Analysis of single-excitation states in quantum optics

TL;DR

This paper investigates the dynamics of single-excitation states in quantum optics, revealing decay behaviors, pole structures, and effects of coupling strength, with extensions to atom distributions in the continuum limit.

Contribution

It refines the pole approximation in the Wigner-Weisskopf model and analyzes decay behaviors for various coupling regimes and atom distributions.

Findings

Atomic amplitudes are sums of decaying exponentials with decay rates and Lamb shifts from poles.

At large times, atomic fields decay as 1/t^3 with a known constant.

Stronger coupling introduces oscillatory exponential behaviors at long times.

Abstract

In this paper we analyze the dynamics of single-excitation states, which model the scattering of a single photon from multiple two level atoms. For short times and weak atom-field couplings we show that the atomic amplitudes are given by a sum of decaying exponentials, where the decay rates and Lamb shifts are given by the poles of a certain analytic function. This result is a refinement of the "pole approximation" appearing in the standard Wigner-Weisskopf analysis of spontaneous emission. On the other hand, at large times, the atomic field decays like $O(1/t^3)$ with a known constant expressed in terms of the coupling parameter and the resonant frequency of the atoms. Moreover, we show that for stronger coupling, the solutions also feature a collection of oscillatory exponentials which dominate the behavior at long times. Finally, we extend the analysis to the continuum limit in which atoms are distributed according to a given density.