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## 物理代写|热力学代写thermodynamics代考|QZE and AZE for Photon Depolarization

We next consider the decay of photon polarization into a noise bath induced by a polarization-randomizing element (polarization rotator) that is controlled by a noisy field (see Fig. 10.11). Although the photon evolution is unitary, it emulates the effects of different types of measurements: selective or nonselective, as well as strong or weak measurements.

We consider the photon polarization evolution at the time instants that are multiples of the round-trip time $\tau_{\mathrm{r}}$ in the cavity. Let us denote by $P_{\mathrm{h}}\left(P_{\mathrm{v}}\right)$ the probability to find a photon with horizontal (vertical) polarization. We first assume that the polarization rotator rotates the polarization by angle $\Delta \varphi$ in each passage of the photon. Coherent, uninterrupted evolution without absorption (transparency $\theta=1$ ) then corresponds to Rabi oscillations of the horizontal polarization probability:
$$P_{\mathrm{h}}(n)=\cos ^2(n \Delta \varphi),$$
where $n$ is the number of photon round trips. In this case, the polarization beam splitter (PBS) in Figure $10.11$ has no effect whatsoever on the polarization evolution.

The opposite limit of a perfect absorber, $\theta=0$, corresponds to decay of the horizontal polarization probability via perfect (projective) measurements resulting in
$$P_{\mathrm{h}}(n)=\cos ^{2 n}(\Delta \varphi),$$
which for small rotation angles, $|\Delta \varphi| \ll 1$, becomes
$$P_{\mathrm{h}}(n)=e^{-n(\Delta \varphi)^2} .$$
This decay is slower than Rabi oscillations and thus signifies the QZE. This case corresponds to selective measurements, since each round trip results in a projection of the polarization state to the horizontal polarization.

The case $0<\theta<1$ corresponds to nonselective, imperfect (weak) measurements, for which the evolution of $P_{\mathrm{h}}$ can be obtained analytically. For small rotation angles and sufficiently strong absorption such that $|\Delta \varphi| \ll 1-\theta$, the evolution is an exponential decay expressed by
$$P_{\mathrm{h}}\left(t=n \tau_{\mathrm{r}}\right)=\exp \left[-\frac{(\Delta \varphi)^2}{\tau_{\mathrm{r}}^2 v} t\right] .$$

## 物理代写|热力学代写thermodynamics代考|Atom Escape from Accelerated Potential

Experimental studies by Raizen’s group considered cold atoms trapped in a “washboard”-shaped potential that is subject to alternating periodic tilting and levelling-off, that is, to periodic acceleration (Fig. 10.15). The finding was that the trapped atoms escape from this periodically accelerated potential nonexponentially, at times $t \lesssim \omega_g /\left(k_L a\right.$ ), where $\hbar \omega_g$ is the band gap, which is roughly half the potential well depth, $a$ the acceleration and $k_L$ the wave number of the laser creating this potential. The present analysis, confirmed by experiment, shows that AZE should arise for measurement (tilting) intervals $\tau \gg 1 / \omega_g$ and QZE for $\tau \ll 1 / \omega_g$. The two trends are plotted in Figures $10.16$ and 10.17, respectively. The two curves in Figure $10.17$ intersect at $\tau_{\mathrm{QZE}}=1 / v_{\mathrm{QZE}}$, and the genuine QZE occurs for $\tau<\tau_{\mathrm{QZE}}$ : In this case, $\tau_{\mathrm{QZE}}\left(\approx 10^{-6} \tau_0\right)$ is much shorter than the boundary of the QZE-scaling region $\tau \sim 1 / \omega_g$ (Fig. 10.16).

Table $10.1$ lists processes wherein the AZE may be observed at accessible measurement intervals $t \sim \tau_{\mathrm{AZE}}$. The QZE ( $\tau_{\mathrm{QZE}} \ll \tau_{\mathrm{AZE}}$ ) may be principally unobservable in open systems or, as a rule, is much less accessible than the AZE ( $\tau_{\mathrm{QZE}} \ll \tau_{\mathrm{AZE}}$ ),as argued in Section 10.4.3. The AZE and QZE are equally accessible in open systems only when the system resonance $\omega_{\mathrm{a}}$ is near the peak of the bath spectrum, and this spectrum is narrow, as in the cases of radiative or photon-polarization decay in a cavity.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|QZE and AZE for Photon Depolarization

$$P_{\mathrm{h}}(n)=\cos ^2(n \Delta \varphi),$$

$$P_{\mathrm{h}}(n)=\cos ^{2 n}(\Delta \varphi),$$

$$P_{\mathrm{h}}(n)=e^{-n(\Delta \varphi)^2}$$

$$P_{\mathrm{h}}\left(t=n \tau_{\mathrm{I}}\right)=\exp \left[-\frac{(\Delta \varphi)^2}{\tau_{\mathrm{I}}^2 v} t\right]$$

## 物理代写|热力学代写thermodynamics代考|Atom Escape from Accelerated Potential

Raizen 小组的实验研究考虑了被困在“搓板”形电位中的冷原子，该电位受到交替的周期性倾斜和平整，即周期性加速（图 10.15）。发现是，被捕 获的原子有时会以非指数的方式从这种周期性加速的电位中逃逸。 $t \lesssim \omega_g /\left(k_L a\right)$ ，在哪里 $\hbar \omega_g$ 是带隙，大约是势阱深度的一半， $a$ 加速度和 $k_L$ 产 生这种势能的激光的波数。经实验证实的当前分析表明，测量（倾斜）间隔应出现 AZE $\tau \gg 1 / \omega_g$ 和 QZE 为 $\tau \ll 1 / \omega_g$. 这两种趋势绘制在图中 10.16和 10.17，分别。图中的两条曲线 $10.17$ 相交于 $\tau_{\mathrm{QZE}}=1 / v_{\mathrm{QZE}}$, 真正的 QZE 发生在 $\tau<\tau_{\mathrm{QZE}}$ ：在这种情况下， $\tau_{\mathrm{QZE}}\left(\approx 10^{-6} \tau_0\right)$ 比 QZE 缩 放区域的边界短得多 $\tau \sim 1 / \omega_g$ (图 10.16) 。

## 有限元方法代写

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## MATLAB代写

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