  Motivated by correlated decay processes producing gain, loss and lasing in
driven semiconductor quantum-dots, we develop a theoretical technique using
Keldysh diagrammatic perturbation theory to derive a Lindblad master equation
that goes beyond the usual second order perturbation theory. We demonstrate the
method on the driven dissipative Rabi model, including terms up to fourth order
in the interaction between the qubit and both the resonator and environment.
This results in a large class of Lindblad dissipators and associated rates
which go beyond the terms that have previously been proposed to describe
similar systems. All of the additional terms contribute to the system behaviour
at the same order of perturbation theory. We then apply these results to
analyse the phonon-assisted steady-state gain of a microwave field driving a
double quantum-dot in a resonator. We show that resonator gain and loss are
substantially affected by dephasing-assisted dissipative processes in the
quantum-dot system. These additional processes, which go beyond recently
proposed polaronic theories, are in good quantitative agreement with
experimental observations.
