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arXiv:1801.02224 · Journal of Mathematical Physics

Spontaneous emission and atomic line shift in causal perturbation theory

Karl-Peter Marzlin, Bryce Fitzgerald

Published Apr 1, 2018• 1 citation
arXivHTMLPDF

Abstract

We derive a spontaneous emission rate and line shift for two-level atoms coupled to the radiation field using causal perturbation theory. In this approach, employing the theory of distribution splitting prevents the occurrence of divergent integrals. Our method confirms the result for an atomic decay rate but suggests that the cutoff frequency for the atomic line shift is determined by the atomic mass, rather than the Bohr radius or electron mass.

DOI
10.1063/1.5002556

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We derive a spontaneous emission rate and line shift for two-level atoms coupled to the radiation field using causal perturbation theory. In this approach, employing the theory of distribution splitting prevents the occurrence of divergent integrals. Our method confirms the result for an atomic decay rate but suggests that the cutoff frequency for the atomic line shift is determined by the atomic mass, rather than the Bohr radius or electron mass.
Abstract “Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.
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.
The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.
The spectrum of spontaneously emitted photons from an atom interacting with a single cavity mode depends on the relative sizes of the mode coupling constant and the cavity loss rate. If the coupling constant is not too large, perturbation theory can he used to show that the atomic linewidth is either increased (enhanced spontaneous emission) or decreased (inhibited spontaneous emission) by the presence of the cavity. When the coupling constant is so large that perturbation theory cannot be used, the spontaneous- emission spectrum forms a doublet, with peaks located at the normal-mode frequencies for the coupled-dipole and cavity-mode oscillators (the so-called "vacuum" Rabi splitting).
An elementary account is given of the absorption and emission of radiation by molecules employing a particular formulation of Schrödinger theory which facilitates the treatment of radiative line widths and level shifts. The familiar Fermi golden rule for transition rates, the Rabi Bloch-Siegert absorption profile, and the Weisskopf-Wigner approximation for spontaneous decay are obtained under appropriate conditions in the context of a common perspective. Comparisons are made with other recently devised perturbative and nonperturbative formulations of electromagnetic transition phenomena which correctly include radiative line widths and level shifts, and the importance of proper treatment of the normalization and phase of the Schrödinger wave function in this connection is emphasized.