  We provide a systematic method for constructing effective dispersive Lindblad
master equations to describe weakly-anharmonic superconducting circuits coupled
by a generic dissipationless nonreciprocal linear system, with effective
coupling parameters and decay rates written in terms of the immittance
parameters characterizing the coupler. This article extends the foundational
work of Solgun et al. (2019) for linear reciprocal couplers described by an
impedance response. Here, we expand the existing toolbox to incorporate
nonreciprocal elements, account for direct stray coupling between immittance
ports, circumvent potential singularities, and include dissipative interactions
arising from interaction with a common bath. We illustrate the use of our
results with a circuit of weakly-anharmonic Josephson junctions coupled to a
multiport nonreciprocal environment and a dissipative port. The results
obtained here can be used for the design of complex superconducting quantum
processors with non-trivial routing of quantum information, as well as analog
quantum simulators of condensed matter systems.
