Frequency Engineered Measurements Connecting Homodyne and Heterodyne

Extracting information from optical and RF fields is an essential part of quantum metrology, quantum networking, and quantum computing protocols. Alongside direct detection, the two most widely used measurements are homodyne and heterodyne which both incorporate mixing with a local oscillator (LO) to retain phase information. Homodyne measures a single quadrature of the field while heterodyne measures both quadratures simultaneously. There are myriad reasons to prefer homodyne or heterodyne for specific applications in quantum optics. We present a fully quantum and measurement theoretic description of the frequency dependent mixed measurement apparatus that is used for both homodyne and heterodyne. By deriving the positive operator valued measure (POVM) for this apparatus we are able to rederive general forms of homodyne and heterodyne and every measurement in between. We demonstrate methods to measure arbitrary combinations of field quadratures and apply these measurements to the important frequency dependent system of frequency comb metrology.