Optical depth dictates universal bounds on many-body decay in atomic ensembles

TL;DR

This paper establishes a universal scaling law for many-body photon emission rates in atomic ensembles, linking optical depth to emission limits across different geometries and disorder levels.

Contribution

It derives a universal law for the maximum photon emission rate based on optical depth, unifying ordered and disordered atomic ensembles in arbitrary dimensions.

Findings

Maximum emission rate scales with atom number and optical depth.

Directional detection affects observed emission scaling.

Optical depth governs many-body cooperative emission limits.

Abstract

Cooperative emission is well understood for idealized symmetric systems, but its limits in spatially extended, free-space ensembles remain an open question. Here, we derive a universal law for the scaling of the maximum photon emission rate with system size that unifies both ordered arrays and disordered atomic clouds in arbitrary dimensions at fixed density. We demonstrate that, for a fixed atomic density, the maximum emission rate scales universally as the product of the atom number and the system's optical depth, with the latter encoding the dimensional scaling across all regimes from independent emission to the Dicke limit. Furthermore, we establish a scaling law for directional detection, revealing that the observed rate depends on the detector's numerical aperture: small apertures yield Dicke-like quadratic scaling, whereas large apertures recover our integrated universal bound. Our results establish optical depth as the parameter governing many-body cooperative emission in both ordered and disordered ensembles, and reveal that directional and total-emission scalings must be carefully distinguished in experimental settings.