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## 物理代写|热力学代写thermodynamics代考|Wigner–Weisskopf Dynamics

We analyze here the evolution of an initially excited two-level atom coupled to an arbitrary electromagnetic (photonic) bath. The bath is characterized by the densityof-modes (DOM) spectrum $\rho(\omega)$ of the electromagnetic field, assumed to be in the vacuum state. The atom-field interaction in the RWA is given by (4.13). The spectral response of the bath, representing, according to Fermi’s Golden Rule, the rate (divided by $2 \pi$ ) of spontaneous emission of the atom at frequency $\omega$, is given hy
$$G(\omega)=\sum_{\Lambda}\left|\eta_{\Lambda}\right|^2 \delta\left(\omega-\omega_{\Lambda}\right),$$
where $\omega_{\Lambda}$ is the frequency of the $\Lambda$ th mode of the bath.
The function $G(\omega)$ is the spectrum of the autocorrelation function of the atombath interaction (cf. Ch. 2) at zero temperature. Namely, the time-domain Fourier transform of $G(\omega)$ (from 0 to $\infty$, since $\omega_{\Lambda} \geq 0$ ) is
$$\Phi(t)=\int_0^{\infty} d \omega G(\omega) e^{-i\left(\omega-\omega_{\mathrm{a}}\right) t}=\sum_{\Lambda}\left|\eta_{\Lambda}\right|^2 e^{i\left(\omega_{\mathrm{a}}-\omega_{\Lambda}\right) t}$$
which can be recast as a correlation function,
$$\Phi(t)=\hbar^{-2}\left\langle e, \operatorname{vac}\left|H_1(t) H_1\right| e, \operatorname{vac}\right\rangle$$

Here |vac $\rangle$ stands for the bath vacuum state and $H_{\mathrm{I}}(t)=e^{i H_0 t} H_{\mathrm{I}} e^{-i H_0 t}$ is the coupling Hamiltonian in the interaction representation, where
$$H_0=\hbar \omega_{\mathrm{a}}|e\rangle\langle e|+\sum_{\Lambda} \hbar \omega_{\Lambda} a_{\Lambda}^{\dagger} a_{\Lambda}$$
is the sum of the free Hamiltonians of the system and the hath. The antocorrelation function $\Phi(t)$ is sometimes referred to as the memory kernel of the bath response. From physical considerations, $G(\omega)$ can be divided into two parts,
$$G(\omega)=G_{\mathrm{s}}(\omega)+G_{\mathrm{b}}(\omega)$$

## 物理代写|热力学代写thermodynamics代考|Photon–Atom Binding and Partial Decay

The evolution of a two-level atom in any zero-temperature (vacuum-state) photonic bath is describable by the joint field-atom wave function, which has the general RWA form,
$$|\Psi(t)\rangle=\alpha(t)\left|e,\left{0_{k \lambda}\right}\right\rangle+\sum_{k, \lambda} \beta_{k \lambda}(t)\left|g, 1_{k \lambda}\right\rangle .$$
Here $\left|\left{0_{k \lambda}\right}\right\rangle$ stands for the totality of field-bath modes in the vacuum state and $\left|1_{k \lambda}\right\rangle$ for the field-bath state with single-photon occupation of the $(\boldsymbol{k}, \lambda)$-mode. The corresponding Schrödinger equation can be solved under the initial condition of an initially excited atom,
$$|\Psi(0)\rangle=\left|e,\left{0_{k \lambda}\right}\right\rangle,$$ in the Laplace-domain form,
$$\hat{\alpha}(s)=\left[s+i \omega_{\mathrm{a}}+\mathcal{G}(s)\right]^{-1}, \quad \hat{\beta}{k \lambda}(s)=-\frac{i \eta{k \lambda}^* \hat{\alpha}(s)}{s+i \omega_{k \lambda}} .$$
Here the Laplace transform is denoted by
$$\hat{\alpha}(s)=\int_0^{\infty} d t \alpha(t) e^{-s t} \quad(\operatorname{Res}>0),$$
where the bath-response (or self-energy) term,
$$\mathcal{G}(s)=\int_0^{\infty} d \omega \frac{G(\omega)}{s+i \omega},$$
is derived from the bath-response (coupling) spectrum
$$G(\omega)=\sum_{k, \lambda}\left|\eta_{k \lambda}\right|^2 \delta\left(\omega-\omega_{k \lambda}\right) \rightarrow|\eta(\omega)|^2 \rho(\omega) .$$
This spectrum is the continuum limit of the $(\boldsymbol{k}, \lambda)$-summation over modes, for a directionally independent (isotropic) DOM $\rho(\omega)$, namely,
$$\sum_{k, \lambda} \rightarrow \int_0^{\infty} d \omega \rho(\omega)$$
We stress that $(5.12 \mathrm{~b})$ is an exact solution of the atom-bath evolution problem under condition (5.12a) for all times, provided we can invert this solution from the Laplace to the time domain. Such inversion is, however, possible in a closed analytical form only for certain spectral bath-response forms $G(\omega)$, as illustrated below.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Wigner–Weisskopf Dynamics

$$G(\omega)=\sum_{\Lambda}\left|\eta_{\Lambda}\right|^2 \delta\left(\omega-\omega_{\Lambda}\right),$$

$$\Phi(t)=\int_0^{\infty} d \omega G(\omega) e^{-i\left(\omega-\omega_{\mathrm{a}}\right) t}=\sum_{\Lambda}\left|\eta_{\Lambda}\right|^2 e^{i\left(\omega_{\mathrm{a}}-\omega_{\Lambda}\right) t}$$

$$\Phi(t)=\hbar^{-2}\left\langle e, \operatorname{vac}\left|H_1(t) H_1\right| e, \operatorname{vac}\right\rangle$$

$$H_0=\hbar \omega_{\mathrm{a}}|e\rangle\langle e|+\sum_{\Lambda} \hbar \omega_{\Lambda} a_{\Lambda}^{\dagger} a_{\Lambda}$$

$$G(\omega)=G_{\mathrm{s}}(\omega)+G_{\mathrm{b}}(\omega)$$

## 物理代写|热力学代写thermodynamics代考|Photon–Atom Binding and Partial Decay

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$\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别

$$\hat{\alpha}(s)=\left[s+i \omega_{\mathrm{a}}+\mathcal{G}(s)\right]^{-1}, \quad \hat{\beta} k \lambda(s)=-\frac{i \eta k \lambda^* \hat{\alpha}(s)}{s+i \omega_{k \lambda}} .$$

$$\hat{\alpha}(s)=\int_0^{\infty} d t \alpha(t) e^{-s t} \quad(\operatorname{Res}>0),$$

$$\mathcal{G}(s)=\int_0^{\infty} d \omega \frac{G(\omega)}{s+i \omega}$$

$$G(\omega)=\sum_{k, \lambda}\left|\eta_{k \lambda}\right|^2 \delta\left(\omega-\omega_{k \lambda}\right) \rightarrow|\eta(\omega)|^2 \rho(\omega)$$

$$\sum_{k, \lambda} \rightarrow \int_0^{\infty} d \omega \rho(\omega)$$

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## MATLAB代写

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