We investigate gain in microwave photonic cavities coupled to voltage-biaseddouble quantum dot systems with an arbitrary strong dot-lead coupling and witha Holstein-like light-matter interaction, by adapting the diagrammatic Keldyshnonequilibrium Green's function approach. We compute out-of-equilibriumproperties of the cavity : its transmission, phase response, mean photonnumber, power spectrum, and spectral function. We show that by the carefulengineering of these hybrid light-matter systems, one can achieve a significantamplification of the optical signal with the voltage-biased electronic systemserving as a gain medium. We also study the steady state current across thedevice, identifying elastic and inelastic tunnelling processes which involvethe cavity mode. Our results show how recent advances in quantum electronicscan be exploited to build hybrid light-matter systems that behave assingle-atom amplifiers and photon source devices. The diagrammatic Keldyshapproach is primarily discussed for a cavity-coupled double quantum dotarchitecture, but it is generalizable to other hybrid light-matter systems.
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