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## 物理代写|热力学代写thermodynamics代考|The Photon-Bound State and Incomplete Decay

In the following analysis of ( $5.12 \mathrm{~b})$, we aim at revealing the peculiar behavior of the atomic excitation decay $\alpha(t)$ and the corresponding emission and I.amb-shift spectra on account of $G(\omega)$ singular features. To this end, we consider a fieldconfining structure [e.g., a waveguide, a cavity or a photonic crystal (PC), as per Ch. 3] wherein $G(\omega)$ has one or several photonic band gaps (PBGs), separated by allowed photonic bands. Let us label each PBG by index $n$ and its lower and upper cutoff frequencies by $\omega_{\mathrm{L} n}$ and $\omega_{\mathrm{U} n}$, respectively:
$$G(\omega)=0 \text { for } \omega_{\mathrm{L} n}<\omega<\omega_{\mathrm{U} n} .$$
From Fermi’s Golden Rule, we may expect an excited atom not to decay, if $\omega_{\mathrm{a}}$ is within a PBG, or decay completely at $t \rightarrow \infty$ if $\omega_{\mathrm{a}}$ is anywhere in an allowed band. Yet, neither conclusion is necessarily true, since the Golden Rule may break down in such scenarins.

In fact, incomplete decay up to long times occurs if $\hat{\alpha}(s)$ in (5.12b) has a purely imaginary pole, $s=-i \omega_n, \hbar \omega_n$ signifying a stable (real) energy level of the total (atom+bath) Hamiltonian. Such a pole requires that $G\left(\omega_n\right)=0$ (i.e., $\omega_n$ is within a PBG). A real level $\hbar \omega_n$ corresponding to such a pole has to satisfy
$$\omega_n=\omega_{\mathrm{a}}+\Delta\left(\omega_n\right) .$$
Here
$$\Delta\left(\omega_n\right)=\int_0^{\omega_{\mathrm{L} n}} d \omega \frac{G(\omega)}{\omega_n-\omega}+\int_{\omega \mathrm{U} n}^{\infty} d \omega \frac{G(\omega)}{\omega_n-\omega},$$
$\hbar \Delta\left(\omega_n\right)$ being the bath-induced Lamb (energy) shift of the atomic resonance $\hbar \omega_{\mathrm{a}}$ (often it has been renormalized to account for the open-space Lamb shift). The two bath-induced Lamb-shift terms in (5.19) arise from the parts of $G(\omega)$ extending, respectively, below and above the $n$th PBG (Fig. 5.2). Whenever (5.18) holds, the corresponding term in $\alpha(t)$ [see $(5.11)$ ] oscillates as $\exp \left(-i \omega_n t\right)$, without decay. This feature signifies a stable, dressed, field-atom state formed by the binding of the photon to the atom or its confinement to the vicinity of the atom. Such confinement is caused by the nearly complete Bragg reflection of the emitted photon at frequencies within the PBG.

## 物理代写|热力学代写thermodynamics代考|Complete Decay: Irreversible (Weisskopf–Wigner) Solutions

As follows from (5.20), incomplete excited-state decay cannot occur for $\omega_{\mathrm{a}}$ well within an allowed band, far from a PBG. We then anticipate that long-time decay should obey, to a good approximation, the (open-space) Weisskopf-Wigner exponential decay law. In this section we discuss higher-order corrections to the standard Weisskopf-Wigner solution and obtain the conditions for its applicability. The inverse Laplace transform of $\hat{\alpha}(s)$ is known to involve contributions from the poles of $\hat{\alpha}(s)$ in (5.12b). The Weisskopf-Wigner approximation applies when the contribution of one of the poles (call it $s_p$ ) is dominant in $\alpha(t)$. Assuming that the self-energy term $\mathcal{G}(s)$ is sufficiently small in the equation $s+i \omega_{\mathrm{a}}+\mathcal{G}(s)=0$ for the poles, $s_p$ can be found perturbatively. Namely, we successively iterate the pole, with $s_p^{(0)}=-i \omega_{\mathrm{a}}$ as the lowest approximation. The first-order iteration yields
$$s_p \approx s_p^{(1)}=-i\left(\omega_{\mathrm{a}}+\Delta_{\mathrm{a}}\right)-\gamma_{\mathrm{a}},$$
where we have abbreviated $\gamma_{\mathrm{a}}=\gamma\left(\omega_{\mathrm{a}}\right)$, and $\Delta_{\mathrm{a}}=\Delta\left(\omega_{\mathrm{a}}\right)$ [see Eqs. (5.29)].

If $\alpha(t)$ can be approximated by the contribution of a single pole $s_p$ (known as the pole approximation), then the inverse Laplace transform of the first Eq. (5.12b) to first-order approximation for the pole yields an exponentially decaying amplitude
$$\alpha(t) \approx \exp \left[-\left(\gamma_{\mathrm{a}}+i \tilde{\omega}{\mathrm{a}}\right) t\right],$$ where $\tilde{\omega}{\mathrm{a}}=\omega_{\mathrm{a}}+\Delta_{\mathrm{a}}$.
Further iterations yield higher-order corrections to the pole. An analysis shows that these corrections can be neglected and hence this regime holds if the function $G(\omega)$ is smooth near $\omega_{\mathrm{a}}$ or, quantitatively, under the conditions
(a) $\left|\gamma_{\mathrm{a}}^{\prime}\right|,\left|\Delta_{\mathrm{a}}^{\prime}\right| \ll 1$;
(b) $\left|\Delta_{\mathrm{a}} \gamma_{\mathrm{a}}^{\prime}\right| \ll \gamma_{\mathrm{a}}$;
(c) $\gamma_{\mathrm{a}},\left|\Delta_{\mathrm{a}}\right| \ll\left|\omega_{\mathrm{a}}-\omega_{\mathrm{g}}\right|$.
(5.39)

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|The Photon-Bound State and Incomplete Decay

$$G(\omega)=0 \text { for } \omega_{\mathrm{L} n}<\omega<\omega_{\mathrm{U} n} .$$

$$\omega_n=\omega_{\mathrm{a}}+\Delta\left(\omega_n\right) .$$

$$\Delta\left(\omega_n\right)=\int_0^{\omega \mathrm{I} n} d \omega \frac{G(\omega)}{\omega_n-\omega}+\int_{\omega \mathrm{U} n}^{\infty} d \omega \frac{G(\omega)}{\omega_n-\omega},$$
$\hbar \Delta\left(\omega_n\right)$ 是原子共振的浴引起的兰姆 (能量) 位移 $\hbar \omega_{\mathrm{a}}$ （通常它已被重新归一化以考虑开放空间的兰姆位移）。（5.19） 中的两个浴诱导兰姆位移项 来自 $G(\omega)$ 分别延伸到下方和上方 $n$ PBG (图 5.2) 。每当 (5.18) 成立时，对应的项 $\alpha(t)$ [看 $(5.11)$ ] 振荡为 $\exp \left(-i \omega_n t\right)$ ，没有衰变。该特征表示由光 子与原子结合或将其限制在原子附近形成的稳定的、修饰的、场原子状态。这种限制是由在 PBG 内的频率处发射的光子的几乎完全布拉格反射引起 的。

## 物理代写|热力学代写thermodynamics代考|Complete Decay: Irreversible (Weisskopf–Wigner) Solutions

$$s_p \approx s_p^{(1)}=-i\left(\omega_{\mathrm{a}}+\Delta_{\mathrm{a}}\right)-\gamma_{\mathrm{a}},$$

$$\alpha(t) \approx \exp \left[-\left(\gamma_{\mathrm{a}}+i \tilde{\omega} \mathrm{a}\right) t\right],$$

(a) $下\left|\gamma_{\mathrm{a}}^{\prime}\right|,\left|\Delta_{\mathrm{a}}^{\prime}\right| \ll 1$
(二) $\left|\Delta_{\mathrm{a}} \gamma_{\mathrm{a}}^{\prime}\right| \ll \gamma_{\mathrm{a}}$;
(C) $\gamma_{\mathrm{a}},\left|\Delta_{\mathrm{a}}\right| \ll\left|\omega_{\mathrm{a}}-\omega_{\mathrm{g}}\right|$.

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