  We investigate gain in microwave photonic cavities coupled to voltage-biased
double quantum dot systems with an arbitrary strong dot-lead coupling and with
a Holstein-like light-matter interaction, by adapting the diagrammatic Keldysh
nonequilibrium Green's function approach. We compute out-of-equilibrium
properties of the cavity : its transmission, phase response, mean photon
number, power spectrum, and spectral function. We show that by the careful
engineering of these hybrid light-matter systems, one can achieve a significant
amplification of the optical signal with the voltage-biased electronic system
serving as a gain medium. We also study the steady state current across the
device, identifying elastic and inelastic tunnelling processes which involve
the cavity mode. Our results show how recent advances in quantum electronics
can be exploited to build hybrid light-matter systems that behave as
single-atom amplifiers and photon source devices. The diagrammatic Keldysh
approach is primarily discussed for a cavity-coupled double quantum dot
architecture, but it is generalizable to other hybrid light-matter systems.
