Theory for the beam splitter in quantum optics: quantum entanglement of photons and their statistics, HOM effect

TL;DR

This paper reviews the theory of beam splitters in quantum optics, emphasizing the differences between conventional and frequency-dependent types, and explores their impact on quantum entanglement, photon statistics, and the HOM effect, especially in waveguide systems.

Contribution

It systematically classifies beam splitters into conventional and frequency-dependent types and develops the theory for the latter, highlighting their unique quantum properties.

Findings

Frequency-dependent beam splitters can significantly alter entanglement and photon statistics.

Waveguide beam splitters exhibit variable reflection coefficients and phase shifts depending on photon frequency.

The Hong-Ou-Mandel effect is notably affected by the frequency dependence of the beam splitter.

Abstract

The theory of the beam splitter (BS) in quantum optics is well developed and based on fairly simple mathematical and physical foundations. This theory has been developed for any type of BS and is based on the constancy of the reflection coefficients $R$ (or the transmission coefficient, where $R+T=1$) and the phase shift $\phi$. It has recently been shown that the constancy of these coefficients cannot always be satisfied for a waveguide BS, where $R$ and $\phi$ depend in a special way on photon frequencies. Based on this, this review systematizes the concept of BS in quantum optics into ``Conventional'' and frequency-dependent BS, and also presents the theory of such BS. It is shown that the quantum entanglement, photon statistics at the output ports, and the Hong-Ou-Mandel (HOM) effect for such BS can be very different. Taking into account the fact that the waveguide BS is currently acquiring an important role in quantum technologies due to the possibility of its miniaturization, this review will be useful not only for theoreticians, but also for experimenters.