Separation of Transitions with Two Quantum Jumps from Cascades

TL;DR

This paper derives general expressions for competing cascade and two-quantum decay processes in quantum systems, showing how complex resonance energies can accurately describe two-photon transition rates near resonance.

Contribution

It provides a theoretical framework within second-order perturbation theory to distinguish cascade contributions and incorporate decay rates into resonance energies.

Findings

Cascade and two-quantum transition processes are quantitatively distinguished.

Including decay rates in resonance energies improves accuracy near resonance poles.

The derived expressions are applicable to atomic transition rate calculations.

Abstract

We consider the general scenario of an excited level |i> of a quantum system that can decay via two channels: (i) via a single-quantum jump to an intermediate, resonant level |bar m>, followed by a second single-quantum jump to a final level |f>, and (ii) via a two-quantum transition to a final level |f>. Cascade processes |i> -> |bar m> -> | f> and two-quantum transitions |i> -> |m> -> |f> compete (in the latter case, |m> 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.