, which allows for a straightforward yet mathematically consistent rederivation of the Weisskopf-Wigner spontaneous emission rate (up to the small correction) within RWA. Importantly, this cutoff is not a mathematical trick to make calculations easier but is related to a real feature of the physical system, the neglect of which leads to inconsistency. More precise analysis demand going beyond RWA and dipole approximation.
Spontaneous emission in dipole approximation is studied theoretically using both source-field theory and a Schrodinger picture approach. Using source-field theory we obtain formal equations for the Poynting vector and energy density without making the rotating wave approximation (RWA) and Weisskopf-Wigner approximation (WWA). The initial condition at t=0 is one in which the atom is in an excited state and the field in the vacuum state. The source-field expressions are evaluated within the the RWA and WWA and are found to satisfy Poynting's theorem. To explore the consequences of not making the RWA and WWA, the Poynting vector and energy density are calculated using perturbation theory. We use a Schrodinger picture approach and essentially reproduce and complement the results of Compagno, Passante, and Persico [J. Mod. Optics 37:8, 1377 (2007)] and those of Power and Thirunamachandran [Phys. Rev. A 45, 54 (1992)] obtained using a Heisenberg picture approach. The theory involves a sum over field mode frequencies and both finite cutoffs and convergence factors are used to carry out the sums. It is shown that the perturbation theory calculation leads to unphysical values for atomic state populations for all times when a sum over all field frequencies is taken, even if a convergence factor is used. It is also proved that the fields calculated using source-field theory always satisfy Poynting's theorem for ct not equal to R, where R is the distance from the atom.
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.