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## 物理代写|热力学代写thermodynamics代考|MQS Formation Conditions

The condition (8.81) for MQS formation is satisfied if $\Delta_{\mathrm{L}}\left(\tau_{\mathrm{MQS}}\right) \gg \gamma\left(\tau_{\mathrm{MQS}}\right)$.
For the satisfaction of condition (8.81), $\tau_{\mathrm{MQS}}$ should exceed the non-Markovian timescale, as explained in this section. At sufficiently low temperatures, $\gamma(t)$ is drastically reduced in the Markovian limit $\left(t \gg t_{\mathrm{c}}\right)$ as opposed to its fast initial non-Markovian increase, where $t_{\mathrm{c}}$, the correlation (memory) time of the bath, is the inverse width of $G_{\mathrm{s}}(\omega)$. Namely,
$$\gamma\left(t \ll t_{\mathrm{c}}\right) \gg \gamma(t \rightarrow \infty)=\gamma$$
[Figs. 8.7(c, d) and 8.8]. The reason for this trend is that $\gamma(t)$ initially has contributions from all the bath modes, $\int G_{\mathrm{s}}(\omega) d \omega$, but subsequently decreases, as the bath-mode excitations at different frequencies cause the mode states to go out of phase as they approach the Markovian regime. On the other hand, $\Delta_{\mathrm{L}}(t)$ increases in the course of the transition from the non-Markovian to the Markovian regime, so that its long-time value satisfies
$$\left|\Delta_{\mathrm{L}}(t \rightarrow \infty)\right| \gg\left|\Delta_{\mathrm{L}}\left(t \ll t_{\mathrm{c}}\right)\right| .$$
Hence, it is beneficial to have $\tau_{\mathrm{MQS}}$ longer than the bath correlation time $t_{\mathrm{c}}$, so that MQS formation encounters a much lower $\gamma$, and much higher $\Delta_{\mathrm{L}}$, than their non-Markovian counterparts, consistently with (8.81).

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

We have shown that resonant dipole-dipole interactions (RDDI) in multiatom systems can be drastically enhanced or suppressed, as well as extended in range, in appropriately designed field-confining structures. Such modifications allow us to engineer hitherto uncontrollable features of energy, entanglement, or information transfer, as summarized below:
(a) In photonic crystals (PC) or periodic (Bragg-grating) waveguides, RDDI suppression results from self-energy modifications due to a sharp cutoff in the density of modes (DOM), and the analytic continuation of the DOM function beyond the cutoff into a band gap. This modification leads to the elimination of the electrostatic $R^{-3}$-divergent limit of this interaction. The RDDI is thereby strongly suppressed at interatomic separations that are much smaller than the emission wavelength (Sec. 8.1).

Such a suppression of the RDDI at interatomic separations characterizing quasimolecules (a few A) would have strong implications on their dynamics, spectroscopy and energy transfer properties. Normally, the symmetrized (ungerade) and antisymmetrized (gerade) states of a dimer,
$$\left|\Psi_{\mathrm{S}(\mathrm{A})}\right\rangle=2^{-1 / 2}\left(\left|e_{1} g_{2}\right\rangle \pm\left|e_{2} g_{1}\right\rangle\right),$$
where 1 and 2 label the atoms, respectively, are shifted by $\pm \Delta_{12}$ due to KDDI from atomic resonance. These shifts would nearly disappear in a band gap.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|MQS Formation Conditions

$$\gamma\left(t \ll t_{\mathrm{c}}\right) \gg \gamma(t \rightarrow \infty)=\gamma$$
[无花果。8.7(c, d) 和 8.8]。这种趋势的原因是 $\gamma(t)$ 最初有来自所有沐浴模式的贡献， $\int G_{\mathrm{s}}(\omega) d \omega$ ，但随后减小，因为不同频率的浴模式激发导致模式状态在接近马尔 可夫状态时异相。另一方面， $\Delta_{\mathrm{L}}(t)$ 在从非马尔可夫体制向马尔可夫体制过渡的过程中增加，因此其长期价值满足
$$\left|\Delta_{\mathrm{L}}(t \rightarrow \infty)\right| \gg\left|\Delta_{\mathrm{L}}\left(t \ll t_{\mathrm{c}}\right)\right| .$$

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

$$\left|\Psi_{\mathrm{S}(\mathrm{A})}\right\rangle=2^{-1 / 2}\left(\left|e_{1} g_{2}\right\rangle \pm\left|e_{2} g_{1}\right\rangle\right),$$

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## MATLAB代写

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