Quantum-optical description of losses in ring resonators based on field-operator transformations

Abstract

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. In addition, we examine the effects of internal and coupling losses on the Hong-Ou-Mandel manifold first discussed in Hach et al. Phys. Rev.A 89, 043805 (2014) that generalizes the destructive interference of two incident photons interfering on a 50:50 beam splitter (HOM effect) to the case of an add/drop double bus ring resonator.

Photon-pair generation in a lossy microring resonator. I. Theory

“…By tracking the infinite number of round trip circulations of the cavity field in the single bus mrr, AH derive the expression (with τ → ρ a and κ → τ a in [22])…”

“…Alsing et al [22] (abbreviated as AH) with ρ a → τ a real) and θ a = ω T a . The lossless case corresponds to α a → 1.…”

“…For the quantum derivation including internal propagation loss (generalizing the the lossless mrr considerations begun in [27]), AH [22] used an expression by Loudon [28,29] for the attenuation loss of a traveling wave, modeled from a continuous set of beams splitters acting as scattering centers [28,29],…”

“…This paper is organized as follows. In Section II we derive and solve the equations of motion for the pump, signal and idler fields within a mrr coupled to a single external bus using a combination of the formalism of Raymer and McKinstrie [21] and Alsing, et al [22]. We consider the weak, non-depleted pump field limit where higher order nonlinear processes such as self-phase and crossphase modulations effects are neglected.…”

“…This formalism is valid in the high cavity Q limit, near cavity resonances, but does not adequately address processes throughout the entire free spectral range of the cavity. In this work we investigate entangled photon pair generation in a microring resonator using a recent input-output formalism based on the work of Raymer and McKinstrie [21] and Alsing, et al [22] that incorporates the circulation factors that account for the multiple round trips of the fields within the cavity. We consider biphoton pair generation within the mrr via both SPDC and SFWM, and compute the generated two-photon signal-idler intra-cavity and output state from a single bus (all-through) microring resonator.…”