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Scientific Reports

Quantum entanglement and statistics of photons on a beam splitter in the form of coupled waveguides

Dmitry N. Makarov, Eugeny S Gusarevich, A. A. Goshev, K. A. Makarova, S. N. Kapustin, Anastasia A. Kharlamova, Yu. V. Tsykareva

Published May 13, 2021• 23 citations
Publisher

Abstract

It is well known that a beam splitter (BS) can be used as a source of photon quantum entanglement. This is due to the fact that the statistics of photons changes at the output ports of the BS. Usually, quantum entanglement and photon statistics take into account the constancy of the reflection coefficient R or the transmission coefficient T of the BS, where [Formula: see text]. It has recently been shown that if BS is used in the form of coupled waveguides, the coefficients R and T will depend on the photon frequencies. In this paper, it is shown that the quantum entanglement and statistics of photons at the output ports of a BS can change significantly if a BS is used in the form of coupled waveguides, where the coefficients R and T are frequency-dependent.

DOI
10.1038/s41598-021-89838-5

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It is well known that a beam splitter (BS) can be used as a source of photon quantum entanglement. This is due to the fact that the statistics of photons changes at the output ports of the BS. Usually, quantum entanglement and photon statistics take into account the constancy of the reflection coefficient R or the transmission coefficient T of the BS, where [Formula: see text]. It has recently been shown that if BS is used in the form of coupled waveguides, the coefficients R and T will depend on the photon frequencies. In this paper, it is shown that the quantum entanglement and statistics of photons at the output ports of a BS can change significantly if a BS is used in the form of coupled waveguides, where the coefficients R and T are frequency-dependent.
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 T, where R+T=1) and the phase shift ϕ. It has recently been shown that the constancy of these coefficients cannot always be satisfied for a waveguide BS, where R and ϕ 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.
It is known that the beam splitter in the form of coupled waveguides (BS) is one of the main devices used in quantum optics and quantum technologies. A BS has two independent parameters: one is the reflection coefficient R or the transmission coefficient T, where [Formula: see text]; the second is the phase shift [Formula: see text]. In various applications of quantum optics, these coefficients are considered constant. This is due to the fact that the frequency dependence of these coefficients is usually not taken into account, or this dependence is such that it cannot affect the constancy of these coefficients. It is shown that the coefficients R, T and phase shift [Formula: see text] are generally values that depend on the frequencies of incoming photons, the interaction time of photons in the BS, and the type of BS. It is established that in general, R, T and [Formula: see text] cannot be considered constant coefficients, and the criteria for when they can be considered constant are defined. The results obtained must be taken into account when analyzing and planning experiments where the beam splitter is presented in the form of coupled waveguides.
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.
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.
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.
Usually in quantum optics, the theory of large- and small-scale waveguide beam splitters is the same. In this paper, it is shown that the theory of the nanoscale waveguide beamsplitter has a significant difference from a similar device, but of a larger scale. It is shown that the previously known theory of the waveguide beam splitter is a particular case of the theory presented here. The wave function at the output ports of the nanoscale beam splitter is analyzed. The results obtained are sensitive to the size of the beam splitter, the coupling parameter of the two waveguides, and the degree of nonmonochromaticity of the photons entering the first and second ports of the beam splitter. The results are important for quantum technologies using a nanosized beam splitter.