## 物理代写|热力学代写thermodynamics代考|Cooperative Self-Energy in Periodic Structures

The $G_{j j^{\prime}}(\omega)$ two-atom terms are written in free space as sums over wave vectors $\boldsymbol{k}$ of the phase-difference factors $\left(\boldsymbol{\wp} \cdot \boldsymbol{\epsilon}\lambda\right)^2 \exp (i \boldsymbol{k} \cdot \boldsymbol{R})$, where $\boldsymbol{\epsilon}\lambda$ is a unit vector of polarization and $\boldsymbol{R}=\boldsymbol{r}j-\boldsymbol{r}{j^{\prime}}$. By contrast, in periodic, dispersive structures, these terms and their contributions to the self-energy can be evaluated in the normalmode basis of the structure described in Chapters 3 and 4 . On separating its real and imaginary parts, we may evaluate in the Markovian limit the principal-value term, $\Delta_{j j^{\prime}}$, that corresponds to the cooperative Lamb (RDDI) shift, and the $\delta$-function term, $\gamma_{j j^{\prime}}$, that represents the cooperative contribution to the line width or rate of fluorescence (spontaneous emission):
\begin{aligned} \gamma_{j j^{\prime}}-i \Delta_{j j^{\prime}}=& \frac{2 \pi}{\mathcal{V}} \sum_\alpha \omega_\alpha \phi_\alpha^*\left(\boldsymbol{r}j\right) \phi\alpha\left(\boldsymbol{r}{j^{\prime}}\right)\left(\boldsymbol{\rho} \cdot \boldsymbol{\epsilon}\lambda\right)^2 \ & \times\left[\delta\left(\omega_\alpha-\omega_{\mathrm{a}}\right)-i P\left(\frac{1}{\omega_\alpha-\omega_{\mathrm{a}}}+\frac{1}{\omega_\alpha+\omega_{\mathrm{a}}}\right)\right], \end{aligned}
where (as in Sec. 8.1.1) $\omega_\alpha$ and $\phi_\alpha\left(\boldsymbol{r}_j\right)$ denote the mode frequency and mode function, respectively, $\alpha=(\boldsymbol{K}, n, \lambda)$, where $\hbar \boldsymbol{K}$ is the mode quasimomentum, $n$ is the Brillouin zone number (Ch. 3), and $\mathcal{V}$ is the quantization volume.

## 物理代写|热力学代写thermodynamics代考|Cooperative Self-Energy in Isotropic Structures

In what follows, (8.15) will be evaluated in the mode-continuum limit for isotropic media in which $\omega$ depends only on the modulus of $\boldsymbol{K}$, so that
$$\sum_{\alpha=(\boldsymbol{K}, n, \lambda)} \rightarrow \mathcal{V} \sum_{n, \lambda} \int d \Omega_{\hat{\boldsymbol{K}}} \int K^2 d K$$

where $\int d \Omega_{\hat{\boldsymbol{K}}}$ denotes solid-angle integration. This assumption can be invoked in three-dimensional (3D) periodic structures (photonic crystals), upon approximating the polyhedral Brillouin surface in $\boldsymbol{K}$ space by a sphere, so that $\int K^2 d K$ extends over the $n$th Brillouin zone.

To evaluate $\Delta_{j j^{\prime}}$ and $\gamma_{j j^{\prime}}$ in such media, we must first integrate (8.15) over all angles and sum over the mode polarizations. Then, for plane wave $\phi_\alpha$.
$$\sum_{\lambda=1}^2 \int d \Omega_{\hat{\boldsymbol{K}}}\left(\hat{\boldsymbol{\rho}} \cdot \boldsymbol{\epsilon}\lambda\right)^2 \phi\alpha^\left(\boldsymbol{r}j\right) \phi\alpha\left(\boldsymbol{r}{j^{\prime}}\right)=\frac{1}{2} \int d \Omega{\hat{\boldsymbol{K}}}\left[1-(\hat{\boldsymbol{\beta}} \cdot \boldsymbol{K})^2\right] e^{i \boldsymbol{K} \cdot \boldsymbol{R}} \equiv f(K R),$$
(8.17a)
where $\hat{\jmath}$ denotes the unit vector along the atomic dipole $\wp$. This integral may be explicitly performed for two-atom quasi-molecular dimer states, such that $\wp | \boldsymbol{R}$ (dimer $\Sigma$ states) or $\wp \perp \boldsymbol{R}$ (dimer $\Pi$ states),
\begin{aligned} &f_{\Sigma}(K R)=3\left[\frac{\sin K R}{(K R)^3}-\frac{\cos K R}{(K R)^2}\right], \ &f_{\Pi}(K R)=-\frac{3}{2}\left[\frac{\sin K R}{(K R)^3}-\frac{\cos K R}{(K R)^2}-\frac{\sin K R}{K R}\right] . \end{aligned}
In periodic structures conforming to the isotropic approximation, these expressions can be generalized to Bloch waves $\phi_\alpha$, yielding, upon averaging over $r_1$ and the orientations of $\boldsymbol{R}$,
$$\begin{gathered} \sum_{\lambda=1}^2 \int d \Omega_{\hat{K}}\left(\hat{\boldsymbol{\rho}} \cdot \boldsymbol{\epsilon}\lambda\right)^2\left\langle\phi\alpha^\left(\boldsymbol{r}j\right) \phi\alpha\left(\boldsymbol{r}{j^{\prime}}\right)\right\rangle=\frac{1}{2} f_n(K R) \Phi_n^{j j^{\prime}}(K), \ \Phi_n^{j j^{\prime}}(K)=\sum_g \frac{\sin g R}{g R}\left(\left|C{g \alpha}\right|^2+\sum_{g^{\prime}} \sum_{\left(g^{\prime} \neq g\right)} C_{g \alpha}^* C_{g^{\prime} \alpha} \frac{\sin \left|g-g^{\prime}\right| r_{j^{\prime}}}{\left|g-g^{\prime}\right| r_{j^{\prime}}}\right), \end{gathered}$$

## 物理代写|热力学代写thermodynamics代考|Cooperative Self-Energy in Periodic Structures

$$\gamma_{j j^{\prime}}-i \Delta_{j j^{\prime}}=\frac{2 \pi}{\mathcal{V}} \sum_\alpha \omega_\alpha \phi_\alpha^*(\boldsymbol{r} j) \phi \alpha\left(\boldsymbol{r} j^{\prime}\right)(\boldsymbol{\rho} \cdot \boldsymbol{\epsilon} \lambda)^2 \quad \times\left[\delta\left(\omega_\alpha-\omega_{\mathrm{a}}\right)-i P\left(\frac{1}{\omega_\alpha-\omega_{\mathrm{a}}}+\frac{1}{\omega_\alpha+\omega_{\mathrm{a}}}\right)\right],$$

## 物理代写|热力学代写thermodynamics代考|Cooperative Self-Energy in Isotropic Structures

$$\sum_{\alpha=(\boldsymbol{K}, n, \lambda)} \rightarrow \mathcal{V} \sum_{n, \lambda} \int d \Omega_{\hat{\boldsymbol{K}}} \int K^2 d K$$

$$\sum_{\lambda=1}^2 \int d \Omega_{\hat{\boldsymbol{K}}}(\hat{\boldsymbol{\rho}} \cdot \boldsymbol{\epsilon} \lambda)^2 \phi \alpha^{(\boldsymbol{r} j)} \phi \alpha\left(\boldsymbol{r} j^{\prime}\right)=\frac{1}{2} \int d \Omega \hat{\boldsymbol{K}}\left[1-(\hat{\boldsymbol{\beta}} \cdot \boldsymbol{K})^2\right] e^{i \boldsymbol{K} \cdot \boldsymbol{R}} \equiv f(K R),$$

$$f_{\Sigma}(K R)=3\left[\frac{\sin K R}{(K R)^3}-\frac{\cos K R}{(K R)^2}\right], \quad f_{\Pi I}(K R)=-\frac{3}{2}\left[\frac{\sin K R}{(K R)^3}-\frac{\cos K R}{(K R)^2}-\frac{\sin K R}{K R}\right] .$$

$$\sum_{\lambda=1}^2 \int d \Omega_{\hat{K}}(\hat{\boldsymbol{\rho}} \cdot \boldsymbol{\epsilon} \lambda)^2\left\langle\phi \alpha^{(\boldsymbol{r} j)} \phi \alpha\left(\boldsymbol{r} j^{\prime}\right)\right\rangle=\frac{1}{2} f_n(K R) \Phi_n^{j j^{\prime}}(K), \Phi_n^{j j^{\prime}}(K)=\sum_g \frac{\sin g R}{g R}\left(|C g \alpha|^2+\sum_{g^{\prime}} \sum_{\left(g^{\prime} \neq g\right)}\right.$$

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