### 物理代写|热力学代写thermodynamics代考|MEC302

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Model

Here we focus on a system of two atoms that are near resonant with a high- $Q$ cavity mode. We treat a narrow spectral band of the electromagnetic bath, which is associated with the high- $Q$ mode, separately from the background mode-density spectrum: The latter bath spectrum is smooth and/or off-resonant and represents bath modes that are weakly coupled to the atoms. The two-atom interaction with the narrow band is evaluated exactly, whereas the interaction with the weakly coupled parts of the bath-mode spectrum is treated by second-order perturbation theory. Accordingly, we obtain the effective Hamiltonian in the following second-quantized form:
\begin{aligned} H=& \hbar \sum_{j=1}^{2} \omega_{j}\left|e_{j}\right\rangle\left\langle e_{j}|| \hbar \sum_{k} \omega_{k} a_{k}^{\dagger} a_{k}\right.\ &+\hbar \Delta_{12}\left(\left|e_{1} g_{2}\right\rangle\left\langle e_{2} g_{1}|+| e_{2} g_{1}\right\rangle\left\langle e_{1} g_{2}\right|\right) \ &+\hbar \sum_{j=1}^{2} \sum_{k}\left(\eta_{k j} a_{k}\left|e_{j}\right\rangle\left\langle g_{j}\right|+\text { H.c. }\right) \end{aligned} Here $\left|e_{j}\right\rangle$ and $\left|g_{j}\right\rangle$ are the excited and ground states of the two TLS. The frequency $\omega_{k}$ and annihilation operator $a_{k}$ pertain to a near-resonant bath mode (within the narrow band) whose dipolar coupling to the $j$ th TLS is given by $\eta_{k j} ; \hbar \omega_{j}$ are the excited-state energies. Those energies include the Lamb shifts caused by single-TLS interactions with the off-resonant bath modes (outside the near-resonant narrow band). Finally, $\hbar \Delta_{12}$ is the matrix element of the interatom RDDI (excluding the contribution of the near-resonant modes). Both $\omega_{j}$ and $\Delta_{12}$ may have imaginary (dissipative) parts. The TLS-bath interaction [the last term in (8.40)] is written in the RWA, since the anti-rotating terms are off-resonant and hence their contributions are assumed to be included in $\omega_{j}$ and $\Delta_{12}$. At near-zone distances, the real part of $\Delta_{12}$ reduces to the usual electrostatic RDDI, which varies with the interatom separation $R$ as the inverse cube $R^{-3}$.

## 物理代写|热力学代写thermodynamics代考|Oscillating Exchange between Atoms in a Cavity

Let us assume that $\eta_{1} \approx \eta_{2}$. The two atomic couplings to the mode are nearly equal for identical atoms with parallel dipoles in the near zone of separations, $\omega_{\mathrm{a}} R / c \ll 1$. Under this assumption, the eigenvalues obtained from (8.48) are given (in descending order) by
$$\omega_{1} \approx \omega_{+}, \quad \omega_{2} \approx\left{\begin{array} { l l } { \omega _ { – } , } & { \text { if } R < R _ { \mathrm { c } } , } \ { \omega _ { \mathrm { A } } , } & { \text { if } R > R _ { \mathrm { c } } , } \end{array} \quad \omega _ { 3 } \approx \left{\begin{array}{ll} \omega_{\mathrm{A}}, & \text { if } RR_{\mathrm{c}}, \end{array}\right.\right.$$

where $\omega_{\mathrm{S}(\mathrm{A})}=\omega_{\mathrm{a}} \pm \Delta_{12}(R), R_{\mathrm{c}}$ is the position of the crossing of $\omega_{-}(R)$ and $\omega_{\mathrm{A}}(R)$, and
$$\omega_{\pm}=\frac{1}{2}\left(\omega_{\mathrm{S}}+\omega_{0} \pm \Omega\right), \quad \Omega=\sqrt{2\left(\eta_{1}+\eta_{2}\right)^{2}+\left(\omega_{\mathrm{S}}-\omega_{0}\right)^{2}} .$$
Here and below we assume without loss of generality that $\Delta_{12}$ is positive in the near zone. These results hold throughout the near-zone (except for the pseudocrossing, $R \approx R_{\mathrm{c}}$, discussed below). For $R \rightarrow 0$, the divergence of $\Delta_{12}(R)$ implies that $\left|\omega_{\mathrm{S}}-\omega_{0}\right| \gg\left|\eta_{1}+\eta_{2}\right|$, hence $\omega_{+} \rightarrow \omega_{\mathrm{S}}$ and $\omega_{-} \rightarrow \omega_{0}$, so that the near-resonant field mode is then decoupled from both RDDI-split (symmetric and antisymmetric) states.

Throughout the range of validity of (8.49), the antisymmetric-state eigenvalue remains uncoupled from the mode, since its coupling is $2^{-1 / 2}\left(\eta_{1}-\eta_{2}\right) \approx 0$ in the near zone. The symmetric state and the single-photon state become increasingly hybridized as $R$ grows, provided the detuning $\left|\omega_{0}-\omega_{\mathrm{a}}\right|$ is not too large. This hybridization gives rise to two eigenvalues that are split by $\pm \Omega$, the vacuum Rabi frequency of the symmetric state. The trends surveyed above are also exhibited by the dressed-state eigenfunctions: As $R \rightarrow 0,\left|\Psi_{+}\right\rangle \rightarrow\left|\Psi_{\mathrm{S}},{0}\right\rangle$ and $\left|\Psi_{-}\right\rangle \rightarrow$ $\left|g_{1} g_{2}, 1_{0}\right\rangle$. Otherwise, the symmetric and single-photon states are strongly mixed in $\left|\Psi_{\pm}\right\rangle .$By contrast, $\left|\Psi_{3}\right\rangle \approx\left|\Psi_{\mathrm{A}},{0}\right\rangle$ as long as $\eta_{1} \approx \eta_{2}$.

## 物理代写|热力学代写thermodynamics代考|Model

H=ℏ∑j=12哦j|和j⟩⟨和j||ℏ∑ķ哦ķ一个ķ†一个ķ +ℏD12(|和1G2⟩⟨和2G1|+|和2G1⟩⟨和1G2|) +ℏ∑j=12∑ķ(这ķj一个ķ|和j⟩⟨Gj|+ HC )这里|和j⟩和|Gj⟩是两个 TLS 的激发态和基态。频率哦ķ和歼灭算子一个ķ属于近谐振浴模式（在窄带内），其偶极耦合到jth TLS 由下式给出这ķj;ℏ哦j是激发态能量。这些能量包括由单 TLS 与非共振浴模式（在近共振窄带之外）相互作用引起的兰姆位移。最后，ℏD12是原子间 RDDI 的矩阵元素（不包括近共振模式的贡献）。两个都哦j和D12可能有虚（耗散）部分。TLS-浴相互作用 [(8.40) 中的最后一项] 写在 RWA 中，因为反旋转项是非共振的，因此假设它们的贡献包含在哦j和D12. 在近区距离处，实部D12减少到通常的静电RDDI，它随原子间分离而变化R作为逆立方体R−3.

## 物理代写|热力学代写thermodynamics代考|Oscillating Exchange between Atoms in a Cavity

$$\omega_{\pm}=\frac{1}{2}\left(\omega_{\mathrm{S}}+\omega_{0} \pm \Omega\right), \quad \Omega=\sqrt{2\left(\eta_{1}+\eta_{2}\right)^{2}+\left(\omega_{\mathrm{S}}-\omega_{0}\right)^{2}} .$$

，对称态的真空拉比频率。上述调查的趋势也表现在穿着状态的特征函数上: $R \rightarrow 0,\left|\Psi_{+}\right\rangle \rightarrow\left|\Psi_{\mathrm{S}}, 0\right\rangle$ 和 $\left|\Psi_{-}\right\rangle \rightarrow\left|g_{1} g_{2}, 1_{0}\right\rangle$. 否则，对称和单光子状态强烈混合在 $\left|\Psi_{\pm}\right\rangle$.相比之下， $\left|\Psi_{3}\right\rangle \approx\left|\Psi_{\mathrm{A}}, 0\right\rangle$ 只要 $\eta_{1} \approx \eta_{2}$.

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## MATLAB代写

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