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

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## 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. At $R=R_{\mathrm{c}}$ and for sufficiently large and positive detuning, such that $\omega_0-\omega_{\mathrm{S}} \approx \Omega$, we obtain the limit
$$\left|\Psi_2\right\rangle \rightarrow\left|e_1 g_2,{0}\right\rangle, \quad\left|\Psi_3\right\rangle \rightarrow\left|e_2 g_1,{0}\right\rangle$$
in which the excited eigenstates 2 and 3 become uncoupled, due to the interference of $\left|\Psi_{\mathrm{S}}\right\rangle$ and $\left|\Psi_{\mathrm{A}}\right\rangle$. The corresponding excitation-transfer probability undergoes strong suppression in the pseudocrossing interval, as shown by the time-averaged values (solid curves in Fig. 8.6): $\bar{P}2\left(R=R{\mathrm{c}}\right)=3 c_{+}^2 / 8 \ll 1$ and $\bar{P}1(R=$ $\left.R{\mathrm{c}}\right)=1-c_{+}+(3 / 8) c_{+}^2$, tending to 1 with the increase of $\omega_0-\omega_{\mathrm{S}}$, where $c_{+}=$ $\left[1+\left(\omega_{\mathrm{S}}-\omega_0\right) / \Omega\right] / 2$. The excitation is then strongly trapped at the initial atom, owing to the decoupling of the excited eigenstates.

## 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 $H_1$ 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). The initial state of the system can then be written as
$$|\Psi(0)\rangle \equiv|\theta, \phi\rangle=\left|\psi_1\right\rangle \otimes\left|\psi_2\right\rangle \cdots\left|\psi_N\right\rangle$$
with
$$\left|\psi_j\right\rangle=\frac{1}{\sqrt{2}}\left(\cos \frac{\theta}{2}|\uparrow\rangle+\sin \frac{\theta}{2} e^{i \phi}|\downarrow\rangle\right)$$
where $\theta$ and $\phi$ are the spherical coordinates of the average particle spin. This state is an eigenstate of the collective spin operator $\hat{\boldsymbol{J}} \cdot \hat{\boldsymbol{n}}, \hat{\boldsymbol{n}}$ being the unit vector corresponding to the angles $\theta$ and $\phi$.

# 热力学代写

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

$\eta_1-\eta_2$ 与分离成线性比例 $R$. 这种效应在 (8.49) 中两个 特征值的伪交叉 (接近相等) 处最为显着 (实线，图 8.5），即 $R$ 接近价值 $R_{\mathrm{c}}$ 这样 $\omega_{-}\left(R_{\mathrm{c}}\right)=\omega_{\mathrm{A}}\left(R_{\mathrm{c}}\right)$. 鉴 于 (8.50)，该等式意味着 $\Delta_{12}\left(R_{\mathrm{c}}\right) \sim\left|\eta_1\right|$ 为了 $\left|\omega_0-\omega_{\mathrm{a}}\right| \lesssim 2\left|\eta_1\right|$ ，或者 $\Delta_{12}\left(R_{\mathrm{c}}\right) \approx 2 \eta_1^2 /\left|\omega_0-\omega_{\mathrm{a}}\right|$ 为了
$\left|\omega_0-\omega_{\mathrm{a}}\right| \gg 2\left|\eta_1\right|$. 在这两种情况下，RDDI 引起的和 腔 QED 的能级偏移（或分裂）变得相当。

R_{\mathrm{c}}咐近 RDDI 和 Rabi 分裂的激烈竞争 $R_{\mathrm{C}}$ 修改 (8.49) 中的特征值，将它们替换为 (8.48) 中更准确 的解，
$$\omega_1 \approx \omega_{+}, \quad \omega_{2,3} \approx \frac{1}{2}\left(\omega_{-}+\omega_A \pm \Omega^{\prime}\right)$$

$$\Omega^{\prime}=\sqrt{V_0^2+\left(\omega_{-}-\omega_{\mathrm{A}}\right)^2}, \quad V_0=\frac{\eta_1^2-\eta_2^2}{\sqrt{\Omega\left(\Omega+\omega_0-\omega_{\mathrm{S}}\right.}}$$

$$\left|\Psi_2\right\rangle \rightarrow\left|e_1 g_2, 0\right\rangle, \quad\left|\Psi_3\right\rangle \rightarrow\left|e_2 g_1, 0\right\rangle$$

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

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

$$H_{\mathrm{S}}=\hbar \omega_x \hat{J} x, \quad H \mathrm{~B}=\hbar \sum_k \omega_k a_k^{\dagger} a_k, \quad H_{\mathrm{I}}=\hbar \hat{J} z$$

$$\hat{J}_i=(1 / 2) \sum_j \sigma_j^i(i=x, y, z) \text {. }$$

$$|\Psi(0)\rangle \equiv|\theta, \phi\rangle=\left|\psi_1\right\rangle \otimes\left|\psi_2\right\rangle \cdots\left|\psi_N\right\rangle$$和
$$\left|\psi_j\right\rangle=\frac{1}{\sqrt{2}}\left(\cos \frac{\theta}{2}|\uparrow\rangle+\sin \frac{\theta}{2} e^{i \phi}|\downarrow\rangle\right)$$

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