### 物理代写|热力学代写thermodynamics代考|PHYS2712

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## 物理代写|热力学代写thermodynamics代考|Decay in Lorentzian Bath

The above analysis pertains to the case of a TLS coupled to a near-resonant Lorentzian bath centered at $\omega_{\mathrm{s}}$, as, for example, in a high- $Q$ cavity mode. In this case (Sec. 5.1),
$$G_{\mathrm{s}}(\omega)=L\left(\omega-\omega_{\mathrm{s}}\right)=\frac{g_{\mathrm{s}}^2 \Gamma_{\mathrm{s}}}{\pi\left[\Gamma_{\mathrm{s}}^2+\left(\omega-\omega_{\mathrm{s}}\right)^2\right]},$$

where $g_{\mathrm{s}}$ is the resonant coupling strength and $\Gamma_{\mathrm{s}}$ is the line width. Here $G_{\mathrm{s}}(\omega)$ represents the sharply varying part of the DOM distribution, associated with the narrow cavity mode. The broad portion of the DOM distribution $G_{\mathrm{b}}(\omega)$ represents the TLS coupling to the unconfined (background) free-space modes. The exponential decay factor in the excited state probability is then additive,
$$\gamma=\gamma_{\mathrm{s}}+\gamma_{\mathrm{b}},$$
where $\gamma_{\mathrm{s}}$ is the contribution to $\gamma$ from the sharply varying modes and $\gamma_{\mathrm{b}}=$ $2 \pi G_{\mathrm{b}}\left(\omega_{\mathrm{a}}\right)$ is the rate of spontaneous emission into the background modes.

Since the Fourier transform of the Lorentzian $G_{\mathrm{s}}(\omega)$ is $\Phi_{\mathrm{s}}(t)=g_{\mathrm{s}}^2 e^{-\Gamma_{\mathrm{s}} t},(10.15)$ becomes at short times (without the background-modes contribution)
$$\alpha(\tau) \approx 1-\frac{g_{\mathrm{s}}^2}{\Gamma_{\mathrm{s}}-i \delta}\left[\tau+\frac{e^{\left(i \delta-\Gamma_{\mathrm{s}}\right) \tau}-1}{\Gamma_{\mathrm{s}}-i \delta}\right],$$
where $\delta=\omega_{\mathrm{a}}-\omega_{\mathrm{s}}$. The QZE condition is then expressed by
$$\tau \ll\left(\Gamma_{\mathrm{s}}+|\delta|\right)^{-1}, g_{\mathrm{s}}^{-1} .$$
On resonance $(\delta=0),(10.18)$ and (10.42) yield
$$\gamma_{\mathrm{s}}=g_{\mathrm{s}}^2 \tau .$$
Whereas the background-DOM contribution cannot be changed by the QZE, the sharply varying DOM contribution $\gamma_{\mathrm{s}}$ may allow for the QZE. Only the $\gamma_{\mathrm{s}}$ term decreases with $\tau$ due to the QZE. However, since $\Gamma_{\mathrm{s}}$ has dropped out of (10.44), the decay rate $\gamma$ is the same for both strong-coupling $\left(g_{\mathrm{s}}>\Gamma_{\mathrm{s}}\right)$ and weak-coupling $\left(g_{\mathrm{s}} \ll \Gamma_{\mathrm{s}}\right)$ regimes. Physically, this comes about since the energy uncertainty of the emitted quantum for $\tau \ll g_{\mathrm{s}}^{-1}$ is too large to allow for distinction between reversible and irreversible evolutions.

The evolution inhibition has a different meaning for the two regimes. In the weak-coupling regime, the excited-state population decays nearly exponentially at the rate $g_{\mathrm{s}}^2 / \Gamma_{\mathrm{s}}+\gamma_{\mathrm{b}}$ (at $\delta=0$ ), so that irreversible decay is inhibited, in the spirit of the original QZE prediction. By contrast, in the strong-coupling regime, the damped Rabi oscillations at the frequency $2 g_{\mathrm{s}}$ of the excited-state population are destroyed by repeated measurements. Yet, in both cases the QZE slows down the evolution, resulting in the same exponential law, with the rate (10.44).

## 物理代写|热力学代写thermodynamics代考|QZE and AZE for Intracavity Radiative Decay

Consider atoms within an open cavity that repeatedly interact with a pump laser, which is resonant with the $|e\rangle \rightarrow|u\rangle$ transition frequency. The resulting $|e\rangle \rightarrow|g\rangle$ spontaneous-emission rate is monitored as a function of the laser-pulse repetition rate $1 / \tau$. Each short pump pulse of duration $t_p$ and Rabi frequency $\Omega_p$ is followed by spontaneous decay from $|u\rangle$ back to $|e\rangle$, at a rate $\gamma_{\mathrm{u}}$. This destroys the coherence of the atomic system, as well as reshuffles the population from $|e\rangle$ to $|u\rangle$ and back (Fig. 10.4). Since the interval between measurements must significantly exceed the measurement time, we impose the inequality $\tau \gg t_{\mathrm{p}}$. This inequality can be reduced to the requirement $\tau \gg \gamma_u^{-1}$ if the “measurements” are performed by $\pi$ pulses, such that $\Omega_{\mathrm{p}} t_{\mathrm{p}}=\pi, t_{\mathrm{p}} \ll \gamma_{\mathrm{u}}^{-1}$. This implies choosing a $|u\rangle \rightarrow|e\rangle$ transition with a much shorter radiative lifetime than that of $|e\rangle \rightarrow|g\rangle$.

Figure 10.9, describing the QZE for a Lorentzian line on resonance ( $\delta=0$ ), assuming feasible cavity parameters, shows that the population of $|e\rangle$ decays nearly exponentially at times well within the interruption intervals $\tau$, but when those intervals become too short, the decay is strongly inhibited.

Figure $10.10$ shows that the detuning $\delta=\omega_{\mathrm{a}}-\omega_{\mathrm{s}}$ renders the decay oscillatory. The interruptions by measurements now enhance the decay, in accordance with the AZE.

## 物理代写|热力学代写thermodynamics代考|Decay in Lorentzian Bath

Gs(哦)=大号(哦−哦s)=Gs2Cs圆周率[Cs2+(哦−哦s)2],

C=Cs+Cb,

Cs=Gs2吨.

## 物理代写|热力学代写thermodynamics代考|QZE and AZE for Intracavity Radiative Decay

$|e\rangle \rightarrow|g\rangle$ 作为激光脉冲重复率的函数监测自发发射率 $1 / \tau$. 每个持续时间的短泵浦脉冲 $t_p$ 和拉比频率 $\Omega_p$ 其次是自 发衰变 $|u\rangle$ 回到 $|e\rangle$ ，在一个速率 $\gamma_{\mathrm{u}}$. 这破坏了原子系统的连贯性，并使人口从 $|e\rangle$ 至 $|u\rangle$ 并返回（图 10.4) 。由于 测量之间的间隔必须大大超过测量时间，我们施加不等式 $\tau \gg t_{\mathrm{p}}$. 这个不等式可以简化为要求 $\tau \gg \gamma_u^{-1}$ 如果“测 量”是由 $\pi$ 脉冲，这样 $\Omega_{\mathrm{p}} t_{\mathrm{p}}=\pi, t_{\mathrm{p}} \ll \gamma_{\mathrm{u}}^{-1}$. 这意味着选择一个 $|u\rangle \rightarrow|e\rangle$ 过渡的辐射寿命比 $|e\rangle \rightarrow|g\rangle$.

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