Diagrammatic Expansion for Solving the Lindblad Master Equation

The Lindblad master equation has been an indispensable tool for understanding the dynamics of open quantum systems. However, the vast majority of physically relevant systems do not admit analytic solutions to their dynamics. In these situations, perturbation theory is often employed to get an approximate dynamical solution. The Feynman diagram approach to perturbation theory has proven instrumental in systematically representing the many terms present at each level of the expansion, though it has only been formulated for the unitary evolution of quantum systems. Our work seeks to devise such a diagrammatic expansion for open quantum system dynamics that includes non-unitary evolution elements, in particular those often encountered in quantum optics settings. Here, the Lindbladian jump operators often take the form of bosonic annihiliation operators, and when the Hamiltonian is given by a weighted sum of number operators an exact set of static operator solutions for the non-unitary time evolution can be derived. These solutions can then serve as a starting point for perturbation theory, where we will demonstrate how to represent the terms of the expansion for a few relevant types of interactions. This formalism allows for the inclusion of driven inputs, and supports phenomena that do not appear in the case of unitary time evolution such as the presence of steady states.