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Simple Expressions for the Quantum Entanglement of Non-Monochromatic Photons

Yu. V. Tsykareva, D. N. Makarov, Ю. В. Цыкарева, Д. Н. Макаров

Published Jan 1, 2022• 0 citations
OpenAlex

Abstract

Simple expressions for the quantum entanglement of monochromatic and non-monochromatic photons on a waveguide beam splitter were found and analyzed using von Neumann entropy. It will be shown the great difference between both cases, which can be used to create large quantum entanglement.

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Europhysics Letters (EPL)•May 1, 2025•0 citations
It is well known that the waveguide beam splitter can be used as a source for the quantum entanglement of photons. The analysis of such quantum entanglement is a difficult problem even for monochromatic photons, since the system under study is multiparametric. This paper will show that quantum entanglement can be represented in a simple form not only for monochromatic photons but also for non-monochromatic ones. It will be shown that quantum entanglement for non-monochromatic photons can be very different from monochromatic photons, which can be used to create large quantum entanglement.
Relatively recently, it became known that the interaction of photons with free electrons leads to quantum entanglement of photons. Such photons can be used in many applications of quantum technologies. Typically, to study quantum entanglement, photons are assumed to be monochromatic, although in reality they are not. This paper shows that if non-monochromatic photons are taken into account, their quantum entanglement, based on von Neumann entropy, can differ significantly from the case of monochromatic photons. It has been shown that the entanglement of non-monochromatic photons can be large and reach the maximum possible value.
Studying beam splitters as sources of entangled photons is a relevant topic in modern physics. In most cases, in quantum optics the theories of waveguide beam splitters of large and small sizes have no difference. In this work, it is shown that a nanosized waveguide beam splitter has a significant difference from a similar device, but on a larger scale. The output wave function of the beam splitter’s ports is analyzed in the arbitrary case of photons entering the first and second ports. The results obtained are sensitive to the size of the waveguide beam splitter, the coupling parameter of two waveguides, the monochromatic photons case of a linear beam splitter and the non-monochromaticity grade for photons entering both input ports of the beam splitter. The results are very important for various quantum technologies (e.g. integrated circuits, optical quantum computers) which use this type of beam splitter and quantum metrology.
In this work, we present a comprehensive analytical study of a lossless quantum beam splitter (BS) with transmission coefficient T and reflection coefficient R. Using the unitary operator representation and binomial expansion, we derive the exact probability distribution P_k for detecting k photons in one output port when the input ports are prepared in Fock states |N>_0 |M>_1. The balanced configuration R = T = 1/2 and N = M is examined in detail: we prove that destructive quantum interference suppresses all odd photon numbers at the outputs, leaving only even photon counts. The special case N = M = 1 reproduces the famous Hong-Ou-Mandel (HOM) dip with P_1 = 0. Beyond photon statistics, we demonstrate that the beam splitter functions as a deterministic, tunable source of photon-number entanglement. By tracing over one output mode, we construct the reduced density matrix rho_2 and compute the von Neumann entropy S = - sum_k P_k log2 P_k, a direct measure of entanglement between the two output ports. For the balanced case with N = M = 1, we obtain S = 1 ebit, i.e., maximal Bell-state entanglement. For larger equal photon numbers, the entropy increases monotonically. Our results transform the passive beam splitter into a deterministic source of photon-number entanglement, with direct applications in optical quantum computing, interferometry, and quantum information processing. Numerical three-dimensional plots of the probability distribution are provided, and all derivations are supported by rigorous algebra and a full appendix.
In non-extensive statistical mechanics [14], it is a nonsense statement to say that the entropy of a system is extensive (or not), without mentioning a law of composition of its elements. In this theory quantum correlations might be perceived through quantum information process. This article, that is an extension of recent work [4], is a comparative study between the entropies of Von Neumann and of Tsallis, with some implementations of the effect of entropy in quantum entanglement, important as a process for transmission of quantum information. We consider two factorized (Fock number) states, which interact through a beam splitter bilinear Hamiltonian with two entries. This comparison showed us that the entropies of Tsallis and Von Neumann behave differently depending on the reflectance of the beam splitter. © 2011 Academic Publications.
Abstract Recently it has been shown that quantum entanglement of photons is possible when they interact with free electrons. In reality, photons are not monochromatic and in many cases it is necessary to take into account the spectral width of such photons. In this paper, the Schmidt expansion for non-monochromatic photons interacting with free electrons is found. Their quantum entanglement is found based on the Schmidt parameter. It is shown that the degree of non-monochromaticity of photons greatly affects the result of quantum entanglement. As a result, the results obtained show that it is necessary to take into account the spectral width of incident photons. The limits of the spectral width of incident photons are established, when it can be neglected and photons can be considered monochromatic.