Quantum and Stochastic Dynamics of Kerr Microcombs

Abstract

Kerr microcombs are sets of discrete, equidistant spectral lines and are typically generated by pumping a high-quality factor optical resonator with a continuous- or pulse-wave resonant laser. They have emerged as one of the most important research topics in photonics nowadays, with applications related to spectroscopy, sensing, aerospace, and communication engineering. A key characteristic of these microcombs is the threshold pump power. Below the threshold, two pump photons are symmetrically up- and down-converted as twin photons via spontaneous four-wave mixing, and they can be entangled across up to a hundred eigenmodes. These chipscale, high-dimensional, and room-temperature systems are expected to play a major role in quantum engineering. Above the threshold, the four-wave mixing process is stimulated, ultimately leading to the formation of various types of patterns in the spatio-temporal domain, which can be extended (such as roll patterns), or localized (bright or dark solitons). The semiclassical dynamics of Kerr microcombs have been studied extensively in the last ten years and the deterministic characteristics are well understood. However, the quantum dynamics of the twin-photon generation process, and the stochastic dynamics led by the noise-driven fluctuations, are still not so clear. In the first part of our investigation, we introduce the theoretical framework to study the semiclassical dynamics of the Kerr microcombs based on the slowly varying envelope of the intracavity electrical fields. Two equivalent models -- the coupled-mode model and the Lugiato-Lefever model are used to analyze the spectro- and spatio-temporal dynamics, respectively. These models can determine the impact of key parameters on the Kerr microcomb generation process, such as detuning, losses, and pump power, as well as critical values of the system, such as threshold power. Various types of patterns and combs can be observed through simulations that follow experimental parameters. Furthermore, we show an eigenvalue analysis method to determine the stability of the microcomb, and this method is applied to an unstable microcomb solution to understand the generation of subcombs surrounding the primary comb. In the second and third parts, we investigate a stochastic model where noise is added to the coupled-mode equations governing the microcomb dynamics to monitor the influence of random noise on the comb dynamics. We find the model with additive Gaussian white noise allows us to characterize the noise-induced broadening of spectral lines and permits us to determine the phase noise spectra of the microwaves generated via comb photodetection. Our analysis indicates that the low-frequency part of the phase noise spectra is dominated by pattern drift while the high-frequency part is dominated by pattern deformation. The dynamics of the Kerr microcomb with multiplicative noises, including thermal and photothermal fluctuations, are also investigated in the end. We propose that the dynamics of the noise can be included in the simulation of stochastic dynamics equations, introduce the methods to solve the dynamics of the noise, and study a quiet point method for phase noise reduction. In the fourth part, we use canonical quantization to obtain the quantum dynamics for Kerr microcombs generated by spontaneous four-wave mixing below the threshold and develop the study of them using frequency-bin quantum states. We introduce a method to find the quantum expansion of the output state and explore the properties of the eigenkets. A theoretical framework is also developed to obtain explicit solutions for density operators of quantum microcombs, which allows us to obtain their complete characterization, as well as for the analytical determination of various performance metrics such as fidelity, purity, and entropy. Finally, we describe a quantum Kerr microcomb generator with a pulse-wave laser and propose the time-bin entangled states generated by it.