In our research, we delved into the behavior of a two-level quantum emitter (QE), often referred to as a qubit, when it is located near plasmonic nanodisks constructed from materials such as Molybdenum disulfide (MoS2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MoS}_2$$\end{document}) and Graphene. Our investigation focused on two pivotal aspects: the impact of distance between the QE and the nanodisk on non-Markovian behavior and the quantum speed limit time measures. Furthermore, we explored how the radius and material composition of the nanodisk influenced these parameters. We considered two distinct orientations for the QE's dipole moment: one aligned with the z<^>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{z}$$\end{document} direction, denoted as QE-zz, and the other in the x<^>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{x}$$\end{document} direction, denoted as QE-xx. Our findings revealed that the non-Markovian behavior exhibited a nuanced pattern as the QE moved away from the nanodisk, challenging the expectation of a consistent decrease with distance. Concerning the quantum speed limit time measure, we observed that in scenarios featuring non-Markovian dynamics, this measure is little than the QE's dynamical time in free space. As the degree of non-Markovianity increased, the qubit flips from state e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| e\right\rangle $$\end{document} to state g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| g\right\rangle $$\end{document} with enhanced swiftness. The dynamics for QE-xx displayed a greater degree of non-Markovian behavior than for QE-zz. Furthermore, Graphene nanodisks demonstrated more pronounced plasmonic effects when contrasted with their MoS2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MoS}_2$$\end{document} counterparts. Utilizing benchmark lines on the plots of non-Markovianity measures, we were able to pinpoint the precise distance from a nanodisk which desirable qubit dynamics, e.g., desired interaction regime, could be achieved.
