Lindbladian approach to $P(E)$ theory

The $P(E)$ theory describes inelastic tunneling of particles through junctions integrated into an electric circuit. The inelasticity stems from the interaction between the photonic field of the circuit and the tunneling particles possessing electric charge. In the conventional approach to the $P(E)$ theory, the tunneling rate and the electric current through the junction are derived using Fermi's golden rule. The derivation is carried out to leading order in the tunnel coupling between the leads and includes an averaging over the environmental photonic degrees of freedom. Tracing out the environmental degrees of freedom is also commonly used for open quantum systems when deriving dynamical equations such as the Lindbladian. In this work, we reveal the relations between the $P(E)$ theory and the Lindbladian dynamics. We demonstrate that the assumptions of the Fermi's golden rule are essentially equivalent with the Born-Markovian approximation used in the Lindbladian formalism. The resulting quantum master equation enables us to obtain not only the electric current but various other quantities, including for instance the heat current parametrized by the same $P(E)$ function, in a systematic and convenient way.