Spectroscopy and topological properties of a Haldane light system

We introduce a local spectroscopic method in real space to probe the topological properties of a circuit quantum electrodynamics (cQED) array generalizing previous approaches from one to two dimensions in the plane. As an application, we develop the theory of microwave light propagating in the local probe capacitively coupled to the cQED array associated to a bosonic Haldane model. Interestingly, we show that the measured reflection coefficient, resolved in frequency through the resonance, reveals the model's geometrical properties and topological phase transition. We discuss the role of physical parameters such as the lifetime of the light modes and stability towards local disorder related to further realizations.