# Fluctuations and photon statistics in quantum metamaterial near the superradiant transition

## TL;DR

This paper investigates photon fluctuations and counting statistics near the superradiant transition in the Dicke model, providing analytical and numerical insights into the behavior of photon number fluctuations and their full counting statistics.

## Contribution

It introduces an effective Matsubara action formalism for analyzing photon fluctuations in the Dicke model at the superradiant transition, including full counting statistics.

## Key findings

- Analytical expressions for photon number fluctuations and Fano factor.
- Numerical simulations confirm the analytical results.
- Identification of the Ginzburg-Levanyuk region near the transition.

## Abstract

The analysis of single-mode photon fluctuations and their counting statistics at the superradiant phase transition is presented. The study concerns the equilibrium Dicke model in a regime where the Rabi frequency, related to a coupling of the photon mode with a finite-number qubit environment, plays a role of the transition's control parameter. We use the effective Matsubara action formalism based on the representation of Pauli operators as bilinear forms with complex and Majorana fermions. Then, we address fluctuations of superradiant order parameter and quasiparticles. The average photon number, the fluctuational Ginzburg-Levanyuk region of the phase transition and Fano factor are evaluated. We determine the cumulant generating function which describes a full counting statistics of equilibrium photon number. Exact numerical simulation of the superradiant transition demonstrates quantitative agreement with analytical calculations.