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arXiv:1307.7247 · Journal of Physics B Atomic Molecular and Optical Physics

Theory of the multiphoton cascade transitions with two photon links: comparison of quantum electrodynamical and quantum mechanical approaches

Timur A. Zalialiutdinov, Yu Baukina, Dmitry A. Solovyev, Leonti N Labzowsky

Published May 20, 2014• 11 citations

Abstract

The theory of multiphoton cascade transitions with two-photon links is considered within two different approaches: quantum electrodynamical (QED) and phenomenological quantum mechanical (QM). A problem of regularization of the cascade contributions is investigated in detail. It is argued that the correct regularization should include both initial and intermediate level widths in the singular energy denominators. This result follows both from the QED and from the QM approach. Particular transitions nl → 1s + 2γ with nl = 3s, 4s, 3d, 4d and nl → 1s + 3γ with nl = 3p, 4p are considered as examples. The importance of the proper cascade regularization is also demonstrated.

DOI: 10.1088/0953-4075/47/11/115007

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The theory of multiphoton cascade transitions with two-photon links is considered within two different approaches: quantum electrodynamical (QED) and phenomenological quantum mechanical (QM). A problem of regularization of the cascade contributions is investigated in detail. It is argued that the correct regularization should include both initial and intermediate level widths in the singular energy denominators. This result follows both from the QED and from the QM approach. Particular transitions nl → 1s + 2γ with nl = 3s, 4s, 3d, 4d and nl → 1s + 3γ with nl = 3p, 4p are considered as examples. The importance of the proper cascade regularization is also demonstrated.

We consider the general scenario of an excited level $|i\ensuremath{\rangle}$ of a quantum system that can decay via two channels: (i) via a single-quantum jump to an intermediate, resonant level $|\overline{m}\ensuremath{\rangle}$, followed by a second single-quantum jump to a final level $|f\ensuremath{\rangle}$, and (ii) via a two-quantum transition to a final level $|f\ensuremath{\rangle}$. Cascade processes $|i\ensuremath{\rangle}\ensuremath{\rightarrow}|\overline{m}\ensuremath{\rangle}\ensuremath{\rightarrow}|f\ensuremath{\rangle}$ and two-quantum transitions $|i\ensuremath{\rangle}\ensuremath{\rightarrow}|m\ensuremath{\rangle}\ensuremath{\rightarrow}|f\ensuremath{\rangle}$ compete (in the latter case, $|m\ensuremath{\rangle}$ can be both a nonresonant as well as a resonant level). General expressions are derived within second-order time-dependent perturbation theory, and the cascade contribution is identified. When the one-quantum decay rates of the virtual states are included into the complex resonance energies that enter the propagator denominator, it is found that the second-order decay rate contains the one-quantum decay rate of the initial state as a lower-order term. For atomic transitions, this implies that the differential-in-energy two-photon transition rate with complex resonance energies in the propagator denominators can be used to good accuracy even in the vicinity of resonance poles.

The 4 s -1 s two-quantum decay in the hydrogen atom is calculated in the framework of quantum electrodynamics with allowance for cascades. It is shown that the pure two-photon contribution cannot be separated from the contribution of cascade processes. Interference between the transitions of these two types is comparable in magnitude with the pure two-photon contribution. The separation of pure two-photon contributions is discussed recently in connection with the role of two-photon decays in the separation of radiation from matter in the early Universe.

The problem of the evaluation of the two-photon decay width of excited states in hydrogen is considered. Two different approaches to the evaluation of the width including cascades channels are employed: the summation of the transition probabilities for various decay channels and the evaluation of the imaginary part of the Lamb shift. As application, the two-photon decay channels for the

3s

level of the hydrogen atom are evaluated, including the cascade transition probability $3s\text{\ensuremath{-}}2p\text{\ensuremath{-}}1s$. An important role is assigned to the two-photon decays in astrophysics context, since processes of this kind provide a possibility for the decoupling of radiation and matter in the early universe. We demonstrate the ambiguity of separation of the ``pure'' two-photon contribution and criticize the existing methods for such a separation.

An ambiguity of the separation of cascades from ``pure'' two-photon decay is confirmed with accurate numerical calculations in a gauge-invariant way. A direct evaluation of the two-photon decay width of excited states in H-like ions via the imaginary part of two-loop self-energy is presented. We demonstrate that there is a fundamental difference between the level width and the transition probability in the presence of cascades. The two-photon widths are shown to be different from the two-photon decay rates for the transitions including cascades.