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## 物理代写|热力学代写thermodynamics代考|Partial and Complete Decay near Cutoff

As per (3.23), the bath spectral response near the upper PBG cutoff $\omega_U$ can be modeled by
$$G(\omega)=\frac{\gamma_{\mathrm{c}}^{3 / 2}}{\pi} \frac{\sqrt{\omega-\omega_{\mathrm{U}}}}{\omega-\omega_{\mathrm{U}}+\epsilon_{\mathrm{c}}} \theta\left(\omega-\omega_{\mathrm{U}}\right),$$
where $\epsilon_{\mathrm{c}}$ is the “cutoff-smoothing” parameter and $\gamma_{\mathrm{c}}$ characterizes the atom-bath coupling strength at cutoff. Depending on whether $\epsilon_{\mathrm{c}} \ll \gamma_{\mathrm{c}}$ or $\epsilon_{\mathrm{c}} \gg \gamma_{\mathrm{c}}$, the bath is close to the isotropic case $\epsilon_{\mathrm{c}}=0(D=2)$ or the anisotropic case $\epsilon_{\mathrm{c}} \rightarrow \infty$ $(D=0)$, respectively.

Upon inserting (5.44) into (5.29), we obtain the bath-induced skewed-Lorentzian linewidth
$$\gamma(\omega)=\frac{\gamma_{\mathrm{c}}^{3 / 2} \sqrt{\omega-\omega_{\mathrm{U}}}}{\omega-\omega_{\mathrm{U}}+\epsilon_{\mathrm{c}}} \theta\left(\omega-\omega_{\mathrm{U}}\right)$$
$\theta()$ being the Heaviside step function and the corresponding spectral shift
$$\Delta(\omega)=-\gamma_{\mathrm{c}}^{3 / 2} \begin{cases}\sqrt{\epsilon_{\mathrm{c}}} /\left(\omega-\omega_{\mathrm{U}}+\epsilon_{\mathrm{c}}\right), & \omega>\omega_{\mathrm{U}}, \ \left(\sqrt{\omega_{\mathrm{U}}-\omega}+\sqrt{\epsilon_{\mathrm{c}}}\right)^{-1}, & \omega<\omega_{\mathrm{U}} .\end{cases}$$
It thus follows that the spectrum (5.30) can be strongly non-Lorentzian. The model (5.44) yields an exact analytical expression for the time dependence of spontaneous decay, from which we can infer the criteria for the two regimes of Sections 5.2.2 and 5.2.3:
(i) Incomplete decay [cf. (5.20)] is now obtained for (Fig. 5.3)
$$\omega_{\mathrm{a}}<\omega_{\mathrm{U}}+\frac{\gamma_{\mathrm{c}}^{3 / 2}}{\sqrt{\epsilon_{\mathrm{c}}}} .$$
Equation (5.47) implies that an abrupt, singular cutoff of the DOM with $D=2$ $\left(\epsilon_r \rightarrow 0\right)$ allows for a discrete state at any $\omega_3$, either inside or outside the PBG. The energy of the discrete state, $\hbar \omega_0$, which must lie within the PBG, is found, upon substituting the second of Frs. (5.46) into (5.18), to he a real and positive root of the equation
$$\omega_0=\omega_{\mathrm{a}}-\frac{\gamma_{\mathrm{c}}^{3 / 2}}{\sqrt{\omega_{\mathrm{U}}-\omega_0}+\sqrt{\epsilon_{\mathrm{c}}}} .$$

## 物理代写|热力学代写thermodynamics代考|Atomic Coupling to a High-Q Defect Mode in the PBG

A defect in the periodic structure can produce a narrow-linewidth local mode in the PBG akin to a high- $Q$ cavity mode. In fact, this is the optimal way to create such a cavity mode. The spectral response is then describable by a Lorentzian,
$$G_{\mathrm{d}}(\omega)=\frac{\gamma_{\mathrm{d}}}{\pi} \frac{\Gamma_{\mathrm{d}}^2}{\Gamma_{\mathrm{d}}^2+\left(\omega-\omega_{\mathrm{d}}\right)^2},$$
where $\gamma_{\mathrm{d}}$ expresses the coupling strength of the atomic dipole with the defect field, whereas $\omega_{\mathrm{d}}$ and $\Gamma_{\mathrm{d}}$ represent the line center and width, respectively. Such a defect line has an additive effect on the parameters of the skewed Lorentzian line shape (5.30), as compared to (5.45) and (5.46),

\begin{aligned} &\gamma(\omega) \rightarrow \gamma(\omega)+\frac{\gamma_{\mathrm{d}} \Gamma_{\mathrm{d}}^2}{\Gamma_{\mathrm{d}}^2+\left(\omega-\omega_{\mathrm{d}}\right)^2}, \ &\Delta(\omega) \rightarrow \Delta(\omega)+\frac{\gamma_{\mathrm{d}} \Gamma_{\mathrm{d}}\left(\omega-\omega_{\mathrm{d}}\right)}{\Gamma_{\mathrm{d}}^2+\left(\omega-\omega_{\mathrm{d}}\right)^2} . \end{aligned}
The nonvanishing DOM in the PBG due to a defect causes spontaneous emission within the PBG and thus broadens the discrete state $\omega_0$, rendering it metastable (see Fig. 5.5).

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Partial and Complete Decay near Cutoff

$$G(\omega)=\frac{\gamma_{\mathrm{c}}^{3 / 2}}{\pi} \frac{\sqrt{\omega-\omega_{\mathrm{U}}}}{\omega-\omega_{\mathrm{U}}+\epsilon_{\mathrm{c}}} \theta\left(\omega-\omega_{\mathrm{U}}\right),$$

$$\gamma(\omega)=\frac{\gamma_{\mathrm{c}}^{3 / 2} \sqrt{\omega-\omega_{\mathrm{U}}}}{\omega-\omega_{\mathrm{U}}+\epsilon_{\mathrm{c}}} \theta\left(\omega-\omega_{\mathrm{U}}\right)$$
$\theta()$ 是 Heaviside 阶跃函数和相应的谱移
$$\Delta(\omega)=-\gamma_{\mathrm{c}}^{3 / 2}\left{\sqrt{\epsilon_{\mathrm{c}}} /\left(\omega-\omega_{\mathrm{U}}+\epsilon_{\mathrm{c}}\right), \quad \omega>\omega_{\mathrm{U}},\left(\sqrt{\omega_{\mathrm{U}}-\omega}+\sqrt{\epsilon_{\mathrm{c}}}\right)^{-1}, \quad \omega<\omega_{\mathrm{U}} .\right.$$

(i) 不完全衰变 [cf. (5.20)] 现在得到 (图 5.3)
$$\omega_{\mathrm{a}}<\omega_{\mathrm{U}}+\frac{\gamma_{\mathrm{c}}^{3 / 2}}{\sqrt{\epsilon_{\mathrm{c}}}} .$$

$$\omega_0=\omega_{\mathrm{a}}-\frac{\gamma_{\mathrm{c}}^{3 / 2}}{\sqrt{\omega_{\mathrm{U}}-\omega_0}+\sqrt{\epsilon_{\mathrm{c}}}} .$$

## 物理代写|热力学代写thermodynamics代考|Atomic Coupling to a High-Q Defect Mode in the PBG

$$G_{\mathrm{d}}(\omega)=\frac{\gamma_{\mathrm{d}}}{\pi} \frac{\Gamma_{\mathrm{d}}^2}{\Gamma_{\mathrm{d}}^2+\left(\omega-\omega_{\mathrm{d}}\right)^2},$$

$$\gamma(\omega) \rightarrow \gamma(\omega)+\frac{\gamma_{\mathrm{d}} \Gamma_{\mathrm{d}}^2}{\Gamma_{\mathrm{d}}^2+\left(\omega-\omega_{\mathrm{d}}\right)^2}, \quad \Delta(\omega) \rightarrow \Delta(\omega)+\frac{\gamma_{\mathrm{d}} \Gamma_{\mathrm{d}}\left(\omega-\omega_{\mathrm{d}}\right)}{\Gamma_{\mathrm{d}}^2+\left(\omega-\omega_{\mathrm{d}}\right)^2}$$

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