Entanglement dynamics of two coupled spins interacting with an adjustable spin bath: effect of an exponential variable magnetic field

The present work is devoted to studying the entanglement dynamics of two central spins coupled in a spin environment and subjected, simultaneously, to an external magnetic field changing with time t as an exponential function B1-e-λt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {B}}\left( 1-\mathrm{e}^{-\lambda t}\right) $$\end{document}. We want to determine whether interaction among central spins with an external magnetic field as well as preparation of bath in an appropriate spin coherent state, |β⟩bath\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\beta \rangle _\mathrm{{bath}}$$\end{document}, is shown to affect the decoherence process in a qualitatively significant manner. We show that the dynamics of the entanglement depends on the initial state of the central spins as well as the bath, the coupling constants and the strength of a magnetic field, B,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {B}}, \lambda $$\end{document}. Compared with some cases already discussed in the literature as magnetic fields of periodic sin(λt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sin (\lambda t)$$\end{document} and cos(λt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos (\lambda t)$$\end{document} functions, we can see that a magnetic field of exponential function e-λt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{e}^{-\lambda t}$$\end{document} plays a very crucial role in the entanglement generation between the two-spin qubits and its protection. To do this, we use an operator technique of the Holstein–Primakoff transformation, and the dynamics of the reduced density matrix of two coupled spin qubits is obtained in both finite and infinite numbers of bath spins. We also derive the concurrence measure to quantify the entanglement of the reduced density matrix of the two coupled central spins and look for conditions that provide information on whether this becomes robust against decoherence. It has been shown that the entanglement distribution can be both amplified, stabilized and protected with B,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {B}}, \lambda $$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. These results motivate developments toward the implementation or simulation of the purely theoretical model employing exponential fields.