Detecting Topological Phases of Microwave Photons in a Circuit Quantum Electrodynamics Lattice

Topology is an important degree of freedom in characterizing electronic systems. Recently, it also brings new theoretical frontiers and many potential applications in photonics. However, the verification of the topological nature is highly nontrivial in photonic systems as there is no direct analog of quantized Hall conductance for bosonic photons. Here we propose a scheme of investigating topological photonics in superconducting quantum circuits by a simple parametric coupling method, the flexibility of which can lead to the effective \textit{in situ} tunable artificial gauge field for photons on a square lattice. We further study the detection of the topological phases of the photons. Our idea employs the exotic properties of the edge state modes which result in novel steady states of the lattice under the driving-dissipation competition. Through the pumping and the photon-number measurements of merely few sites, not only the spatial and the spectral characters, but also the momentums and even the integer topological quantum numbers with arbitrary values of the edge state modes can be directly probed, which reveal unambiguously the topological nature of photons on the lattice.