Photon Statistics for Fock and Coherent States Interfering in a Beamsplitter

TL;DR

The paper addresses how different quantum and classical input states transform at a beamsplitter and how to quantify the resulting photon statistics. It uses the lossless beamsplitter model with standard transformations to derive explicit output states for Fock–Fock, Fock–coherent, and coherent–coherent inputs, and then computes mean photon numbers, variances, Mandel $Q$, and $g^{(2)}(0)$ for the outputs. The key contributions include closed-form expressions for output states and detailed analysis of how quantum features like antibunching and Hong–Ou–Mandel interference manifest in local statistics and how coherent components mediate a quantum-to-classical transition. This work provides a clear, pedagogical framework for understanding interference-induced statistics in linear optics, with implications for state characterization and quantum information applications.

Abstract

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.

Figures (5)

• Figure 1: Scheme of a lossless beamsplitter. Input modes $a$ and $b$ carry input states $\ket{in}_a$ and $\ket{in}_b$, respectively, while output modes $c$ and $d$ are linear combinations of the inputs. The BS is characterized by its complex reflection and transmission amplitudes, $(r, t)$ for input $a$ and $(r', t')$ for input $b$, which satisfy the unitarity conditions for a lossless device.
• Figure 2: Mandel $Q_c$ parameter for mode $c$ of a 50:50 BS with Fock-Fock state input of the type $|n\rangle_a|m\rangle_b$, where $|r|^2 = |t|^2 = 1/2$.
• Figure 3: $g^{(2)}(0)$ for mode $c$ of a 50:50 BS with Fock-Fock input of the type $|n\rangle_a|m\rangle_b$, where $|r|^2=|t|^2=1/2$.
• Figure 4: Mandel $Q_c$ parameter for mode $c$ of a 50:50 BS with Fock-coherent state input of the type $|n\rangle_a|\alpha\rangle_b$, where $|r|^2 = |t|^2 = 1/2$. Numerical parameters: $\alpha \in [0,3]$, $n \in \{0,1,2,5,10\}$.
• Figure 5: $g^{(2)}(0)$ for mode $c$ of a 50:50 BS with Fock-coherent input of the type $|n\rangle_a|\alpha\rangle_b$, where $|r|^2=|t|^2=1/2$.