Photon Statistics for Fock and Coherent States Interfering in a Beamsplitter

We present a straightforward yet comprehensive theoretical study of different quantum states emerging from a bi-modal beamsplitter when various input states interfere. Specifically, we analyze the output states for different combinations of input fields, including Fock states $|n\rangle|m\rangle$, hybrid states $|n\rangle|α\rangle$, and coherent states $|α\rangle|β\rangle$. We derive explicit expressions for the output state vectors, calculate the mean photon number, photon number variance, Mandel Q parameter, and secondorder coherence function to characterize the statistical properties of the output fields. Our results are intended as a pedagogical resource, serving as an introductory reference for students and researchers aiming to understand basic photon statistics using beamsplitters.