### 物理代写|热力学代写thermodynamics代考|SEM202

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## 物理代写|热力学代写thermodynamics代考|Electron–Phonon Interaction

The electron-phonon interaction can be shown to bear analogy to the optomechanical interaction in Section 4.2.1. In media where the electron-phonon interaction is weak, the deformation-potential method may be applied to long-wavelength phonons. Whereas in an unstrained (cubic) covalent crystal the electron energy band may be assumed spherical,
$$E_0(\boldsymbol{k})=\frac{\hbar^2 k^2}{2 m^},$$ $m^$ being the effective mass of the conduction electron, a small (uniform) static deformation yields for low $k$,
$$E(\boldsymbol{k}) \simeq E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V},$$
where $\delta \mathcal{V}$ is the dilation (relative volume change) given by the trace of the strain tensor, and
$$C_{\mathrm{d}}=\frac{\partial E(\mathbf{0})}{\partial(\delta \mathcal{V})}=-\frac{2}{3} E_{\mathrm{F}}$$
for a free electron gas, $E_{\mathrm{F}}$ being the Fermi energy.
For long-wavelength acoustic phonons, we have, instead of (4.23),
$$E(\boldsymbol{k}, \boldsymbol{x})=E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V}(\boldsymbol{x})$$
We then expand the Hamiltonian of the dilation perturbation in phonon operators $(3.26)$
\begin{aligned} \tilde{H}{\mathrm{d}}= & \int d^3 x \hat{\Psi}^{\dagger}(\boldsymbol{x}) C{\mathrm{d}} \delta \mathcal{V}(\boldsymbol{x}) \hat{\Psi}(\boldsymbol{x})=\sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}}\left\langle\boldsymbol{k}^{\prime}\left|C_{\mathrm{d}} \delta \mathcal{V}\right| \boldsymbol{k}\right\rangle \ = & i C_{\mathrm{d}} \sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}} \sum_{\boldsymbol{q}}|\boldsymbol{q}| \sqrt{\frac{\hbar}{2 \rho \omega_q}}\left[a_{\boldsymbol{q}} \int d^3 x u_{\boldsymbol{k}^{\prime}}^* u_{\boldsymbol{k}} e^{i\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}+\boldsymbol{q}\right) \cdot \boldsymbol{x}}\right. \ & \left.-a_{\boldsymbol{q}}^{\dagger} \int d^3 x u_{\boldsymbol{k}^{\prime}}^* u_{\boldsymbol{k}} e^{i\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}-\boldsymbol{q}\right) \cdot x}\right] \end{aligned}

## 物理代写|热力学代写thermodynamics代考|Polaronic Interaction of a Two-Level System with a Phonon Bath

This model consists of a driven two-level system (TLS) whose $\sigma_z$ operator is coupled to a (dephasing) bath, while its $\sigma_x$ operator is coupled to another (energy-exchange) bath. The Hamiltonian is then
$$H=H_{\mathrm{S}}+H_{\mathrm{SB}}+H_{\mathrm{B}}$$
where
\begin{aligned} H_{\mathrm{S}} & =\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right), \ H_{\mathrm{SB}} & =\sigma_z \otimes \hbar \sum_k\left(g_k a_k^{\dagger}+g_k^* a_k\right)+\sigma_x \otimes \hbar \sum_k\left(\eta_k b_k^{\dagger}+\eta_k^* b_k\right), \ H_{\mathrm{B}} & =\hbar \sum_k \omega_k a_k^{\dagger} a_k+\hbar \sum_k \tilde{\omega}_k b_k^{\dagger} b_k . \end{aligned}
Here $\Omega$ is the Rabi frequency of the driving field, $g_k$ and $\eta_k$ are coupling strengths of the TLS to the mode $k$ of the corresponding bath, and $a_k^{\dagger}, a_k\left(b_k^{\dagger}, b_k\right)$ are the creation and annihilation operators of the mode $k$ of the corresponding bath. Due to the driving, the system Hamiltonian is not diagonal in the $\sigma_z$ basis, thereby allowing energy exchange with the dephasing bath.

The system dynamics can be studied upon applying to (4.35) the polaron transformation $e^{\mathcal{T}}$, where
$$\mathcal{T}=\sigma_Z \otimes \sum_k\left(\alpha_k a_k^{\dagger}-\alpha_k^* a_k\right), \quad \alpha_k=\frac{g_k}{\omega_k} .$$
This transformation shifts the equilibrium position of the dephasing bath oscillators by a factor proportional to the TLS energy. The transformed Hamiltonian has the form
$$\widetilde{H}=e^{\mathcal{T}} H e^{-\mathcal{T}}=\tilde{H}{\mathrm{S}}+\widetilde{H}{\mathrm{SB}}+\widetilde{H}{\mathrm{B}}$$ where $$\begin{gathered} \widetilde{H}{\mathrm{S}}=\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega_{\mathrm{r}}}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right) \ \widetilde{H}{\mathrm{SB}}=\frac{\hbar \Omega}{2}\left[e^{-i \omega_l t} \sigma{+} \otimes\left(A_{+}-A\right)+e^{i \omega_l t} \sigma_{-} \otimes\left(A_{-}-A\right)\right] \end{gathered}$$ $\begin{gathered}+\left(\sigma_{+} \otimes A_{+}+\sigma_{-} \otimes A_{-}\right) \otimes \hbar \sum_k\left(\eta_k b_k^{\dagger}+\eta_k^* b_k\right), \ \tilde{H}_{\mathrm{B}}=\hbar \sum_k \omega_k a_k^{\dagger} a_k+\hbar \sum_k \tilde{\omega}_k b_k^{\dagger} b_k .\end{gathered}$

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Electron–Phonon Interaction

$$\text { E_O(lboldsymbol }{\mathrm{k}})=\mid f \mathrm{frac}\left{\backslash \mathrm{hbar} \wedge 2 \mathrm{k}^{\wedge} 2\right}\left{2 \mathrm{~m}^{\wedge}\right},$$

$$E(\boldsymbol{k}) \simeq E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V}$$

$$C_{\mathrm{d}}=\frac{\partial E(\mathbf{0})}{\partial(\delta \mathcal{V})}=-\frac{2}{3} E_{\mathrm{F}}$$

$$E(\boldsymbol{k}, \boldsymbol{x})=E_0(\boldsymbol{k})+C_{\mathrm{d}} \delta \mathcal{V}(\boldsymbol{x})$$

$$\tilde{H} \mathrm{~d}=\int d^3 x \hat{\Psi}^{\dagger}(\boldsymbol{x}) C \mathrm{~d} \delta \mathcal{V}(\boldsymbol{x}) \hat{\Psi}(\boldsymbol{x})=\sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}}\left\langle\boldsymbol{k}^{\prime}\left|C_{\mathrm{d}} \delta \mathcal{V}\right| \boldsymbol{k}\right\rangle=\quad i C_{\mathrm{d}} \sum_{\boldsymbol{k}^{\prime}, \boldsymbol{k}} c_{\boldsymbol{k}^{\prime}}^{\dagger} c_{\boldsymbol{k}} \sum_{\boldsymbol{q}}|\boldsymbol{q}| \sqrt{\frac{\hbar}{2 \rho \omega}}$$

## 物理代写|热力学代写thermodynamics代考|Polaronic Interaction of a Two-Level System with a Phonon Bath

$$H=H_{\mathrm{S}}+H_{\mathrm{SB}}+H_{\mathrm{B}}$$

$$H_{\mathrm{S}}=\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right), H_{\mathrm{SB}} \quad=\sigma_z \otimes \hbar \sum_k\left(g_k a_k^{\dagger}+g_k^* a_k\right)+\sigma_x \otimes \hbar \sum_k$$

$$\mathcal{T}=\sigma_Z \otimes \sum_k\left(\alpha_k a_k^{\dagger}-\alpha_k^* a_k\right), \quad \alpha_k=\frac{g_k}{\omega_k} .$$

$$\widetilde{H}=e^{\mathcal{T}} H e^{-\mathcal{T}}=\tilde{H} \mathrm{~S}+\widetilde{H} \mathrm{SB}+\widetilde{H} \mathrm{~B}$$

\begin{aligned} & \widetilde{H} \mathrm{~S}=\frac{\hbar \omega_0}{2} \sigma_z+\frac{\hbar \Omega_{\mathrm{r}}}{2}\left(\sigma_{+} e^{-i \omega_l t}+\sigma_{-} e^{i \omega_l t}\right) \widetilde{H} \mathrm{SB}=\frac{\hbar \Omega}{2}\left[e^{-i \omega_l t} \sigma+\otimes\left(A_{+}-A\right)+e^{i \omega_l t} \sigma_{-} \otimes(A\right. \ & +\left(\sigma_{+} \otimes A_{+}+\sigma_{-} \otimes A_{-}\right) \otimes \hbar \sum_k\left(\eta_k b_k^{\dagger}+\eta_k^* b_k\right), \tilde{H}_{\mathrm{B}}=\hbar \sum_k \omega_k a_k^{\dagger} a_k+\hbar \sum_k \tilde{\omega}_k b_k^{\dagger} b_k \end{aligned}

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## MATLAB代写

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