The Polarization of Plasmonic Field and Depolarization Factor for The Ellipsoid in Semiconductor Quantum Dot-metallic Nano Ellipsoid System

Abstract

We theoretically study the polarization of plasmonic fields in a hybrid nanosystem composed of three different metallic nano ellipsoid and a semiconductor quantum dot. The components of the hybrid nanosystem interact with three electromagnetic fields and one another via dipole-dipole interactions. We derive the density matrix equations at a steady state for the description of the optical properties of the hybrid nanosystem. The polarization of the plasmonic fields induced on the prolate, spherical, and oblate nano ellipsoid is calculated. We find that the polarization of the plasmonic fields depends on the number of the metallic nanoparticles (ellipsoid), the depolarization factor of the metallic nano ellipsoid, the direction of the plasmonic field, and the metallic nano ellipsoid parameters of the hybrid nanosystem. The phenomena of electromagnetically induced transparency (EIT) and electromagnetically induced transparency with amplification (EITA) are obtained in this work.

Introduction

A plasmonic response of the metallic nanoparticle (MNP) to the excitons of the semiconductor quantum dot (SQD) in a hybrid nanostructures system is recently a very active area of research. These structures stimulate the processes of the dipole-dipole coupling between (MNP) and (SQD), and display a variety of novel optical properties that are required in the Field of Optoelectronics [1,2,3]. The processes of the multi-dipole and plasmon-exciton dipole creation in the nanosystem (quantum dot-metallic nanoparticles) was studied in [4,5,6,7,8,9], where the interaction in the nanosystem between propagating surface plasmons in silver nanowires and excitons generated in quantum dots in which the energy is directly transferred from the propagating surface plasmons to the excitons without converting to photons, was investigated [10]. It was taken a novel double hole structure to improve the single-emitter emission rate, where the numerical investigations show that the structure possesses a large local electric field [11]. The energy absorption rate spectrum of an asymmetric double SQD-MNP hybrid structure depends on several parameters to obtain the double nonlinear Fano effects and the Autler-Townes splitting [12]. In addition to these properties, the effect of the shape, size of the plasmonic MNP and the depolarization field in polarizable objects of general shape was investigated [13,14,15,16,17,18,19]. In the most theoretical publications, studies of the properties of metallic nano ellipsoid (MNE) and some phenomena in the hybrid nanosystem (SQD-MNE) are fundamental for more improvement of nanotechnology [20,21,22,23,24,25,26]. Such as the exciton-plasmon coupling which enhance the second-harmonic generation (SHG), for example SHG photons and SHG surface plasmon polaritons emitted by the quantum dot and metallic nanoparticle respectively, were investigated [27]. It was determined that the coherent molecular resonances generated by a coupling process (between the quantum dots and metallic nanoparticles) which is influenced by the self-renormalization of the plasmonic fields and the structural parameters of the systems particularly the size and shape of the metallic nanoparticle [28]. It was important to study the modification of the linear third-order and fifth-order susceptibilities of a quantum dot that coupled to a spherical metallic nanoparticle, and calculate the susceptibilities and the spontaneous emission rate of the quantum dot as a result of the Purcell effect next to the metallic nanosphere using a boundary element method [29]. The behavior of the plasmonic mode at mid-infrared wavelenght, generation of tunable gain without inversion, and the states of polarization of coherent plasmonic field has also been demonstrated in SQD-MNP system using the dynamics of these states evolved with time when this system was interacting with a time-dependent laser field [30,31,32,33]. The influence of the strength of the plasmon-exciton dipole interaction for probe and two control fields for different parameters of the hybrid SQD-MNP nanosystem was studied, where the direction and detunings of the three electromagnetic fields played an important role in the characterization of the SQD-MNP nanosystem [34]. The study of the properties associated with the first-order optical susceptibility of a two-level SQD coupled to a core-shell metal nanoparticle which is dependent on the geometrical characteristics of the metal nanoshell on the material of both the dielectric environment and the dielectric core as well as on the polarization direction of the incident electric field and the distance separating the two nanoparticles was investigated [35].

In this work, we theoretically study the polarization of the three distinct metallic nano ellipsoid MNE set in close vicinity of a single spherical semiconductor quantum dot (SQD). MNE are taken to be a prolate, spherical and oblate ellipsoid. The hybrid nanosystem interact with three external electromagnetic fields which induce the dipole moments on MNE. Specifically we modify the density matrix equation at a steady-state for the hybrid SQD-MNE nanosystem. We find that the polarization induced by the plasmonic field depends on the number of the metallic nanoparticles (ellipsoid), the depolarization factor of the metallic nano ellipsoid and the direction of the plasmonic field. The polarization is calculated for each MNE, which is affected by the other MNE, as well for all MNE together. The outline of this paper is organized as in the “Theoretical Model and Methods” section, we give the model under the consideration in this work and present the calculation of the polarization of MNE under the interaction of the SQD-MNE hybrid nanosystem. We derive the expressions of the effective Rabi frequencies with terms of the induced dipole moments and subsequently derive the associated density matrix. Our numerical results are presented in the “Numerical Results and Discussion” section. Finally, the conclusions are drawn from our results in the “Conclusion” section.

Theoretical Model and Methods

Fig. 1

A schematic diagram of the SQD and three different MNE hybrid nanosystem. The SQD has a four-levels of two quasi lambda-type and coupling with three electromagnetic fields

In this section, we theoretically study the hybrid nanosystem composed of three distinct metallic nano Ellipsoid (MNE) situated adjacent to small size of a single spherical semiconductor quantum dot SQD. A schematic diagram for the nanosystem is illustrated in Fig. 1. We consider that the SQD (two quasi-$\Lambda$-type) involves four level atomic system $\left| 1\right\rangle$, $\left| 2\right\rangle$, $\left| 3\right\rangle$ and $\left| 4\right\rangle$ whose energy levels $\hbar \omega _{n}$ $(n=1$, 2, 3, 4). The interband transition from level $\left| 4\right\rangle$ to levels $\left| 1\right\rangle$, $\left| 2\right\rangle$ and $\left| 3\right\rangle$ have frequency $\omega _{4n}=\omega _{4}-\omega _{n}$ $(n=1$, 2, 3) respectively. The nanosystem interacts with three applied electromagnetic fields, $\textbf{E}_{j}\left( t\right) =\textbf{E}_{j}e^{-i\upsilon _{j}t}$ where $\textbf{E}_{j}$ and $\upsilon _{j}$ are the amplitude and frequency respectively $(j=1,\ 2,\ 3)$. The three electromagnetic fields possess the Rabi frequencies $\Omega _{j}=(\mathbf {\mu }_{4j}\cdot \textbf{E}_{j})/\hbar$, $\mu _{4j}$ represent the transition dipole moment elements between the levels from $\left| 4\right\rangle$ to $\left| j\right\rangle$. The probe field excites the interband transition $\left| 4\right\rangle \rightarrow \left| 1\right\rangle$ while the pump and control fields excite the interband transition $\left| 4\right\rangle \rightarrow \left| 2\right\rangle$ and $\left| 4\right\rangle \rightarrow \left| 3\right\rangle$ respectively. The dielectric constant of SQD is represented by $\varepsilon _{s}$. On the other hand, the three distinct metallic nano ellipsoid are denoted by MNE$_{i}$ where $(i=1$, 2, 3) are treated as classical Ellipsoid with specific dielectric function: $\varepsilon _{i}\left( \omega \right) =1-\omega _{pi}^{2}/\omega (\omega +i\gamma _{pi})$, and determined from the Drude model [36], where $\omega _{pi}$ and $\gamma _{pi}$ $(i=1$, 2, 3) are the plasmon frequency and the damping constant for the three different MNE$_{i}$ respectively. We consider the MNE$_{1}$ has the shape of prolate spheroid (ellipsoid with two equal semi-minor axes $b_{1}=c_{1}$) and the semi-major axis would be $a_{1}$. The MNE$_{2}$ has shape of spherical with radius $a_{2}$ (ellipsoid with $a_{2}=b_{2}=c_{2}$ ). The MNE$_{3}$ has shape of oblate spheroid (ellipsoid with two equal the semi-major axes $a_{3}=b_{3}$) and the semi-minor axis would be $c_{3}$. We consider the distance from the midpoint of the semi-major axis of MNE$_{1}$ to the center of SQD is $r_{1}$, the center-to-center distance between MNE$_{2}$ and SQD donated by $r_{2}$, and the distance from the midpoint of the semi-minor axis of MNE$_{3}$ to the center of SQD is $r_{3}$. Where $r_{1}$ is parallel to the X-axis, $r_{2}$ tilted at angle $\theta$ on the X-axis and $r_{3}$ parallel to the Y-axis. All these centers are in the plane (YOX), where the center of the SQD is situated at the origin O. The whole nanosystem is surrounded by a background with a dielectric constant $\varepsilon _{b}$. The interaction between the three distinct MNE and a single SQD is characterized by the large electric dipole-dipole moments of the SQD and MNE. The total Hamiltonian for the SQD-MNE system is as follows:

$$\begin{aligned} H=\underset{n=1}{\overset{4}{\sum }}\hbar \omega _{n}\sigma _{nn}-\sum _{j=1}^{3}\mathbf {\mu }_{4j}\cdot \textbf{E}_{SQD}^{j}\sigma _{4j}e^{-i\nu _{j}t}+h.c \end{aligned}$$

(1)

where $\sigma _{4j}=\left| 4\right\rangle \, \left\langle j\right|$ is the dipole transition operator between $\left| 4\right\rangle$ and $\left| j\right\rangle$ of the SQD, $\textbf{E}_{SQD}^{j}$ represent the field which are falling on the SQD due to the contributions of the nanosystem components induced by the applied field $\textbf{E}_{j}$. So $\textbf{E}_{SQD}^{j}$ can be written as follows:

$$\begin{aligned} \begin{aligned} \textbf{E}_{SQD}^{j}&= (\textbf{E}_{j} + \sum _{i=1}^{3}\textbf{E} _{SQD}^{ji})/\varepsilon _{effs} \\ \textbf{E}_{SQD}^{ji}&=(3\left( \textbf{p}_{ji}\mathbf {\cdot \hat{r}} _{i}\right) \mathbf {\hat{r}}_{i} - \textbf{p}_{ji})/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{i}^{3} \end{aligned} \end{aligned}$$

(2)

where $\varepsilon _{effs}=(2\varepsilon _{b}+\varepsilon _{s})/3\varepsilon _{b}$ is the screening factor of SQD. The external applied fields of $(\textbf{E}_{1},\textbf{E}_{2})$ are along the X-direction, and the $\textbf{E}_{3}$ is along the Y-direction. $\textbf{E}_{SQD}^{ji}$ $(i=1$, 2, 3) are the electric field induced on the SQD due to the dipole induced on the three different MNE. Where the unit vector $\mathbf {\hat{r}}_{i}$ is along the vector $\textbf{r}_{i}$. $\textbf{p}_{ji}$ are the vector dipole moments that induce on the surface of the three different MNE$_{i}$ (j stand for the number of the field, and i stand for the number of MNE$_{i}$). Consider the dipole of MNE$_{1}$ and MNE$_{2}$ lie in the X- direction while the dipole of MNE$_{3}$ lies in the Y- direction, where $\textbf{p}_{ji}$ is given by the following:

$$\begin{aligned} \textbf{p}_{ji}=\alpha _{ki}\textbf{E}_{ji}, \, \alpha _{ki}=\frac{V_{i}\varepsilon _{0}\varepsilon _{b}\left( \varepsilon _{i}\left( \omega \right) -\varepsilon _{b}\right) }{\varepsilon _{b}+\zeta _{ki}\left( \varepsilon _{i}\left( \omega \right) -\varepsilon _{b}\right) } \end{aligned}$$

(3)

where $\alpha _{ki}$ is the polarizability of the MNE$_{i}$ and $V_{i}=\frac{4\pi }{3}a_{i}b_{i}c_{i}$ is the volume of MNE$_{i}$, $\zeta _{ki}$ is called the depolarization factor (k donate to the direction of the field X or Y). If the applied field is parallel to the X-axis [37], then,

$$\begin{aligned} \zeta _{xi}=0.5\left( 1-\zeta _{yi}\right) \end{aligned}$$

(4)

where $\zeta _{xi}$ is the depolarization factor in the X-direction. $\zeta _{yi}$ is the depolarization factor in the Y-direction, which its value for prolate spheroid is as follows:

$$\begin{aligned} \zeta _{y1}=\frac{1-e_{1}^{2}}{e_{1}^{2}}\left( -1+\frac{1}{2e_{1}}\ln \frac{1+e_{1}}{1-e_{1}}\right) \end{aligned}$$

(5)

and for oblate spheroid:

$$\begin{aligned} \zeta _{y3}=\frac{g\left( e_{3}\right) }{2e_{3}^{2}}\left( \frac{\pi }{2}-\tan ^{-1}g\left( e_{3}\right) \right) -\frac{g^{2}\left( e_{3}\right) }{2} \end{aligned}$$

(6)

where $e_{1}^{2}=(1-b_{1}^{2}/a_{1}^{2})$, $\ g\left( e_{3}\right) =\left( (1-e_{3}^{2})/e_{3}^{2}\right) ^{1/2}$, $e_{3}^{2}=(1-c_{3}^{2}/a_{3}^{2}).$ At the value of $\ \zeta _{k2}=1/3$, we will get the polarizability of the sphere ($\alpha _{2}$). $\textbf{E}_{ji}$ are the electric field acting on the three MNE, which are given by the following:

$$\begin{aligned} \textbf{E}_{ji}=\left( \textbf{E}_{j} + \textbf{E}_{ji}^{SQD} + \textbf{E} _{ji}^{jm}+\textbf{E}_{ji}^{jl}\right) /\varepsilon _{effi} \end{aligned}$$

(7)

where $\varepsilon _{effi}=(2\varepsilon _{b}+\varepsilon _{i}\left( \omega \right) )/3\varepsilon _{b}$ is the screening factor of MNE$_{i}$. The field from the SQD on the three different MNE are calculated from the relationship [38]:

$$\begin{aligned} \begin{aligned} \textbf{E}_{ji}^{SQD}&=(3\left( \textbf{p}_{j}^{SQD}\mathbf {\cdot \hat{r}} _{i}\right) \, \mathbf {\hat{r}}_{i}\mathbf {-p}_{j}^{SQD})/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{i}^{3} \\ p_{j}^{SQD}&=\mu _{4j}\rho _{4j}+h.c \end{aligned} \end{aligned}$$

(8)

Also, the field $\textbf{E}_{ji}^{jm}$ and $\textbf{E}_{ji}^{jl}$ are the result of the interaction between every two polarized MNEs for $(i,\ m,\ l =1,2,3$ and $i\ne m\ne l)$ and given by the following:

$$\begin{aligned} \begin{aligned} \textbf{E}_{ji}^{jm}&= (3\left( \textbf{p}_{ji}\mathbf {\cdot \hat{r}}_{im}\right) \mathbf {\hat{r}}_{im}\mathbf {-p}_{ji})/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{im}^{3}\\ \textbf{E}_{ji}^{jl}&= (3\left( \textbf{p}_{ji}\mathbf {\cdot \hat{r}}_{il}\right) \mathbf {\hat{r}}_{il}\mathbf {-p}_{ji})/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{il}^{3} \end{aligned} \end{aligned}$$

(9)

where $\textbf{r}_{im}$($\textbf{r}_{il}$) is the distance between the MNE$_{i}$ and MNE$_{m}$(or MNE$_{i}$ and MNE$_{l}$). So, we get the induced dipole moments for the three MNE if the field in the X-direction from Eq. (3): $(j=1,\ 2)$

$$\begin{aligned} p_{j1}=\alpha _{x1}\left( G_{2}E_{j}+G_{3}p_{j}^{SQD}\right) /G_{1} \end{aligned}$$

(10)

$$\begin{aligned} p_{j2}=\alpha _{2}\left( L_{1}E_{j}+L_{2}p_{j}^{SQD}+L_{3}p_{j1}\right) /L_{4} \end{aligned}$$

(11)

$$\begin{aligned} p_{j3}=\alpha _{x3}\left( E_{j}-0.5D_{3}p_{j}^{SQD}+N_{13}p_{j1}+N_{23}p_{j2}\right) /\varepsilon _{eff3} \end{aligned}$$

(12)

where

$$\begin{aligned} \begin{aligned} L_{1}&=\left( \varepsilon _{eff3}+C_{23}\alpha _{x3}\right) \\ L_{2}&=\left( N_{2}\varepsilon _{eff3}-0.5D_{3}C_{23}\alpha _{x3}\right) \\ L_{3}&=\left( N_{12}\varepsilon _{eff3}+C_{23}N_{13}\alpha _{x3}\right) \\ L_{4}&=\left( \varepsilon _{eff2}\varepsilon _{eff3}-C_{23}N_{23}\alpha _{2}\alpha _{x3}\right) \end{aligned} \end{aligned}$$

(13)

$$\begin{aligned} \begin{aligned} G_{1}&=\left\{ L_{4}\varepsilon _{eff1}\varepsilon _{eff3}-\alpha _{x1}\left( L_{3}N_{12}\alpha _{2}\varepsilon _{eff3}+ L_{4}C_{13}N_{13}\alpha _{x3}+L_{3}C_{13}N_{23}\alpha _{2}\alpha _{x3}\right) \right\} \\ G_{2}&=\left\{ L_{4}\varepsilon _{eff3}+L_{1}\varepsilon _{eff3}N_{12}\alpha _{2}+L_{4}C_{13}\alpha _{x3}+L_{1}C_{13}N_{23}\alpha _{2}\alpha _{x3}\right\} \\ G_{3}&=\left\{ -2D_{1}L_{4}\varepsilon _{eff3}+L_{2}\varepsilon _{eff3}N_{12}\alpha _{2} - 0.5D_{3}L_{4}C_{13}\alpha _{3x}+L_{2}C_{13}N_{23}\alpha _{2}\alpha _{x3}\right\} \end{aligned} \end{aligned}$$

(14)

$$\begin{aligned} \begin{aligned} C_{23}&=3 \, \left( -r_{2}\cos \theta \right) \left( r_{3}-r_{2}\sin \theta \right) /4\pi \varepsilon {\circ }\varepsilon _{b}r_{23}^{5} \\ C_{13} &=-3r_{1}r_{3}/4\pi \varepsilon {\circ }\varepsilon _{b}r_{13}^{5} \\ N_{2}&=\left( 3\cos ^{2}\theta -1\right) /4\pi \varepsilon {\circ }\varepsilon _{b}r_{2}^{3}\\ N_{12}&=(3\left( r_{2}\cos \theta -r_{1}\right) ^{2}-r_{12}^{2})/4\pi \varepsilon {\circ }\varepsilon _{b}r_{12}^{5} \\ N_{23}&=(3 \ \left( r_{2}\cos \theta \right) ^{2}-r_{23}^{2})/4\pi \varepsilon {\circ }\varepsilon _{b}r_{23}^{5} \\ N_{13}&=\left( 3r_{1}^{2}-r_{13}^{2}\right) /4\pi \varepsilon {\circ }\varepsilon _{b}r_{13}^{5} \end{aligned} \end{aligned}$$

(15)

and the induced dipole moments $p_{3i}$ for the three MNE if the field in the Y-direction as follows:

$$\begin{aligned} p_{31}=\alpha _{y1}\left\{ (K_{1}M_{1}+\alpha _{2}K_{2}K_{3})E_{3} +(K_{1}M_{3}+\alpha _{2}M_{2}K_{3})p_{3}^{SQD})\right\} /K_{1}M_{4} \end{aligned}$$

(16)

$$\begin{aligned} p_{32}=\alpha _{2}\left\{ K_{2}E_{3}+M_{2} \, p_{3}^{SQD}\right\} /K_{1} \end{aligned}$$

(17)

$$\begin{aligned} p_{33}=\alpha _{y3}\left\{ E_{3}+D_{3}p_{3}^{SQD}+C_{13}p_{31}+C_{23}p_{32}\right\} /\varepsilon _{eff3} \end{aligned}$$

(18)

where

$$\begin{aligned} \begin{aligned} M_{1}&=\varepsilon_ {eff3}+D_{13}\alpha_ {y3} \\ M_{2}&=M_{4}\left( C_{2}\varepsilon_ {eff3}+D_{23}D_{3}\alpha_ {y3}\right) \\&\quad+\alpha _{y1}M_{3}\left( C_{12}\varepsilon_ {eff3}+D_{23}C_{13}\alpha _{y3}\right) \\ M_{3}&=D_{1}\varepsilon_ {eff3}+D_{13}D_{3}\alpha_ {y3}\\ M_{4}&=\varepsilon_{eff1}\varepsilon _{eff3}-C_{13}D_{13}\alpha _{y3}\alpha _{y1} \end{aligned} \end{aligned}$$

(19)

$$\begin{aligned} \begin{aligned} K_{1}&=M_{4} \, \left( \varepsilon _{eff2}\varepsilon _{eff3}-C_{23}D_{23}\alpha_ {y3}\alpha _{2}\right)\\&\quad -\alpha _{2}\alpha _{y1}K_{3}\left( C_{12}\varepsilon_ {eff3}+D_{23}C_{13}\alpha _{y3}\right) \\ K{2}&=M_{4} \, \left( \varepsilon _{eff3}+D_{23}\alpha _{y3}\right)\\&\quad+\alpha _{y1}M_{1}\left( C_{12}\varepsilon _{eff3}+D_{23}C_{13}\alpha _{y3}\right) \\ K_{3}&=C_{12}\varepsilon _{eff3}+C_{23}D_{13}\alpha _{y3} \end{aligned} \end{aligned}$$

(20)

$$\begin{aligned} \begin{aligned} D_{13}&=\left( 3r_{3}^{2}-r_{13}^{2}\right) /4\pi \varepsilon _{\circ }\varepsilon _{b}r_{13}^{5}, \ D_{23}=(3\left( r_{3}-r_{2}\sin \theta \right) ^{2}-r_{23}^{2})/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{23}^{5} \\ D_{1}&=-1/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{1}^{3}, \ D_{2}=3\cos \theta \sin \theta /4\pi \varepsilon _{\circ }\varepsilon _{b}r_{2}^{3}, \ D_{3}=2/4\pi \varepsilon _{\circ }\varepsilon _{b}r_{3}^{3} \\ C_{2}&=\left( 3\sin ^{2}\theta -1\right) /4\pi \varepsilon _{\circ }\varepsilon _{b}r_{2}^{3}, \ C_{12}=3r_{2}\left( r_{2}\cos \theta -r_{1}\right) \sin \theta /4\pi \varepsilon _{\circ }\varepsilon _{b}r_{12}^{5} \end{aligned} \end{aligned}$$

(21)

The polarization $P_{ji},$ $P_{jT}$ and $P_{TT}$ for the induced dipole moments $p_{jT},$ $p_{jT}$ and $p_{TT}$, respectively, of the three different MNE can be defined by using the following expression [36]: $(j=1,\ 2,\ 3)$

$$\begin{aligned} P_{ji}=Np_{ji}, \ P_{jT}=Np_{jT}, \ P_{TT}=Np_{TT} \end{aligned}$$

(22)

where

$$\begin{aligned} p_{jT}=p_{j1}+p_{j2}+p_{j3}, \, p_{TT}=p_{1T}+p_{2T}+p_{3T} \end{aligned}$$

(23)

and N is the number of atoms per unit volume. We define $P_{jT}$ as the polarization of all the three different MNE under the effect field (j), and $P_{TT}$ as the polarization of all the three different MNE under the effect of the three fields. The relation between $\Omega _{j}^{eff}$ and $\textbf{E}_{SQD}^{j}: \, (j=1, \ 2,\ 3)$

$$\begin{aligned} \Omega _{j}^{eff}=\mathbf {\mu }_{4j}\cdot \textbf{E}_{SQD}^{j}/\hbar \end{aligned}$$

(24)

So that the total Hamiltonian of the SQD-MNE hybrid nanosystem can be written as follows:

$$\begin{aligned} H=\underset{n=1}{\overset{4}{\sum }}\hbar \omega _{n}\sigma _{nn}-\hbar \sum _{j=1}^{3}\left( \Omega _{j}^{eff}\sigma _{4j}e^{-i\nu _{j}t}+h.c\right) \end{aligned}$$

(25)

The expressions for the effective Rabi frequencies $\Omega _{j}^{eff}$ with terms of the induced dipole moments have been obtained as follows:

$$\begin{aligned} \Omega _{1}^{eff}=\left( \Omega _{1}-2D_{1}\overset{\sim }{p_{11}} +N_{2}\overset{\sim }{p_{12}}\right) /\varepsilon _{eff} \end{aligned}$$

(26)

$$\begin{aligned} \Omega _{2}^{eff}=\left( \Omega _{2}-2D_{1}\overset{\sim }{p_{21}} +N_{2}\overset{\sim }{p_{22}}\right) /\varepsilon _{eff} \end{aligned}$$

(27)

$$\begin{aligned} \Omega _{3}^{eff}=\left( \Omega _{3} + D_{2}\overset{\sim }{p_{32}} + D_{3}\overset{\sim }{p_{33}}\right) /\varepsilon _{eff} \end{aligned}$$

(28)

where

$$\begin{aligned} \overset{\sim }{p_{j1}}=\mu _{4j} \, p_{j1}/\hbar , \ \overset{\sim }{p_{j2}}=\mu _{4j} \, p_{j2}/\hbar , \ \overset{\sim }{p_{j3}}=\mu _{4j} \, p_{j3}/\hbar \end{aligned}$$

(29)

and $(j=1, \ 2, \ 3)$, the equations of motion for the SQD-MNE hybrid nanosystem after the rotating wave approximation, and the electric-dipole approximation can be written as [39] follows:

$$\begin{aligned} \frac{\partial \rho _{22}\left( t\right) }{\partial t}=i\Omega _{2}^{eff} \rho _{24}-i\Omega _{2}^{eff^{*}}\rho _{42}+2\gamma _{2}\rho _{44}+2\gamma _{00}\rho _{11}-2\gamma _{0}\rho _{22} \end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial \rho _{33}\left( t\right) }{\partial t}=i\Omega _{3}^{eff}\rho _{34}-i\Omega _{3}^{eff^{*}}\rho _{43}+2\gamma _{3}\rho _{44} \end{aligned}$$

(31)

$$\begin{aligned} \begin{aligned} \frac{\partial \rho _{44}\left( t\right) }{\partial t}&=\mathop{-}i\Omega _{1}^{eff}\rho _{14}+i\Omega _{1}^{eff^{*}}\rho _{41}-i\Omega _{2}^{eff}\rho _{24}+i\Omega _{2}^{eff^{*}}\rho _{42} \\&\quad \mathop{-}i\Omega _{3}^{eff}\rho _{34}+i\Omega _{3}^{eff^{*}}\rho _{43}-2\left( \gamma _{1}+\gamma _{2}+\gamma _{3}\right) \rho _{44} \end{aligned} \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial \rho _{12}\left( t\right) }{\partial t}=-\beta _{1}\rho _{12}-i\Omega _{1}^{eff^{*}}\rho _{42}+i\Omega _{2}^{eff}\rho _{14} \end{aligned}$$

(33)

$$\begin{aligned} \frac{\partial \rho _{13}\left( t\right) }{\partial t}=-\beta _{2}\rho _{13}-i\Omega _{1}^{eff^{*}}\rho _{43}+i\Omega _{3}^{eff}\rho _{14} \end{aligned}$$

(34)

$$\begin{aligned} \frac{\partial \rho _{14}\left( t\right) }{\partial t}=-\beta _{3}\rho _{14}-i\Omega _{1}^{eff^{*}}\left( \rho _{44}-\rho _{11}\right) +i\Omega _{2}^{eff^{*}}\rho _{12}+i\Omega _{3}^{eff^{*}}\rho _{14} \end{aligned}$$

(35)

$$\begin{aligned} \frac{\partial \rho _{23}\left( t\right) }{\partial t}=-\beta _{4}\rho _{23}-i\Omega _{2}^{eff^{*}}\rho _{43}\ +i\Omega _{3}^{eff}\rho _{24} \end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial \rho _{24}\left( t\right) }{\partial t}=-\beta _{5}\rho _{24}+i\Omega _{1}^{eff^{*}}\rho _{21}-i\Omega _{2}^{eff^{*}}\left( \rho _{44}-\rho _{22}\right) +i\Omega _{3}^{eff^{*}}\rho _{23} \end{aligned}$$

(37)

$$\begin{aligned} \frac{\partial \rho _{34}\left( t\right) }{\partial t}=-\beta _{6}\rho _{34}+i\Omega _{1}^{eff^{*}}\rho _{31}+i\Omega _{2}^{eff^{*}}\rho _{32}-i\Omega _{3}^{eff^{*}}\left( \rho _{44}-\rho _{33}\right) \end{aligned}$$

(38)

The beyond equations follow from the constraints: $\sum _{n=1}^{4}\rho _{nn}=1;$ $\rho _{ji}=\rho _{ij}^{*}$, where

$$\begin{aligned} \begin{aligned} \beta _{1}&=\left[ \gamma _{0}+\gamma _{00}+i\left( \Delta _{2}-\Delta _{1}\right)\right] \\ \beta_{2}&=\left[ i\left( \Delta _{3}-\Delta _{1}\right) +\gamma _{00}\right] \\ \beta _{3}&=\left[ \gamma _{00}+\gamma _{1}+\gamma _{2}+\gamma _{3}-i\Delta _{1}\right] \\ \beta _{4}&=\left[ \gamma _{0}+i\left( \Delta _{3}-\Delta _{2}\right) \right] \\ \beta _{5}&=\left[ \gamma _{0}+\gamma _{1}+\gamma _{2}+\gamma _{3}-i\Delta _{2}\right] \\ \beta _{6}&=\left[ \gamma _{1}+\gamma _{2}+\gamma _{3}-i\Delta _{3}\right] \end{aligned} \end{aligned}$$

(39)

where $\Delta _{j}$ is the detuning of applied fields: $\Delta _{j}=\omega _{4j}-\nu _{j}$, $(j=1,2,3)$, and $\gamma _{1},\gamma _{2}$ and $\gamma _{3}$ represent the radiative decay rates of the excitation states $\left| 2\right\rangle$, $\left| 3\right\rangle$ and $\left| 4\right\rangle$ due to spontaneous emission, respectively. $\gamma _{0}$ and $\gamma _{00}$ are the nonradioactive decay.

Numerical Results and Discussion

In this section, we present our results for the polarization induced on the three MNE ($P_{ji}$, $P_{jT}$, $P_{TT}$ ) when the exciton-plasmon coupling occurs in the hybrid nanosystem. The three different MNE are gold nano ellipsoid with $\omega _{pi}=9.02$ ev and $\gamma _{0i}=0.026$ ev [40]. The parameters of the MNE-SQD are specified as follows: $\mu _{41}=\mu _{42}=\mu _{43}=2.4e$ nm, $\gamma _{0}=0.001ns^{-1}$, $\gamma _{00}=0.01$ $ns^{-1}$, $\gamma _{1}=\gamma _{3}=0.02ns^{-1}$and $\gamma _{2}=0.2ns^{-1}$. $\varepsilon _{s}=8$, $\varepsilon _{b}=12$, $a_{1}=20$ nm, $b_{1}=8$ nm, $a_{3}=10$ nm, $b_{3}=5nm$, $a_{2}=9$ nm and $\theta =\frac{\pi }{4}$. $\left( \Delta _{1}=\Delta _{2}=\Delta \right)$ and $\left( \Delta _{3}=0\right)$. The distances $\left( r_{1}, \ r_{2}, \ r_{3}\right) =\left( 20, \ 30, \ 18\right)$ nm,  and $\Omega _{1}=0.5$ ns$^{-1}$, $\Omega _{2}=11$ ns$^{-1}$, $\Omega _{3}=9$ ns$^{-1}$. Other parameters are indicated in the figure captions and described in what follows. The quantum coherence terms ($\rho _{ij}$ and $\rho _{ji}$) are obtained (analytical and numerical) via solving equations $(30-38)$ in the steady state. We plot the spectra of the imaginary part of the polarization ($P_{ji}$, $P_{jT}$, $P_{TT}$ ) versus the detuning $\left( \Delta \right)$ in the two dimensional, and the spectra of the three-dimensional (3-D) of $\text {Im }\left( P_{jT}\right)$ as a function of the detuning ($\Delta$) and the depolarization factor of MNE$_{1}\left( \zeta _{x1},\zeta _{y1}\right)$ and MNE$_{3}\left( \zeta _{x3},\zeta _{y3}\right)$.

Fig. 2

a is taken for $\text {Im }\left( P_{11}\right)$, the three values of $\left( \zeta _{x1}\right)$ are 0.3923 (blue curve), 0.4324 (red curve), and 0.4590 (black curve). b is taken for $\text {Im }\left( P_{21}\right)$, the three various values of ($\zeta _{x3}=0.4017$ (blue curve), 0.4113 (red curve) and 0.4347 (black curve)). c is taken for $\text {Im }\left( P_{31}\right)$. the three various values of $\zeta _{y1}=0.2154$ (blue curve), 0.1351 (red curve), and 0.0820 (black curve). d is taken for $\text {Im }\left( P_{31}\right)$, the three various of $\zeta _{y3}=0.2152$ (blue curve), 0.1775 (red curve) and 0.1307 (black curve)

Figure 2 presents an investigation of the imaginary part of the polarization induced of MNE$_{1}$(prolate spheroid) under the effect of the three fields E$_{j}$ and the interaction with other MNE$_{2}$ and MNE$_{3}$. Figure 2a represents $\text {Im }\left( P_{11}\right)$ under the effect of the field E$_{1}$ for the three various values of the depolarization factor of MNE$_{1}$ $\left( \zeta _{x1}\right)$: 0.3923 (blue curve), 0.4324 (red curve), and 0.4590 (black curve). Figure 2a exhibits an electromagnetically induced transparency with amplification (EITA) and has the amplification of the spectra small when the depolarization factor is small $\left( \zeta _{x1}=0.3923\right)$. Figure 2b shows $\text {Im }\left( P_{21}\right)$ under the effect of the field E$_{2}$, which have the same direction as E$_{1}$, and the depolarization factor of MNE$_{3}\left( \zeta _{x3}\right)$, the three various values of the depolarization factor of MNE$_{3}$ ($\zeta _{x3}=0.4017$ (blue curve), 0.4113 (red curve) and 0.4347 (black curve)). For all values of $\left( \zeta _{x3}\right)$, there is electromagnetic-induced transparency accompanied by amplification (EITA). The spectra of $\text {Im }\left( P_{21}\right)$ have large (EITA) at increasing the depolarization factor of MNE$_{3}$ ($\zeta _{x3}=0.4347$). Figure 2c describes $\text {Im }\left( P_{31}\right)$ which under the effect of the field E$_{3}$ and the depolarization factor of MNE$_{1}$ $\left( \zeta _{y1}\right)$ which have three distinct values: 0.2154 (blue curve), 0.1351 (red curve), and 0.0820 (black curve). In Fig. 2c, the optical spectra have the spectra EITA for all values of $\zeta _{y1}$, the depth of the spectra, in negative scale, increases with decreasing the depolarization factor of MNE$_{1}$ $\left( \zeta _{y1}\right)$. Figure 2d exhibits $\text {Im }\left( P_{31}\right)$ under the effect of the depolarization factor of MNE$_{3}$ $\zeta _{y3}=0.2152$ (blue curve), 0.1775 (red curve) and 0.1307 (black curve).The optical spectra have the spectra EITA in the negative $\text {Im }\left( P_{31}\right)$ for all values of $\zeta _{y3}$. We concluded that the imaginary part of the polarization induced on MNE$_{1}$ is affected by the depolarization factor of MNE$_{1}$and MNE$_{3}$ and the direction of the fields. We notice the difference between Fig. 2c and d caused by the various depolarization factors of MNE$_{1}$ ($\zeta _{y1}$) and MNE$_{3}$ ($\zeta _{y3}$).

Fig. 3

a is taken for $\text {Im }\left( P_{12}\right)$, the value of three values of $\left( \zeta _{x1}\right)$ as in (Fig. 2a). b is taken for $\text {Im }\left( P_{22}\right)$, the three various values of ($\zeta _{x3}$) as in (Fig. 2b). c is taken for $\text {Im }\left( P_{32}\right)$, the three various values of $\zeta _{y1}$ as in (Fig. 2c). d is taken for $\text {Im }\left( P_{32}\right)$, the three various of $\zeta _{y3}$ as in (Fig. 2d)

In Fig. 3, we investigate the imaginary part of the polarization induced on MNE$_{2}$ (spherical) under the effect of the three fields E$_{j}$ and the interaction with other MNE$_{1}$ and MNE$_{3}$. Figure 3a represents $\text {Im }\left( P_{12}\right)$ under the effect of the field E$_{1}$ for the three various values of the depolarization factor of MNE$_{1}$ $\left( \zeta _{x1}\right)$ (as in Fig. 2a). We have $\text {Im }\left( P_{22}\right)$ is shown in Fig. 3b under the effect of the field E$_{2}$ for the three various values of the depolarization factor of MNE$_{3}$ ($\zeta _{x3}$) (as in Fig. 2b). The optical spectra $\text {Im }\left( P_{12}\right)$ and $\text {Im }\left( P_{22}\right)$ have electromagnetic-induced transparency accompanied by amplification (EITA), the spectra of $\text {Im }\left( P_{21}\right)$ and $\text {Im }\left( P_{22}\right)$ have high (EITA) at increasing the depolarization factor of MNE$_{1}\left( \text {at }\zeta _{x1}=0.4590\right)$ and MNE$_{3}$ (at $\zeta _{x3}=0.4347$) respectively. The height of EIT window of $\text {Im }\left( P_{12}\right)$ increases when the depolarization factor of MNE$_{1}$ ($\zeta _{x1}$) increase, but the EIT window of $\text {Im }\left( P_{22}\right)$ is wider when decreasing the MNE$_{3}$ ($\zeta _{x3}$). Figure 3c represent $\text {Im }\left( P_{32}\right)$ under the effect of the third field E$_{3}$ and the value of depolarization factor of MNE$_{1}$ $\left( \zeta _{y1}\right)$ (as in Fig. 2c), Fig. 3d illustrates $\text {Im }\left( P_{32}\right)$ under the effect of the depolarization factor of MNE$_{3}$ ($\zeta _{y3}$) (as in Fig. 2d). In Fig. 3c, d, show that as the value of ($\zeta _{y1}$ and $\zeta _{y3}$) decreases, the optical spectra increase in height and the EIT window becomes longer, we notice the spectra of an EIT have characteristics of zero absorption. The difference between Figs. 3c, d and 2c, d, goes back to the difference of MNE, where in Fig. 2, the polarization induced on MNE$_{1}$ and in Fig. 3, the polarization induced on MNE$_{2}$.

Fig. 4

a is taken for $\text {Im }\left( P_{13}\right)$, the three values of $\left( \zeta _{x1}\right)$ as in (Fig. 2a). b is taken for $\text {Im }\left( P_{23}\right)$, the three various values of $\zeta _{x3}$ as in (Fig. 2b). c is taken for $\text {Im }\left( P_{33}\right)$, the three various values of $\zeta _{y1}$ as in (Fig. 2c). d is taken for $\text {Im }\left( P_{32}\right)$, the three various of $\zeta _{y3}$ as in (Fig. 2d)

Figure 4 demonstrates the spectra of the imaginary part of the polarization induced on MNE$_{3}$ (oblate spheroid) under the effect of the three fields E$_{j}$ and the influence of the interaction with other MNE$_{1}$ and MNE$_{2}$. Figure 4a represents $\text {Im }\left( P_{13}\right)$ under the effect of the field E$_{1}$ and the three various values of the depolarization factor of MNE$_{1}$ ($\zeta _{x1}$) (as in Fig. 2a). Figure 4b shows $\text {Im }\left( P_{23}\right)$ for three various values of the depolarization factor of MNE$_{3}$ ($\zeta _{x3}$) (as in Fig. 2b), and under the effect of the field E$_{2}$. The spectra of $\text {Im }\left( P_{13}\right)$ and $\text {Im }\left( P_{23}\right)$ show (EITA) but every spectrum have different properties according to the value of depolarization factor of MNE$_{1}$ and MNE$_{3}$, especially the spectra of $\text {Im }\left( P_{23}\right)$. At increasing the depolarization factor of MNE$_{1}\left( \text {at }\zeta _{x1}=0.4590\right)$ and MNE$_{3}$ (at $\zeta _{x3}=0.4347$) the spectra of (EITA) increase respectively. Figure 4c are almost symmetrical with Fig. 4d, where Fig. 4c exhibits the depolarization factor of MNE$_{1}$ $\left( \zeta _{y1}\right)$ with three distinct values (as in Figs. 2c), and 4d shows the depolarization factor of MNE$_{3}$: ($\zeta _{y3}$) (as in Fig. 2d). We notice that when the field Influencing is the third field on the MNE$_{3}$, EIT spectra only appear, where the direction of the third field and the dipole moment of MNE$_{3}$ in the Y-direction.

Fig. 5

${\textbf {a}}_{1}$ is taken for $\text {Im }\left( P_{1T}\right)$, the three values of $\left( \zeta _{x1}\right)$ as in (Fig. 2a). ${\textbf {b}}_{1}$ is taken for $\text {Im }\left( P_{3T}\right)$, the three various values of $\zeta _{y1}$ as in (Fig. 2c). ${\textbf {a}}_{2}$ and ${\textbf {b}}_{2}$ represent a ($3-D$) of $\text {Im }\left( P_{1T}\right)$ and $\text {Im }\left( P_{3T}\right)$ as a function of the detuning ($\Delta$) and the depolarization factor $\left( \zeta _{x1}\text { and }\zeta _{y1}\right)$ respectively

Figure 5 presents an investigation into the imaginary part of the polarization induced on the MNE under the effect of the field E$_{1}$ and E$_{3}$. Figure 5$\text{a}_{1}$, $\text{b}_{1}$ represents $\text {Im }\left( P_{1T}\right)$ and $\text {Im }\left( P_{3T}\right)$ respectively, in the two- dimensional, Fig. 5$\text{a}_{2}$ and $\text{b}_{2}$ represent a three-dimensional (3-D) of $\text {Im }\left( P_{1T}\right)$ and $\text {Im }\left( P_{3T}\right)$ as a function of the detuning ($\Delta$) and the depolarization factor $\left( \zeta _{x1}\text { and }\zeta _{y1}\right)$ respectively.The values of the depolarization factor of MNE$_{1}\left( \zeta _{x1}\right)$ (as in Fig. 2a) affect the two-dimensional and three-dimensional (3-D) of $\text {Im }\left( P_{1T}\right)$. The depolarization factor of MNE$_{1}\left( \zeta _{y1}\right)$ exhibits three distinct values (as in Fig. 2c) effect on the two dimensional and three-dimensional (3-D) of $\text {Im }\left( P_{3T}\right)$. Figure 5$\text{a}_{1}$ exhibits an electromagnetically induced transparency with amplification (EITA) for $\text {Im }\left( P_{1T}\right)$, we notice, when the amplification of the spectra small, the depolarization factor is small also $\left( \zeta _{x1}=0.3923\right)$. Also, we have when the depolarization factor $\left( \zeta _{x1}\right)$ increases, the spectra (EITA) also increase gradually under the effect of field E$_{1}$ in the X-direction. Figure 5$\text{a}_{2}$ represents Fig. 5$\text{a}_{1}$, in the form of three-dimensional (3-D) for the polarization of the three MNE when changing the depolarization factor of MNE$_{1}$ $\left( \zeta _{x1}\right)$, we notice the spectra increase in height with increasing the depolarization factor of MNE$_{1}$until for ($\zeta _{x1}\succ 0.4590$) in Fig. 5$\text{a}_{2}$. So, the effect of the depolarization factor of MNE$_{1}$ on the polarization induced on the total MNE depends on the number of the metal nano Ellipsoid (prolate, spherical, oblate), the direction of the field and the dipole moment of MNE. The depolarization factor of MNE$_{1}$ affects not just MNE$_{1}$ but the total of MNE. Figure 5$\text{b}_{1}$ represents the three MNE and reveals the EIT gain spectrum in the negative of $\text {Im }\left( P_{3T}\right)$ and at a small value of $\zeta _{y1}=\left( 0.0820\right)$. We observe the absorption spectra at $\zeta _{y1}=\left( 0.2154, \ 0.1351\right)$, where at a large value of $\zeta _{y1}=\left( 0.2154\right)$ the EIT window becomes longer. The spectra of absorption and gain have met at zero detuning $\left( \Delta \right)$. Figure 5$\text{b}_{2}$ explains Fig. 5$\text{b}_{1}$. In Fig. 5$\text{b}_{2}$, we observe the gain spectra at a small value of $\zeta _{y1}$ and the absorption spectra when increasing $\zeta _{y1}$. This figure explains the importance of the role of the total three MNE and the interaction between the three metal nano Ellipsoid (prolate, spherical, oblate), which consider more addition for the exciton-plasmon coupling that occurs in the hybrid nanosystem.

Fig. 6

${\textbf {a}}_{1}$ is taken for $\text {Im }\left( P_{2T}\right)$, the values of $\left( \zeta _{x3}\right)$ as in (Fig. 2b). ${\textbf {b}}_{1}$ is taken for $\text {Im }\left( P_{3T}\right)$, the values of $\left( \zeta _{y3}\right)$ as in (Fig. 2d). ${\textbf {a}}_{2}$ and ${\textbf {b}}_{2}$ represent a ($3-D$) of $\text {Im }\left( P_{2T}\right)$ and $\text {Im }\left( P_{3T}\right)$ as a function of the detuning ($\Delta$) and the depolarization factor $\left( \zeta _{x3}\text { and }\zeta _{y3}\right)$ respectively

Figure 6 illustrates an investigation of the imaginary part of the polarization induced on the MNE under the effect of the field E$_{2}$ and E$_{3}$. Figure 6$\text{a}_{1}$, $\text{a}_{2}$ represent the optical spectra $\text {Im }\left( P_{2T}\right)$ in the two and three dimensional respectively, when subjected to the influence of the second field $E_{2}$ for the three various values of the depolarization factor of MNE$_{3}$ ($\zeta _{x3}$) (as in Fig. 2b). Figure 6$\text{b}_{1}$, $\text{b}_{2}$ represent the optical spectra $\text {Im }\left( P_{3T}\right)$ in the two and three dimensional respectively, under the effect of $E_{3}$ for the various values of the depolarization factor of MNE$_{3}$: ($\zeta _{y3}$) (as in Fig. 2d). In Fig. 6$\text{a}_{1}$, The deep EIT of spectra ($\zeta _{x3}=0.4017$ and 0.4113) and spectrum ($\zeta _{x3}=0.4347$) are meeting at zero detuning $\left( \Delta \right)$. Figure 6$\text{a}_{2}$ demonstrates Fig. 6$\text{a}_{1}$ that the spectra show (EITA) for the average values of $\zeta _{x3}$. Figure 6$\text{a}_{2}$ shows at the large values of $\zeta _{x3}$, the spectra show (EIT) only. Figure 6$\text{b}_{1}$, $\text{b}_{2}$ demonstrate the optical spectra $\text {Im }\left( P_{3T}\right)$ in the two and three dimensional respectively, when subjected to the influence of the third field $E_{3}$ for three various values of the depolarization factor of MNE$_{3}$: $\zeta _{y3}$ (as in Fig. 2d). Figure 6$\text{b}_{1}$ shows the EIT spectra, which have zero absorption and increase in height when the depolarization factor of MNE$_{3}$ $\left( \zeta _{y3}\right)$ decreases. Figure 6$\text{b}_{2}$ describes Fig. 6$\text{b}_{1}$, we notice, in Fig. 6$\text{b}_{2}$, for large values $\left( \zeta _{y3}\succ 0.2152\right)$, the EIT spectra decrease gradually. So, the polarization induced on the three different nanoparticles or more is affected by the depolarization factor of MNE$_{i}$ ($i=1$, 3,..) with distinct properties.

Fig. 7

a–d are taken for $\text {Im }\left( P_{1T}\right)$, $\text {Im }\left( P_{2T}\right)$, $\text {Im }\left( P_{3T}\right)$ and $\text {Im }\left( P_{TT}\right)$ respectively. The three values of $\left( q\right)$ are 0.7143(blue curve), 0.5(red curve) and 0.3846 (black curve)

Figure 7 illustrates the spectra of the imaginary part of the polarization ($\text {Im }\left( P_{jT}\right)$) ($j=1,2,3,$ T) for the three different values of the ratio (q) that is equal: $q=0.7143$ nm (blue curve), $q=0.5nm$ (red curve) and $q=0.3846$ nm (black curve), where the ratio of the minor axis of MNE$_{3}$ to the minor axis of MNE$_{1}$ is ($q=\frac{c_{3}}{b_{1}}$). Figure 7a–d are taken for $\text {Im }\left( P_{1T}\right)$, $\text {Im }\left( P_{2T}\right)$, $\text {Im }\left( P_{3T}\right)$ and $\text {Im }\left( P_{TT}\right)$ respectively. Figure 7a, b have the EITA phenomenon for different values ratio (q). $\text {Im }\left( P_{3T}\right)$ has the positive optical EIT spectrum with zero absorption when $q=0.7143$ nm, and the negative EIT spectra for $q=0.5$ nm and $q=0.3846$ nm. Figure 7d shows the influence of the three fields on the three nanoparticles. The EIT and EITA phenomena were observed when $q=0.7143$ nm and ($q=0.5$ nm and $q=0.3846$ nm) respectively. Figure 7 demonstrates the importance of the parameter (q) that connects the three nanoparticles and plays an important role in illustrating various phenomena for the three nanoparticles.

Fig. 8

a–d are taken for $\text {Im }\left( P_{j1}\right)$, $\text {Im }\left( P_{j2}\right)$, $\text {Im }\left( P_{j3}\right)$ and $\text {Im }\left( P_{jT}\right)$ respectively for ($j=1,\ 2,\ 3$). blue curve for ($\text {Im }\left( P_{1i}\right) x10^{3}$), red curve for $\text {Im }\left( P_{2i}\right)$, and black curve for $\text {Im }\left( P_{31}\right) x10$, $\text {Im }\left( P_{32}\right)$, $\text {Im } \left( P_{33}\right)$ and $\text {Im }\left( P_{3T}\right) x10$ ($i=1,\ 2,\ 3,\ T$). $\theta =\frac{3\pi }{2}$ and $\Omega _{1}=0.01$ns$^{-1}$

In Fig. 8, we examine the influence of the angle ($\theta$) and the Rabi frequency ($\Omega _{1}$) on the polarization of MNE$_{i}$ when changing $\theta$ to $\theta =\frac{3\pi }{2}$, and the Rabi frequency $\Omega _{1}$ to 0.01ns$^{-1}$ in this figure. Figure 8a–d are taken for $\text {Im }\left( P_{j1}\right)$, $\text {Im }\left( P_{j2}\right)$, $\text {Im }\left( P_{j3}\right)$, and $\text {Im }\left( P_{jT}\right)$, respectively, where j refers to the field 1, 2, 3 inside each plot. We take the blue curve for $\text {Im }\left( P_{1i}\right) x10^{3}$, red curve for $\text {Im }\left( P_{2i}\right)$, and black curve for $\text {Im }\left( P_{31}\right) x10$, $\text {Im }\left( P_{32}\right)$, $\text {Im }\left( P_{33}\right)$ and $\text {Im }\left( P_{3T}\right) x10$ ($i=1,\ 2,\ 3,\ T$). The spectra of $\text {Im }\left( P_{1i}\right)$ under the influence of $\left( E_{1}\right)$ have the EITA phenomenon. The spectra of $\text {Im }\left( P_{2i}\right)$ under the influence of $\left( E_{2}\right)$ have large window and narrow EIT phenomenon, which the spectra is taking the form Y or inverted Y. The spectra of $\text {Im }\left( P_{3i}\right)$ under the influence of $\left( E_{3}\right)$ have small windows at the top and long narrow EIT phenomenon. Figure 8d shows the effect of every field on the total nanoparticles. The behavior of the polarization of MNE$_{i}$ under the influence of the ($\theta$) and ($\Omega _{1}$)(when $\Omega _{1}$ is weak) is distinct when compared with the previous figures. Therefore, the nanosystem can switch from (EITA) to (EIT) and vice versa by controlling the Rabi frequency ($\Omega _{1}$), the angle ($\theta$), and the direction of the fields.

Conclusion

We have studied the hybrid nanosystem composed of three different metallic nano ellipsoid (MNE) and a single spherical semiconductor quantum dot SQD, where SQD has two quasi-lambda type four-level atomic structure. We have solved the density matrix equations for the MNE-SQD hybrid nanosystem to obtain the coherence terms. The expressions for the effective Rabi frequencies $\Omega _{j}^{eff}$ with terms of the induced dipole moments have been obtained. We have derived the analytical expressions for the polarization for all three different metallic nano ellipsoid via the three electromagnetic fields felt on the MNE-SQD hybrid nanosystem. We discussed the importance of studying three different metallic nano ellipsoid together instead of one. We found that the spectra of the imaginary part of polarization depended on the number of metallic nanoparticles (ellipsoid) and the depolarization factor of MNE$_{1}$ and MNE$_{3}$. The depolarization factor and the ratio of the minor (or major) axis of MNE$_{3}$ to the minor (or major) axis of MNE$_{1}$ play a severe role in illustrating the optical properties of the hybrid nanosystem. The phenomena of zero absorption, (EIT) and (EITA) have been achieved in the spectra of the imaginary part of polarization of each MNE$_{i}$ via controlling the angle ($\theta$), the Rabi frequency ($\Omega _{1}$), the direction of the field and other nanosystem parameters. The effect of the geometrical properties of the MNE on our results is considered.