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Generalized projected dynamics for non-system observables of non-equilibrium quantum impurity models

The reduced dynamics formalism has recently emerged as a powerful tool to study the dynamics of non-equilibrium quantum impurity models in strongly correlated regimes. Examples include the non-equilibrium Anderson impurity model near the Kondo crossover temperature and the non-equilibrium Holstein model, for which the formalism provides an accurate description of the reduced density matrix of the system for a wide range of timescales. In this work, we generalize the formalism to allow for non-system observables such as the current between the impurity and leads. We show that the equation of motion for the reduced observable of interest can be closed with the equation of motion for the reduced density matrix and demonstrate the new formalism for a generic resonant level model.