Localized surface plasmons supported by plasmonic nanostructures can dramatically alter the photonic density of states and cause a significant enhancement of the light-matter interaction [1], [2]. The enhanced light-matter interaction has potential applications ranging from nanoscale coherent light source [3], [4], biological nanoprobe [5], [6] to advanced quantum technologies, such as ultrasensitive quantum sensing [7]. Therefore, a lot of experimental and theoretical research works have been devoted to studying the strong coupling between the quantum emitters (QEs) and various metallic nanoparticles [8], [9], [10]. To achieve the strong coupling, QEs are often placed near the structures since light is confined in the sub-wavelength region around the metallic surface [11], [12], [13], [14], [15], [16], [17]. Besides the metallic nanoparticles, two-dimensional (2D) materials, such as graphene disk [18], [19], [20] and molybdenum disulfide nanodisk (MoS2) [21], [22], [23], [24], emerge as another promising platform for tuning light-matter interactions into the strong-coupling regime with the advantages of in situ control and unprecedented degree of freedom [25], and demonstrating the compelling applications like the infrared photodetector [26] with extraordinary optoelectronic properties and integrated neuromorphic hardware [27].
Recently, extensive theoretical works have also been done to extend the cavity quantum electrodynamics (cQED) concept to study the strongly coupled plasmon-QE hybrid system at nanoscale, and demonstrated some interesting phenomena like the quantum interference effect due to the anisotropic Purcell effect of 2D materials [28], [29], [30], [31]. However, to the best of our knowledge, a simple and physically transparent picture of the coherent energy exchange between the QE and the 2D materials is still lacking, due to the broadband and non-Lorentzian Purcell factor. The complicated optical response of 2D materials leads to the failure in directly applying the cQED treatments into such hybrid systems, since the 2D materials cannot be considered as a conventional Lorentzian resonantor. The effective mode theory describes the plasmon-QE interaction in the language of cQED where the creation and annihilation operators no longer have a continuous spectral dependence. Instead, the complicated plasmonic response is reconstructed by a set of separated, lossy field operator associated with the respective resonance peaks of given structure. The connection between the effective modes and the original quantization is well defined, so that all the required parameters in the effective models can be easily obtained from the Green's tensor of the system [28], [29].
In this work, we study the population dynamics of a V-type QE strongly coupled to a MoS$_{2}$ nanodisk by developing an effective multi-mode approach, which can precisely approximate the complicated response of MoS$_{2}$ by a dozen of Lorentzian cavity modes. Subsequently, the quantum dynamics of the hybrid system is described by an effective model in full analogy with multi-mode Jaynes–Cummings Hamiltonian. We show that our method can successfully capture the non-Markovian dynamics of strongly coupled QE-MoS$_{2}$ system.
The quantum system we study is depicted in Fig. 1, which consists of a V-type QE closely located to a MoS$_{2}$ nanodisk with radius R. The origin of the coordinate system is chosen at the center of nanodisk, and the plane of nanodisk lies in the $xy$ plane. A QE with two degenerate excited states $|1\rangle$, $|2\rangle$ and one ground state $|0\rangle$ is located at distance $z$ from the surface of nanodisk. The transition frequency between the two degenerate excited states and the ground state is characterized by $\omega _{0}$, where the energy of ground state $|0\rangle$ has been set to zero. The dipole moment can be written as $\mathbf {d}_{10}=d\mathbf {e}_-$ and $\mathbf {d}_{20}=d\mathbf {e_{+}}$, where $\mathbf {e_{\pm }}=(1/\sqrt{2}) (\mathbf {e}_{z}\pm i\mathbf {e}_{x})$ denotes the right- and left-rotating unit vectors, and $d$ is taken as a real number [32].
The Hamiltonian of the system is given by
\begin{align*}
\hat{\boldsymbol{H}}\!=\!&\sum _{i=1,2}\hbar \left(\omega _{0}\!-\!i\frac{\gamma _{0}}{2}\right)\hat{\sigma _{ii} } \!+\! \int d\mathbf {r}\int _{0}^{+\infty }d\omega \hbar \omega \hat{\mathbf {f}_{\omega }^{\dagger }}\left(\mathbf {r}\right)\cdot \hat{\mathbf {f}_{\omega }}\left(\mathbf {r}\right) \\
&-\sum _{i=1,2}\left[\hat{\sigma }_{i0}\int _{0}^{+\infty }d\omega \mathbf {d}_{i0} \cdot \hat{\mathbf {E}}_{\omega }^{+}\left(\mathbf {r}_{d}\right)+\mathbf {H.c.} \right], \tag{1}
\end{align*}
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\begin{align*}
\hat{\boldsymbol{H}}\!=\!&\sum _{i=1,2}\hbar \left(\omega _{0}\!-\!i\frac{\gamma _{0}}{2}\right)\hat{\sigma _{ii} } \!+\! \int d\mathbf {r}\int _{0}^{+\infty }d\omega \hbar \omega \hat{\mathbf {f}_{\omega }^{\dagger }}\left(\mathbf {r}\right)\cdot \hat{\mathbf {f}_{\omega }}\left(\mathbf {r}\right) \\
&-\sum _{i=1,2}\left[\hat{\sigma }_{i0}\int _{0}^{+\infty }d\omega \mathbf {d}_{i0} \cdot \hat{\mathbf {E}}_{\omega }^{+}\left(\mathbf {r}_{d}\right)+\mathbf {H.c.} \right], \tag{1}
\end{align*}
where the atomic operator $\hat{\sigma }_{ii}=|i\rangle \langle i|$ with $|i\rangle (i=1,2)$ being the two excited states of QE located at $\mathbf {r}_{d}$. $\hat{\mathbf {f}_{\omega }}$ is the bosonic annihilation for the electric field of MoS$_{2}$ nanodisk, which is represented by a continuum of harmonic oscillators. In (1), the first term stands for the energy of the V-type QE and the decay rate $\gamma _{0}$ of the two excited states. The second term refers to the energy of vacuum electromagnetic field, and the last term describes the QE-electromagnetic field interaction, where the electric field operator $\hat{\mathbf {E}}_{\omega }^{+}(\mathbf {r}_{d})$ is given by
\begin{align*}
\hat{\mathbf {E}}_{\omega }^{+}\left(\mathbf {r}_{d}\right)= i\sqrt{\frac{\hbar }{\pi \varepsilon _{0} } } \frac{\omega ^{2} }{c^{2} } \int d{\mathbf {r}}^{\prime } \sqrt{\text{Im}\left[{\varepsilon _{{\omega }}^{\prime }}\left({\mathbf {r}}^{\prime }\right) \right]} \mathbf {G}_{\omega }\left(\mathbf {r}_{d},{\mathbf {r}}^{\prime } \right) \hat{\mathbf {f}_{\omega }}\left({\mathbf {r}}^{\prime } \right), \\
\tag{2}
\end{align*}
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\begin{align*}
\hat{\mathbf {E}}_{\omega }^{+}\left(\mathbf {r}_{d}\right)= i\sqrt{\frac{\hbar }{\pi \varepsilon _{0} } } \frac{\omega ^{2} }{c^{2} } \int d{\mathbf {r}}^{\prime } \sqrt{\text{Im}\left[{\varepsilon _{{\omega }}^{\prime }}\left({\mathbf {r}}^{\prime }\right) \right]} \mathbf {G}_{\omega }\left(\mathbf {r}_{d},{\mathbf {r}}^{\prime } \right) \hat{\mathbf {f}_{\omega }}\left({\mathbf {r}}^{\prime } \right), \\
\tag{2}
\end{align*}
where $\varepsilon _{{\omega }}^{\prime }({\mathbf {r}}^{\prime })$ denotes the complex dielectric function of MoS$_{2}$ nanodisk. $\omega$ is the angular frequency and $c$ denotes the speed of light in the vacuum. $\mathbf {G}_{\omega }(\mathbf {r}_{d},{\mathbf {r}}^{\prime })$ denotes the classical electromagnetic Green's tensor with frequency $\omega$ at ${\mathbf {r}_{d}}$ and a point source located at ${\mathbf {r}}^{\prime }$. Taking into account the electrostatic approximation [20], the total Green's tensor for the MoS$_{2}$ nanodisk is given as $\mathbf {G}_{\omega }^{\text{tot} } (\mathbf {r}_{d},{\mathbf {r}}^{\prime }) = \mathbf {G}_{\omega }^{0 } (\mathbf {r}_{d},{\mathbf {r}}^{\prime }) + \mathbf {G}_{\omega }^{\text{ind} } (\mathbf {r}_{d},{\mathbf {r}}^{\prime })$, where $\mathbf {G}_{\omega }^{0 } (\mathbf {r}_{d},{\mathbf {r}}^{\prime })$ is the homogeneous part and $\mathbf {G}_{\omega }^{\text{ind} } (\mathbf {r}_{d},{\mathbf {r}}^{\prime })$ is the induced part associated with the interaction between the QE and the MoS$_{2}$ nanodisk.
For studying the spontaneous emission and considering the one-excitation approximation, the wave function of our system at time $t$ can be written as
\begin{align*}
|\psi (t) \rangle =& c_{1} (t) e^{-i\omega _{0}t } |1;\varnothing \rangle + c_{2} (t) e^{-i\omega _{0}t } |2;\varnothing \rangle \\
&+\int d\mathbf {r}\int d\omega \mathbf {C} \left(\mathbf {r},\omega, t \right) e^{-i\omega t } |0;1_{\mathbf {r},\omega } \rangle, \tag{3}
\end{align*}
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\begin{align*}
|\psi (t) \rangle =& c_{1} (t) e^{-i\omega _{0}t } |1;\varnothing \rangle + c_{2} (t) e^{-i\omega _{0}t } |2;\varnothing \rangle \\
&+\int d\mathbf {r}\int d\omega \mathbf {C} \left(\mathbf {r},\omega, t \right) e^{-i\omega t } |0;1_{\mathbf {r},\omega } \rangle, \tag{3}
\end{align*}
where $|{n}^{\prime }; a \rangle =|{n}^{\prime } \rangle \otimes |a \rangle$, with $|{n}^{\prime } \rangle ({n}^{\prime } =0,1,2)$ stands for the quantum states of the QE, and $|a \rangle (a =\varnothing,1_{\mathbf {r},\omega })$ denotes the photon states of the system. $|1;\varnothing \rangle$ ($|2;\varnothing \rangle$) represents that the QE is in the excited state $|1 \rangle$ ($|2 \rangle$) and there is no photon in the system, while $|0;1_{\mathbf {r},\omega } \rangle$ indicates that the QE has returned to the ground state and there is one photon in the system with energy $\hbar \omega$. The elementary excitation of the light field is defined through the action of the bosonic vector field operator on the vacuum state $\hat{\mathbf {f}_{\omega }^{\dagger }}(\mathbf {r})|\varnothing \rangle =|1_{\mathbf {r},\omega } \rangle$.
A. Induced Electrostatic Green's Tensor
In the following, we develop the effective modes theory based on Green's function to describe the hybrid system. We first give the calculation method of Green's function. In our model, the QE is located at the center of the MoS$_{2}$ nanodisk, $\mathbf {r}_{d}=(0,0,z)$, thus the induced part of the Green's tensor is given by [21], [33]
\begin{align*}
\mathbf {G}_{\omega }^{xx } \left(\mathbf {r}_{d},\mathbf {r}_{d} \right) = -\frac{c^{2} }{2\omega ^{2} } \sum _{n=0}^{\infty }c_{n}^{1}\left(z,\omega \right) \frac{ \left[ \sqrt{\left(z/R \right) ^{2}+1 } -z/R \right] ^{2n+2} }{\sqrt{\left(z/R \right) ^{2}+1 }}, \\
\tag{4}
\end{align*}
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\begin{align*}
\mathbf {G}_{\omega }^{xx } \left(\mathbf {r}_{d},\mathbf {r}_{d} \right) = -\frac{c^{2} }{2\omega ^{2} } \sum _{n=0}^{\infty }c_{n}^{1}\left(z,\omega \right) \frac{ \left[ \sqrt{\left(z/R \right) ^{2}+1 } -z/R \right] ^{2n+2} }{\sqrt{\left(z/R \right) ^{2}+1 }}, \\
\tag{4}
\end{align*}
and
\begin{align*}
\mathbf {G}_{\omega }^{zz } \left(\mathbf {r}_{d},\mathbf {r}_{d} \right) = \pm \frac{c^{2} }{2\omega ^{2} } \sum _{n=1}^{\infty }c_{n}^{0}\left(z,\omega \right) \frac{ \left[ \sqrt{\left(z/R \right) ^{2}+1 } -z/R \right] ^{2n+1} }{\sqrt{\left(z/R \right) ^{2}+1 }}, \\
\tag{5}
\end{align*}
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\begin{align*}
\mathbf {G}_{\omega }^{zz } \left(\mathbf {r}_{d},\mathbf {r}_{d} \right) = \pm \frac{c^{2} }{2\omega ^{2} } \sum _{n=1}^{\infty }c_{n}^{0}\left(z,\omega \right) \frac{ \left[ \sqrt{\left(z/R \right) ^{2}+1 } -z/R \right] ^{2n+1} }{\sqrt{\left(z/R \right) ^{2}+1 }}, \\
\tag{5}
\end{align*}
where the minus (plus) sign applies to $z> 0$ ($z< 0$) in (5). The $c_{n}^{l}(z,\omega)$ $(l=0,1)$ represents the expansion coefficients with $l=0$ and $l=1$ denoting the dipole moment of the QE are along $z$- and $x$-orientations, respectively. In fact, the expansion coefficients $c_{n}^{l}(z,\omega)$ form a vector $\mathbf {c}^{l} =(c_{1}^{l}(z,\omega), c_{2}^{l}(z,\omega),\ldots,c_{n}^{l}(z,\omega)) ^{T}$ satisfying the solution of the matrix equation which can be expressed as [19]
\begin{align*}
\left[\Omega _{0}^{2} (\omega) \mathbf {K }^{l} -\omega ^{2} \mathbf {G}^{l}\right]\mathbf {c }^{l} =i \frac{\omega \sigma (\omega)}{R^{2}} \mathbf {D }^{l}\mathbf { d}^{l}, \tag{6}
\end{align*}
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\begin{align*}
\left[\Omega _{0}^{2} (\omega) \mathbf {K }^{l} -\omega ^{2} \mathbf {G}^{l}\right]\mathbf {c }^{l} =i \frac{\omega \sigma (\omega)}{R^{2}} \mathbf {D }^{l}\mathbf { d}^{l}, \tag{6}
\end{align*}
where $\Omega _{0}^{2} (\omega)=-i\omega \sigma _{\text{MoS} _{2} } (\omega)/2\epsilon _{0}R$ with $\sigma _{\text{MoS} _{2} } (\omega)$ given in (12). The matrices $\mathbf {D}^{l}$, $\mathbf {G}^{l}$ and $\mathbf {K}^{l}$ have the form:
\begin{align*}
\ D_{qk}^{l}=&\frac{\delta _{qk}}{2(l+2 q+1)},& \tag{7}
\\
G_{qk}^{l}\!=\!& \frac{\delta _{q0} \delta _{k 0}}{8 l(l\!+\!1)^{2}}\!+\!\frac{1}{8} \frac{1}{(l\!+\!2 q\!+\!3)(l\!+\!2 q\!+\!2)(l\!+\!2 q\!+\!1)} \delta _{q\!+\!1, k} \\
&+\frac{1}{4} \frac{1}{(l+2 q+2)(l+2 q+1)(l+2 q)} \delta _{qk} \\
&+\frac{1}{8} \frac{1}{(l+2 k+3)(l+2 k+2)(l+2 k+1)} \delta _{q, k+1}, \tag{8}
\\
K_{qk}^{l}\!=\!&\frac{(-1)^{k-q+1}}{\pi \left[4(k-q)^{2}\!-\!1\right](l\!+\!k\!+\!q\!+\!1 / 2)(l+k+q+3 / 2)}, \tag{9}
\end{align*}
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\begin{align*}
\ D_{qk}^{l}=&\frac{\delta _{qk}}{2(l+2 q+1)},& \tag{7}
\\
G_{qk}^{l}\!=\!& \frac{\delta _{q0} \delta _{k 0}}{8 l(l\!+\!1)^{2}}\!+\!\frac{1}{8} \frac{1}{(l\!+\!2 q\!+\!3)(l\!+\!2 q\!+\!2)(l\!+\!2 q\!+\!1)} \delta _{q\!+\!1, k} \\
&+\frac{1}{4} \frac{1}{(l+2 q+2)(l+2 q+1)(l+2 q)} \delta _{qk} \\
&+\frac{1}{8} \frac{1}{(l+2 k+3)(l+2 k+2)(l+2 k+1)} \delta _{q, k+1}, \tag{8}
\\
K_{qk}^{l}\!=\!&\frac{(-1)^{k-q+1}}{\pi \left[4(k-q)^{2}\!-\!1\right](l\!+\!k\!+\!q\!+\!1 / 2)(l+k+q+3 / 2)}, \tag{9}
\end{align*}
where $q,k=1,2,3\dots$ for $l=0$ and $q,k=0,1,2\dots$ for $l=1$. Then, $d_{k}^{l}$ are the components of the vector $\mathbf {d}^{l}$,which are given by
\begin{align*}
d_{k}^{1}=&(2 k+2) \frac{1}{ R^{2}} \int _{0}^{1} d r P_{k}^{(1,0)}\left(1-2 r^{2}\right) \frac{r^{3}}{\left[r^{2}+\left(z / R\right)^{2}\right]^{3 / 2}}, \tag{10}
\\
d_{k}^{0}=&2(2 k+1) \frac{z}{ R^{3}} \int _{0}^{1} d r P_{k}^{(0,0)}\left(1\!-\!2 r^{2}\right) \frac{r}{\left[r^{2}\!+\!\left(z / R\right)^{2}\right]^{3 / 2}}, \tag{11}
\end{align*}
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\begin{align*}
d_{k}^{1}=&(2 k+2) \frac{1}{ R^{2}} \int _{0}^{1} d r P_{k}^{(1,0)}\left(1-2 r^{2}\right) \frac{r^{3}}{\left[r^{2}+\left(z / R\right)^{2}\right]^{3 / 2}}, \tag{10}
\\
d_{k}^{0}=&2(2 k+1) \frac{z}{ R^{3}} \int _{0}^{1} d r P_{k}^{(0,0)}\left(1\!-\!2 r^{2}\right) \frac{r}{\left[r^{2}\!+\!\left(z / R\right)^{2}\right]^{3 / 2}}, \tag{11}
\end{align*}
where $P_{k}^{(l,0)}$ is the Jacobi polynomials.
B. MoS$_{2}$ Conductivity
From $\Omega _{0}^{2} (\omega)$, it can be seen that in order to obtain the Green's function we need to acquire the MoS$_{2}$ conductivity. The optical response of the MoS$_{2}$ nanodisk is described through its surface conductivity, called $\sigma _{\mathrm{MoS_{2}} }$. Taking into account the interaction of light with the lowest energy $A$ and $B$ excitons [34], the MoS$_{2}$ nanodisk surface conductivity is given by [22]
\begin{align*}
\sigma _{\mathrm{MoS_{2}} }= \frac{4\alpha _{0}cv^{2} }{\pi a_{ex}^{2}\omega } \sum _{k=A,B}\frac{-i}{E_{k}-\hbar \omega -i\hbar \gamma _{k} }. \tag{12}
\end{align*}
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\begin{align*}
\sigma _{\mathrm{MoS_{2}} }= \frac{4\alpha _{0}cv^{2} }{\pi a_{ex}^{2}\omega } \sum _{k=A,B}\frac{-i}{E_{k}-\hbar \omega -i\hbar \gamma _{k} }. \tag{12}
\end{align*}
where $\alpha _{0}$ is the fine structure constant; $a_{ex}=0.8$ $\text{nm}$ stands for the exciton Bohr radius. $\hbar \gamma _{A}$ and $\hbar \gamma _{B}$ are the damping parameters with $\hbar \gamma _{A}=0.5$ $\text{meV}$ and $\hbar \gamma _{B}=1.1$ $\text{meV}$; $E_{A}=1.9$ $\text{eV}$ and $E_{B}=2.1$ $\text{eV}$ are the exciton energies. Also, $v$ stands for a velocity parameter, and we use $v=0.55$ $\mathrm{nm/fs}$ [35], [36].
C. Localized Exciton Mode Quantization and Effective Model
The photonic density of states of MoS$_{2}$ nanodisk presents multi-resonance [34] as Fig. 2 shows. Accordingly, we propose an effective multi-mode model to study the quantum dynamics of strong light-matter interaction between the QE and the MoS$_{2}$ nanodisk [37], [38]. The coupled QE-MoS$_{2}$ nanodisk system can be described as a V-type QE coupled to $N$ localized exciton (LE) modes supported by the MoS$_{2}$ nanodisk [23] with different coupling strength, while there is no interaction between the LE modes. Therefore, the effective Hamiltonian of hybrid system takes the same form as multi-mode Jaynes–Cummings model, which is given in (13), shown at the bottom of this page.
\begin{align*}
\hat{H}_{\text{eff}}=& \hbar \left(\omega _{0}-i \frac{\gamma _{\mathrm{0}}}{2}\right)\left(\hat{\sigma }_{11}+\hat{\sigma }_{22}\right) +\sum _{\bar{n}} \hbar \left(\omega _{\bar{n}}-i \frac{\gamma _{\bar{n}}}{2}\right) \left(\hat{b}_{\bar{n}}^{(1)\dagger } \hat{b}_{\bar{n}}^{(1)}+\hat{b}_{\bar{n}}^{(2)\dagger } \hat{b}_{\bar{n}}^{(2)}\right) \\
&-i \hbar \sum _{\bar{n}}\left(\tilde{g}_{\bar{n}}^{(11)} \hat{b}_{\bar{n}}^{(1)} \hat{\sigma }_{+}^{(1)}+\tilde{g}_{\bar{n}}^{(22)} \hat{b}_{\bar{n}}^{(2)} \hat{\sigma }_{+}^{(2)}+\tilde{g}_{\bar{n}}^{(12)} \hat{b}_{n}^{(1)} \hat{\sigma }_{+}^{(2)}+\tilde{g}_{\bar{n}}^{(21)} \hat{b}_{\bar{n}}^{(2)} \hat{\sigma }_{+}^{(1)}-\mathbf {H.c.}\right). \tag{13}
\end{align*}
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\begin{align*}
\hat{H}_{\text{eff}}=& \hbar \left(\omega _{0}-i \frac{\gamma _{\mathrm{0}}}{2}\right)\left(\hat{\sigma }_{11}+\hat{\sigma }_{22}\right) +\sum _{\bar{n}} \hbar \left(\omega _{\bar{n}}-i \frac{\gamma _{\bar{n}}}{2}\right) \left(\hat{b}_{\bar{n}}^{(1)\dagger } \hat{b}_{\bar{n}}^{(1)}+\hat{b}_{\bar{n}}^{(2)\dagger } \hat{b}_{\bar{n}}^{(2)}\right) \\
&-i \hbar \sum _{\bar{n}}\left(\tilde{g}_{\bar{n}}^{(11)} \hat{b}_{\bar{n}}^{(1)} \hat{\sigma }_{+}^{(1)}+\tilde{g}_{\bar{n}}^{(22)} \hat{b}_{\bar{n}}^{(2)} \hat{\sigma }_{+}^{(2)}+\tilde{g}_{\bar{n}}^{(12)} \hat{b}_{n}^{(1)} \hat{\sigma }_{+}^{(2)}+\tilde{g}_{\bar{n}}^{(21)} \hat{b}_{\bar{n}}^{(2)} \hat{\sigma }_{+}^{(1)}-\mathbf {H.c.}\right). \tag{13}
\end{align*}
The $\hat{\sigma }_{+}^{(1)}=|1\rangle \langle 0|$ is the raising operator of excited state $|1\rangle$, with the transition electric dipole moments $\mathbf {d}_{10}$. $\hat{\sigma }_{+}^{(2)}=|2\rangle \langle 0|$ is the raising operator of excited state $|2\rangle$, with the transition electric dipole moments $\mathbf {d}_{20}$. $\bar{n}$ is compact index notation, which can be interpreted as a spatial coordinate under a column coordinate system description. $\hat{b}_{\bar{n}}^{(1)}$ and $\hat{b}_{\bar{n}}^{(2)}$ are the two new bosonic field operators of LE$_{\bar{n}}$ mode ($\bar{n}$-th LE mode) of MoS$_{2}$ nanodisk, as a function of the coupling between the field and excited states $|1\rangle$, $|2\rangle$ of the V-type QE, respectively [39]. $\omega _{\bar{n}}$ and $\gamma _{\bar{n}}$ are the corresponding resonance frequency and decay rate, respectively. $\tilde{g}_{\bar{n}}^{(ji)}(j=1,2;i=1,2)$ is the coupling strength for the interaction between the excited state $|i\rangle$ of QE and the electric field of LE$_{\bar{n}}$ mode. The specific expressions of $\tilde{g}_{\bar{n}}^{(ji)}$ are given by
\begin{align*}
\tilde{g}_{\bar{n}}^{(11)}=&g_{\bar{n}}^{(1) }, \tag{14}
\\
\tilde{g}_{\bar{n}}^{(12)}=&\mu _{21}^{\left(\bar{n} \right) } g_{\bar{n}}^{(2) }, \tag{15}
\\
\tilde{g}_{\bar{n}}^{(22)}=&\sqrt{1-\left|\mu _{\bar{n}}^{(21) } \right|^{2} } g_{\bar{n}}^{(2) }, \tag{16}
\\
\tilde{g}_{\bar{n}}^{(21)}=&0, \tag{17}
\end{align*}
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\begin{align*}
\tilde{g}_{\bar{n}}^{(11)}=&g_{\bar{n}}^{(1) }, \tag{14}
\\
\tilde{g}_{\bar{n}}^{(12)}=&\mu _{21}^{\left(\bar{n} \right) } g_{\bar{n}}^{(2) }, \tag{15}
\\
\tilde{g}_{\bar{n}}^{(22)}=&\sqrt{1-\left|\mu _{\bar{n}}^{(21) } \right|^{2} } g_{\bar{n}}^{(2) }, \tag{16}
\\
\tilde{g}_{\bar{n}}^{(21)}=&0, \tag{17}
\end{align*}
where $g_{\bar{n}}^{(1) }$ and $g_{\bar{n}}^{(2) }$ are the coupling strength for the interaction between the excited states $\left.| 1 \right\rangle$, $\left.| 2 \right\rangle$ of the QE and the electric field of LE$_{\bar{n}}$ mode when the excited states $\left.| 1 \right\rangle$, $\left.| 2 \right\rangle$ are assumed to exist individually, respectively. $\mu _{21}^{(\bar{n}) }$ stands for the mode overlap and is given by
\begin{equation*}
\mu _{21}^{(\bar{n}) }=(-1)^{l}. \tag{18}
\end{equation*}
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\begin{equation*}
\mu _{21}^{(\bar{n}) }=(-1)^{l}. \tag{18}
\end{equation*}
For nanodisk structures, the plasmon eigenmodes can be classified by angular momentum $l$ and contain a subset of radial indexes $n$ to describe the various types of modes in radial constraints [40], which we can obtain $\bar{n}=(l,n)$. For the hybrid systems studied in this paper, the mode overlap takes a value of either $\mu _{21}^{(\bar{n}) }=1$ or $\mu _{21}^{(\bar{n}) }=-1$, corresponding to when the angular momentum is $l=0$ or $l=1$ in our model, respectively. Now, we can take $n$ instead of the compact index $\bar{n}$ under the condition that $\mu _{21}^{(\bar{n}) }=1$ or $\mu _{21}^{(\bar{n}) }=-1$.
Therefore, we can express the effective multi-mode Hamiltonian ((13)) in the matrix form in the basis $\lbrace \left. | 1,\varnothing \right\rangle, \left.| g,1_{1} \right\rangle,{\ldots },\left. | g,1_{N} \right\rangle,\left.| 2,\varnothing \right\rangle \rbrace$[28], [37]
\begin{align*}
&{\hat{H}}_{\text{eff}}=\\
& \hbar \!\begin{pmatrix}\!-i\frac{\gamma _{0} }{2} & g_{1}^{(1)} &g_{2}^{(1)} &\cdots &g_{N}^{(1)} &0 \\
\\
g_{1}^{(1)}& \bigtriangleup _{1}-i\frac{\gamma _{1} }{2} &0 &\cdots & 0 &g_{1}^{(2)} \\
g_{2}^{(1)}& 0 & \bigtriangleup _{2}-i\frac{\gamma _{2} }{2} &\ddots &\vdots &g_{2}^{(2)} \\
\vdots & \vdots & \ddots &\ddots &0 &\vdots \\
g_{N}^{(1)}& 0 & 0 & \cdots & \bigtriangleup _{n}-i\frac{\gamma _{N} }{2} &g_{N}^{(2)} \\
0 & g_{1}^{(2)} & g_{2}^{(2)} &\cdots & g_{N}^{(2)} &-i\frac{\gamma _{0} }{2}\!\! \end{pmatrix}\!, \tag{19}
\end{align*}
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\begin{align*}
&{\hat{H}}_{\text{eff}}=\\
& \hbar \!\begin{pmatrix}\!-i\frac{\gamma _{0} }{2} & g_{1}^{(1)} &g_{2}^{(1)} &\cdots &g_{N}^{(1)} &0 \\
\\
g_{1}^{(1)}& \bigtriangleup _{1}-i\frac{\gamma _{1} }{2} &0 &\cdots & 0 &g_{1}^{(2)} \\
g_{2}^{(1)}& 0 & \bigtriangleup _{2}-i\frac{\gamma _{2} }{2} &\ddots &\vdots &g_{2}^{(2)} \\
\vdots & \vdots & \ddots &\ddots &0 &\vdots \\
g_{N}^{(1)}& 0 & 0 & \cdots & \bigtriangleup _{n}-i\frac{\gamma _{N} }{2} &g_{N}^{(2)} \\
0 & g_{1}^{(2)} & g_{2}^{(2)} &\cdots & g_{N}^{(2)} &-i\frac{\gamma _{0} }{2}\!\! \end{pmatrix}\!, \tag{19}
\end{align*}
where $\bigtriangleup _{n}=\omega _{n}-\omega _{0}$ is the detuning between the $\text{LE} _{n}$ mode and the QE.
SECTION III.
Results and Discussion
In this section, we discuss the population dynamics of the coupled system. We first find and determine all the frequencies of plasmonic resonance in MoS$_{2}$ nanodisks. Then we use a multi-Lorentzian model to derive the coupling strength $g_{n }$ for each $\text{LE}_{n }$ mode. Finally, we present the results for the excited states of the V-type QE and partial $\text{LE}_{n }$ modes for two initial states and various transition frequency.
The MoS$_{2}$ nanodisk supports LE modes at the resonance frequencies $\omega _{n}^{l}$ obtained by numerically solving the equation
\begin{align*}
\sigma _{\mathrm{MoS_{2}} }(\omega _{n}^{l})/\omega _{n}^{l}=2i\varepsilon _{0} R/ \zeta _{n}^{l}, \tag{20}
\end{align*}
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\begin{align*}
\sigma _{\mathrm{MoS_{2}} }(\omega _{n}^{l})/\omega _{n}^{l}=2i\varepsilon _{0} R/ \zeta _{n}^{l}, \tag{20}
\end{align*}
where $ \zeta _{n}^{l}$ denotes the disk geometry eigenmodes which correspond to the plasmon resonances in individual 2D nanodisk [19], [41]. The $ \zeta _{n}^{l}$ is obtained numerically by solving the equation which is give by
\begin{align*}
\mathbf {K }^{l}\mathbf {c }_{n }^{l} =\zeta _{n}^{l}\mathbf {G}^{l}\mathbf {c }_{n }^{l}. \tag{21}
\end{align*}
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\begin{align*}
\mathbf {K }^{l}\mathbf {c }_{n }^{l} =\zeta _{n}^{l}\mathbf {G}^{l}\mathbf {c }_{n }^{l}. \tag{21}
\end{align*}
To determine the coupling strength $g_{n}$ for each $\text{LE}_{n}$ mode, we introduce a multi-Lorentzian model for the photonic density of states [30]
\begin{align*}
\kappa (\omega)\equiv \sum _{n}\left|g_{n}(\omega)\right|^{2}=\frac{\omega ^{2}}{\hbar \pi \varepsilon _{0} c^{2}} \mathbf {d} \cdot \operatorname{Im}\left[\mathbf {G}\left(\mathbf {r}_{d}, \mathbf {r}_{d}, \omega \right)\right] \cdot \mathbf {d}^{*}, \tag{22}
\end{align*}
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\begin{align*}
\kappa (\omega)\equiv \sum _{n}\left|g_{n}(\omega)\right|^{2}=\frac{\omega ^{2}}{\hbar \pi \varepsilon _{0} c^{2}} \mathbf {d} \cdot \operatorname{Im}\left[\mathbf {G}\left(\mathbf {r}_{d}, \mathbf {r}_{d}, \omega \right)\right] \cdot \mathbf {d}^{*}, \tag{22}
\end{align*}
where $g_{n}(\omega)=\sqrt{\frac{\gamma _{n}}{2\pi }} \frac{g_{n}(\mathbf {r}_{d})}{\omega -\omega _{n}+i\frac{\gamma _{n}}{2}}$ is the coupling parameter. $\omega _{n}$ can be obtained from the (20) depending on the vaule of $\mu _{21}^{(\bar{n})}$. According to the dipole moment of the V-type QE and the (22), it can be seen that $g_{n}^{(1)}$ is equal to $g_{n}^{(2)}$. Thus, we can evaluate the values of coupling strength and decay rate through $\mathbf {d}=\mathbf {d}_{20}$. The mode overlap takes the value as $\mu _{21}^{(\bar{n})}=-1$ corresponding to $\sum _{n}|g_{n}(\omega)|^{2}=\frac{\omega ^{2}}{\hbar \pi \varepsilon _{0} c^{2}} \mathbf {d}_{20}\cdot \operatorname{Im}[\mathbf {G}_{\omega }^{xx } (\mathbf {r}_{d},\mathbf {r}_{d})] \cdot \mathbf {d}_{20}^{*}$, and the mode overlap takes the value as $\mu _{21}^{(\bar{n})}=1$ corresponding to $\sum _{n}|g_{n}(\omega)|^{2}=\frac{\omega ^{2}}{\hbar \pi \varepsilon _{0} c^{2}} \mathbf {d}_{20} \cdot \operatorname{Im}[\mathbf {G}_{\omega }^{zz }(\mathbf {r}_{d}, \mathbf {r}_{d})] \cdot \mathbf {d}_{20}^{*}$.
We consider a dipole moment of $d=24\mathrm{\,D}$ of the V-type QE placed at $\mathbf {r}_{\mathrm{d}}=(0,0,2 \mathrm{\,nm})$ from the MoS$_{2}$ nanodisk with radius $R=30\mathrm{\,nm}$, and then we can obtain the parameters of the multi-Lorentzian model by simple curve fitting. The obtained parameters are listed in Tabel 1, the fitting results are presented in Fig. 2.
Finally, we discuss the population dynamics for the excited states of the V-type QE and partial $\text{LE}_{n}$ modes. In the following study, we consider that the QE excited states have a decay rate of $\gamma _{0}=0.05$ $\text{meV}$ [21]. For the effective Hamiltonian (19), we define the right and left eigenvectors $|\Pi _{m}^{R}\rangle$ and $|\Pi _{m}^{L}\rangle$, respectively, with $\hat{H}_{\text{eff }}|\Pi _{m}^{R}\rangle =\lambda _{m}|\Pi _{m}^{R}\rangle$ and $\hat{H}_{\text{eff} }^{\dagger } |\Pi _{m}^{L}\rangle =\lambda _{m}^{\ast }|\Pi _{m}^{L}\rangle$. The eigenvectors satisfy the relation $\langle \Pi _{m}^{L} | \Pi _{m}^{R}\rangle =\delta _{mn}$. The eigenvectors can be written as [42]
\begin{align*}
\left|\Pi _{m}^{R}\right\rangle =&m_{01}|1\rangle |\varnothing \rangle +\sum _{n=1}^{N} m_{n}|g\rangle |1_{n}\rangle +m_{02}|2\rangle |\varnothing \rangle, \tag{23}
\\
\left|\Pi _{m}^{L}\right\rangle =&-m_{01}^{\star } |1\rangle |\varnothing \rangle +\sum _{n=1}^{N} m_{n}^{\ast }|g\rangle |1_{n}\rangle -m_{02}^{\star }|2\rangle |\varnothing \rangle, \tag{24}
\end{align*}
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\begin{align*}
\left|\Pi _{m}^{R}\right\rangle =&m_{01}|1\rangle |\varnothing \rangle +\sum _{n=1}^{N} m_{n}|g\rangle |1_{n}\rangle +m_{02}|2\rangle |\varnothing \rangle, \tag{23}
\\
\left|\Pi _{m}^{L}\right\rangle =&-m_{01}^{\star } |1\rangle |\varnothing \rangle +\sum _{n=1}^{N} m_{n}^{\ast }|g\rangle |1_{n}\rangle -m_{02}^{\star }|2\rangle |\varnothing \rangle, \tag{24}
\end{align*}
where the expansion coefficients $m_{01}$, $m_{n}(n=1,2,\ldots,N)$ and $m_{02}$ are the values of the elements of the normalized eigenvector $|\Pi _{m}^{R}\rangle$ in the basis $\lbrace | 1,\varnothing \rangle, | g,1_{1} \rangle,{\ldots }, | g,1_{N} \rangle,| 2,\varnothing \rangle \rbrace ^{T}$, which are also called the Hopfield coefficients [43], [44]. Accordingly, the wavefunction of at time $t$ is written as
\begin{align*}
|\psi _{\text{eff}}(t)\rangle =&C_{e 1}(t)|1\rangle |\varnothing \rangle \!+\!C_{e 2}(t)|2\rangle |\varnothing \rangle \!+\!\sum _{n=1}^{N} C_{n}(t)\cdot |g\rangle |1_{n}\rangle \\
=&\sum _{m=1}^{N+2} \eta _{m}|\Pi _{m}^{R}\rangle e^{-i \lambda _{m}t }. \tag{25}
\end{align*}
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\begin{align*}
|\psi _{\text{eff}}(t)\rangle =&C_{e 1}(t)|1\rangle |\varnothing \rangle \!+\!C_{e 2}(t)|2\rangle |\varnothing \rangle \!+\!\sum _{n=1}^{N} C_{n}(t)\cdot |g\rangle |1_{n}\rangle \\
=&\sum _{m=1}^{N+2} \eta _{m}|\Pi _{m}^{R}\rangle e^{-i \lambda _{m}t }. \tag{25}
\end{align*}
The estimation of the coefficient $\eta _{m}$ depends on the initial state of the system. We consider two initial states, $|\psi _{\text{eff}}(0)\rangle =|1\rangle |\varnothing \rangle$ for QE excited state $|1\rangle$ and the superposition state $|\psi _{\text{eff}}(0)\rangle =\sqrt{\frac{1}{2}} |1\rangle |\varnothing \rangle +\sqrt{\frac{1}{2}} |2\rangle |\varnothing \rangle$. For the former case, the coefficient $\eta _{m}$ is expressed as $\eta _{m}=\left\langle \Pi _{m}^{L} \mid \psi _{\text{eff }}(0)\right\rangle =-m_{01}$, and the wavefunction is written as $|\psi _{\text{eff}}(t)\rangle =\sum _{m=1}^{N+2} \eta _{m}|\Pi _{m}^{R}\rangle e^{-i\lambda _{m} t}=\sum _{m=1}^{N+2}-m_{01} | \Pi _{m}^{R}\rangle e^{-i\lambda _{m} t}$. The time evolution of the excited state $|1\rangle$ and $|2\rangle$ are given by $| C_{e 1}(t) |^{2}=| \left\langle 1,\varnothing | \psi _{\text{eff}}(t) \right\rangle |^{2}=| \sum _{m=1}^{N+2}m_{01}^{2} e^{-i\lambda _{m} t} |^{2}$ and $| C_{e 2}(t) |^{2}=| \left\langle 2,\varnothing | \psi _{\text{eff}}(t) \right\rangle |^{2}=| \sum _{m=1}^{N+2}m_{01} m_{02} e^{-i\lambda _{m} t} |^{2}$, respectively. On the other hand, the population of $n\text{th}$ $\text{LE}$ mode is expressed as $| C_{n}(t) |^{2}=| \left\langle 0,1_{n} | \psi _{\text{eff}}(t) \right\rangle |^{2}=| \sum _{m=1}^{N+2}m_{01} m_{n} e^{-i\lambda _{m} t} |^{2}$, where $\lambda _{m}$ denotes the eigenvalue of the effective Hamiltonian, which is given by
\begin{align*}
\lambda _{m}=\omega _{m}-i\frac{\gamma _{m} }{2}, \tag{26}
\end{align*}
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\begin{align*}
\lambda _{m}=\omega _{m}-i\frac{\gamma _{m} }{2}, \tag{26}
\end{align*}
where we take $m=1,\ldots, N+2$ for LE modes $N=11$ and 2.
While for the case of superposition state as the initial state, we obtain the coefficient $\eta _{m}=\left\langle \Pi _{m}^{L} \mid \psi _{\text{eff }}(0)\right\rangle =-\sqrt{\frac{1}{2}}m_{01}-\sqrt{\frac{1}{2}}m_{02}$, and the wavefunction is expreesed as $|\psi _{\text{eff}}(t)\rangle =\sum _{m=1}^{N+2} \eta _{m}|\Pi _{m}^{R}\rangle e^{-i\lambda _{m} t}=\sum _{m=1}^{N+2}\left(-\sqrt{\frac{1}{2}}m_{01}-\sqrt{\frac{1}{2}}m_{02}\right) | \Pi _{m}^{R}\rangle e^{-i\lambda _{m} t}$. We can obtain the time evolution of the excited states $|1\rangle$ and $|2\rangle$ as $| C_{e 1}(t) |^{2}=| \left\langle 1,\varnothing | \psi _{\text{eff}}(t) \right\rangle |^{2}=\frac{1}{2}| \sum _{m=1}^{N+2}(m_{01}+m_{02})m_{01}e^{-i\lambda _{m} t} |^{2}$ and $| C_{e 2}(t) |^{2}=| \left\langle 2,\varnothing | \psi _{\text{eff}}(t) \right\rangle |^{2}=\frac{1}{2}| \sum _{m=1}^{N+2}(m_{01}+m_{02})m_{02}e^{-i\lambda _{m} t} |^{2}$, respectively. While the population of $\text{LE} _{n}$ modes is $| C_{n}(t) |^{2}=| \left\langle 0,1_{n} | \psi _{\text{eff}}(t) \right\rangle |^{2}=\frac{1}{2}| \sum _{m=1}^{N+2}(m_{01}+m_{02}) m_{n} e^{-i\lambda _{m} t} |^{2}$.
In Fig. 3, we present the population dynamics of a V-type QE and $\text{LE}$ with transition frequency $\omega _{0}=1.94892\mathrm{\,eV}$[21]. The population evolution of $| C_{e 1}(t) |^{2}$, $| C_{e2}(t) |^{2}$ and partial $\text{LE} _{n}$ modes for initial state $|\psi _{\text{eff}}(0)\rangle =|1\rangle |\varnothing \rangle$ are shown in Fig. 3(a) for time $t< 3$ ps.The population dynamics exhibits evident oscillation, a strong characteristic of non-Markovian dynamics. It can be seen that the population of the initial state $|\psi _{\text{eff}}(0)\rangle =|1\rangle |\varnothing \rangle$ decays rapidly within about 50 fs. Specifically, we observe that the oscillation of population transfers between the excited states $|1\rangle$ and $|2\rangle$, indicating that the V-type QE couples to $\text{LE} _{n}$ modes supported by the MoS$_{2}$ nanodisk. The excited states population of the V-type QE decays to about 20$\%$ of its initial value within the first 3 ps and remains a period of time. Furthermore, we find that the $\text{LE} _{2}$, $\text{LE} _{4}$ and $\text{LE}_{11}$ modes primarily govern the population transfer between the states of the QE and the nanodisk. The inset in Fig. 3(a) shows that the effective mode have the same oscillation period as the excited state population of the QE. We thus observe the multiple channels energy transfer process. The energy transfer direction are from initial state $|1\rangle$ to effective modes ($\text{LE} _{2}$, $\text{LE} _{3}$, $\text{LE} _{4}$, $\text{LE} _{5}$ and $\text{LE}_{11}$), and then from $\text{LE}$ effective modes to QE states $|1\rangle$ and $|2\rangle$.
In Fig. 3(b), the time evolution of $| C_{e 1}(t) |^{2}$, $| C_{e 2}(t) |^{2}$, and partial $\text{LE} _{n}$ modes with initial state $|\psi _{\text{eff}}(0)\rangle =\sqrt{\frac{1}{2}} |1\rangle |\varnothing \rangle +\sqrt{\frac{1}{2}} |2\rangle |\varnothing \rangle$ are shown during the first 3 ps. It can be seen that the evolution of $| C_{e 1}(t) |^{2}$ is equal to $| C_{e 2}(t) |^{2}$ at any time. We note that the population of the V-type QE states $|1\rangle$ and $|2\rangle$ decays totally within about 3 ps. Moreover, we observe that the energy transfer between the QE and nanodisk mainly occurs in the $\text{LE} _{2}$, $\text{LE} _{3}$, $\text{LE} _{4}$, $\text{LE} _{5}$ and $\text{LE}_{11}$ modes. Compared to Fig. 3(a), there are more effective modes taking the lead in the energy transfer process. Under with initial state $|\psi _{\text{eff}}(0)\rangle =\sqrt{\frac{1}{2}} |1\rangle |\varnothing \rangle +\sqrt{\frac{1}{2}} |2\rangle |\varnothing \rangle$, the population dynamics of the QE and effective modes exhibits the characteristic of higher-frequency oscillation. In addition, the channel of the energy transfer is either from the two upper states of the QE to effective modes or the reversible process as shown in the inset of Fig. 3(b).
Lastly, we consider the case of the V-type QE with transition frequency $\omega _{0}=1.92128\mathrm{\,eV}$[21]. The population dynamics of a V-type QE and LE is presented in Fig. 3(c) and (d). Under these conditions, we obtain $N=12$ from (22) and numerically time evolution $| C_{e 1}(t) |^{2}$, $| C_{e 2}(t) |^{2}$ and $| C_{n}(t) |^{2}$. In Fig. 3(c), we show the population evolution $| C_{e 1}(t) |^{2}$, $| C_{e 2}(t) |^{2}$, and partial $\text{LE} _{n}$ modes with initial state $|\psi _{\text{eff}}(0)\rangle = |1\rangle |\varnothing \rangle$ during the first 3 ps. The strong non-Markovian character can be observed in the inset of Fig. 3(c). We find that the oscillation of population transfer between states $|1\rangle$ and $|2\rangle$ decreases over time. The result indicates that the V-type QE interacts with the $n\text{th}$ mode. The excited states population of the V-type QE decays to about 30$\%$ of its initial value within the first 3 ps. Furthermore, we notice that the energy transfer between the QE and nanodisk mainly occurs in the $\text{LE} _{4}$ and $\text{LE}_{12}$ modes. Compared to Fig. 3(a), we find that the effective modes ($\text{LE} _{3}, \text{LE} _{4}, \text{LE}_{12}$) have a lower amplitude and oscillation frequency, which leads to a smooth change of population during the dynamics of QE at early stage. In addition, the population dynamics of QE exhibits the small oscillation and longer oscillation period during the first 0.5 ps due to the relatively weak coupling to the effective modes.
In Fig. 3(d), we present the time evolution of $| C_{e 1}(t) |^{2}$, $| C_{e 2}(t) |^{2}$, and partial $\text{LE} _{n}$ modes with initial state $|\psi _{\text{eff}}(0)\rangle =\sqrt{\frac{1}{2}} |1\rangle |\varnothing \rangle +\sqrt{\frac{1}{2}} |2\rangle |\varnothing \rangle$ during the first 3 ps. We also notice the non-Markovian characteristic in the population dynamics of the V-type QE. We note that the population of the V-type QE states $|1\rangle$ and $|2\rangle$ decays rapidly with high-frequency oscillation and completely decreases within about 7 ps (not shown here). Moreover, we notice that the energy transfer between the QE and the nanodisk mainly occurs in the $\text{LE} _{3}$, $\text{LE} _{4}$ and $\text{LE}_{12}$ modes. In contrast to the results shown in Fig. 3(c), the effective modes ($\text{LE} _{3}$, $\text{LE} _{4}$ and $\text{LE}_{12}$) have higher frequency and larger oscillation amplitude, leading to a more rapid decay of QE within the first 3 ps. Furthermore, during the first 3 ps, the QE decay presented in Fig. 3(b) is more drastic compared to Fig. 3(d). The reason is that both the number and oscillation strength involved in the energy transfer from the effective modes to the QE are larger.
In conclusion, we study the population dynamics of a QE near a MoS$_{2}$ nanodisk with different initial states based on the proposed effective multi-mode model. An effective Hamiltonian for the hybrid system is derived from our model, which is in the same form as the multi-mode Jaynes–Cummings model, and thus corresponding to the conventional cQED description. By quantifying the photonic density of states of MoS$_{2}$ nanodisk through a multi-Lorentzian model, we can determine the parameters of the effective Hamiltonian to investigate the quantum dynamics of the system in the manner of cQED. We find that oscillatory population dynamics between the two excited states of the V-type QE, which clearly exhibits the non-Markovian feature. We also find that the energy transfer between the V-type QE and the $\text{LE} _{n}$ modes associated with different initial states. For spontaneous emission process with OE initially in one of the excited state, the resultant decay of QE population is observably slower than that of QE initially in the superposition state since the latter case involves more effective modes that contribute to the coupling of QE to MoS$_{2}$ nanodisk and these effective modes have higher frequency and greater oscillation. We believe that the proposed effective multi-mode model can provide insight into the strong light-matter interaction between the QE and the 2D materials and is helpful to develop the advanced quantum technologies.
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