# 物理代写|热力学代写thermodynamics代考|MECH3720

## 物理代写|热力学代写thermodynamics代考|Symmetry Considerations

From the $N$ linearly independent states $\left|e_j\right\rangle$, we construct, by orthogonalization to $\left|\phi^{(N]}}\right\rangle$, the $N-1$ states
\begin{aligned} \left|\phi_l^{\mid N}{ }^{1]}\right\rangle &=-\frac{1}{\sqrt{N}}\left|e_N\right\rangle+\sum_{j=1}^{N-1}\left[\frac{1+(1 / \sqrt{N})}{N-1}-\delta_{j l}\right]\left|e_j\right\rangle \ & \equiv \sum_{j=1}^N f_j^l\left|e_j\right\rangle, \quad l=1,2, \ldots, N-1 . \end{aligned}
The states (7.51) are normalized to 1 and orthogonal to each other and to $\left|\phi^{[N]}\right\rangle$. They comprise the basis of the irreducible representation characterized by the Young tableau ${N-1,1}$.

The wave function of the singly excited multiatom state can be expanded into a sum of states comprising the fully ( $N$-fold) symmetric state and the lower $(N-1)$ fold symmetry states,
$$\left|\psi_{\text {exc }}\right\rangle=c^{[N]}\left|\phi^{[N]}\right\rangle+\sum_{l=1}^{N-1} c_l^{[N-1]}\left|\phi_l^{[N-1]}\right\rangle,$$
yielding the set of equations for their probability amplitudes,
$$\begin{gathered} \dot{c}^{(N)}=-\gamma_{\mathrm{col}} c^{[N]}-\sum_{l=1}^{N-1} s_l^* c_l^{[N-1]} \ \dot{c}l^{(N-1)}=-s_l c^{(N)}-\sum{l^{\prime}=1}^{N-1} Q_{l^{\prime}} c_{l^{\prime}}^{(N-1)} \end{gathered}$$
with the initial conditions
$$\dot{c}^{[N]}(0)=1, \quad c_l^{[N-1}}(0)=0, \quad l=1,2, \ldots N-1 .$$
The mixing in (7.53), between the fully symmetric state and the $l$ th state of lower symmetry, occurs at the rate
$$s_l=\frac{\gamma_1}{\sqrt{N}} \sum_{j=1}^N \sum_{j^{\prime}=1}^N f_j^l \mathcal{F}\left(\boldsymbol{r}j-\boldsymbol{r}{j^{\prime}}\right)$$

## 物理代写|热力学代写thermodynamics代考|Markovian Theory of Two-Atom Self-Energy

We consider a pair of identical TLS with energy levels $|g\rangle$ and $|e\rangle$ and transition frequency $\omega_{\mathrm{a}}$. These TLS are coupled to the vacuum field of the structure (photonic bath), and we neglect their coupling to modes outside the structure. The TLSbath dipole couplings are $\eta_{k j}=-\wp_{e g} \cdot \boldsymbol{\phi}k\left(\boldsymbol{r}_j\right), \wp{e g}$ denoting [cf. (4.7)] the dipole matrix element of the $|g\rangle \leftrightarrow|e\rangle$ transition (taken to be real), $\omega_k$ and $\boldsymbol{\phi}_k(\boldsymbol{r})$ being, respectively, the $k$ th mode frequency and the spatial function at $r_j$, the location of the $j$ th TLS $(j=1,2)$. The interaction Hamiltonian in the dipole approximation has the following form (without the RWA) in the interaction picture,
$$H_{\mathrm{I}}=\hbar \sum_{j=1}^2 \sum_k\left(\eta_{k j} \hat{a}k e^{-i \omega_k t}+\text { H.c. }\right)\left(\hat{\sigma}_j^{-} e^{-i \omega_a t}+\text { H.c. }\right),$$ where $\hat{a}_k$ and $\hat{\sigma}_j^{-}$are the field-mode and the TLS lowering operators, respectively. The two-atom dissipative linewidth and dispersive energy-shift, $\gamma{j j j^{\prime}}$ and $\Delta_{j j^{\prime}}$, respectively, are here calculated in the weak-coupling limit to second order in the coupling strength. To this end, we evaluate the two-atom transition amplitude from the state where atom $j$ is excited to the state where atom $j^{\prime}$ is excited, which is then given by
\begin{aligned} U_{j j^{\prime}} &=\delta_{j j^{\prime}}+U_{j j^{\prime}}^{(2)} \ U_{j j^{\prime}}^{(2)} &=\frac{1}{2 \pi i} \int_{-t / 2}^{t / 2} d t_1 \int_{-t / 2}^{t / 2} d t_2 \int_{-\infty}^{\infty} d \omega e^{i\left(\omega_{\mathrm{a}}-\omega\right)\left(t_2-t_1\right)} W_{j j^{\prime}}(\omega) \end{aligned}
in terms of the two-atom self-energy
$$W_{j j^{\prime}}(\omega)=\sum_k \frac{\eta_{k j} \eta_{k j^{\prime}}^}{\omega-\omega_k+i 0^{+}}+\sum_k \frac{\eta_{k j^{\prime}} \eta_{k j}^}{\omega-2 \omega_{\mathrm{a}}-\omega_k+i 0^{+}} .$$

## 物理代写|热力学代写thermodynamics代考|Symmetry Considerations

$$\left|\phi_l^{\left|N_1\right|}\right\rangle=-\frac{1}{\sqrt{N}}\left|e_N\right\rangle+\sum_{j=1}^{N-1}\left[\frac{1+(1 / \sqrt{N})}{N-1}-\delta_{j l}\right]\left|e_j\right\rangle \quad \equiv \sum_{j=1}^N f_j^l\left|e_j\right\rangle, \quad l=1,2, \ldots, N-1 .$$

$$\left|\psi_{\text {exc }}\right\rangle=c^{[N]}\left|\phi^{[N]}\right\rangle+\sum_{l=1}^{N-1} c_l^{[N-1]}\left|\phi_l^{[N-1]}\right\rangle,$$

$$\dot{c}^{(N)}=-\gamma_{c o l} c^{[N]}-\sum_{l=1}^{N-1} s_l^* c_l^{[N-1]} \dot{c} l^{(N-1)}=-s_l c^{(N)}-\sum l^{\prime}=1^{N-1} Q_{l^{\prime}} c_{l^{\prime}}^{(N-1)}$$

$$\backslash \text { dot }{c}^{\wedge}{[N]}(0)=1 \text {, \quad c_ }\left.\right|^{\wedge}{[N-1}}(0)=0 \text {, \quad I }=1,2 \text {, } \backslash \text { Idots } N-1 \text { 。 }$$
（7.53) 中的混合，在完全对称状态和l较低对称性的第 th 状态，发生在速率
$$s_l=\frac{\gamma_1}{\sqrt{N}} \sum_{j=1}^N \sum_{j^{\prime}=1}^N f_j^l \mathcal{F}\left(\boldsymbol{r} j-\boldsymbol{r} j^{\prime}\right)$$

## 物理代写|热力学代写thermodynamics代考|Markovian Theory of Two-Atom Self-Energy

$$U_{j j^{\prime}}=\delta_{j j^{\prime}}+U_{j j^{\prime}}^{(2)} U_{j j^{\prime}}^{(2)}=\frac{1}{2 \pi i} \int_{-t / 2}^{t / 2} d t_1 \int_{-t / 2}^{t / 2} d t_2 \int_{-\infty}^{\infty} d \omega e^{i\left(\omega_{\mathrm{a}}-\omega\right)\left(t_2-t_1\right)} W_{j j^{\prime}}(\omega)$$

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