On the entanglement swapping by using the beam splitter

In this paper, we outline three different setups for the entanglement swapping process. The common aspect of these setups is generation of the atom-field entangled state in the optical cavity and using a 50 : 50 beam splitter to swap the entanglement to the new subsystems. The first scheme contains two distinct atom-field entangled states (|ψ〉AjFj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \vert\psi\rangle_{{\rm A}_{j}{\rm F}_{j}}$\end{document}, j = 1, 2), each previously generated via the Jaynes-Cummings model. According to our proposal, once the two field states are emitted into the beam splitter and when the field F1 is detected, the two participating atoms, without detecting F2 , form an entangled state. Therefore, the entanglement swapping from “Aj -Fj ”, j = 1, 2 to the two atoms appropriately occurs. The second scheme contains an entangled state of “ A1 - F1 ” as well as a cavity field F2 , in which the entangled state is generated by using two Ramsey zones and the atom-field interaction in the large detuning regime. Injecting the fields F1 and F2 into the beam splitter, operating an inverse Hadamard gate on the atom A1 and the atom-field interacted subsystem “ A1 - F1 ” in the large detuning regime, helps us transfer the entanglement to the “ A1 - F2 ” subsystem. In the final scheme, the atom A1 is initially entangled with the cavity field F1 via the atom-field interaction in the optical cavity. Also, the other field F2 is considered in the coherent state |α〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert\alpha\rangle$\end{document}. Injecting the state of the two fields into the 50 : 50 beam splitter, one arrives at an entangled state between A1 and F2 for small and large values of |β|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \vert\beta\vert$\end{document} (a measure of the coherent field intensity). Accordingly, again the entanglement swapping takes place properly.