Digital homodyne and heterodyne detection for stationary bosonic modes

Homodyne and heterodyne detection techniques are extensively used to characterize and herald dynamics of propagating modes within the field of quantum optics. However, implementing these methods on stationary modes confined within high-quality superconducting cavities is challenging. Here, we show a protocol tailored for stationary cavities that yields measurement statistics identical to those from homo- or heterodyne detectors. This protocol is based on repeated interactions between the cavity and a qubit, followed by qubit measurements and a weighted integration of the binary measurement outcomes. Our approach enables both balanced and unbalanced heterodyne detection on stationary modes, thereby granting access to a comprehensive toolbox, which for example includes quantum verification protocols.