An exact analytical solution for Dicke superradiance

TL;DR

This paper provides a new, compact analytical solution for the dynamics of Dicke superradiance, enabling explicit calculations of state populations and insights into spectral properties of collective decay.

Contribution

It introduces a closed-form, computationally efficient solution for Dicke superradiance dynamics, generalizing previous results and revealing structural spectral insights.

Findings

Explicit populations of Dicke states at any time

Solution expressed as finite sum over residues or contour integral

Enhanced understanding of spectral degeneracies and Lindbladian modes

Abstract

We revisit the Dicke superradiance problem, where an ensemble of N identical two-level systems undergoes collective spontaneous decay. While an exact analytical solution has been known since 1977, its algebraic complexity has hindered practical use. Here we present a compact, closed-form solution that expresses the dynamics as a finite sum over residues or, equivalently, a complex contour integral. The method yields explicit populations of all Dicke states at arbitrary times and system sizes, and generalizes naturally to arbitrary initial conditions. Our formulation is computationally efficient and offers structural insights into the role of spectral degeneracies and Lindbladian eigenmodes in collective decay.