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Journal of Experimental and Theoretical Physics Letters

Quantum Entanglement of Photon-Added Coherent States on a Beam Splitter

Dmitry N. Makarov

Published May 1, 2025• 0 citations
Publisher

Abstract

The study of new quantum states and methods for their generation is an important part of the current development of quantum technologies. One of the most important characteristics of such states is their quantum entanglement. Photon-added coherent states, which are defined as $${\text{|}}\alpha ,n\rangle = {{\mathcal{N}}_{n}}{{({{\hat {a}}^{\dag }})}^{n}}{\text{|}}\alpha \rangle $$ , where $${{\mathcal{N}}_{n}}$$ is the normalization constant, $${\text{|}}\alpha \rangle $$ is the coherent state, $${{\hat {a}}^{\dag }}$$ is the creation operator, and n is the number of added photons, have been introduced relatively recently. The quantum entanglement of such states passing through a beam splitter has been considered for the first time in this work. A simple analytical representation of the wavefunction at the outputs of the beam splitter has been found and the quantum entanglement of the resulting two-qubit states has been considered. It has been shown that new quantum states have good prospects for their use in quantum technologies.

DOI
10.1134/s0021364025605986

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The quantum entanglement of photons is one of the interesting and important manifestations of the quantum world, which is already being used in quantum technologies. The study of new methods of quantum entanglement of photons is one of the promising areas of quantum optics. In this work, it is shown that photons, in a two-mode electromagnetic field, can be quantum entangled when they interact with free electrons. It is shown that the quantum entanglement of such photons can be very large and such photons can be a good source of high quantum entanglement. In addition, it is shown that free electrons can be a beam splitter for a two-mode electromagnetic field.
We prove that a beam splitter, one of the most common optical components, fulfills several classes of majorization relations, which govern the amount of quantum entanglement that it can generate. First, we show that the state resulting from kkk photons impinging on a beam splitter majorizes the corresponding state with any larger photon number ${k}^{\ensuremath{'}}>k$, implying that the entanglement monotonically grows with kkk. Then we examine parametric infinitesimal majorization relations as a function of the beam-splitter transmittance and find that there exists a parameter region where majorization is again fulfilled, implying a monotonic increase of entanglement by moving towards a balanced beam splitter. We also identify regions with a majorization default, where the output states become incomparable. In this latter situation, we find examples where catalysis may nevertheless be used in order to recover majorization. The catalyst states can be as simple as a path-entangled single-photon state or a two-mode squeezed vacuum state.
We propose an experimentally feasible scheme to generate various types of entangled states of light fields by using beam splitters and single-photon detectors. Two beams of light fields are incident on two beam splitters respectively with each beam being asymmetrically split into two parts in which one part is supposed to be so weak that it contains at most one photon. We let the two weak output modes interfere at a third beam splitter. A conditional joint measurement on both weak output modes may result in an entanglement between the other two output modes. The conditions for the maximal entanglement are discussed based on the concurrence. Several specific examples are also examined.
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.
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