### 物理代写|热力学代写thermodynamics代考|CL3010

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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|热力学代写thermodynamics代考|QZE Scaling

The QZE scaling is marked by a decrease of the decay rate $\gamma$ with an increase of $v$. It is obtained when the measurement (or dephasing) rate $v$ is much larger than the spectral width and the detuning of the bath (Fig. 10.7):
$$v \gg \Gamma_{\mathrm{B}}, \quad v \gg\left|\omega_{\mathrm{a}}-\omega_{\mathrm{M}}\right| .$$

Here we have assumed that the main part of the integral $\int_0^{\infty} d \omega G(\omega)$ is concentrated in an interval of the width of order of $\Gamma_{\mathrm{B}}$ and $\omega_{\mathrm{M}}$ is a frequency within this interval. In the special case of a peak-shaped $G(\omega), \omega_{\mathrm{M}}$ can be replaced by the position of the maximum. In the limit (10.28), one can approximate the spectrum $G(\omega)$ by a $\delta$-function $\left(\Gamma_{\mathrm{B}}=0\right)$.

There is, however, a caveat associated with the limit (10.28). When $G(\omega)$ is too narrow, the evolution (in the absence of measurements or dephasing) is generally non-monotonic and hence cannot be described by means of a positive decay rate. For example, for $G(\omega) \sim \delta\left(\omega-\omega_{\mathrm{a}}\right)$, the decay rate is undefined, because the population of state $|e\rangle$ demonstrates resonant Rabi oscillations without decay, periodically exchanging energy with an infinitely narrow band of modes, in accordance with the Jaynes-Cummings model. Namely, condition (10.28) for the QZE presumes the weak-coupling regime of the system and the bath. This regime always holds for a sufficiently broad and smooth $G(\omega)$. Even when $G(\omega)$ is very narrow or has sharp features, so that the weak-coupling regime does not hold in the absence of measurements, this regime is still valid for a sufficiently high rate $v$ of repeated measurements (or dephasing). This comes about since, in the latter case, the energy level $|e\rangle$ is broadened, acquiring spectral density $F\left(\omega-\omega_{\mathrm{a}}\right)$, so that its coupling with the bath is described by the spectrum $G(\omega)$, which is smoothed out by its convolution with $F\left(\omega-\omega_{\mathrm{a}}\right)$ [cf. (10.23)].

Assuming that (10.28) holds, the universal formula (10.23) yields the characteristic form of QZE,
$$\gamma \approx C / \nu \text {. }$$
Here $C$ is the integrated bath-coupling spectrum or, equivalently, the variance of the coupling Hamiltonian in the state $|e\rangle$,
$$C=\int G(\omega) d \omega=\left\langle e\left|V^2\right| e\right\rangle,$$
and we have introduced the general definition,
$$v=[2 \pi F(0)]^{-1}$$

## 物理代写|热力学代写thermodynamics代考|Intermediate Scaling

More subtle behavior occurs in the intermediate range between the QZE and AZE regimes. Let us assume, for simplicity, that $G(\omega)$ is single-peaked and satisfies condition (10.32). When $v$ increases up to the range $v \gg\left|\omega_{\mathrm{m}}-\omega_{\mathrm{a}}\right|$, then condition (10.28) implies the QZE scaling of (10.29) or (10.34). Yet $\gamma$ remains larger than the Golden Rule rate $\gamma_{\mathrm{GR}}(10.37)$, up to much higher $v$, according to the following condition for “genuine QZE,”
$$\gamma<\gamma_{\mathrm{GR}} \text { for } v>v_{\mathrm{QZE}} \text {, }$$
where in the case of a finite $C$
$$v_{\mathrm{QZE}}=\frac{C}{\gamma_{\mathrm{GR}}}=\frac{C}{2 \pi G\left(\omega_{\mathrm{a}}\right)},$$
or in the case of (10.33)
$$v_{\mathrm{QZE}}=\left(\frac{B}{\gamma_{\mathrm{GR}}}\right)^{1 / \beta}=\left[\frac{B}{2 \pi G\left(\omega_{\mathrm{a}}\right)}\right]^{1 / \beta} .$$
The rate $v_{\text {QZE }}$ may be much greater than the minimal $v$ value of the QZE-scaling regime, as shown in Section 10.4.3.

The value of $v_{\text {QzE }}$ given by (10.39a) was termed the reciprocal “jump time,” (i.e., the longest time interval between measurements for which the decay rate is appreciably changed). However, the reciprocal jump time is in fact $\delta_{\mathrm{a}}$ in (10.36a), which may be smaller by many orders of magnitude than $v_{\mathrm{QZE}}$.

## 物理代写|热力学代写thermodynamics代考|QZE Scaling

QZE 缩放以衰减率的降低为标志 $\gamma$ 随着增加 $v$. 它是在测量 (或移相) 速率时获得的 $v$ 远大于光谱宽度和槽的失谐 (图 10.7) :
$$v \gg \Gamma_{\mathrm{B}}, \quad v \gg\left|\omega_{\mathrm{a}}-\omega_{\mathrm{M}}\right| .$$

$$\gamma \approx C / \nu .$$

$$C=\int G(\omega) d \omega=\left\langle e\left|V^2\right| e\right\rangle,$$

$$v=[2 \pi F(0)]^{-1}$$

## 物理代写|热力学代写thermodynamics代考|Intermediate Scaling

$\gamma<\gamma_{\mathrm{GR}}$ for $v>v_{\mathrm{QZE}}$,

$$v_{\mathrm{QZE}}=\frac{C}{\gamma_{\mathrm{GR}}}=\frac{C}{2 \pi G\left(\omega_{\mathrm{a}}\right)},$$

$$v_{\mathrm{QZE}}=\left(\frac{B}{\gamma_{\mathrm{GR}}}\right)^{1 / \beta}=\left[\frac{B}{2 \pi G\left(\omega_{\mathrm{a}}\right)}\right]^{1 / \beta} .$$

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