### 物理代写|热力学代写thermodynamics代考|NEM2201

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Here we consider transitions between two electronic states of an atom, $|e\rangle$ and $|g\rangle$, resulting in the emission or absorption of one photon, via the interaction (4.1). The matrix element for single-photon emission that corresponds to the term linear in $\boldsymbol{A}$ in (4.1), expanded as per (3.5), is given by
\begin{aligned} \left\langle g, n_\lambda(\boldsymbol{k})+1\left|H_{\mathrm{I}}\right| e, n_\lambda(\boldsymbol{k})\right\rangle= & -\frac{e}{m}\left(\frac{2 \pi \hbar}{\mathcal{V} \omega_{\boldsymbol{k}}}\right)^{1 / 2}\left[n_\lambda(\boldsymbol{k})+1\right]^{1 / 2} \ & \times\left\langle g\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot \sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r}i} \boldsymbol{p}_i\right| e\right\rangle, \end{aligned} where $n\lambda(\boldsymbol{k})$ is the initial photon number in the mode $(\boldsymbol{k}, \lambda), \boldsymbol{k}$ being the wave vector and $\lambda$ the polarization, $\omega_{\boldsymbol{k}}$ is the photon frequency at a given $\boldsymbol{k}, \mathcal{V}$ is the quantization volume, $\boldsymbol{\epsilon}{\boldsymbol{k} \lambda}$ is a unit polarization vector of the photon, $e$ and $m$ are the electron charge and mass, whereas $\boldsymbol{r}_i$ and $\boldsymbol{p}_i$ are the position and momentum of the atomic electron $i$. The corresponding transition probability per unit time is then $$w\lambda d \Omega_{\mathrm{s}}=\frac{e^2 \omega_{\mathrm{a}} d \Omega_{\mathrm{s}}}{2 \pi m^2 \hbar c^3}\left[n_\lambda(\boldsymbol{k})+1\right]\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot\left\langle g\left|\sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r}i} p_i\right| e\right\rangle\right|^2,$$ where $\omega{\mathrm{a}}=\left(E_e-E_g\right) / \hbar$ and $E_e$ and $E_g$ are the energies of the excited and ground atomic (electronic) states, $\Omega_{\mathrm{s}}$ being the solid angle of the emission. Equation (4.3) can be adapted to the absorption of a photon upon replacing the factor $\left[n_\lambda(\boldsymbol{k})+1\right]$ by $n_\lambda(\boldsymbol{k})$.

The electric dipole approximation is valid provided we can approximate the exponential factors in Eqs. (4.2) and (4.3) by unity: $$e^{-i k \cdot \boldsymbol{r}i} \approx 1 .$$ This holds if the wavelength $2 \pi / k$ of the photon is very large compared to the size $R$ of the atom, as in the case of optical atomic transitions. Then the wave functions of $|e\rangle$ and $|g\rangle$ restrict the values of $\boldsymbol{r}_i$ to $\left|\boldsymbol{k} \cdot \boldsymbol{r}_i\right| \lesssim k R \ll 1$. The equation of motion $$i \hbar \dot{\boldsymbol{r}}_i=\left[\boldsymbol{r}_i, H{\mathrm{a}}\right]$$
where $H_{\mathrm{a}}$ is the atomic Hamiltonian, yields
$$\left\langle g\left|\boldsymbol{p}i\right| e\right\rangle=m\left\langle g\left|\dot{\boldsymbol{r}}_i\right| e\right\rangle=-i m \omega{\mathrm{a}}\left\langle g\left|\boldsymbol{r}i\right| e\right\rangle$$ The electric dipole operator $\boldsymbol{d}=e \sum_i \boldsymbol{r}_i$ in this two-state basis may be represented by $$\boldsymbol{d}=\sum{j, l=e, g}|j\rangle\langle j|\boldsymbol{d}| l\rangle\langle l|=\sum_{j, l=e, g} \boldsymbol{\wp}{j l} \sigma{j l}$$
where $\wp_{j l}=\langle j|d| l\rangle$ is the electric-dipole transition matrix element. The transition operators $\sigma_{j l}=|j\rangle\langle l|$ form the set
\begin{aligned} \sigma_z & =|e\rangle\langle e|-| g\rangle\langle g|, \ \sigma_{+} & =|e\rangle\langle g|, \ \sigma_{-} & =|g\rangle\langle e|, \end{aligned}
where $\sigma_{+}, \sigma_{-}$, and $\sigma_z$ satisfy the spin-1/2 algebra of the Pauli matrices, that is,
$$\begin{gathered} {\left[\sigma_{-}, \sigma_{+}\right]=-\sigma_z,} \ {\left[\sigma_{-}, \sigma_z\right]=2 \sigma_{-} .} \end{gathered}$$

## 物理代写|热力学代写thermodynamics代考|Polaronic System–Bath Interactions

We consider the basic opto-mechanical Hamiltonian that governs an optical cavity mode (denoted by O) that is coupled to a photonic bath and to a mechanical oscillator (denoted by M). The total Hamiltonian then has the form
\begin{aligned} H_{\text {Tot }} & =H_{\mathrm{O}+\mathrm{M}}+\left(O^{\dagger}+O\right) \otimes B ; \ H_{\mathrm{O}+\mathrm{M}} & =\omega_{\mathrm{O}} O^{\dagger} O+\Omega_{\mathrm{M}} M^{\dagger} M+g O^{\dagger} O\left(M+M^{\dagger}\right) . \end{aligned}
Here $O^{\dagger}, O$ and $M^{\dagger}, M$ are the creation and annihilation operators of the cavity mode and the oscillator, respectively; $\omega_{\mathrm{O}}, \Omega_{\mathrm{M}}$ and $g$ are their respective frequencies and coupling rate; and $B$ is the photonic-bath operator (Fig. 4.1).

We transform these operators to the basis of hybridized optical-mechanical modes that diagonalize $H_{\mathrm{O}+\mathrm{M}}$ without changing their frequency. Namely,
\begin{aligned} H_{\mathrm{O}+\mathrm{M}} & =\widetilde{H}{\mathrm{O}}+\widetilde{H}{\mathrm{M}}, \quad \widetilde{H}{\mathrm{O}}=\omega{\mathrm{O}} \tilde{O}^{\dagger} \tilde{O}-\left(g \widetilde{O}^{\dagger} \widetilde{O}\right)^2 \frac{1}{\Omega_{\mathrm{M}}}, \quad \widetilde{H}{\mathrm{M}}=\Omega{\mathrm{M}} \tilde{M}^{\dagger} \tilde{M}, \ \tilde{M} & =M+\frac{g}{\Omega_{\mathrm{M}}} O^{\dagger} O, \quad \widetilde{O}=O e^{\frac{g}{\Omega_{\mathrm{M}}^{\mathrm{M}}}\left(M^{\dagger}-M\right)} . \end{aligned}
The new variables can be expressed in terms of the unitary (“polaron”) transformation $$U=e^{\frac{g}{\frac{g}{2 \mathrm{M}}\left(M^{+}-M\right) O^{\dagger} O}} .$$
as $\tilde{O}=U^{\dagger} O U$ and $\tilde{M}=U^{\dagger} M U$. Then, the interaction between the optical mode and the photonic bath is found to indirectly affect the mechanical oscillator.

We shall restrict ourselves to low excitations of the transformed number operators $\hat{n}{\tilde{O}}=\widetilde{O}^{\dagger} \tilde{O}$ and $\hat{n}{\tilde{M}}=\tilde{M}^{\dagger} \tilde{M}$ and to the weak optomechanical-coupling regime. Namely, we shall assume
$$\left(\frac{g}{\Omega_{\mathrm{M}}}\right)^2\left\langle n_{\tilde{M}}\right\rangle \ll 1, \quad \frac{g^2}{\Omega_{\mathrm{M}}}\left\langle n_{\tilde{O}}\right\rangle^2 t \ll 1,$$
where $\left\langle n_{\tilde{M}}\right\rangle$ and $\langle n \tilde{O}\rangle$ are the mean numbers of quanta in the $\tilde{M}$ and $\widetilde{O}$ degrees of freedom, respectively.

# 热力学代写

$$\left\langle g, n_\lambda(\boldsymbol{k})+1\left|H_{\mathrm{I}}\right| e, n_\lambda(\boldsymbol{k})\right\rangle=-\frac{e}{m}\left(\frac{2 \pi \hbar}{\mathcal{V} \omega_{\boldsymbol{k}}}\right)^{1 / 2}\left[n_\lambda(\boldsymbol{k})+1\right]^{1 / 2} \times\left\langle g\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot \sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r i}} \boldsymbol{p}i\right| e\right\rangle$$ 在哪里 $n \lambda(\boldsymbol{k})$ 是模式中的初始光子数 $(\boldsymbol{k}, \lambda), \boldsymbol{k}$ 是波矢量和 $\lambda$ 极化， $\omega_k$ 是给定的光子频率 $\boldsymbol{k}, \mathcal{V}$ 是量化体 积， $\boldsymbol{\epsilon k} \lambda$ 是光子的单位偏振矢量， $e$ 和 $m$ 是电子电荷和质量，而 $\boldsymbol{r}_i$ 和 $\boldsymbol{p}_i$ 是原子电子的位置和动量 $i$. 对应的 单位时间转移概率为 $$w \lambda d \Omega{\mathrm{s}}=\frac{e^2 \omega_{\mathrm{a}} d \Omega_{\mathrm{s}}}{2 \pi m^2 \hbar c^3}\left[n_\lambda(\boldsymbol{k})+1\right]\left|\epsilon_{\boldsymbol{k} \lambda}^* \cdot\left\langle g\left|\sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r} i} p_i\right| e\right\rangle\right|^2$$

$$e^{-i k \cdot r i} \approx 1$$

$$\langle g|\boldsymbol{p} i| e\rangle=m\left\langle g\left|\dot{\boldsymbol{r}}i\right| e\right\rangle=-i m \omega \mathrm{a}\langle g|\boldsymbol{r} i| e\rangle$$ 电偶极算子 $\boldsymbol{d}=e \sum_i \boldsymbol{r}_i$ 在这个两国基础上可以表示为 $$\boldsymbol{d}=\sum j, l=e, g|j\rangle\langle j|\boldsymbol{d}| l\rangle\langle l|=\sum{j, l=e, g} \wp j l \sigma j l$$

$$\sigma_z=|e\rangle\langle e|-| g\rangle\left\langle g\left|, \sigma_{+}=\right| e\right\rangle\left\langle g\left|, \sigma_{-}=\right| g\right\rangle\langle e|,$$

$$\left[\sigma_{-}, \sigma_{+}\right]=-\sigma_z,\left[\sigma_{-}, \sigma_z\right]=2 \sigma_{-}$$

## 物理代写|热力学代写thermodynamics代考|Polaronic System–Bath Interactions

$$H_{\mathrm{Tot}}=H_{\mathrm{O}+\mathrm{M}}+\left(O^{\dagger}+O\right) \otimes B ; H_{\mathrm{O}+\mathrm{M}}=\omega_{\mathrm{O}} O^{\dagger} O+\Omega_{\mathrm{M}} M^{\dagger} M+g O^{\dagger} O\left(M+M^{\dagger}\right) \text {. }$$

$$H_{\mathrm{O}+\mathrm{M}}=\widetilde{H} \mathrm{O}+\widetilde{H} \mathrm{M}, \quad \widetilde{H} \mathrm{O}=\omega \mathrm{O} \tilde{O}^{\dagger} \tilde{O}-\left(g \widetilde{O}^{\dagger} \widetilde{O}\right)^2 \frac{1}{\Omega_{\mathrm{M}}}, \quad \widetilde{H} \mathrm{M}=\Omega \mathrm{M} \tilde{M}^{\dagger} \tilde{M}, \tilde{M} \quad=M$$

$$U=e^{\frac{g}{\frac{g}{2 M}\left(M^{+}-M\right) o \dagger o}} .$$

$$\left(\frac{g}{\Omega_{\mathrm{M}}}\right)^2\left\langle n_{\tilde{M}}\right\rangle \ll 1, \quad \frac{g^2}{\Omega_{\mathrm{M}}}\left\langle n_{\tilde{O}}\right\rangle^2 t \ll 1$$

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