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In this work we examine loss in ring resonator networks from an "operatorvalued phasor addition" approach (or OVPA approach) which considers themultiple transmission and cross coupling paths of a quantum field traversing aring resonator coupled to one or two external waveguide buses. We demonstratethe consistency of our approach by the preservation of the operator commutationrelation of the out-coupled bus mode. We compare our results to those obtainedfrom the conventional quantum Langevin approach which introduces noiseoperators in addition to the quantum Heisenberg equations in order to preservecommutation relations in the presence of loss. It is shown that the twoexpressions agree in the neighborhood of a cavity resonance where the Langevinapproach is applicable, whereas the operator valued phasor addition expressionwe derive is more general, remaining valid far from resonances.

A quantum optical description of losses in ring resonators based on field operator transformations