
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39294337
72776
10.1038/s41598-024-72776-3
Article
Nonlinear parametric generation and optical vortex transfer in graphene ensemble under Landau quantization
Mehdinejad Ali Arashmdva1986@gmail.com

https://ror.org/01jw2p796 grid.411748.f 0000 0001 0387 0587 Department of Physics, Sharif University of Science and Technology, Tehran, Iran
18 9 2024
18 9 2024
2024
14 2183627 4 2024
10 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
We explore the dynamics of nonlinear parametric generation and light beam propagation in a Landau-quantized graphene structure with three energy levels interacting with two laser pulses, utilizing the Maxwell-Bloch equations. By applying a laser field to one transition of the graphene sample while keeping the second beam initially absent, the distinctive preparation of the graphene sample, coupled with its weak interaction with laser radiation, results in the parametric generation of a new laser beam in a different transition. We investigate the influence of diverse system parameters on both the efficiency of the generated beam and the propagation dynamics of both beams. Our findings reveal that manipulating these parameters can induce oscillations in the intensity of propagated beams, mitigate absorption losses during propagation allowing for earlier relaxation, and enhance the efficiency of energy transfer from the initial to the generated beam. Additionally, we demonstrate the transfer of optical vortices within the graphene ensemble by introducing an optical vortex to the initial beam. This scheme holds promise for applications in high-dimensional quantum information processing.

Keywords

Light propagation
Parametric generation
Optical vortex
Graphene
Subject terms

Optics and photonics
Physics
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

In the realm of optical research, there has been a surge of interest in understanding and manipulating light propagation within coherently driven media1–4. Central to this exploration is electromagnetically induced transparency (EIT)5,6, a phenomenon that significantly influences how light behaves in atomic media, granting transparency that allows light beams to travel over extended distances. At the core of this is the impact of quantum coherence and interference, providing a range of exciting possibilities. The use of quantum coherence and interference has led to a diverse range of propagation effects. From the storage and retrieval of light pulses7,8 to the emergence of slow optical solitons9,10, and from the practicality of all-optical switches11 to the efficiency of frequency conversion12–14, matched pulses15,16, and the intricacies of parametric nonlinear generation17–20, these phenomena offer promising applications in the expansive field of optoelectronics.

Having proven its success in atomic systems, the study of nonlinear optical properties has smoothly transitioned into the domain of semiconductor quantum wells and quantum dots21–33. Theoretical and experimental endeavors in these semiconductor materials mark a new chapter in understanding of controlled light propagation, departing from the familiar landscapes of atomic media. This shift presents a compelling interplay of challenges and opportunities, encouraging researchers to unravel the principles governing nonlinear optical behavior in semiconductor nanostructures. This evolving field invites a thorough exploration, uncovering the unique characteristics and potential applications arising from the nonlinear optical properties within semiconductor nanostructures.

Exploring the dynamics of light propagation in ultra-thin materials, particularly single-atom-thick substances like graphene, adds an intriguing dimension to the understanding of the corresponding phenomena34,35. Graphene, an allotrope of carbon, distinguishes itself with an abnormal two-dimensional (2D) Dirac-like electronic structure34,35. This unique feature gives rise to a plethora of relativistic quantum phenomena that are seldom observed in conventional materials, owing to the distinctive optical transition selection rules in graphene36. Governed by Dirac’s equation, the transport of massless Dirac quasiparticle electrons in graphene exhibits a linear energy-momentum dispersion relation37,38. Notably, graphene displays a substantial nonlinear response within its optical spectrum39,40. The behavior of Dirac electrons can be finely tuned through the application of external electromagnetic fields41. In the presence of a strong magnetic field, numerous nonlinear optical properties of graphene have been explored, encompassing the generation of entangled photons42, the manifestation of large optical nonlinearities43,44, wave mixing (MWM)45, the formation of optical solitons46–49, temporal dynamics50, slow light propagation51, and controllable optical bistability and multistability52,53. This venture into the nonlinear optical realm of graphene provides interesting phenomena, each unveiling the material’s unique response under distinct conditions. The investigation into graphene’s nonlinear properties not only broadens our fundamental understanding of optical interactions in 2D materials but also opens avenues for potential applications in advanced photonics and optoelectronics.

Building upon recent advances in graphene research, our paper explores the dynamics of a three-level graphene sample interacting with two weak probe fields. Drawing inspiration from the unique band structure and selection rules for optical transitions near the Dirac point, resulting in extreme nonlinearity, we navigate the propagation dynamics of the system by solving the Maxwell-Bloch equations with a focus on a scenario where the graphene sample is initially prepared in a superposition of lower levels. Notably, we explore the interaction of graphene with weak light beams, aiming to explore the interplay of quantum states and optical responses within this material. We assume the presence of one laser beam at the outset of sample, while the other remains absent. The coupled propagation dynamics reveal the generation of a new weak probe field. Numerically exploring the propagation of both beams across varying distances, we investigate the influence of critical system parameters, including decay rates, initial probability amplitudes, and detunings of applied laser fields. Our findings shed light on the graphene sample’s interesting characteristics, showcasing its potential to enhance energy transfer efficiency or suppress absorption losses at initial propagation stages. Remarkably, for the case with large detunings, both beams exhibit oscillatory behavior instead of exponential decay, necessitating substantially larger propagation distances for relaxation. We also introduce an optical vortex to the initial beam. Our results demonstrate that nonlinear parametric generation facilitates the transfer of optical vortices between different frequencies within the three-level graphene sample. This study, inspired by recent advances in graphene research, introduces a novel dimension to the understanding of optical phenomena in graphene material. The incorporation of optical vortices, known for carrying orbital angular momentum (OAM)54, aligns with the current surge of interest in their spatially dependent interaction with quantum systems55–60. This research not only deepens the comprehension of graphene’s nonlinear behavior but also paves the way for innovative applications in quantum optics and information processing.

The extraordinary optical and electronic properties of graphene, stemming from the linear, massless dispersion of electrons near the Dirac point and the chiral nature of electron states, set the stage for a study with promising implications. Additionally, the high capacity offered by optical vortices for high-dimensional optical information processing adds another layer of significance to our research. The unique characteristics of graphene, owing to its distinctive electron dispersion and chiral electron states, position it as a remarkable platform for exploring advanced optical phenomena. The study’s incorporation of optical vortices, known for their potential in high-dimensional information processing, enhances the versatility of the findings.

Model and equations

In our work, we focus on a 2D graphene crystal characterized by three energy levels, operating within the influence of a robust magnetic field, resulting in the formation of a Lambda-type configuration, as illustrated in Fig. 1. The presence of this magnetic field introduces an intriguing feature to the graphene crystal, where the linear dispersion relation leads to unequally spaced Landau levels (LLs) with transition energies proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt B$$\end{document}43. The exotic selection rules of graphene\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \left| n \right|= \pm 1$$\end{document}, denoted by the energy quantum number (n), dictate dipole-allowed transitions. In the presence of both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \left| n \right|= - 1$$\end{document}and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \left| n \right|=+1$$\end{document}, the absorption of right-hand circularly polarized light (RHC) and left-hand circularly polarized light (LHC) occurs independently61.

The optical transitions between neighboring LLs in graphene occur within the infrared to terahertz (THz) frequency range when subjected to a magnetic field ranging from 0.01 to 10 Tesla: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar {\omega _c}=36\sqrt {B(Tesla)} \,meV$$\end{document}. The electric fields\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {E}={\vec {E}_p}+{\vec {E}_c},\,$$\end{document}associated with these transitions can be expressed as follows:1a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\vec {E}}_p}={{\vec {e}}_p}{E_p}\exp ( - i{\omega _p}t+i{{\vec {k}}_p}.\,\vec {r})+c.\,c.$$\end{document}

1b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\vec {E}}_c}={{\vec {e}}_c}{E_c}\exp ( - i{\omega _c}t+i{{\vec {k}}_c}.\,\vec {r})+c.\,c.$$\end{document}

In this context, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vec {e}_j}$$\end{document}and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vec {k}_j}$$\end{document}represent unit vectors corresponding to the polarization field and the wave vector, respectively, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E_j}(j=p,c)$$\end{document} denotes the slowly varying envelope. A weak, tunable left-hand circularly polarized incident probe pulse with amplitude \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vec {E}_p}$$\end{document} and carrier frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega _p}$$\end{document} facilitates the inter-LL transition\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| 1 \right\rangle \leftrightarrow \left| 3 \right\rangle$$\end{document}. Simultaneously, the optical excitation through a LHC polarized weak field induces the intra-LL transition\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| 2 \right\rangle \leftrightarrow \left| 3 \right\rangle$$\end{document}.

Fig. 1 (a) Landau levels in proximity to the Dirac point overlaid on the linear electron dispersion in the absence of a magnetic field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E= \pm {v_F}\left| p \right|$$\end{document}. (b) An energy level diagram illustrating optical transitions in graphene under the influence of two weak probe and signal fields. The states correspond to the LLs with energy quantum numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n= - 1,0,1$$\end{document}, respectively.

In the absence of an optical field, the effective-mass Hamiltonian near the K point62,63 or a single-layer graphene situated in the x-y plane and subject to a magnetic field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\hat {z}$$\end{document} can be expressed as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_0}={\nu _F}\hat {\vec {\sigma }}.\,\hat {\vec {\pi }}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${v_F}=3{\gamma _0}/(2\hbar a) \approx {10^6}\,m/s\,$$\end{document}and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat {\vec {\pi }}=\hat {\vec {p}}+e\vec {A}/c$$\end{document} indicate the Fermi velocity (band parameter) and the generalized momentum operator, respectively (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _0}=2.8\,eV$$\end{document}stands for the nearest-neighbor hopping energy, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=1.42\,\mathop A\limits^{o}$$\end{document} illustrates the C-C spacing, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat {\vec {p}}$$\end{document} represents the electron momentum operator, e indicates the electron charge, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {A}$$\end{document} denotes the vector potential, is equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\,Bx)$$\end{document} for a static magnetic field).

To examine the interaction with incident optical fields, it is necessary to incorporate the vector potential of the optical field (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vec {A}_{opt}}=ic\vec {E}/\omega$$\end{document}) to the vector potential of the magnetic field in the generalized momentum operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat {\vec {\pi }}$$\end{document} in the Hamiltonian. This results in the interaction Hamiltonian:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat {H}_{\operatorname{int} }}={v_F}\vec {\sigma }.\,\frac{e}{c}\,{\vec {A}_{opt}}.$$\end{document} This equation illustrates that, in contrast to an electron with a parabolic dispersion relation, there are no higher-order terms for the interaction Hamiltonian\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\operatorname{int} }}$$\end{document}in graphene near the Dirac point. Consequently, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\operatorname{int} }}$$\end{document} remains linear with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vec {A}_{opt}}$$\end{document} even in the presence of a relatively strong optical field. Notably, the interaction Hamiltonian exclusively depends on the Pauli matrix vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\sigma }$$\end{document}and it does not involve the momentum operator.

By leveraging Liouville’s equation:2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\partial \hat {\rho }}}{{\partial t}}= - \frac{i}{\hbar }[{\hat {H}_{\operatorname{int} }},\,\hat {\rho }]+\hat {\Re }\{ \hat {\rho }\}$$\end{document}

the resulting density matrix equations of motion for Dirac electrons in graphene interacting with laser fields can be derived as follows: 3a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot {\rho }}_{11}}=i{\Omega _p}{\rho _{13}} - i\Omega _{p}^{{}}{\rho _{31}},$$\end{document}

3b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot {\rho }}_{22}}= - {\gamma _2}{\rho _{22}}+i{\Omega _c}{\rho _{23}} - i\Omega _{c}^{{}}{\rho _{32}},$$\end{document}

3c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot {\rho }}_{12}}=i({\Delta _p} - {\Delta _c}+\frac{{i{\gamma _2}}}{2}){\rho _{12}} - i{\Omega _p}{\rho _{32}}+i{\Omega _c}{\rho _{13}},$$\end{document}

3d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot {\rho }}_{13}}=i({\Delta _p}+\frac{{i{\gamma _3}}}{2}){\rho _{13}}+i{\Omega _p}({\rho _{11}} - {\rho _{33}})+i\Omega _{c}^{{}}{\rho _{12}},$$\end{document}

3e \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot {\rho }}_{23}}=i({\Delta _c}+i\frac{{{\gamma _2}+{\gamma _3}}}{2}){\rho _{23}}+i{\Omega _p}{\rho _{21}}+i{\Omega _c}({\rho _{22}} - {\rho _{33}}),$$\end{document}

3f \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho _{11}}+{\rho _{22}}+{\rho _{33}}=1,$$\end{document}

where in Eq. 2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat {\Re }\{ \hat {\rho }\}$$\end{document} represents incoherent relaxation, which could arise from various sources such as disorder, interaction with phonons, and carrier-carrier interactions. Moreover, in Eq. 3, the corresponding detunings are defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}=({\varepsilon _{n=2}} - {\varepsilon _{n= - 1}})/\hbar - {\omega _p},$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}=({\varepsilon _{n=2}} - {\varepsilon _{n=1}})/\hbar - {\omega _c}$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon _n}=\operatorname{sgn} (n)\hbar {\omega _c}\sqrt {\left| n \right|}$$\end{document} shows the energy of the LL for electrons near the Dirac point, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0,\, \pm 1,\, \pm 2,\,…,\,{\omega _c}=\sqrt 2 {v_F}/{l_c}$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${l_c}=\sqrt {\hbar c/eB}$$\end{document} stands for the magnetic length. The corresponding one- half Rabi- frequencies are defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p}=({\vec {\mu }_{31}}.{\vec {e}_p}){E_p}/(2\hbar ),\,$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}=({\vec {\mu }_{32}}.{\vec {e}_c}){E_c}/(2\hbar )$$\end{document}. The dipole matrix elements for the corresponding optical transition is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vec {\mu }_{mn}}=\left\langle m \right|\vec {\mu }\left| n \right\rangle =e.\,\left\langle m \right|\vec {r}\left| n \right\rangle =\frac{{i\hbar e}}{{{\varepsilon _n} - {\varepsilon _m}}}\left\langle m \right|{v_F}\vec {\sigma }\left| n \right\rangle$$\end{document}. The decay rate of the state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| j \right\rangle$$\end{document} is denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _j}$$\end{document}.

The Maxwell equations that govern the propagation of both probe beams in our study are expressed as follows: 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\partial {\Omega _p}}}{{\partial z}}=i{\kappa _{13}}{\rho _{13}}, \hfill \\ \frac{{\partial {\Omega _c}}}{{\partial z}}=i{\kappa _{23}}{\rho _{23}},$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa _{13}}=N{\omega _p}|{\vec {\mu }_{13}}.{\vec {e}_p}{|^2}/2\hbar {\varepsilon _r}{\nu _F}$$\end{document}and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa _{23}}=N{\omega _c}|{\vec {\mu }_{23}}.{\vec {e}_c}{|^2}/2\hbar {\varepsilon _r}{\nu _F}$$\end{document}represent the coupling constants. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=5 \times {10^{12}}\,c{m^{ - 2}}$$\end{document} represents the sheet electron concentration of graphene, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon _r}=4.5$$\end{document} signifies the substrate dielectric constant64,65.

Let us assume a condition wherein the system assumes an initial state of superposition, concurrently occupying both lower states: 5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (0)=\alpha \left| 1 \right\rangle +\beta \left| 2 \right\rangle,$$\end{document}

Under the condition of weak matter-light interaction (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p},{\Omega _c}<<{\gamma _{2,3}}$$\end{document}), we can employ the first-order approximation to get\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho _{33}}=0,{\rho _{12}}=\alpha {\beta ^*},{\rho _{11}}={\left| \alpha \right|^2},\,{\rho _{22}}={\left| \beta \right|^2}.$$\end{document} Subsequently, from Eq. 3(d) and 3(f), we obtain steady-state solutions as follows: 6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho _{13}}=\frac{{{{\left| \alpha \right|}^2}{\Omega _p}+\alpha {\beta ^*}{\Omega _c}}}{{{\Delta _p}+i\frac{{{\gamma _3}}}{2}}},$$\end{document}

7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho _{23}}=\frac{{{{\left| \beta \right|}^2}{\Omega _c}+{\alpha ^*}\beta {\Omega _p}}}{{{\Delta _c}+i\frac{{{\gamma _2}+{\gamma _3}}}{2}}}.$$\end{document}

This approximation enables us to derive expressions 6 and 7 capturing the system’s behavior under the specified weak matter-light interaction conditions. The steady-state solutions obtained will shed light on the evolution of the propagation of both beams in the system and provide valuable insights into the influence of the initial superposition state on the subsequent dynamics.

Upon substituting Eqs. (6) and (7) into the Maxwell Eq. (4), we derive the following coupled equations, elucidating the propagation dynamics of both weak fields: 8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\partial {\Omega _p}}}{{\partial z}}= - i{S_1}({\left| \alpha \right|^2}{\Omega _p}+\alpha {\beta ^*}{\Omega _c}),$$\end{document}

9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\partial {\Omega _{{p_2}}}}}{{\partial z}}= - i{S_2}({\alpha ^*}\beta {\Omega _p}+{\left| \beta \right|^2}{\Omega _c}),$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_1}=\frac{{{\kappa _{13}}}}{{{\Delta _p}+i\frac{{{\gamma _3}}}{2}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_2}=\frac{{{\kappa _{23}}}}{{{\Delta _c}+i\frac{{{\gamma _2}+{\gamma _3}}}{2}}}$$\end{document}. Assuming the incident first probe field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _{{p_{}}}}$$\end{document} is initially present at the beginning of the medium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p}(z=0)=\Omega$$\end{document}, and the second probe beam is absent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}(z=0)=0$$\end{document}, the situation can be described as follows: 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p}(z)=\frac{\Omega }{W}\left[ {{S_1}{{\left| \alpha \right|}^2}{e^{ - iWz}}+{S_2}{{\left| \beta \right|}^2}} \right],$$\end{document}

11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}(z)=\frac{\Omega }{W}{\alpha ^*}\beta {S_2}\left[ {{e^{ - iKz}} - 1} \right],$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W={S_1}|\alpha {|^2}+{S_2}|\beta {|^2}$$\end{document}. The observed emergence of the second probe beam (Eq. (11)), initially absent, signifies a remarkable outcome arising from the intricate interplay between the beams and the graphene sample. This phenomenon is not only a consequence of the interaction of the beams with the graphene sample but is also intricately linked to the initial preparation of the medium given in Eq. (3). Leveraging this observation, we will employ Eqs. (10) and (11) in the next section to embark on a numerical exploration. Our goal is to delve into the nonlinear parametric generation of a new weak probe field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}$$\end{document}and investigate the propagation dynamics of both beams within the graphene sample. This will allow us to discern the influence of various system parameters on the evolution of the system, shedding light on the conditions that govern the generation and propagation of weak probe fields in this unique graphene setup.

Results: nonlinear parametric generation and propagation dynamics

In this section, we perform on a numerical exploration to delve into the propagation dynamics of weak probe fields within the graphene system. Our aim is to investigate how different parameters of the system impact this dynamic evolution.

Impact of decay rates

Figure 2 vividly illustrates the intensities of both beams, denoted as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _p}(z){|^2}/|\Omega {|^2}$$\end{document}(solid line) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _c}(z){|^2}/|\Omega {|^2}$$\end{document}(dashed line) throughout the propagation under resonant conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}={\Delta _c}=0$$\end{document}. Specifically, we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta =1/\sqrt 2$$\end{document}and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa _{13}}={\kappa _{23}}={\gamma _2}\,\,\mu {m^{ - 1}}$$\end{document}. At the onset of propagation, the first probe beam \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p}$$\end{document} is present within the graphene medium, while the second probe beam \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}$$\end{document}is initially absent. However, as the propagation ensues, an intriguing phenomenon happens. The second probe beam is generated and begins to intensify, concomitantly with a decrease in the intensity of the first probe beam. Eventually, both probe beams reach a steady-state value through an exponential behavior.

Examining Fig. 2(a), it is evident that for smaller values of the decay rate\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}=0.5{\gamma _2}$$\end{document}, and the graphene sample’s efficiency in transferring energy from the first probe beam to the second beam remains below 0.2% (see Fig. 2(a)). This efficiency increases with a larger decay rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}$$\end{document}, as illustrated in Fig. 2(c)-(d). Notably, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}=3{\gamma _2}$$\end{document} the efficiency reaches 0.2% (Fig. 2(d)). However, it is crucial to note that the increase in the decay rate also amplifies absorption effects within the system. Consequently, the graphene sample takes longer to complete exponential decay and achieve a steady-state value, as observed in Fig. 2(c)-(d). This trade-off between efficiency and absorption effects underscores the delicate balance of decay rate in influencing the propagation dynamics of weak probe fields within the graphene sample.

Fig. 2 The evolution of field intensities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _p}(z){|^2}/|\Omega {|^2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _c}(z){|^2}/|\Omega {|^2}$$\end{document} of both beams during the propagation for different values of decay rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}$$\end{document}: (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}=0.5{\gamma _2}$$\end{document}, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}={\gamma _2}$$\end{document}, (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}=2{\gamma _2}$$\end{document}, and (d) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _3}=3{\gamma _2}$$\end{document}. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}={\Delta _c}=0$$\end{document},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta =1/\sqrt 2$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa _{13}}={\kappa _{23}}={\gamma _2}\,\,\mu {m^{ - 1}}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma _2}=3 \times {10^{13}}\,{s^{ - 1}}$$\end{document}.

Impact of initial probability amplitude

In Fig. 3, we delve into the influence of different initial values of probability amplitudes, denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta$$\end{document}, showcasing the graphene sample’s initial preparation with distinct coefficients of populations in each level. Specifically, we consider four cases: (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1/\sqrt 3 ,\beta =\sqrt 2 /\sqrt 3$$\end{document}, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\sqrt 2 /\sqrt 3 ,\beta =1/\sqrt 3$$\end{document}, (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1/\sqrt 7 ,\beta =\sqrt 6 /\sqrt 7$$\end{document}, and (d) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\sqrt 6 /\sqrt 7 ,\beta =1/\sqrt 7$$\end{document}. The exploration reveals that the initial preparation of the graphene sample significantly influences the propagation of both probe beams. In cases where the population initially in the second state\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| 2 \right\rangle$$\end{document} is higher than in the first state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| 1 \right\rangle$$\end{document}(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <\beta$$\end{document}), (e.g., Fig. 3(a) and 3(c)), less energy is transferred to the generated probe beam. Consequently, the intensity of the initial probe beam dominates, and the generated beam appears comparatively weaker. Conversely, when the majority of the population initially resides in the first state\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| 1 \right\rangle$$\end{document}(i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >\beta$$\end{document}),(see Fig. 3(b) and 3(d)), the transfer of energy to the generated probe beam is more pronounced. Consequently, the intensity of the generated probe beam dominates over that of the initial probe beam throughout the propagation. This indicates the pivotal role of the initial population distribution within the graphene sample in determining the power balance between the initial and generated probe beams during the propagation process.

Fig. 3 The evolution of field intensities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega_{p} (z)|^{2} /|\Omega |^{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega_{c} (z)|^{2} /|\Omega |^{2}$$\end{document} of both beams during the propagation. Here we take (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1/\sqrt 3 ,\beta = \sqrt 2 /\sqrt 3$$\end{document}, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \sqrt 2 /\sqrt 3 ,\beta = 1/\sqrt 3$$\end{document}, (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1/\sqrt 7 ,\beta = \sqrt 6 /\sqrt 7$$\end{document}, and (d) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \sqrt 6 /\sqrt 7 ,\beta = 1/\sqrt 7$$\end{document}. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 0$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa_{13} = \kappa_{23} = \gamma_{2} \,\,\mu m^{ - 1}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma_{2} = \gamma_{3}$$\end{document} and the other parameters are the same as Fig. 2.

Individual impact of probe detunings

In Figs. 4 and 5, we investigate the distinct impact of the first and second probe detunings on the propagation dynamics of both beams, keeping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta =1/\sqrt 2$$\end{document}. In Fig. 4, we consider the second probe beam to be in resonance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}=0$$\end{document}, while in Fig. 5, we focus on the first probe beam being in resonance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}=0$$\end{document}. In Fig. 4, where the second probe beam is in resonance, increasing the first probe field detuning \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}$$\end{document}results in both probe beams reaching steady state at later propagation distances. Notably, the efficiency of energy transfer to the generated probe beam is significantly increased by raising \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}$$\end{document}. This enhancement in efficiency, however, comes at the cost of heightened absorption effects in the graphene sample, manifesting in the exponential decay behavior observed in Fig. 4. This decay behavior extends to larger distances before reaching a steady non-decay state.

Conversely, in Fig. 5, with the first probe beam in resonance, increasing the second probe field detuning\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}$$\end{document} has an opposite effect on the propagation dynamics of the beams. The rise in\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}$$\end{document} results in a reduction of absorption effects in the graphene sample, leading both beams to reach steady-state behavior at shorter propagation distances. However, this reduction in absorption effects is accompanied by a decrease in the efficiency of energy transfer to the second probe beam. As\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}$$\end{document} increases, the intensity of the first probe beam dominates over the generated probe beam during their propagation, highlighting the trade-off between absorption effects and energy transfer efficiency in the graphene sample. These results underscore the sensitivity of the system’s behavior to the detuning parameters, indicating the need for a careful balance in selecting these parameters to achieve optimal performance in the propagation dynamics of weak probe fields within the graphene sample. This suggests that detunings can serve as a crucial parameter to optimize the propagation of probe beams within the system according to experimental objectives. When the primary aim is to enhance the efficiency of transfer to the generated beam, adjusting the detuning of the first probe beam can be advantageous. Conversely, if the goal is to suppress absorption effects in the graphene sample over shorter propagation distances, manipulating the detuning of the second probe beam\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}$$\end{document} while keeping the detuning of the first probe field at zero (resonance) can be an effective strategy.

Fig. 4 The evolution of field intensities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _p}(z){|^2}/|\Omega {|^2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _c}(z){|^2}/|\Omega {|^2}$$\end{document} of both beams during the propagation. Here we take (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}=0.5{\gamma _2}$$\end{document}, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}={\gamma _2}$$\end{document}, (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}=1.5{\gamma _2}$$\end{document}, and (d) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}=2{\gamma _2}$$\end{document}. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}=0$$\end{document},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta =1/\sqrt 2$$\end{document}, and the other parameters are the same as Fig. 2.

Fig. 5 The evolution of field intensities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _p}(z){|^2}/|\Omega {|^2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\Omega _c}(z){|^2}/|\Omega {|^2}$$\end{document} of both beams during the propagation. Here we take (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}=0.5{\gamma _2}$$\end{document}, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}={\gamma _2}$$\end{document}, (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}=1.5{\gamma _2}$$\end{document}, and (d) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}=2{\gamma _2}$$\end{document}. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}=0$$\end{document},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta =1/\sqrt 2$$\end{document}, and the other parameters are the same as Fig. 2.

Simultaneous impact of both probe detunings

In Fig. 6, we explore the simultaneous impact of both probe field detunings on the intensity of both initial and generated beams over their propagation. Here, we introduce nonzero detunings for both beams, shedding light on the intricate interaction between the two. Observing Fig. 6, the interaction between the beams becomes more complex with the introduction of nonzero detunings for both probes. An intriguing phenomenon emerges as both beams exhibit oscillatory behavior during their propagation within the graphene sample. The intensity oscillations are notably influenced by the values of the detunings. As both detunings are increased, the oscillatory behavior becomes more pronounced. Additionally, the graphene sample reaches a steady state with increased difficulty, requiring larger propagation distances. This is shown by comparing Fig. 6(d) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}={\Delta _c}=4{\gamma _2}$$\end{document}with Fig. 6(a) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}={\Delta _c}={\gamma _2}$$\end{document}). The complex oscillatory behavior and the dependence on detuning values indicates the rich dynamics inherent in the interaction between the initial and generated beams within the graphene sample. This insight is crucial for experimental design and suggests that adjusting both probe detunings can lead to intricate and potentially controllable behaviors in the propagation dynamics of weak probe fields within the graphene system.

Fig. 6 The evolution of field intensities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega_{p} (z)|^{2} /|\Omega |^{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega_{c} (z)|^{2} /|\Omega |^{2}$$\end{document} of both beams during the propagation. Here we take (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = \gamma_{2}$$\end{document}, (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 2\gamma_{2}$$\end{document}, (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 3\gamma_{2}$$\end{document}, and (d) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 4\gamma_{2}$$\end{document}. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \beta = 1/\sqrt 2$$\end{document}, and the other parameters are the same as Fig. 2.

OAM transfer in graphene sample

Consider the scenario where the initial probe beam is a vortex and carries OAM. This vortex beam can be defined as: 12 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p}(z=0)=\Omega =\left| \Omega \right|{(\frac{r}{w})^{\left| l \right|}}{e^{ - {r^2}/{w^2}}}{e^{il\phi }},$$\end{document}

where r shows the cylindrical radius, w stands for the waist of beam, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \Omega \right|$$\end{document}denote the strengths of the vortex beams. 1 is the orbital angular momentum (OAM) of the probe field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _p}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}represents the azimuthal angle. Equation 12, along with Eqs. 10 and 11, now reveals that when the initial probe beam is a vortex, the generated probe beam \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}$$\end{document} acquires the same OAM as the initial vortex beam. This result signifies the transfer of OAM from one frequency to another within the graphene sample, making this medium suitable for OAM transfer purposes.

Figure 7 (Fig. 8) shows intensity distributions (corresponding helical phase patterns) of the generated second probe vortex beams \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega _c}$$\end{document}with different OAM numbers l = 1–4 at the end of graphene sample for the resonance condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}={\Delta _c}=0$$\end{document}. In the case where OAM number l equals 1, an intriguing doughnut-shaped intensity distribution emerges, characterized by a central dark hollow region (Fig. 7(a)). Figure 8(a) visually represents the corresponding phase pattern for this scenario, unveiling a distinctive phase shift from 0 to 2π surrounding the singularity point. As we advance to higher OAM values, specifically l = 2, 3 and 4, the dimensions of the central dark hollow region expand, evident in Fig. 7(b)-(c). Simultaneously, the phase undergoes substantial jumps, transitioning from 0 to 4π, 6π, and 8π around the singularity point, respectively (Fig. 8(b)-(d)). This progression in both intensity and phase characteristics underscores the dynamic behavior of the vortex beam as the OAM number varies, revealing a versatile platform for manipulating orbital angular momentum within the graphene sample.

Fig. 7 Intensity distributions of the generated second probe vortex beam \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega_{c}$$\end{document} with different OAM numbers (a) l = 1, (b), l = 2, (c) l = 3 and (d) l = 4 at the end of graphene sample. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 0$$\end{document}, and the rest of parameters are the same as Fig. 2.

Fig. 8 Helical phase patterns of the generated second probe vortex beam \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega_{c}$$\end{document} with different OAM numbers (a) l = 1, (b), l = 2, (c) l = 3 and (d) l = 4 at the end of graphene sample. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 0$$\end{document} the parameters are the same as Fig. 7.

In the context of nonzero detunings, depicted in Fig. 9, it is observed that the phase patterns undergo a clockwise rotation when compared to the resonance case shown in Fig. 8. While the intensity distributions are not explicitly shown (as they exhibit similarities to Fig. 7), the focus here is on the distinctive rotation of the phase patterns. When both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _p}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _c}$$\end{document} are nonzero, the exponential term in Eq. (11) induces a modulation of the phase patterns through the variations in the terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_2}$$\end{document}. This modulation results in a clockwise rotation of the phase patterns. This rotation introduces an additional layer of complexity to the vortex beam’s behavior within the graphene sample under the influence of nonzero detunings.

Fig. 9 Helical phase patterns of the generated second probe vortex beam \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega_{c}$$\end{document} with different OAM numbers (a) l = 1, (b), l = 2, (c) l = 3 and (d) l = 4 at the end of graphene sample. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{p} = \Delta_{c} = 4\gamma_{2}$$\end{document} the parameters are the same as Fig. 8.

Conclusions

In conclusion, in this study, we have studied the nonlinear parametric generation and propagation dynamics of light beams within a Landau-quantized graphene structure featuring three energy levels. The presence of an initially weak probe field initiated the generation of a new laser in a distinct transition within the graphene medium. Our thorough investigation into the impact of various system parameters showed a spectrum of outcomes, including the induction of intensity oscillations in propagated beams, the suppression of absorption losses during propagation, and the enhancement of energy transfer efficiency from the initial to the generated beam. Notably, the introduction of an optical vortex to the initial beam illustrates the potential for transferring optical vortices within the graphene ensemble. This finding holds significant promise for applications in high-dimensional quantum information processing. Our comprehensive study of the graphene system’s behavior under different conditions not only contributes to the fundamental understanding of light-matter interactions in Landau-quantized structures but also opens avenues for practical applications. The controllability demonstrated in manipulating propagation dynamics and transferring optical vortices underscores the versatility of the proposed scheme. Future research directions may involve further optimization of parameters for specific applications and the exploration of additional quantum information processing possibilities within this intriguing graphene framework.

Author contributions

Ali Mehdinejad wrote the paper.

Data availability

“The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request”.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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