Nanorod Size Dependence of Coherent Coupling between Individual and Collective Excitations via Transverse Electromagnetic Field

The Hamiltonian for electrons in an electromagnetic field is described as

	$\displaystyle\hat{H}$	$\displaystyle=\sum_{j}\frac{\left\{\hat{\bm{p}}_{j}-e\int d\bm{r}\bm{A}(\bm{r},t)\delta(\bm{r}-\hat{\bm{r}}_{j})\right\}^{2}}{2m^{*}}$
		$\displaystyle+e\sum_{j}\int d\bm{r}\phi(\bm{r},t)\delta(\bm{r}-\hat{\bm{r}}_{j}),$
		$\displaystyle+\frac{1}{2}\sum_{i\neq j}\frac{e^{2}}{4\pi\epsilon_{0}|\hat{\bm{r}}_{i}-\hat{\bm{r}}_{j}|},$		(1)

where $m^{*}$ and $e$ are the electron effective mass and charge, respectively, and $\epsilon_{0}$ is the dielectric constant of the vacuum. For the Coulomb gauge, the vector potential satisfies ${\rm div}\bm{A}=0$ and describes the T field. The scalar potential gives the L field, which is divided into the nuclei and external origins, $\phi(\bm{r},t)=\phi_{\rm ncl}({\bm{r}})+\phi_{\rm ext}(\bm{r},t)$. We assume that the nucleis are fixed. Note that the Coulomb interaction in the third therm in Eq. (1) is a part of the L component.

We consider the second quantization for the Hamiltonian $\hat{H}$. In the absence of external fields, $\bm{A}=0$ and $\phi_{\rm ext}=0$, we obtain

	$\displaystyle\hat{H}_{0}$	$\displaystyle=\sum_{n,n^{\prime}}{\psi_{n^{\prime}}^{*}(\bm{x})\left(-\frac{\hbar^{2}}{2m^{*}}\bm{\nabla}_{x}^{2}\right)\psi_{n}(\bm{x})\hat{a}_{n^{\prime}}^{\dagger}}\hat{a}_{n}$
		$\displaystyle+\int d\bm{x}\sum_{n,n^{\prime}}\hat{a}_{n^{\prime}}^{\dagger}{\psi_{n^{\prime}}^{*}(\bm{x})}\left[\phi_{\rm ncl}({\bm{x}})+\hat{\phi}_{\rm e-e}({\bm{x}})\right]{\psi_{n}(\bm{x})}\hat{a}_{n},$		(2)

with

$$\hat{\phi}_{\rm e-e}({\bm{x}})=\int d\bm{x}^{\prime}{e\sum_{m,m^{\prime}}\frac{\psi_{m^{\prime}}^{*}(\bm{x}^{\prime})\psi_{m}(\bm{x}^{\prime})}{4\pi\epsilon_{0}|\bm{x}-\bm{x}^{\prime}|}\hat{a}_{m^{\prime}}^{\dagger}\hat{a}_{m},}$$

(3)

describing the “inner” Coulomb interaction. Here, $\hat{a}_{n}^{\dagger}$ and $\hat{a}_{n}$ are the creation and annihilation operators of the electrons of the state $n$, respectively. $\psi_{n}(\bm{x})$ is the wavefunction of the state $n$. We treat $\hat{H}_{0}$ as a nonperturbative Hamiltonian, which has no time dependence. Note that for an electrically neutral of material, $\langle\phi_{\rm ncl}+\hat{\phi}_{\rm e-e}\rangle_{0}=0$ must be satisfied. Here, $\langle\cdots\rangle_{0}$ denotes the statistical average of the system without an external field. The electron density in a static situation is given as $\rho_{0}(\bm{x})/e=\sum_{n}\psi_{n}^{*}(\bm{x})\psi_{n}(\bm{x})\langle\hat{a}_{n}^{\dagger}\hat{a}_{n}\rangle_{0}$. Hence, the charge nutral condition is rewritten as $\phi_{\rm ncl}(\bm{x})=-\int d\bm{x}^{\prime}\frac{\rho_{0}(\bm{x}^{\prime})}{4\pi\epsilon_{0}|\bm{x}-\bm{x}^{\prime}|}$.

Let us consider a perturbative Hamiltonian $\hat{H}^{\prime}$ when an external field is applied,

$$\hat{H}=\hat{H}_{0}+\hat{H}^{\prime}.$$

(4)

We assume a monochromatic electromagnetic field, $\bm{A}(\bm{x},t)=\bm{A}(\bm{x};\omega)e^{-i\omega t}$. Then,

$$\hat{H}^{\prime}\simeq-\int d\bm{x}\left[\hat{\bm{j}}{(\bm{x})}\cdot\bm{A}(\bm{x};\omega)-\delta\hat{\rho}{(\bm{x})}\phi_{\rm ext}(\bm{x};\omega)\right]e^{-i\omega t}.$$

(5)

Here, $\hat{\bm{j}}(\bm{x})$ and $\delta\hat{\rho}(\bm{x})=\hat{\rho}(\bm{x})-\rho_{0}(\bm{x})$ are the induced current and the (deviation of) charge density operators, respectively,

	$\displaystyle\hat{\bm{j}}(\bm{x})$	$\displaystyle=\sum_{n,n^{\prime}}\bm{j}_{n^{\prime}n}(\bm{x})\hat{a}_{n^{\prime}}^{\dagger}\hat{a}_{n},$		(6)
	$\displaystyle\delta\hat{\rho}(\bm{x})$	$\displaystyle=\sum_{n,n^{\prime}}\rho_{n^{\prime}n}(\bm{x})\hat{a}_{n^{\prime}}^{\dagger}\hat{a}_{n}-\rho_{0}(\bm{x}).$		(7)

The matrix elements for the current and charge are

	$\displaystyle\rho_{n^{\prime}n}(\bm{x})$	$\displaystyle=e\psi_{n^{\prime}}^{*}(\bm{x})\psi_{n}(\bm{x})$		(8)
	$\displaystyle\bm{j}_{n^{\prime}n}(\bm{x})$	$\displaystyle=-\frac{e\hbar}{2im^{*}}\Big{[}\left\{\bm{\nabla}_{x}\psi_{n^{\prime}}^{*}(\bm{x})\right\}\psi_{n}(\bm{x})-\psi_{n^{\prime}}^{*}(\bm{x})\left\{\bm{\nabla}_{x}\psi_{n}(\bm{x})\right\}\Big{]}$
		$\displaystyle-\frac{e^{2}}{m^{*}}\psi_{n^{\prime}}^{*}(\bm{x})\bm{A}(\bm{x};\omega)\psi_{n}(\bm{x}).$		(9)

In the following, we consider a self-consistent treatment of the induced densities and the external fields with the Maxwell’s equations. In the second term in Eq. (5), the interaction between the external field and the positive background charge due to the nuclei is included. For the self-consistent treatment, the interaction between the induced charge densities should be obtained. We consider an additional scalar potential

$$\hat{\phi}_{\rm pol}(\bm{x})=\int d\bm{x}^{\prime}\frac{\delta\hat{\rho}(\bm{x}^{\prime})}{4\pi\epsilon_{0}|\bm{x}-\bm{x}^{\prime}|}$$

(10)

by the induced polarized charge due to the light incidence, which originates from the electron-electron interaction. Then, the perturbative Hamiltonian is modified as

	$\displaystyle\hat{H}^{\prime}\simeq-\int d\bm{x}\left[\hat{\bm{j}}(\bm{x})\cdot\bm{A}(\bm{x};\omega)-\delta\hat{\rho}(\bm{x})\Big{(}\phi_{\rm ext}(\bm{x};\omega)+\hat{\phi}_{\rm pol}(\bm{x})\Big{)}\right]$
	$\displaystyle\hskip 184.9429pt\times e^{-i\omega t}.$		(11)

From the Hamiltonian $\hat{H}=\hat{H}_{0}+\hat{H}^{\prime}$, a nonlocal susceptibility is deduced. In the formulation, we use a four-vector representation, $\bm{\mathcal{A}}=(\bm{A},-\phi/c)^{\rm t}$ and $\hat{\bm{\mathcal{J}}}=(\hat{\bm{j}},c\delta\hat{\rho})^{\rm t}$. By taking the statistical average of the four-vector current, $\langle\hat{\bm{\mathcal{J}}}(\bm{x},t)\rangle=\bm{\mathcal{J}}(\bm{x};\omega)e^{-i\omega t}$, we obtain the constitutive equation,

$$\bm{\mathcal{J}}(\bm{x};\omega)=\bm{\mathcal{J}}_{0}(\bm{x};\omega)+\int d\bm{x}^{\prime}\bar{\mathcal{X}}(\bm{x},\bm{x}^{\prime};\omega)\cdot\bm{\mathcal{A}}(\bm{x}^{\prime};\omega).$$

(12)

The first term is $\bm{\mathcal{J}}_{0}=-(e\rho_{0}/m)(\bm{A},0)^{\rm t}$. The nonlocal susceptibility is described in terms of $\bm{\mathcal{J}}_{\nu\mu}=\langle\nu|\hat{\bm{\mathcal{J}}}|\mu\rangle$, [23]

	$\displaystyle\bar{\mathcal{X}}(\bm{x},\bm{x}^{\prime};\omega)=\sum_{\mu,\nu}$	$\displaystyle\left[f_{\nu\mu}(\omega)\bm{\mathcal{J}}_{\mu\nu}(\bm{x})\left(\bm{\mathcal{J}}_{\nu\mu}(\bm{x}^{\prime})\right)^{\rm t}\right.$
		$\displaystyle\left.+h_{\nu\mu}(\omega)\bm{\mathcal{J}}_{\nu\mu}(\bm{x})\left(\bm{\mathcal{J}}_{\mu\nu}(\bm{x}^{\prime})\right)^{\rm t}\right].$		(13)

Here, we consider the bases $|\mu\rangle$ and $|\nu\rangle$ from the eigenstates of $\hat{H}_{0}$ including the electron-electron interaction. The factors are $f_{\nu\mu}(\omega)=\eta_{\mu}/(\hbar\omega_{\nu\mu}-\hbar\omega-i\gamma)$ and $h_{\mu\nu}(\omega)=\eta_{\mu}/(\hbar\omega_{\nu\mu}+\hbar\omega+i\gamma)$ with the difference of eigenenergies $\hbar\omega_{\nu\mu}=E_{\nu}-E_{\mu}$. The imaginary part $\gamma$ is an infinitesimal value owing to the causality or the assumed nonradiative damping constant. The numerator is $\eta_{\mu}=\langle\mu|e^{-\beta\hat{H}_{0}}|\mu\rangle/Z_{0}$ with the partition function $Z_{0}={\rm Tr}\{e^{-\beta\hat{H}_{0}}\}$.

The current and charge densities $\bm{\mathcal{J}}(\bm{r};\omega)$ become the sources of response fields in the Maxwell’s equations. In the vector and scalar potential representations, the Maxwell’s equations are summarized as

$$\bar{\mathcal{D}}(\bm{x};\omega)\bm{\mathcal{A}}(\bm{x};\omega)=-\mu_{0}\bm{\mathcal{J}}(\bm{x};\omega)$$

(14)

with

$$\bar{\mathcal{D}}(\bm{x};\omega)=\left(\begin{matrix}\bm{\nabla}^{2}+(\omega/c)^{2}&-i(\omega/c)\bm{\nabla}\\ 0&-\bm{\nabla}^{2}\end{matrix}\right).$$

(15)

The formal solution of Eq. (14) is

$$\bm{\mathcal{A}}(\bm{x};\omega)=\bm{\mathcal{A}}_{0}(\bm{x};\omega)-\mu_{0}\int d\bm{x}^{\prime}\bar{\mathcal{G}}(\bm{x},\bm{x}^{\prime};\omega)\bm{\mathcal{J}}(\bm{x}^{\prime};\omega)$$

(16)

with the Green’s function satisfying

$$\bar{\mathcal{D}}(\bm{x};\omega)\bar{\mathcal{G}}(\bm{x},\bm{x}^{\prime};\omega)=\delta(\bm{x}-\bm{x}^{\prime}).$$

(17)

The first term denotes an “incident field” as $\bar{\mathcal{D}}(\bm{x};\omega)\bm{\mathcal{A}}_{0}(\bm{x};\omega)=0$.

The constitutive equation (12) and the solution of Maxwell’s equations (16) are in a self-consistent relation. To make it solvable form, we construct a matrix form of the self-consistent equation by multiplying $\left(\bm{\mathcal{J}}_{\nu^{\prime}\mu^{\prime}}(\bm{x})\right)^{\rm t}$, $\left(\bm{\mathcal{J}}_{\mu^{\prime}\nu^{\prime}}(\bm{x})\right)^{\rm t}$, and $\left(\varphi_{j^{\prime}}(\bm{x})\bm{e}_{\beta}\right)^{\rm t}$ from the left and integrating with respect to $\bm{x}$. Here, $\varphi_{j^{\prime}}(\bm{x})$ is a function for the expansion of delta function, $\delta(\bm{x}-\bm{x}^{\prime})=\sum_{j}\varphi_{j}^{\ast}(\bm{x})\varphi_{j}(\bm{x}^{\prime})$, which is applied to $\bm{\mathcal{J}}_{0}(\bm{x};\omega)$ term. For $\left(\bm{\mathcal{J}}_{\nu^{\prime}\mu^{\prime}}(\bm{x})\right)^{\rm t}$, the equation becomes

	$\displaystyle\left(\hbar\omega_{\nu^{\prime}\mu^{\prime}}-\hbar\omega-i\gamma\right)X_{\nu^{\prime}\mu^{\prime}}^{(-)}$	$\displaystyle=Y_{\nu^{\prime}\mu^{\prime}}^{(0)}+\sum_{j,\alpha}U_{\nu^{\prime}\mu^{\prime},j\alpha}X_{j\alpha}^{(A)}$
		$\displaystyle-\sum_{\mu,\nu}\left[K_{\nu^{\prime}\mu^{\prime},\mu\nu}X_{\nu\mu}^{(-)}+L_{\nu^{\prime}\mu^{\prime},\nu\mu}X_{\mu\nu}^{(+)}\right]$		(18)

with

	$\displaystyle X_{\nu\mu}^{(-)}(\omega)$	$\displaystyle=\frac{1}{\hbar\omega_{\nu\mu}-\hbar\omega-i\gamma}\int d\bm{x}\left(\bm{\mathcal{J}}_{\nu\mu}(\bm{x})\right)^{\rm t}\bm{\mathcal{A}}(\bm{x};\omega),$		(19)
	$\displaystyle X_{\mu\nu}^{(+)}(\omega)$	$\displaystyle=\frac{1}{\hbar\omega_{\nu\mu}+\hbar\omega+i\gamma}\int d\bm{x}\left(\bm{\mathcal{J}}_{\mu\nu}(\bm{x})\right)^{\rm t}\bm{\mathcal{A}}(\bm{x};\omega),$		(20)
	$\displaystyle X_{j\alpha}^{(A)}(\omega)$	$\displaystyle=\int d\bm{x}\left(\varphi_{j}(\bm{x})\bm{e}_{\alpha}\right)^{\rm t}\bm{\mathcal{A}}(\bm{x};\omega),$		(21)
	$\displaystyle Y^{\rm(0)}_{\nu^{\prime}\mu^{\prime}}(\omega)$	$\displaystyle=\int d\bm{x}\left(\bm{\mathcal{J}}_{\nu^{\prime}\mu^{\prime}}(\bm{x})\right)^{\rm t}\bm{\mathcal{A}}_{0}(\bm{x};\omega)$		(22)

and

	$\displaystyle K_{\nu^{\prime}\mu^{\prime},\mu\nu}(\omega)$	$\displaystyle=\mu_{0}\eta_{\mu}\iint d\bm{x}d\bm{x}^{\prime}\left(\bm{\mathcal{J}}_{\nu^{\prime}\mu^{\prime}}(\bm{x})\right)^{\rm t}\bar{\mathcal{G}}(\bm{x},\bm{x}^{\prime};\omega)\bm{\mathcal{J}}_{\mu\nu}(\bm{x}^{\prime}),$		(23)
	$\displaystyle L_{\nu^{\prime}\mu^{\prime},\nu\mu}(\omega)$	$\displaystyle=\mu_{0}\eta_{\mu}\iint d\bm{x}d\bm{x}^{\prime}\left(\bm{\mathcal{J}}_{\nu^{\prime}\mu^{\prime}}(\bm{x})\right)^{\rm t}\bar{\mathcal{G}}(\bm{x},\bm{x}^{\prime};\omega)\bm{\mathcal{J}}_{\nu\mu}(\bm{x}^{\prime}),$		(24)
	$\displaystyle U_{\nu^{\prime}\mu^{\prime},j\alpha}(\omega)$	$\displaystyle=\left(\frac{\omega_{\rm p}}{c}\right)^{2}\iint d\bm{x}d\bm{x}^{\prime}\left(\bm{\mathcal{J}}_{\nu^{\prime}\mu^{\prime}}(\bm{x})\right)^{\rm t}\bar{\mathcal{G}}(\bm{x},\bm{x}^{\prime};\omega)$
		$\displaystyle\hskip 119.50157pt\times\left(\varphi_{j}^{\ast}(\bm{x}^{\prime})\bm{e}_{\alpha}\right).$		(25)

$K_{\nu^{\prime}\mu^{\prime},\mu\nu}$, $L_{\nu^{\prime}\mu^{\prime},\nu\mu}$, and $U_{\nu^{\prime}\mu^{\prime},j\alpha}$ are the matrix elements in the matrix form of the self-consistent equation. Here, $\omega_{\rm p}=\sqrt{e^{2}n_{0}/(\epsilon_{0}m^{*})}$ is the plasma frequency for bulk materials with $n_{0}=(2m^{*}\varepsilon_{\rm F}/\hbar^{2})^{3/2}/(3\pi^{2})$ being the density of electrons. The factors $X_{\nu\mu}^{(\mp)}$, $X_{j\alpha}^{(A)}$, and $Y^{\rm(0)}_{\nu^{\prime}\mu^{\prime}}$ are treated in vector forms as $\bm{X}$ and $\bm{Y}$. For multiplying $\left(\bm{\mathcal{J}}_{\mu^{\prime}\nu^{\prime}}(\bm{x})\right)^{\rm t}$ and $\left(\varphi_{m^{\prime}}(\bm{x})\bm{e}_{\beta}^{\rm t}\right)$, we can introduce other matrix elements. By combining the three equations, we built a matrix $\bar{\Xi}(\omega)$ in the matrix form of the self-consistent equation

$$\bar{\Xi}(\omega)\bm{X}(\omega)=\bm{Y}(\omega).$$

(26)

The matrix $\bar{\Xi}(\omega)$ includes the information of electronic excitations in a microscopic description and the electromagnetic field acting on the excitations. Hence, the roots of $\bar{\Xi}(\omega)$ give the spectrum of the individual and collective excitations of electrons and holes. The detailed formulation is summarized in our previous study [22].

The evaluation of the matrix elements for $\bar{\Xi}(\omega)$ is implemented by numerical calculations. The factors $K_{\nu^{\prime}\mu^{\prime},\mu\nu}$ and $L_{\nu^{\prime}\mu^{\prime},\nu\mu}$ with the Green’s function $\bar{\mathcal{G}}$ consists of the current-current, current-charge, and charge-charge interactions, which are mediated by the T-, T-L hybridization, and L-components of the electromagnetic field in $\bar{\mathcal{G}}$. Then, we introduce a tuning parameter $\zeta$ to switch between the presence and absence of the T field contribution in $\bar{\mathcal{G}}$. At $\zeta=0(1)$, the T field is neglected (completely considered). In the following, we take account of the effect of the background refractive index, $n_{\rm b}$. In Eqs. (23)–(25), multiple integrals for the space coordinates are included. By applying the Fourier transformation, the number of integrals can be reduced [22],

	$\displaystyle K_{\mu^{\prime}0,0\mu}$	$\displaystyle=\mu_{0}\int d\bm{k}\left[{\zeta^{2}}\left(\tilde{\bm{j}}_{\mu^{\prime}0}(-\bm{k})\right)^{\rm t}\left(\frac{1}{-\bm{k}^{2}+(n_{\rm b}\omega/c)^{2}}\right)\tilde{\bm{j}}_{0\mu}(\bm{k})\right.$
		$\displaystyle\hskip 8.53581pt+\zeta\left(\tilde{\bm{j}}_{\mu^{\prime}0}(-\bm{k})\right)^{\rm t}\left(-\frac{1}{\bm{k}^{2}}\frac{n_{\rm b}\omega/c}{-\bm{k}^{2}+(n_{\rm b}\omega/c)^{2}}\bm{k}\right)(c/n_{\rm b})\tilde{\rho}_{0\mu}(\bm{k})$
		$\displaystyle\hskip 8.53581pt\left.+(c/n_{\rm b})\tilde{\rho}_{\mu^{\prime}0}(-\bm{k})\left(\frac{1}{\bm{k}^{2}}\right)(c/n_{\rm b})\tilde{\rho}_{0\mu}(\bm{k})\right]$		(27)

Here, $n_{\rm b}\omega/c$ denotes the wavenumber of light in the nanorod and the environment with the refractive index $n_{\rm b}$. We examine the numerical calculations for the Fourier transferred coordinates.

We consider a rectangular nanorod as a plasmon-hosting nanostructure. The electron and hole wavefunctions in a rectangular nanorod are given as

	$\displaystyle\psi_{({\rm e}\mu)=(n_{x},n_{y},n_{z})}(\bm{x})$
	$\displaystyle=\sqrt{\frac{2}{L_{x}}}\sin\left(\frac{n_{x}\pi}{L_{x}}x\right)\sqrt{\frac{2}{L_{y}}}\sin\left(\frac{n_{y}\pi}{L_{y}}y\right)\sqrt{\frac{2}{L_{z}}}\sin\left(\frac{n_{z}\pi}{L_{z}}z\right)$		(28)

and

	$\displaystyle\psi_{({\rm h}\bar{\mu})=(\bar{n}_{x},\bar{n}_{y},\bar{n}_{z})}(\bm{x})$
	$\displaystyle=\sqrt{\frac{2}{L_{x}}}\sin\left(\frac{\bar{n}_{x}\pi}{L_{x}}x\right)\sqrt{\frac{2}{L_{y}}}\sin\left(\frac{\bar{n}_{y}\pi}{L_{y}}y\right)\sqrt{\frac{2}{L_{z}}}\sin\left(\frac{\bar{n}_{z}\pi}{L_{z}}z\right)$		(29)

with $L_{x}$, $L_{y}$, and $L_{z}$ being the lengths of nanorod in the $x$, $y$, and $z$ directions, respectively. As a demonstration, we suppose that $\langle\bm{x}|\mu=({\rm e}\mu,{\rm h}\bar{\mu})\rangle=\psi_{({\rm e}\mu)=(n_{x},n_{y},n_{z})}(\bm{x})\psi_{({\rm h}\bar{\mu})=(\bar{n}_{x},\bar{n}_{y},\bar{n}_{z})}(\bm{x})$ is given as the basis of the Hamiltonian $\hat{H}_{0}$ including the electron-electron interaction when the state $|\mu=(\rm{e}\mu,\rm{h}\bar{\mu})\rangle$ satisfies $\varepsilon_{{\rm e}\mu}>\varepsilon_{\rm F}\geq\varepsilon_{{\rm h}\bar{\mu}}$. Here, $\varepsilon_{\rm F}$ is the Fermi energy in the nanostructure. By applying Eqs. (28) and (29) to the self-consistent equation, we can discuss the effect of the spatial correlation between the T field and wavefunctions on the coherent coupling.

We calculate the eigenvalues of the system, where the complex eigenvalues are provided by the roots of ${\rm det}[\bar{\Xi}(\omega)]=0$. The spatial correlation between the electromagnetic field and the wavefunction of the excitations is an essential factor for the coherent coupling. Then, we examine the modulation of the background refractive index $n_{\rm b}=\sqrt{\epsilon_{\rm b}/\epsilon_{0}}$ and the nanorod length $L_{z}$ to tune their spatial correlation, whereas the nanorod thickness is fixed at $L_{x}=10\,\mathrm{nm}$ and $L_{y}=15\,\mathrm{nm}$ to hold the subband structures by the confinement. The Fermi energy of conduction electrons is set as $\varepsilon_{\rm F}=3$ to $5\,\mathrm{eV}$. The effective mass is $m^{*}=0.07m_{\rm e}$ with $m_{\rm e}$ being the electron mass in vacuum [25]. As a typical size scale, we use $L_{0}=100\,\mathrm{nm}$. Then, the order of the confinement energy is $E_{0}=\hbar^{2}\pi^{2}/(2m^{*}L_{0}^{2})\simeq 0.537\,\mathrm{meV}$. We put $\gamma=0.1\times E_{0}$ as the nonradiative damping constant in Eqs. (18)-(20). The plasmon excitation is mainly formed by the electron excitations near the Fermi surface with much smaller excitation energies than the plasmon energy [26]. Hence, we focus on bare individual excitations for $(n_{x},n_{y},n_{z})$ and $(\bar{n}_{x},\bar{n}_{y},\bar{n}_{z})=(n_{x},n_{y},n_{z}-1)$. This yields the smallest wavenumber in the plasmon and electron-hole pair excitation spectra, $|\bm{q}|=\pi\sqrt{(\frac{n_{x}-\bar{n}_{x}}{L_{x}})^{2}+(\frac{n_{y}-\bar{n}_{y}}{L_{y}})^{2}+(\frac{n_{z}-\bar{n}_{z}}{L_{z}})^{2}}=\frac{\pi}{L_{z}}$. For our consideration with $\varepsilon_{\rm F}=3\,\mathrm{eV}$ and $L_{z}=500\,\mathrm{nm}$, the number of bases for the bare individual excitations is 114 including the spin degrees of freedom. They are distributed in the range of $2.2\,\mathrm{meV}\leq\varepsilon_{{\rm eh},\mu}\leq 15.8\,\mathrm{meV}$, where $\varepsilon_{{\rm eh},\mu}=\varepsilon_{{\rm e}\mu}-\varepsilon_{{\rm h}\bar{\mu}}$ means the excitation energy of single electron-hole pair.

It is worthy to note that the three-dimensional bulk plasma frequency is evaluated as $\hbar\omega_{\rm p}\approx 2.93\,\mathrm{eV}$ when $\varepsilon_{\rm F}=3\,\mathrm{eV}$, $m^{*}=0.07m_{\rm e}$, and $n_{\rm b}=1$. For the three-dimensional case, the plasma frequency at $|\bm{q}|\to 0$ is finite and constant for $|\bm{q}|$. However, for one-dimensional nanorod we consider, the plasmon excitation also exhibits a one-dimensional-like behavior, where the plasma frequency is proportional to $|\bm{q}|$ [27, 28]. Hence, the evaluated collective excitation energy for the nanorod at $|\bm{q}|=\pi/L_{z}$ is lower than the bulk plasma frequency.

Here, we assume a light effective mass of electrons, such as narrow gap semiconductors. Plasmon excitation in the narrow gap semiconductor nanostructures has been examined to extend the plasmonics to THz region [29]. Such nanostructures can be fabricated with high accuracy and the plasmon excitation is modulated by the size and shape strongly [30]. In the following, we consider a modulation of the collective plasmon excitation by the nanorod length $L_{z}$ to tune an influence of the T field.

We consider a modulation of the refractive index $n_{\rm b}$ to tune the wavelength of the electromagnetic field. We assume that the nanorod and surrounding medium have the same $n_{\rm b}$. Then, we avoid a cavity effect due to light reflection at the surfaces of the materials and can focus on the effect of the pure spatial correlation of electronic wavefunctions and electromagnetic fields.

Figure 1 exhibits the real parts of the eigenvalues of the matrix $\bar{\Xi}(\omega)$ in Eq. (26) with only the L field contribution ($\zeta=0$). If the self-consistent treatment with the Maxwell’s equations is not considered, the eigenvalues are exactly equal to the electron-hole pair excitations. In Fig. 1, almost all the eigenvalues are found as the individual excitations in $2.2\,\mathrm{meV}\leq\hbar\omega\leq 15.8\,\mathrm{meV}$. An isolated curve on the higher energy side is interpreted as the collective excitation, which is denoted as $E_{\rm c}^{\rm(L)}$. This collective excitation is formed by the Coulomb interaction between the individual excitations.

The light reflection and the cavity effect can occur even by the resonant part of the refractive index. Such effects should be significant if the material size is larger than or the same order of magnitude as the light wavelength. However, in the present consideration, the reflection and cavity effect are not our scope and we focus on the coherent coupling by the spatial correlation between the T field and the excitation wavefunctions. The overall behaviors of the collective excitation against $n_{\rm b}$ seen in Fig. 1 are explained in the discussion for Fig. 2.

Figure 2 represents $E_{\rm c}^{\rm(L)}$ for several nanorod lengths from $L_{z}=50\,\mathrm{nm}$ to $1000\,\mathrm{nm}$. Equation (27) reads that the effects of $n_{\rm b}$ are the modulation of the wavelength and the screening of Coulomb interaction. When the T field is not considered, the important effect of $n_{\rm b}$ appears only through the screening. The energy of the collective excitation, which originates from the Coulomb interaction, might be reduced monotonically by $1/n_{\rm b}^{2}$ with the increase of $n_{\rm b}$. In $L_{z}=500\,\mathrm{nm}$ nanorod at $n_{\rm b}=1$, the collective excitation energy is $E_{\rm c}^{\rm(L)}\simeq 144\,\mathrm{meV}$. When $L_{z}=100\,\mathrm{nm}$, it is $E_{\rm c}^{\rm(L)}\simeq 518\,\mathrm{meV}$.

As stated in section 3.1, we have considered those electronic transition processes in which the wavenumber in the $z$ direction changes by $\pi/L_{z}$. The collective excitations with $\pi/L_{z}$ correspond to the standing waves with the longest wavelength, which are considered to have induced charges of opposite sign near both the nanorod ends. Our results show that, with a decrease in $L_{z}$, namely, with an increase in wavenumber, the excitation energy ascends remarkably. This is quite similar to the one-dimensional plasmon dispersion in which the energy rises up from the origin with increasing wavenumber (see, for example, Fig. 6 in Ref. 27).

The collective excitation approaches the distributed region of the individual excitations when $n_{\rm b}$ increases. Hence, they are indistinguishable at larger $n_{\rm b}$, which means that the collective mode could not be formed owing to weak Coulomb interaction at large $n_{\rm b}$. In a short nanorod, the confinement of electrons is strong. Thus, the Coulomb interaction is effectively enlarged and the collective excitation energy is higher for a shorter $L_{z}$. The above behaviors of $E_{\rm c}^{\rm(L)}$ against $n_{\rm b}$ and $L_{z}$ indicate that our small electronic system reproduces reasonable properties of collective excitations in metals.

Figure 2(b) shows the translated wavelength of the excitation energy as $\lambda_{\rm c}=hc/(n_{\rm b}E_{\rm c}^{\rm(L)})$. The refractive index $n_{\rm b}$ shortens the wavelength. At the same time, it suppresses the collective excitation energy $E_{\rm c}^{\rm(L)}(n_{\rm b})$ as shown in Fig. 2(a). Therefore, $\lambda_{\rm c}$ does not change largely with $n_{\rm b}$.

In the present electronic system, the T field contribution to the collective excitation energy is not prominent if it is seen in the scale of Fig. 2. Thus, to see the contribution of the T field clearly, the modulation ratio $(E_{\rm c}^{\rm(L)}-E_{\rm c}^{\rm(LT)})/E_{\rm c}^{\rm(L)}$ is shown in Fig. 3. Here, $E_{\rm c}^{\rm(LT)}$ means the collective excitation energy when the T field contribution is fully considered ($\zeta=1$). Figure 3(a) shows $L_{z}$ dependence of the ratio for various values of $n_{\rm b}$. Because the calculation including the T field is considerably heavy, we plot limited points as functions of $n_{\rm b}$. When the nanorod length $L_{z}$ is longer, the distribution of individual excitations and the collective excitation energy become lower. (No figure for the distribution of individual excitations.) The ratio on the ordinate is always positive, and the T field contribution decreases the collective excitation energy. As the length $L_{z}$ is longer, the modulation ratio is larger. From this figure, we see that a significant T field contribution to the collective excitation appears from several hundred nanometers of $L_{z}$ despite the mismatch of length scale between the nanorod length ($L_{z}\sim 100\,\mathrm{nm}$) and the translated wavelength ($\lambda_{\rm c}=hc/(n_{\rm b}E_{\rm c}^{\rm(L)})\sim 10000\,\mathrm{nm}$) for the plasmon ($\sim 100\,\mathrm{meV}$). Considering the situation that a small basis set of electronic system is used in the present calculation, we can expect that the prominent effect of T field would appear in realistic systems. However, we see that with an increase of $n_{\rm b}$, the modulation ratio decreases slightly for every $L_{z}$. This tendency indicates that the effect of the large $n_{\rm b}$ making the light wavelength shorter is outweighed by the effect of that reducing the collective excitation energy leading to a longer wavelength exciting the collective excitation.

We replot the data in Fig. 3(a) as their dependence on $n_{\rm b}$ for the various $L_{z}$ as shown in Fig. 3(b). The decrease of the modulation ratio with the increase of $n_{\rm b}$ reflects the $n_{\rm b}$ dependence of the translated wavelength $\lambda_{\rm c}$. From this result, we understand that to enlarge the coherent coupling by the T field, the system size is an important parameter because of a spatial correlation between the T field and the wavefunctions of excitations. Also, we can expect that if considering large $|\bm{q}|$ and $\varepsilon_{\rm F}$, we might obtain a situation for $L_{z}\sim\lambda_{\rm c}$, where the effect by $n_{\rm b}$ modulation would be enlarged.

Next, we consider the Fermi energy dependence of the T field effect. Figure 4 shows the modulation ratio for several Fermi energies for the electrons in the nanorod with $L_{z}=500\,\mathrm{nm}$. When $\varepsilon_{\rm F}$ is increased, the number of individual excitations contributing to form the collective excitation is increased because the “area” of the Fermi surface is enlarged. Then, the collective excitation energy becomes larger and the corresponding wavelength is shorter. As mentioned in the explanation of Fig. 2, the corresponding wavelength is much longer than the rod length $L_{z}$ in the present situation. Therefore, an increase of $\varepsilon_{\rm F}$ enhances the coherent coupling by the T field. Although the modulation of $\varepsilon_{\rm F}$ is artificial in the present demonstration, if much larger number of the individual excitations in realistic samples is considered, we understand that this result gives an important insight into the role of T field-mediated coherent coupling between the collective and individual excitations.