  In this work we examine loss in ring resonator networks from an "operator
valued phasor addition" approach (or OVPA approach) which considers the
multiple transmission and cross coupling paths of a quantum field traversing a
ring resonator coupled to one or two external waveguide buses. We demonstrate
the consistency of our approach by the preservation of the operator commutation
relation of the out-coupled bus mode. We compare our results to those obtained
from the conventional quantum Langevin approach which introduces noise
operators in addition to the quantum Heisenberg equations in order to preserve
commutation relations in the presence of loss. It is shown that the two
expressions agree in the neighborhood of a cavity resonance where the Langevin
approach is applicable, whereas the operator valued phasor addition expression
we derive is more general, remaining valid far from resonances.
