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Physical Review A

Spontaneous parametric down-conversion in waveguides: A backward Heisenberg picture approach

Zhenshan S. Yang, Marco Liscidini, John E. Sipe

Published Mar 5, 2008• 120 citations
Publisher

Abstract

We develop a fully quantum-mechanical approach for the description of spontaneous parametric down-conversion in waveguides. Our method allows us to write an expression for the quantum state easily and directly, even when multiple photon pairs are generated in the process. The role of material dispersion in the normalization of the modes of the system, usually neglected, is properly treated here. The example of an ${\text{Al}}_{y}{\text{O}}_{x}/{\text{Al}}_{0.5}{\text{Ga}}_{0.5}\text{As}$ ridge waveguide is discussed to illustrate the application of the formalism.

DOI
10.1103/physreva.77.033808

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Conference on Lasers and Electro-Optics•Jan 1, 2018•0 citations
We develop a fully quantum-mechanical approach for the description of spontaneous parametric down-conversion in waveguides. Our method allows us to write an expression for the quantum state easily and directly, even when multiple photon pairs are generated in the process. The role of material dispersion in the normalization of the modes of the system, usually neglected, is properly treated here. The example of an AlyOx/Al0.5Ga0.5As{\text{Al}}_{y}{\text{O}}_{x}/{\text{Al}}_{0.5}{\text{Ga}}_{0.5}\text{As}Aly​Ox​/Al0.5​Ga0.5​As ridge waveguide is discussed to illustrate the application of the formalism.
In this paper, a configuration consisting in a pump propagating in an orthogonal direction with respect to the surface of a nanowaveguide is studied. It is possible to show that the generation by spontaneous parametric down-conversion of a signal and an idler photons inside the nanowaveguide is strongly dependent on the waveguide thickness, because the high refractive index of the waveguide core creates also an optical cavity for the pump field. Furthermore, it was evaluated the best condition for photon-pair generation in case of a nanowaveguide in gallium indium phosphide, which has a −43m symmetry.For the same waveguide geometry, but orientation parallel to the [110] plane, it is possible to use spontaneous parametric down-conversion to generate counter-propagating photon-pair with the same polarization. When the efficiency of the generated photon-pairs in TM0 and TE0 modes is the same, then it is possible to obtain polarization entangled photons, by using this configuration. Furthermore, by adding a mirror at one of the end of the waveguide, the system can be used to produce a squeezed-state.
The ability to engineer pairs of entangled photons is essential to quantum information science, and generating these states using spontaneous parametric down-conversion (SPDC) in nano- and micrometer-scale materials offers numerous advantages. To properly engineer such sources, a reliable model describing nano- and micrometer-scale SPDC is necessary; however, such a theoretical description remains a challenge. Here, we propose and derive a simplified model to describe SPDC in resonant structures, which considers the generation of photon pairs and the resonant enhancement of spectral bands to be separate processes, even though they actually occur simultaneously. We compare our simplified model to both the rigorous theory of SPDC in an etalon - a simple example of a resonant structure - and our experiments on SPDC in etalons and find agreement for low-gain SPDC. By simplifying the calculations required to generate photon pairs, our model promises to make designing complex resonant structures easier, and it promises to hasten the iteration of designs across the field of quantum state engineering.
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.