Characterizing photon number statistics using conjugate optical homodyne detection

We study the problem of determining the photon number statistics of an unknown quantum state using conjugate optical homodyne detection. We quantify the information gain in a single-shot measurement and show that the photon number statistics can be recovered in repeated measurements on an ensemble of identical input states without scanning the phase of the input state or randomizing the phase of the local oscillator used in homodyne detection. We demonstrate how the expectation maximization algorithm and Bayesian inference can be utilized to facilitate the reconstruction and illustrate our approach by conducting experiments to study the photon number distributions of a weak coherent state and a thermal state source.