  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.
