Cavity quantum material correlations from Keldysh field theory

In recent years, strong light-matter coupling has emerged as a promising mechanism to tune many-body correlations in electronic and spin systems. Notable examples and theoretical proposals include realizing dressed light-matter states, tuning superconducting order parameters, and stabilizing exotic magnetic orders. However, a key question remains in how to probe these light-matter entangled states. Here, we propose to use quantum optical photon counting measurements to deduce the phase and correlation functions of the material. We construct a theoretical relation between the input photon state, the correlation functions of the material in the cavity, and the correlation functions of the light output by a leaky cavity. We use Keldysh field theory to both generalize quantum optical input-output relations to extended quantum many-body systems, and to relate correlation functions to photon coherences. Our work builds a bridge between theoretically proposed cavity quantum materials and experimental quantum photon spectroscopy.