物理代写|热力学代写thermodynamics代考|MECH3024

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Open-TLS Continuous Dephasing

As mentioned above, the QZE is obtained via both selective and nonselective measurements, since all measurements yield state reduction, which corresponds to the destruction of coherence between the initial state and all other states. However, the coherent TLS evolution may be disrupted not only by measurements. In particular, effects of nonselective measurements can be emulated by means of coherence disruption due to random fluctuations of the TLS frequency via, for example, random ac-Stark shifts of the level $|e\rangle$ or $|g\rangle$, caused by an off-resonant intensity-fluctuating field. When the population of the level $|e\rangle$ is averaged over the noise realizations, it satisfies Eq. (10.20), where $\gamma$ in (10.19) is now replaced by
$$\gamma=2 \pi \int G(\omega) L\left(\omega-\omega_{\mathrm{a}}\right) d \omega .$$
In this formula, $L\left(\omega-\omega_{\mathrm{a}}\right)$ is the Lorentzian-shaped (normalized to 1) relaxation spectrum of the coherence element $\rho_{e g}(t)$, which is the Fourier transform of the exponentially decaying $\rho_{e g}(t)$. This behavior represents the common dephasing model. The width of this Lorentzian relaxation spectrum is $\tau_{\mathrm{d}}^{-1}=\left\langle\Delta \omega^2\right\rangle \tau_c$, which is the product of the mean-square Stark shift and the noisy-field correlation time. For (10.21) to be valid, $\gamma$ should be much less than this spectral width, $\gamma \tau_{\mathrm{d}} \ll 1$. A necessary condition for the QZE is that the noise-induced width $\tau_d^{-1}$ be larger than the width of the spectral response $G(\omega)$, as detailed below.

The random ac-Stark shifts both shift and broaden the spectral transition. In order to avoid the shifting, we may employ a continuous driving field that is resonant (or nearly resonant) with the $|e\rangle \leftrightarrow|u\rangle$ transition. This process is described by the same scheme as in Figure 10.4, the only difference being that the impulsive field $\Omega_{\mathrm{d}}(t)$ is replaced by a continuous field. Provided the decay rate of this transition, $\gamma_{\mathrm{u}}$, is larger than the Rabi frequency $\Omega_{\mathrm{d}}$ of the driving field, $\gamma$ can be shown to be given by (10.21), with a Lorentzian (dephasing) width
$$\frac{1}{\tau_{\mathrm{d}}}=\frac{\Omega_{\mathrm{d}}^2}{2 \gamma_{\mathrm{u}}}$$

物理代写|热力学代写thermodynamics代考|Universal Formula

The decay rate $\gamma$ [cf. (10.19), (10.21)] in both of the above schemes is seen to conform to the same universal formula (Fig. 10.5),
$$\gamma=2 \pi \int G(\omega) F\left(\omega-\omega_{\mathrm{a}}\right) d \omega,$$
where $G(\omega)$ is the spectral bath response, whereas $F(\omega)$ (normalized to 1) is the spectrum of the coherence fluctuations due to the measurement- or noise-induced dephasing: $F(\omega)$ may be, for example, sinc-shaped,
$$F(\omega)=\frac{\tau}{2 \pi} \operatorname{sinc}^2 \frac{\omega \tau}{2},$$
or Lorentzian-shaped,
$$F(\omega)=L(\omega) \equiv \frac{1}{\pi} \frac{\tau_{\mathrm{d}}}{\omega^2 \tau_{\mathrm{d}}^2+1} .$$
The (universal) result (10.23) can be rewritten in the form
$$\gamma=\int \gamma_{\mathrm{GR}}(\omega) F\left(\omega-\omega_{\mathrm{a}}\right) d \omega,$$
where $\gamma_{\mathrm{GR}}(\omega)=2 \pi G(\omega)$ is the unperturbed (“Golden Rule”) decay rate of $|e\rangle$ whose energy is shifted to $\hbar \omega$. Equation (10.26) allows us to interpret the modification of the decay rate as resulting from the energy broadening (uncertainty) $\Delta E$ of the level $|e\rangle$, the shape of the level broadening being described by $F(\omega)$.

物理代写|热力学代写thermodynamics代考|Open-TLS Continuous Dephasing

$$\gamma=2 \pi \int G(\omega) L\left(\omega-\omega_{\mathrm{a}}\right) d \omega .$$

$$\frac{1}{\tau_{\mathrm{d}}}=\frac{\Omega_{\mathrm{d}}^2}{2 \gamma_{\mathrm{u}}}$$

物理代写|热力学代写thermodynamics代考|Universal Formula

$$\gamma=2 \pi \int G(\omega) F\left(\omega-\omega_{\mathrm{a}}\right) d \omega,$$

$$F(\omega)=\frac{\tau}{2 \pi} \operatorname{sinc}^2 \frac{\omega \tau}{2}$$

$$F(\omega)=L(\omega) \equiv \frac{1}{\pi} \frac{\tau_{\mathrm{d}}}{\omega^2 \tau_{\mathrm{d}}^2+1}$$
(通用) 结果 (10.23) 可以改写为
$$\gamma=\int \gamma_{\mathrm{GR}}(\omega) F\left(\omega-\omega_{\mathrm{a}}\right) d \omega$$

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MATLAB代写

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