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Integrated optical devices will play a central role in the future development of nonlinear quantum photonics. Here we focus on the generation of high-gain nonclassical light within them. 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 experiments.

Growing interest in quantum computing for practical applications has led to a surge in the availability of programmable machines for executing quantum algorithms. Present day photonic quantum computers have been limited either to non-deterministic operation, low photon numbers and rates, or fixed random gate sequences. Here we introduce a full-stack hardware-software system for executing many-photon quantum circuits using integrated nanophotonics: a programmable chip, operating at room temperature and interfaced with a fully automated control system. It enables remote users to execute quantum algorithms requiring up to eight modes of strongly squeezed vacuum initialized as two-mode squeezed states in single temporal modes, a fully general and programmable four-mode interferometer, and genuine photon number-resolving readout on all outputs. Multi-photon detection events with photon numbers and rates exceeding any previous quantum optical demonstration on a programmable device are made possible by strong squeezing and high sampling rates. We verify the non-classicality of the device output, and use the platform to carry out proof-of-principle demonstrations of three quantum algorithms: Gaussian boson sampling, molecular vibronic spectra, and graph similarity.

Photonic molecules are composed of two or more optical resonators, arranged such that some of the modes of each resonator are coupled to those of the other. Such structures have been used for emulating the behaviour of two-level systems, lasing, and on-demand optical storage and retrieval. Coupled resonators have also been used for dispersion engineering of integrated devices, enhancing their performance for nonlinear optical applications. Delicate engineering of such integrated nonlinear structures is required for developing scalable sources of non-classical light to be deployed in quantum information processing systems. In this work, we demonstrate a photonic molecule composed of two coupled microring resonators on an integrated nanophotonic chip, designed to generate strongly squeezed light uncontaminated by noise from unwanted parasitic nonlinear processes. By tuning the photonic molecule to selectively couple and thus hybridize only the modes involved in the unwanted processes, suppression of parasitic parametric fluorescence is accomplished. This strategy enables the use of microring resonators for the efficient generation of degenerate squeezed light: without it, simple single-resonator structures cannot avoid contamination from nonlinear noise without significantly compromising pump power efficiency, and are thus limited to generating only weak degenerate squeezing. We use this device to generate 8(1) dB of broadband degenerate squeezed light on-chip, with 1.65(1) dB directly measured, which is the largest amount of squeezing yet reported from any nanophotonic source., Comment: Significantly updated: improved data, results, and experimental details

We experimentally demonstrate stimulated four-wave mixing in two linearly uncoupled integrated Si$_3$N$_4$ micro-resonators. In our structure the resonance combs of each resonator can be tuned independently, with the energy transfer from one resonator to the other occurring in the presence of a nonlinear interaction. This method allows flexible and efficient on-chip control of the nonlinear interaction, and is readily applicable to other third-order nonlinear phenomena.

We experimentally demonstrate the generation of a three-photon discrete-energy-entangled W state using multi-photon-pair generation by spontaneous four-wave mixing in an optical fiber. We show that by making use of prior information on the photon source we can verify the state produced by this source without resorting to frequency conversion.

We report demonstrations of both quadrature squeezed vacuum and photon number difference squeezing generated in an integrated nanophotonic device. Squeezed light is generated via strongly driven spontaneous four-wave mixing below threshold in silicon nitride microring resonators. The generated light is characterized with both homodyne detection and direct measurements of photon statistics using photon number-resolving transition edge sensors. We measure $1.0(1)$~dB of broadband quadrature squeezing (${\sim}4$~dB inferred on-chip) and $1.5(3)$~dB of photon number difference squeezing (${\sim}7$~dB inferred on-chip). Nearly-single temporal mode operation is achieved, with measured raw unheralded second-order correlations $g^{(2)}$ as high as $1.95(1)$. Multi-photon events of over 10 photons are directly detected with rates exceeding any previous quantum optical demonstration using integrated nanophotonics. These results will have an enabling impact on scaling continuous variable quantum technology., Comment: Minor revisions, and Fig. 5 corrected; Now published in Science Advances

We demonstrate a system composed of two resonators that are coupled solely through a nonlinear interaction, and where the linear properties of each resonator can be controlled locally. We show that this class of dynamical systems has peculiar properties with important consequences for the study of classical and quantum nonlinear optical phenomena. As an example we discuss the case of dual-pump spontaneous four-wave mixing., Comment: 4 Figures

We propose a novel squeezed light source capable of meeting the stringent requirements of continuous variable quantum sampling. Using the effective $\chi_2$ interaction induced by a strong driving beam in the presence of the $\chi_3$ response in an integrated microresonator, our device is compatible with established nanophotonic fabrication platforms. With typical realistic parameters, squeezed states with a mean photon number of 10 or higher can be generated in a single consistent temporal mode at repetition rates in excess of 100MHz. Over 15dB of squeezing is achievable in existing ultra-low loss platforms.

We study the generation of Greenberger-Horne-Zeilinger (GHZ) states of three path-encoded photons. Inspired by the seminal work of Bouwmeester et al. [1] on polarization-entangled GHZ states, we find a corresponding path representation for the photon states of an optical circuit, identify the elements required for the state generation, and propose a possible implementation of our strategy. Besides the practical advantage of employing an integrated system that can be fabricated with proven lithographic techniques, our example suggests that it is possible to enhance the generation efficiency by using microring resonators., Comment: 8 pages, 4 figures

We demonstrate that an integrated silicon microring resonator is capable of efficiently producing photon pairs that are completely unentangled; such pairs are a key component of heralded single photon sources. A dual-channel interferometric coupling scheme can be used to independently tune the quality factors associated with the pump and signal and idler modes, yielding a biphoton wavefunction with Schmidt number arbitrarily close to unity. This will permit the generation of heralded single photon states with unit purity., Comment: 5 pages, 3 figures

Tripartite entangled states, such as GHZ and W states, are typically generated by manipulating two pairs of polarization-entangled photons in bulk optics. Here we propose a scheme to generate W states that are entangled in the energy degree of freedom in an integrated optical circuit. Our approach employs photon pairs generated by spontaneous four-wave mixing (SFWM) in a micro- ring resonator. We also present a feasible procedure for demonstrating the generation of such a state, and we compare polarization-entangled and energy-entangled schemes for the preparation of W states.