Separation of transitions with two quantum jumps from cascades

Abstract

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.