assignmentutor-lab™ 为您的留学生涯保驾护航 在代写热力学thermodynamics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写热力学thermodynamics代写方面经验极为丰富，各种代写热力学thermodynamics相关的作业也就用不着说。

• 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
assignmentutor™您的专属作业导师

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

Let us now consider the effect of a small near-zone difference $\eta_{1}-\eta_{2}$ that scales linearly with the separation $R$. This effect is most salient at the pseudocrossing (near-equality) of two eigenvalues in (8.49) (solid curves, Fig. 8.5), namely for $R$ close to the value $R_{\mathrm{c}}$ such that $\omega_{-}\left(R_{\mathrm{c}}\right)=\omega_{\mathrm{A}}\left(R_{\mathrm{c}}\right)$. This equality implies, in view of (8.50), that $\Delta_{12}\left(R_{\mathrm{c}}\right) \sim\left|\eta_{1}\right|$ for $\left|\omega_{0}-\omega_{\mathrm{a}}\right| \lesssim 2\left|\eta_{1}\right|$, or $\Delta_{12}\left(R_{\mathrm{c}}\right) \approx 2 \eta_{1}^{2} /\left|\omega_{0}-\omega_{\mathrm{a}}\right|$ for $\left|\omega_{0}-\omega_{\mathrm{a}}\right| \gg 2\left|\eta_{1}\right|$. In both cases the RDDI-induced and cavity-QED level shifts (or splittings) become comparable.

The strong competition of RDDI and Rabi splittings near $R_{\mathrm{c}}$ modifies the eigenvalues in (8.49), replacing them with the more accurate solutions of (8.48),
$$\omega_{1} \approx \omega_{+}, \quad \omega_{2,3} \approx \frac{1}{2}\left(\omega_{-}+\omega_{\mathrm{A}} \pm \Omega^{\prime}\right),$$
where
$$\Omega^{\prime}=\sqrt{V_{0}^{2}+\left(\omega_{-}-\omega_{\mathrm{A}}\right)^{2}}, \quad V_{0}=\frac{\eta_{1}^{2}-\eta_{2}^{2}}{\sqrt{\Omega\left(\Omega+\omega_{0}-\omega_{\mathrm{S}}\right)}} .$$
Here $\left|V_{0}\right|$, the minimal splitting between $\omega_{2}$ and $\omega_{3}$, determines the width of the pseudocrossing interval, $\left|R_{1}-R_{2}\right|$, where $\omega_{a}\left(R_{1,2}\right)-\omega_{-}\left(R_{1,2}\right)=\pm V_{0}$. For two atoms far from a node of a sinusoidal mode, $\left|V_{0}\right| \sim\left|\eta_{1}\left(\eta_{1}-\eta_{2}\right)\right| /\left(\sqrt{8}\left|\eta_{1}\right|+\right.$ $\left.\left|\omega_{0}-\omega_{\mathrm{S}}\right|\right)$

Whereas the eigenfunction $\left|\Psi_{1}\right\rangle=\left|\Psi_{+}\right\rangle$is not affected by the pseudocrossing, $\left|\Psi_{-}\right\rangle$and $\left|\Psi_{\mathrm{A}}\right\rangle$ are strongly mixed near $R_{\mathrm{c}}$. This mixing signifies the complete breaking of the symmetry [Eq. (8.46)], that characterizes the two-atom system subject to RDDI in open space.

## 物理代写|热力学代写thermodynamics代考|Model and Dynamics

We here consider $N$ noninteracting spin- $1 / 2$ particles or atomic TLS that are identically, linearly coupled to a bosonic (oscillator) bath via $\sigma_{z}$ (unlike $\sigma_{x}$ in the Dicke model). In the collective basis, the many-body Hamiltonian has the following form, without the RWA,
$$H=H_{\mathrm{S}}+H_{\mathrm{B}}+H_{\mathrm{I}},$$

where
$$H_{\mathrm{S}}=\hbar \omega_{x} \hat{J}{x}, \quad H{\mathrm{B}}=\hbar \sum_{k} \omega_{k} a_{k}^{\dagger} a_{k}, \quad H_{\mathrm{I}}=\hbar \hat{J}{z} \sum{k} \eta_{k}\left(a_{k}+a_{k}^{\dagger}\right) .$$
Here the notation is as in Chapter 7 , particularly, $a_{k}^{\dagger}$ and $a_{k}$ are the creation and annihilation bosonic operators of the $k$ th bath mode, and the collective spin operators in $H_{\mathrm{S}}$ and $H_{\mathrm{I}}$ are, as before, $\hat{J}{i}=(1 / 2) \sum{j} \sigma_{j}^{i}(i=x, y, z)$.

The bath interacts separately with each subspace of the system labeled by the total-spin value $J$, since $H$ commutes with $\hat{J}^{2}=\sum_{i} \hat{J}_{i}^{2}$. It is thus sufficient to study the interaction of the bath with a $(2 J+1)$-dimensional system.

The noncommutativity of $\hat{J}{x}$ and $\hat{J}{z}$ in (8.59) renders the dynamics of the system insolvable. In order to circumvent this difficulty, we prepare the system in an eigenstate of $\hat{J}{x}=(1 / 2) \sum{k} \sigma_{k}^{x}$ (a superposition of $\hat{J}{z}$ eigenstates) and then switch off $H{\mathrm{S}}=\omega_{x} \hat{J}{x}$. Equivalently, at time $t=0$ each spin is prepared in a superposition of its $\sigma{k}^{z}$ (energy) eigenstates, so that the total system is initially in a product of such superposition states. The individual spins are then uncorrelated (unentangled).

# 热力学代写

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

RDDI和Rabi分裂的激烈竞争接近 $R_{\mathrm{c}}$ 修改 (8.49) 中的特征值，用 (8.48) 的更精确解替换它们，
$$\omega_{1} \approx \omega_{+}, \quad \omega_{2,3} \approx \frac{1}{2}\left(\omega_{-}+\omega_{\mathrm{A}} \pm \Omega^{\prime}\right)$$

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

## 物理代写|热力学代写thermodynamics代考|Model and Dynamics

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

$$H_{\mathrm{S}}=\hbar \omega_{x} \hat{J} x, \quad H \mathrm{~B}=\hbar \sum_{k} \omega_{k} a_{k}^{\dagger} a_{k}, \quad H_{\mathrm{I}}=\hbar \hat{J} z \sum k \eta_{k}\left(a_{k}+a_{k}^{\dagger}\right)$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。