# 物理代写|热力学代写thermodynamics代考|Analysis

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

The time-dependent wave function for the system of a single photon shared by the field and the atoms (TLS) is written as
$$\left|\Psi_{j^{\prime}}(t)\right\rangle=\sum_{j=1}^2 V_{j j^{\prime}}(t)\left|e_j g_{3-j},\left{0_k\right}\right\rangle+\sum_k \beta_{k j^{\prime}}(t)\left|g_1 g_2, 1_k\right\rangle,$$
where $j^{\prime}$ indicates the atom that is excited initially, $\left|\left{0_k\right}\right\rangle$ and $\left|1_k\right\rangle$ being the vacuum and $k$-mode single-quantum states, respectively. In the subspace $\left{\left|e_1 g_2\right\rangle,\left|e_2 g_1\right\rangle\right}$ of the collective atomic states, an initially pure, normalized, state remains pure thereafter (though generally not normalized) and has the form $\alpha_1(t)\left|e_1 g_2\right\rangle+\alpha_2(t)\left|e_2 g_1\right\rangle$ with the coefficients
$$\alpha_j(t)=\sum_{j^{\prime}=1}^2 V_{j j^{\prime}}(t) \alpha_{j^{\prime}}(0)$$
From the Schrödinger equation we obtain the following exact Laplace-transform solution for the excited-state evolution operator
$$\hat{V}(s)=D^{-1}(s) U(s)$$
with $D(s)$ being the determinant of the $2 \times 2$ matrix $U(s)$. The diagonal and off-diagonal elements of the matrix $U(s)$ represent single-atom and interatomic contributions, respectively,
\begin{aligned} U_{j j}(s) & =s+i \omega_j+i J_{j j}(s), \ U_{j j^{\prime}}(s) & =-i \Delta_{12}-i J_{j j^{\prime}}(s) \quad\left(j \neq j^{\prime}\right), \end{aligned}

where the integration over the narrow band of near-resonant modes yields
\begin{aligned} J_{j j^{\prime}}(s) & =\int \frac{G_{j j^{\prime}}(\omega) d \omega}{i s-\omega}, \ G_{j j j^{\prime}}(\omega) & =\sum_k \eta_{k j} \eta_{k j^{\prime}}^* \delta\left(\omega-\omega_k\right) . \end{aligned}
The roots of the determinant equation $D(s)=0$ correspond to the levels (eigenvalues) of the two-atom system.

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

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 $\eta_1 \approx \eta_2$.

These eigenfunctions can be used to calculate the probability of excitation transfer from the initially excited atom 1 to the initially unexcited atom $2, P_2(t)$, and the corresponding excitation-trapping probability $P_1(t)$,
$$P_j(t)=\left|\sum_{i=1}^3\left\langle\varphi_j \mid \Psi_i\right\rangle\left\langle\Psi_i \mid \varphi_1\right\rangle e^{-i \omega_i t}\right|^2,$$
where $\Psi_i$ are the dressed-state eigenfunctions (labelled by $i=1,2,3$ ) and $\left|\varphi_j\right\rangle=$ $\left|e_j g_{3-j},{0}\right\rangle$. We find three distinct atomic-state eigenvalues, causing aperiodic oscillations of $P_j(t)$, instead of the sinusoidal oscillations in Chapter 7 . The timeaveraged probabilities,
$$\bar{P}j=\sum{i=1}^3\left|\left\langle\varphi_j \mid \Psi_i\right\rangle\left\langle\Psi_i \mid \varphi_1\right\rangle\right|^2$$
are approximately equal, $\bar{P}1 \approx \bar{P}_2$. They vary from $3 / 8$ at $\left|\omega{\mathrm{S}}-\omega_0\right| \ll\left|\eta_1+\eta_2\right|$ to the free-space value $1 / 2$ at $\left|\omega_{\mathrm{S}}-\omega_0\right| \gg\left|\eta_1+\eta_2\right|$. Because of the nonzero probability of the field-mode excitation, we have
$$P_1(t)+P_2(t)<1$$
for the sum of the excitation probabilities of the two atoms.

# 热力学代写

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

Weft{Neft |e_1 g_2|right|rangle, left|e_2 g_1|right|Iangle|right} 在集体原子状态中，一个最初纯的、归一化的状态此后 保持纯 (尽管通常末归一化) 并且具有以下形式 $\alpha_1(t)\left|e_1 g_2\right\rangle+\alpha_2(t)\left|e_2 g_1\right\rangle$ 与系数
$$\alpha_j(t)=\sum_{j^{\prime}=1}^2 V_{j j^{\prime}}(t) \alpha_{j^{\prime}}(0)$$

$$\hat{V}(s)=D^{-1}(s) U(s)$$

$$U_{j j}(s)=s+i \omega_j+i J_{j j}(s), U_{j j^{\prime}}(s) \quad=-i \Delta_{12}$$

$$J_{j j^{\prime}}(s)=\int \frac{G_{j j^{\prime}}(\omega) d \omega}{i s-\omega}, G_{j j j^{\prime}}(\omega)=\sum_k \eta_{k j} \eta_{k j^{\prime}}^* \delta$$

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

$$P_j(t)=\left|\sum_{i=1}^3\left\langle\varphi_j \mid \Psi_i\right\rangle\left\langle\Psi_i \mid \varphi_1\right\rangle e^{-i \omega_i t}\right|^2$$

$$\bar{P} j=\sum i=1^3\left|\left\langle\varphi_j \mid \Psi_i\right\rangle\left\langle\Psi_i \mid \varphi_1\right\rangle\right|^2$$大致相等, $\bar{P} 1 \approx \bar{P}2$. 他们从 $3 / 8$ 在 $\left|\omega \mathrm{S}-\omega_0\right| \ll \mid \eta_1+\eta_2$ |到自由空间值 $1 / 2$ 在 $\left|\omega{\mathrm{S}}-\omega_0\right| \gg\left|\eta_1+\eta_2\right|$. 由于场模激发的非零概率， 我们有
$$P_1(t)+P_2(t)<1$$

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## MATLAB代写

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