A quantum optical description of losses in ring resonators based on   field operator transformations
  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. 
Paul M. Alsing and Edwin E. Hach III and Christopher C. Tison and A.   Matthew Smith
quant-ph