# Photon statistics of a double quantum dot micromaser: Quantum treatment

## TL;DR

This paper provides a comprehensive quantum analysis of photon statistics in a double quantum dot micromaser, revealing the microscopic principles of photon gain and threshold conditions, and extending laser theory to hybrid quantum systems.

## Contribution

It introduces a rigorous quantum treatment of photon statistics in a semiconductor micromaser with a double quantum dot, extending laser theory to mesoscopic gain media.

## Key findings

- Derived the masing threshold condition at various temperatures and biases.
- Analyzed photon statistics below and above the threshold.
- Extended Scully-Lamb quantum laser theory to quantum dot systems.

## Abstract

A semiconductor single-atom micromaser consists of a microwave cavity coupled to a gain medium, a double quantum dot driven out of equilibrium by a bias voltage. The masing threshold of this system was recently probed by measuring photon statistics in the cavity [Y-Y. Liu \textit{et al}, Phys. Rev. Lett. 119, 097702 (2017)]. In this paper, we develop an in-depth, rigorous understanding of this experiment and related works. First, we use a semiclassical theory and study transmission spectroscopy. This approach allows us to derive the masing threshold condition for arbitrary temperature and voltage bias, and expose microscopic principles required for realizing photon gain and thereby a photon amplifier. Next, by employing the quantum master equation approach we extend the Scully-Lamb quantum theory of a laser to the present setup, and investigate the statistics of emitted photons below and above the masing threshold as a function of experimentally tunable parameters. Although our focus is primarily on hybrid quantum dot circuit - quantum electrodynamics systems, our approach is adaptable to other light-matter systems where the gain medium consists of a mesoscopic structure.