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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.

## 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

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

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

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