### 物理代写|热力学代写thermodynamics代考|MECH3720

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|热力学代写thermodynamics代考|Cold-Atom Optical-Polariton Baths

We consider a medium composed of cold alkali atoms with level configuration as shown in Figure $3.6 .$

The atoms, taken to be optically pumped to the ground states $|b\rangle$, resonantly interact with a running-wave classical field that drives the atomic transition $|c\rangle \rightarrow|a\rangle$ with the Rabi frequency $\Omega_{\mathrm{d}}$. Near the two-photon (Raman) resonance
$|b\rangle \rightarrow|c\rangle$, the atomic medium then becomes transparent through an effect known as electromagnetically induced transparency (EIT) for a weak (quantum) signal field $\hat{\mathcal{E}}$ that acts on the transition $|b\rangle \rightarrow|a\rangle$.

A classical signal pulse of duration $t_{\mathrm{s}}$ in the atomic medium, under EIT conditions, is slowed down to group velocity $v_{\mathrm{s}}$ and spatially compressed, by a factor of $v_{\mathrm{s}} / c \ll 1$, to the length $z_{\text {loc }} \approx v_{\mathrm{s}} t_{\mathrm{s}}$. In a medium of length $L$, such that $z_{\text {loc }}<L$, the signal pulse is converted into a standing-wave polaritonic excitation (Fig. 3.6), provided the driving field is adiabatically switched off and the signal pulse is stopped in the medium. The atoms then dispersively interact with a standing-wave classical field having the Rabi frequency $\Omega_{\mathrm{s}}(z)=2 \Omega_{\mathrm{s}} \cos \left(k_{\mathrm{s}} z\right)$ and detuning $\delta \gg \Omega_{\mathrm{s}}$ from the atomic transition $|c\rangle \rightarrow|d\rangle$. This field induces a spatially periodic ac Stark shift of level $|c\rangle$ and a corresponding modulation of the refractive index for the signal field,
$$\delta n_{\mathrm{s}}(z)=\frac{c}{v_{\mathrm{s}}} \frac{4 \delta_{\mathrm{s}}}{\omega_{a b}} \cos ^{2}\left(k_{\mathrm{s}} z\right),$$
where $\delta_{\mathrm{s}}=\Omega_{\mathrm{s}}^{2} / \delta$ is the ac Stark-shift amplitude, $v_{\mathrm{s}} \propto\left|\Omega_{\mathrm{d}}\right|^{2}$, and $\omega_{a b}$ is the $|a\rangle \leftrightarrow$ $|b\rangle$ transition frequency.

## 物理代写|热力学代写thermodynamics代考|Magnon Baths

The low-lying energy states of spin systems coupled by exchange interactions give rise to quantized spin waves. The spin-wave quanta are known as magnons.

The simplest bath Hamiltonian that gives rise to magnons is that of $N$ identical spin- $S$ particles with nearest-neighbor interactions in a ferromagnetic spin lattice. It has the form,$$H_{\mathrm{B}}=-J \sum_{j, j^{\prime}} \hat{\boldsymbol{S}}{j} \cdot \hat{\boldsymbol{S}}{j^{\prime}}-2 \mu_{0} \mathcal{B}{0} \sum{j} \hat{S}{j z}$$ where $\hat{S}{j}$ is the $j$ th spin operator, $\hat{S}{j^{\prime}}$ are the spin operators of its nearest neighbors on a lattice, $J$ is the positive exchange integral, $2 \mu{0}$ is the magnetic moment of a particle, and $\mathcal{B}_{0} \geq 0$ is the static magnetic field that aligns the spins along the $\mathrm{z}$ axis.

To study this bath, it is convenient to resort to the Holstein-Primakoff transformation of the spin operators to bosonic creation and annihilation operators $a_{j}^{\hat{\dagger}}, a_{j}$, which has the form
\begin{aligned} &\hat{S}{j}^{+}=\hat{S}{j x}+i \hat{S}{j y}=(2 S)^{1 / 2}\left(1-\frac{a{j}^{\dagger} a_{j}}{2 S}\right)^{1 / 2} a_{j} \ &\hat{S}{j}^{-}=\hat{S}{j x}-i \hat{S}{j y}=(2 S)^{1 / 2} a{j}^{\dagger}\left(1-\frac{a_{j}^{\dagger} a_{j}}{2 S}\right)^{1 / 2} \ &\hat{S}{j z}=S-a{j}^{\dagger} a_{j} \end{aligned}
We next introduce the collective spin-wave (magnon) operators $a(\boldsymbol{k}), a^{\dagger}(\boldsymbol{k})$, that satisfy
$$a_{j}=\frac{1}{\sqrt{N}} \sum_{k} e^{-i \boldsymbol{k} \cdot \boldsymbol{x}{j}} a(\boldsymbol{k}),$$ where $\boldsymbol{x}{j}$ is the position vector of the $j$ th spin particle. We can rewrite the bath Hamiltonian in terms of these collective operators as
$$H_{\mathrm{B}}=H_{\mathrm{M}}+H_{\mathrm{M}}^{(1)} .$$

## 物理代写|热力学代写thermodynamics代考|Cold-Atom Optical-Polariton Baths

$|b\rangle \rightarrow|c\rangle$ ，然后原子介质通过称为电磁感应透明 (EIT) 的效应变得透明，用于弱（量子) 信号场 $\hat{\mathcal{E}}$ 作用于过渡 $|b\rangle \rightarrow|a\rangle$

$$\delta n_{\mathrm{s}}(z)=\frac{c}{v_{\mathrm{s}}} \frac{4 \delta_{\mathrm{s}}}{\omega_{a b}} \cos ^{2}\left(k_{\mathrm{s}} z\right)$$

## 物理代写|热力学代写thermodynamics代考|Magnon Baths

$$H_{\mathrm{B}}=-J \sum_{j, j^{\prime}} \hat{\boldsymbol{S}} j \cdot \hat{\boldsymbol{S}} j^{\prime}-2 \mu_{0} \mathcal{B} 0 \sum j \hat{S} j z$$

$$\hat{S} j^{+}=\hat{S} j x+i \hat{S} j y=(2 S)^{1 / 2}\left(1-\frac{a j^{\dagger} a_{j}}{2 S}\right)^{1 / 2} a_{j} \quad \hat{S} j^{-}=\hat{S} j x-i \hat{S} j y=(2 S)^{1 / 2} a j^{\dagger}\left(1-\frac{a_{j}^{\dagger} a_{j}}{2 S}\right)$$

$$a_{j}=\frac{1}{\sqrt{N}} \sum_{k} e^{-i \boldsymbol{k} \cdot \boldsymbol{x} j} a(\boldsymbol{k}),$$

$$H_{\mathrm{B}}=H_{\mathrm{M}}+H_{\mathrm{M}}^{(1)} .$$

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## MATLAB代写

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