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Journal of the Optical Society of America B

How does it scale? Comparing quantum and classical nonlinear optical processes in integrated devices

L. G. Helt, Marco Liscidini, John E. Sipe

Published Aug 1, 2012• 153 citations
Publisher

Abstract

We present equations for the power generated via spontaneous (quantum) and stimulated (classical) nonlinear optical processes in integrated devices. Equations for the same structure and same order process are derived from the same Hamiltonian, allowing for direct and easy comparison including the ability to estimate the efficiency of a quantum process based solely on experimental data from a classical process in the same device. We show that, in the CW limit and under the undepleted pump approximation, the average energy of a generated photon divided by a characteristic time plays the role of the classical “seed” signal in a quantum process, and that extending the length of a structure or taking advantage of a resonant cavity does not enhance spontaneous processes the same way as stimulated processes.

DOI
10.1364/josab.29.002199

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We present equations for the power generated via spontaneous (quantum) and stimulated (classical) nonlinear optical processes in integrated devices. Equations for the same structure and same order process are derived from the same Hamiltonian, allowing for direct and easy comparison including the ability to estimate the efficiency of a quantum process based solely on experimental data from a classical process in the same device. We show that, in the CW limit and under the undepleted pump approximation, the average energy of a generated photon divided by a characteristic time plays the role of the classical “seed” signal in a quantum process, and that extending the length of a structure or taking advantage of a resonant cavity does not enhance spontaneous processes the same way as stimulated processes.
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Integrated optical devices will play a central role in the future development of nonlinear quantum photonics. Here we consider the generation of nonclassical states of light within them with a focus on Gaussian states beyond the low-gain, single photon pair regime accurately described by perturbation theory. Starting from the solid foundation provided by Maxwell’s equations, we then move to applications by presenting a unified formulation that allows for a comparison of stimulated and spontaneous experiments in ring resonators and nanophotonic waveguides and leads directly to the calculation of the quantum states of light generated in high-gain nonlinear quantum photonic experiments.
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Photonic integrated circuits with second-order ( χ (2) ) nonlinearities are rapidly scaling to remarkably low powers. At this time, state-of-the-art devices achieve saturated nonlinear interactions with thousands of photons when driven by continuous-wave lasers, and further reductions in these energy requirements enabled by the use of ultrafast pulses may soon push nonlinear optics into the realm of single-photon nonlinearities. This tutorial reviews these recent developments in ultrafast nonlinear photonics, discusses design strategies for realizing few-photon nonlinear interactions, and presents a unified treatment of ultrafast quantum nonlinear optics using a framework that smoothly interpolates from classical behaviors to the few-photon scale. These emerging platforms for quantum optics fundamentally differ from typical realizations in cavity quantum electrodynamics due to the large number of coupled optical modes. Classically, multimode behaviors have been well studied in nonlinear optics, with famous examples including soliton formation and supercontinuum generation. In contrast, multimode quantum systems exhibit a far greater variety of behaviors, and yet closed-form solutions are even sparser than their classical counterparts. In developing a framework for ultrafast quantum optics, we identify what behaviors carry over from classical to quantum devices, what intuition must be abandoned, and what new opportunities exist at the intersection of ultrafast and quantum nonlinear optics. Although this article focuses on establishing connections between the classical and quantum behaviors of devices with χ (2) nonlinearities, the frameworks developed here are general and are readily extended to the description of dynamical processes based on third-order χ (3) nonlinearities.