Photon statistics in waveguide QED: II Exact solutions in a thermodynamic limit

TL;DR

This paper derives exact analytical solutions for photon emission statistics in waveguide QED systems in the thermodynamic limit, revealing superradiance, subradiance, and coherence properties as the number of atoms becomes very large.

Contribution

It introduces a second order mean field method that becomes exact in the thermodynamic limit, providing new insights into collective emission phenomena in waveguide QED.

Findings

Exponential superradiance occurs before a specific time proportional to the atom lifetime.

Emission approaches independent ensemble behavior as the number of atoms increases.

Shot-to-shot fluctuations diminish and vanish in the large N limit, with coherence effects emerging.

Abstract

Waveguide quantum electrodynamics (WQED) offers a powerful framework for controlling light-matter interactions and realizing collective phenomena such as super- and subradiance. In general waveguide settings, the quantum dynamics spans the full Hilbert space, rendering exact theoretical treatments exponentially difficult and currently out of reach, and only a few models have exact, analytical solutions. Motivated by recent experiments, we treat the thermodynamic limit of the number of atoms, $N \rightarrow \infty$, while the homogeneous atom-waveguide coupling $\beta \rightarrow 0$ keeping the optical depth $4N\beta$ fixed. In this limit, a second order mean field method is exact, giving analytical solutions for the statistics of the photons emitted in the waveguide both for chiral and symmetric configurations starting from full inversion. As $N \rightarrow \infty$, the emission in freespace approaches that of an independent ensemble. However, until a special time, $\approx 1.59 \times$ the lifetime of a single-atom, we show an exponentially enhanced superradiance in the waveguide as the optical depth increases. After the special time, the emission into the waveguide exhibits subradiance. We also show that the initial shot-to-shot fluctuations in the rate of emission into the waveguide diminish in a chiral system and vanish in a symmetric system as $N$ approaches $\infty$. Additionally, the equal-time second-order correlation becomes trivial, showing that finite-size effects are essential to observe the emergence of second-order coherence. Finally, going beyond the thermodynamic limit requires higher order mean field methods. Our results illustrate finite- and infinite-body collective effects in symmetric and symmetry-lacking systems.