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Modeling nonlinear optics in lossy microring systems: two strategies

Milica Banić, Luca Zatti, Marco Liscidini, John E. Sipe

Published Nov 29, 2021• 0 citations
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Abstract

We present two complementary strategies for modeling nonlinear quantum optics in realistic resonant integrated optical devices, where scattering loss is present. In the first strategy, we model scattering loss as an effective absorption; in the second, we employ a Hamiltonian treatment, with the loss modelled as a coupling to a 'phantom' channel. As an example, we use these two approaches to model spontaneous four-wave mixing in (i) a ring-channel system and (ii) an add-drop system. We present the rates of photon pairs, broken pairs, and lost pairs for both systems, as well as the full biphoton wavefunction (BWF) including the effects of scattering. We find that for a high-finesse resonator coupled to an arbitrary number of channels, the full BWF can be extracted from the BWF associated with photon pairs in an experimentally accessible output channel.

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We present two complementary strategies for modeling nonlinear quantum optics in realistic resonant integrated optical devices, where scattering loss is present. In the first strategy, we model scattering loss as an effective absorption; in the second, we employ a Hamiltonian treatment, with the loss modelled as a coupling to a 'phantom' channel. As an example, we use these two approaches to model spontaneous four-wave mixing in (i) a ring-channel system and (ii) an add-drop system. We present the rates of photon pairs, broken pairs, and lost pairs for both systems, as well as the full biphoton wavefunction (BWF) including the effects of scattering. We find that for a high-finesse resonator coupled to an arbitrary number of channels, the full BWF can be extracted from the BWF associated with photon pairs in an experimentally accessible output channel.
We present two complementary strategies for modeling nonlinear quantum optics in realistic integrated optical devices, where scattering loss is present. In the first strategy, we model scattering loss as an attenuation; in the second, we employ a Hamiltonian treatment that includes a mechanism for scattering loss, such as a phantom waveguide. These strategies can be applied to a broad range of structures and processes. As an example, we use these two approaches to model spontaneous four-wave mixing in (i) a ring-channel system and (ii) an add-drop system. Even for these well-understood systems, our strategies yield some additional results. We show the rates of photon pairs, broken pairs, and lost pairs and their dependence on system parameters. We show that the properties of lost and broken photon pairs in such structures can be related to those of the unscattered photon pairs, which are relatively simple to measure.
We develop a general Hamiltonian treatment of spontaneous four-wave mixing in a microring resonator side-coupled to a channel waveguide. The effect of scattering losses in the ring is included, as well as parasitic nonlinear effects including self- and cross-phase modulation. A procedure for computing the output of such a system for arbitrary parameters and pump states is presented. For the limit of weak pumping an expression for the joint spectral intensity of generated photon pairs, as well as the singles-to-coincidences ratio, is derived.
This thesis presents an analysis of photon pairs generated via either spontaneous parametric downconversion or spontaneous four-wave mixing in channel waveguides as well as in microring resonators side-coupled to channel waveguides. The state of photons exiting a particular device is calculated within a general Hamiltonian formalism that simplifies the link between quantum nonlinear optics experiments and classical nonlinear optics experiments. This state contains information regarding photon pair production efficiency as well as modal and spectral correlations between the two photons, characterized by a two-dimensional spectral distribution function called the biphoton wave function.\n\nIn the limit of a low probability of pair production, photon pair production fficiencies are cast into forms resembling corresponding well-known classical nonlinear optical frequency conversion efficiencies, making it easy to see what plays the role of a classical “seed” field in an un-seeded (quantum) process. This also allows photon pair production efficiencies to be calculated based on the results of classical nonlinear optical experiments. It is further calculated that, unless enerated photons are collected over a very narrow frequency range, their generation efficiency does not scale the same way with device length in a channel waveguide, or resonance quality factor in a microring resonator, as might be expected from the corresponding classical frequency conversion efficiency. Although calculations do not include self- or cross-phase modulation, nor two-photon absorption\nor free-carrier absorption, it is calculated that their neglect is justified in the low pair production probability limit. Linear (scattering) loss is also neglected, though partially addressed in the final chapter of this thesis.\n\nBiphoton wave functions are calculated explicitly, such that their shape and orientation, including approximate analytic expressions for their widths, can easily be determined. This further allows estimation of the suitability of their associated photon pairs for various quantum information processing applications. As an alternative to dispersion engineering a channel waveguide photon pair source, it is calculated that microring resonators can very naturally produce nearly spectrally uncorrelated photon pairs, which behave very much like idealized single-mode photons and are thus useful for applications involving the interference of photons from multiple sources.
We present a self-contained analytical model for biphoton generation in microring resonators. Encompassing both all-pass and add-drop geometries, identical and distinct pump and biphoton coupling coefficients, and continuous-wave and pulsed pumping, our interaction-picture-based approach reveals time-frequency biphoton correlations while also predicting absolute generation rates. Under continuous-wave excitation, we find critical coupling of both the pump and biphoton to maximize the rate of single photons extracted from the microring, whereas critical coupling of the pump but overcoupling of the biphoton maximize the two-photon rate. Under pulsed pumping, overcoupling of both pump and biphoton (to different degrees) maximizes photon extraction probabilities, albeit under a tradeoff with spectral factorability that we quantify via parameter scans over a range of coupler pairings. As a whole, our formalism should prove valuable for the practical design of integrated photon sources, merging a flexible and intuitive biphoton-centric depiction with quantitative predictions closely tied to experimental parameters.
Microring resonators (MRR) are becoming a standard resource for highly efficient entangled photon pair generation. We examine the nonlinear processes in MRR using an input-output formalism that explicitly accounts for the contribution of the round trip circulation of the fields inside the MRR. This formalism reduces to the standard Langevin operator input-output formalism for cavities in the high cavity Q limit, near cavity resonances. Our approach is more general, applicable for arbitrary internal and bus-ring coupling losses, and away from resonances. We compare/contrast these two formalisms, and apply our approach to the output quantum state resulting from spontaneous parametric down conversion and spontaneous four wave mixing within the MRR. We compare our results to similar calculations employing the standard IO formalism, and note the role of the commutators of the noise operators on the undesired accidental singles rate.