超导量子电路与拓扑能带理论的结合有望给出奇异的光子拓扑现象和一系列的新奇应用。然而，当前这一领域的实验研究实质上仍被局限于处在Abelian人工规范场下的单粒子图像。我们在本项目拟超越这一现状，从理论上研究在超导量子电路阵列中构造Non-Abelian规范场并观测由此导致的光子拓扑物理。我们将首先发展参量转换机制以实现阵列格点间的非平凡连接矩阵。这一方法具有其它物理系统难以达到的灵活性，可望实现具有任意形式且局域可调的Non-Abelian规范场。进一步的，我们将探索如何利用这一人工规范场在一维和二维超导量子电路阵列上展示非平凡的光学拓扑效应，包括探测表征体能带和边缘模式的整数拓扑不变量和研究人工规范场—光子非线性竞争下的光子量子随机游走动力学。我们将以当前的实验技术为出发点，既为超导量子电路体系的相关实验进展提供及时的理论支持，又为拓扑光子学的理论研究提供新的方法与思路。

The combination of superconducting quantum circuit (SQC) and topological band theory is expected to result in a variety of exotic photonic topological phenomena and novel practical applications. However, experimental research in this direction is still restricted in the realm of single-particle dynamics under synthetic Abelian gauge fields. In this project, we go beyond this situation by investigating theoretically the synthesization of artificial Non-Abelian gauge fields and the consequent demonstration of nontrivial photonic topological physics in SQC lattices. We study in the first step the realization of non-trivial link matrices between the sites on the lattices by the parametric frequency conversion formalism. The distinct merit of this method is the unprecedented flexibility it enjoys, which can hardly be achieved by other physical systems and can provide artificial Non-Abelian gauge fields with arbitrary form and in situ tunability. We then turn to explore the application of the synthesized artificial gauge fields to one-dimensional and two-dimensional SQC lattices. Especially, we focus on the realization and the measurement of the consequent non-trivial photonic topological effects, including the integer topological invariants of the bulk bands and the edge modes, and the quantum random walk dynamics of the photons under the competition between the synthetic gauge fields and the nonlinearity of the SQC lattice. Starting from current level of technology, our research is expected to provide immediate support to the related experimental progress in SQC and offer new ideas to the theory of topological photonics.