### 物理代写|热力学代写thermodynamics代考|CL3010

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 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
$$\sum_{\alpha=(K, n, \lambda)} \rightarrow \mathcal{V} \sum_{n, \lambda} \int d \Omega_{\hat{K}} \int K^{2} d K,$$ where $\int d \Omega_{\hat{K}}$ 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 $\phi_{\alpha}$,
$$\sum_{\lambda=1}^{2} \int d \Omega_{\hat{\boldsymbol{K}}}\left(\hat{\boldsymbol{\beta}} \cdot \boldsymbol{\epsilon}{\lambda}\right)^{2} \phi{\alpha}^{*}\left(\boldsymbol{r}{j}\right) \phi{\alpha}\left(\boldsymbol{r}{j^{\prime}}\right)=\frac{1}{2} \int d \Omega{\hat{\boldsymbol{K}}}\left[1-(\hat{\boldsymbol{\beta}} \cdot \boldsymbol{K})^{2}\right] e^{i \boldsymbol{K} \cdot \boldsymbol{R}} \equiv f(K R),$$
where $\hat{\boldsymbol{\wp}}$ denotes the unit vector along the atomic dipole $\wp$. This integral may be explicitly performed for two-atom quasi-molecular dimer states, such that $\wp | R$ (dimer $\Sigma$ states) or $\wp \perp R$ (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}

## 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

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

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

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

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## MATLAB代写

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