# 物理代写|热力学代写thermodynamics代考|MEC302

## 物理代写|热力学代写thermodynamics代考|Cooperative Self-Energy in Band Gaps

Let us consider photonic crystals free of dissipation or disorder, where the edge of a band gap is a singularity of the DOM (i.e., $\rho_n(\omega)$ vanishes abruptly). Then, within a bandgap,
$$\gamma_{i j^{\prime}} \propto \gamma \propto \rho_n\left(\omega_{\mathrm{a}}\right)=0 .$$
The evaluation of $\Delta_{j j^{\prime}}$ in a band gap is more involved. The frequency is expanded about the singularity $\omega_{\mathrm{c}}$, the lower or upper edge of the bandgap (Fig. 3.3),
$$\omega=\omega_{\mathrm{c}}+b \kappa^2+\ldots,$$
$\kappa$ being the deviation from $K_{\mathrm{c}}$, the wave vector of the band edge. The firstderivative term in the expansion vanishes at the cutoff. Just above the cutoff frequency or the lower edge of a band gap $\omega_{\mathrm{c}}$, this expansion corresponds to
$$\rho_n(\omega) \simeq \frac{\omega\left(\omega^2-\omega_{\mathrm{c}}^2\right)^{1 / 2}}{\left(2 b \omega_{\mathrm{c}}\right)^{3 / 2}} \theta\left(\omega^2-\omega_{\mathrm{c}}^2\right),$$
$\theta$ being the Heaviside step function. An analogous expression is obtained for $\omega_{\mathrm{c}}$ at the upper edge of a band gap.

For this form of $\rho_n(\omega), \Delta_{j j^{\prime}}$ can be evaluated for $\omega_{\mathrm{a}}$ just below $\omega_{\mathrm{c}}$ upon extending the integral in (8.21) over the domain $\omega_{\mathrm{c}}<\omega<\infty$ and $-\infty<\omega<-\omega_{\mathrm{c}}$. We consider the integrand to be symmetric in $\omega$, exclude the bandgap and its edges, $-\omega_{\mathrm{c}} \leq \omega \leq \omega_{\mathrm{c}}$, by branch cuts, and close the contour by a circle of infinite radius (Fig. 8.2). The $\pm \omega_{\mathrm{a}}$ residues that contribute to the integral are the product of two factors: (i) the analytic continuation of $\rho_n(\omega)$ [Eq. (8.28)] into the band gap that has the form,
$$\rho_n\left(\pm \omega_{\mathrm{a}}\right)=\chi^2 \frac{d \chi}{d \omega}=\pm i \frac{\omega_{\mathrm{a}}\left(\omega_{\mathrm{a}}^2-\omega_{\mathrm{c}}^2\right)^{1 / 2}}{\left(2 b \omega_{\mathrm{c}}\right)^{3 / 2}}$$
where $\chi$ is the imaginary part of the wave vector; and (ii) the analytic continuation of $f_n(K R) \Phi_n^{j j^{\prime}}(K)$, which amounts to the replacement of the angular integral in (8.18) by an integral over evanescent modes with complex wave vectors:
$$\int d \Omega_{\hat{\boldsymbol{K}}}\left(\hat{\boldsymbol{\gamma}} \cdot \boldsymbol{\epsilon}\lambda\right)^2 \phi\alpha^*\left(\boldsymbol{r}j\right) \phi\alpha\left(\boldsymbol{r}{j^{\prime}}\right) \rightarrow \int d \Omega{\hat{\boldsymbol{K}}}\left(\hat{\boldsymbol{\gamma}} \cdot \boldsymbol{\epsilon}_\lambda\right)^2 \exp (i \boldsymbol{K} \cdot \boldsymbol{R}-|\boldsymbol{\chi} \cdot \boldsymbol{R}|)$$

## 物理代写|热力学代写thermodynamics代考|Non-Markovian Theory of RDDI in Waveguides

Here we outline a nonperturbative, non-Markovian, theory for RDDI in a fieldconfining structure, such as a waveguide. From Hamiltonian (8.1), if atom 1 is initially excited, the state of the combined (atoms+bath) system is, in the RWA,
$$|\psi(t)\rangle=\alpha_1(t)\left|e_1, g_2, 0\right\rangle+\alpha_2(t)\left|g_1, e_2, 0\right\rangle+\sum_k \beta_k(t)\left|g_1, g_2, 1_k\right\rangle$$
Upon inserting this state into the Schrödinger equation, we obtain dynamical equations for $\alpha_1(t), \alpha_2(t)$, and $\beta_k(t)$. Taking their Laplace transform for the initial conditions, $\alpha_1(0)=1, \alpha_2(0)=\beta_k(0)=0$, we then obtain the Laplace transform of $\alpha_1(t)$
$$\tilde{\alpha}1(s)=\left[s+J{11}(s)+i \omega_{\mathrm{a}}-\frac{J_{12}(s) J_{21}(s)}{s+J_{22}(s)+i \omega_{\mathrm{a}}}\right]^{-1},$$
where $J_{j j^{\prime}}(s)=\sum_k \frac{\eta_{k, j}^* \eta_{k, j^{\prime}}}{s+i \omega_k}$. It can be shown that $J_{j j^{\prime}}\left(-i \omega_{\mathrm{a}}\right)=-i \Delta_{j j^{\prime},-}$ [cf. (8.12)]. As before, we consider only the MWG transverse mode $m=1, n=1$, here for $\omega_{\mathrm{a}}$ close to the cutoff $\omega_{11}$, such that the denominator of the spectrum (8.34) is approximated by $\sqrt{\left(\omega / \omega_{11}\right)^2-1} \approx \sqrt{2} \sqrt{\omega / \omega_{11}-1}$. Upon evaluating the integrals in $J_{j j^{\prime}}(s)$ for the approximated spectrum, we invert the Laplace transform $(8.38)$ to find
$$\alpha_1(t)=\sqrt{i} e^{-i \omega_{11} t} \sum_{l=1}^5 c_l\left[\frac{1}{\sqrt{\pi t}}+\sqrt{i} u_l e^{i u_l^2 t} \operatorname{erfc}\left(-u_l \sqrt{i t}\right)\right]$$
Although the explicit form of the constants $c_l, u_l$ is complicated, the $1 / \sqrt{\pi t}$ dependence of the first term in the square brackets indicates that $\alpha_1(t)$ does not decay exponentially at long times, thus deviating drastically from the Markovian (GR) decay.

## 物理代写|热力学代写thermodynamics代考|Cooperative Self-Energy in Band Gaps

$$\gamma_{i j^{\prime}} \propto \gamma \propto \rho_n\left(\omega_{\mathrm{a}}\right)=0 .$$

$$\omega=\omega_{\mathrm{c}}+b \kappa^2+\ldots,$$
$\kappa$ 偏离 $K_{\mathrm{c}}$ ，波段边缘的波矢量。展开式中的一阶导数项在截止处消失。刚好高于截止频率或带隙的下边缘 $\omega_{\mathrm{c}}$ ，这个展开对应于
$$\rho_n(\omega) \simeq \frac{\omega\left(\omega^2-\omega_{\mathrm{c}}^2\right)^{1 / 2}}{\left(2 b \omega_{\mathrm{c}}\right)^{3 / 2}} \theta\left(\omega^2-\omega_{\mathrm{c}}^2\right),$$
$\theta$ 是 Heaviside 阶跃函数。得到一个类似的表达式 $\omega_c$ 在带隙的上边缘。

$$\rho_n\left(\pm \omega_{\mathrm{a}}\right)=\chi^2 \frac{d \chi}{d \omega}=\pm i \frac{\omega_{\mathrm{a}}\left(\omega_{\mathrm{a}}^2-\omega_{\mathrm{c}}^2\right)^{1 / 2}}{\left(2 b \omega_{\mathrm{c}}\right)^{3 / 2}}$$

$$\int d \Omega_{\hat{\boldsymbol{K}}}(\hat{\boldsymbol{\gamma}} \cdot \boldsymbol{\epsilon} \lambda)^2 \phi \alpha^*(\boldsymbol{r} j) \phi \alpha\left(\boldsymbol{r} j^{\prime}\right) \rightarrow \int d \Omega \hat{\boldsymbol{K}}\left(\hat{\boldsymbol{\gamma}} \cdot \boldsymbol{\epsilon}_\lambda\right)^2 \exp (i \boldsymbol{K} \cdot \boldsymbol{R}-|\boldsymbol{\chi} \cdot \boldsymbol{R}|)$$

## 物理代写|热力学代写thermodynamics代考|Non-Markovian Theory of RDDI in Waveguides

$$|\psi(t)\rangle=\alpha_1(t)\left|e_1, g_2, 0\right\rangle+\alpha_2(t)\left|g_1, e_2, 0\right\rangle+\sum_k \beta_k(t)\left|g_1, g_2, 1_k\right\rangle$$

$$\tilde{\alpha} 1(s)=\left[s+J 11(s)+i \omega_{\mathrm{a}}-\frac{J_{12}(s) J_{21}(s)}{s+J_{22}(s)+i \omega_{\mathrm{a}}}\right]^{-1},$$

$\sqrt{\left(\omega / \omega_{11}\right)^2-1} \approx \sqrt{2} \sqrt{\omega / \omega_{11}-1}$. 在评估积分时 $J_{j j^{\prime}}(s)$ 对于近似谱，我们反转拉普拉斯变换(8.38)去 寻找
$$\alpha_1(t)=\sqrt{i} e^{-i \omega_{11} t} \sum_{l=1}^5 c_l\left[\frac{1}{\sqrt{\pi t}}+\sqrt{i} u_l e^{i u_i^2 t} \operatorname{erfc}\left(-u_l \sqrt{i t}\right)\right]$$

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