We study the problem of determining the photon number statistics of an unknown quantum state by simultaneously measuring conjugate quadratures with double homodyne detectors. Classically, the sum of the squared outputs of the two homodyne detectors is proportional to the intensity (thus the photon number) of the input light. Quantum mechanically, due to vacuum noise, the above photon number measurement is intrinsically noisy. We quantify the information gain in a single-shot measurement and discuss potential applications of this technology in quantum key distribution. We also 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.
@article{arxiv.1702.02558,
title = {Characterizing photon number statistics using conjugate optical homodyne detection},
author = {Bing Qi and Pavel Lougovski and Brian P. Williams},
journal= {arXiv preprint arXiv:1702.02558},
year = {2020}
}