# 物理代写|热力学代写thermodynamics代考|Open-System Decay Modified by Measurements

## 物理代写|热力学代写thermodynamics代考|Open-System Decay Modified by Measurements

The probability amplitude $\alpha(t)$ of survival in $|e\rangle$, at the energy $\hbar \omega_{\mathrm{a}}$, satisfies the following exact integrodifferential equation
$$\dot{\alpha}=-\int_0^t d t^{\prime} e^{i \omega_2\left(t-t^{\prime}\right)} \Phi\left(t-t^{\prime}\right) \alpha\left(t^{\prime}\right),$$
where $\alpha(t)=\langle e \mid \Psi(t)\rangle e^{i \omega_{\mathrm{a}} t}$ and
$$\Phi(t)=\hbar^{-2}\left\langle e\left|V^{-i H_0 t / \hbar} V\right| e\right\rangle=\hbar^{-2} \sum_n\left|V_{e n}\right|^2 e^{-i \omega_n t}$$
is the bath autocorrelation function (Ch. 2), expressed by $V_{e n}=\langle e|V| n\rangle$, where $|n\rangle(\neq|e\rangle)$ are $H_0$ eigenstates with energies $\hbar \omega_n$, which we treat as the bath.

Equation (10.13) is exactly soluble, but here we investigate its short-time behavior, which is obtainable by setting $\alpha\left(t^{\prime}\right) \approx \alpha(0)=1$ in (10.13). This results in the expression
$$\alpha(t)=1-\int_0^t d t^{\prime} e^{i \omega_{\mathrm{a}} t^{\prime}}\left(t-t^{\prime}\right) \Phi\left(t^{\prime}\right),$$
which encompasses all powers of $t$ (in the phase factors!) and allows for interference between various decay channels. By contrast, the standard quadratic expansion in $t$ yields the population time-dependence [cf. (10.3)],
$$\rho_{e e}(t)=|\alpha(t)|^2 \approx 1-t^2 / \tau_Z^2$$
where the Zeno time
$$\tau_Z=\hbar / \Delta E_e$$
is the inverse variance of the energy in $|e\rangle$ [cf. Eq. (10.4)]. Yet the estimate (10.16) may fail, as discussed below, since (10.15) may not only yield the QZE but also its inverse, the anti-Zeno effect (AZE), depending on the short-time behavior of $|\alpha(t)|^2$.

## 物理代写|热力学代写thermodynamics代考|Decay Modification by Realistic Measurements

Thus far, we have assumed in this section instantaneous measurements of $|e\rangle$. Does a more realistic description of measurements support these foregoing results? The answer is affirmative for the two possible types of measurements of $|e\rangle$ :
(a) Impulsive measurements are realizable, as in Cook’s scheme (Fig. 10.4): The decay process is then repeatedly interrupted by short pulses, each pulse transferring the population of $|e\rangle$ to a higher state $|u\rangle$, which then decays back to $|e\rangle$ through photon emission. The basic requirement for impulsive measurements is that the duration of the $|e\rangle \rightarrow|u\rangle$ transfer followed by $|u\rangle \rightarrow|e\rangle$ decay is much shorter than all other timescales in the process, notably the memory time of the bath as discussed below. The same procedure may be carried out by similarly acting on the initially unoccupied state $|g\rangle$.
(b) Continuous measurements that act weakly on the system, causing only partial state reduction at any given time. Such measurements should not be misconstrued as the limit of successive projections separated by vanishingly short intervals, $\tau \rightarrow 0$ : Such infinitely frequent projections are unphysical, corresponding to an infinite energy spread $\hbar / \tau$. Continuous measurements are feasible when the state is monitored incessantly, but they still require a finite time $\tau$ for completing an observation (i.e., they have a finite effective rate $1 / \tau_{\mathrm{d}}$ ).

Measurements can be either selective or nonselective, regardless of whether they are impulsive or continuous. Thus, measurements in Cook’s scheme are selective if the photons emitted at the $|u\rangle \rightarrow|e\rangle$ transition are detected and the detection results are read out. The measurements are nonselective if the results are unread. The same is true for continuous measurements. Finally, if the photons are not detected, then the process is unitary but has the same effect on the TLS evolution as nonselective measurements. Such a unitary process that emulates nonselective measurements was implemented in the pioneering experiment by Itano et al. (1990).

# 热力学代考

## 物理代写|热力学代写thermodynamics代考|Open-System Decay Modified by Measurements

$$\dot{\alpha}=-\int_0^t d t^{\prime} e^{i \omega_2\left(t-t^{\prime}\right)} \Phi\left(t-t^{\prime}\right) \alpha\left(t^{\prime}\right)$$

$$\Phi(t)=\hbar^{-2}\left\langle e\left|V^{-i H_0 t / \hbar} V\right| e\right\rangle=\hbar^{-2} \sum_n\left|V_{e n}\right|^2 e^{-i \omega_n t}$$

$$\alpha(t)=1-\int_0^t d t^{\prime} e^{i \omega_{\mathrm{a}} t^{\prime}}\left(t-t^{\prime}\right) \Phi\left(t^{\prime}\right)$$

$$\rho_{e e}(t)=|\alpha(t)|^2 \approx 1-t^2 / \tau_Z^2$$

$$\tau_Z=\hbar / \Delta E_e$$

## 物理代写|热力学代写thermodynamics代考|Decay Modification by Realistic Measurements

（a）脉冲测量是可以实现的，如 Cook 的方案 (图 10.4)：然后衰变过程被短脉冲 复打断，每个脉冲转移 $|e\rangle$ 到更高的境界 $|u\rangle$ ，然后衰减回 $|e\rangle$ 通过光子发射。脉冲测量的 基本要求是持续时间 $|e\rangle \rightarrow|u\rangle\rangle$ 转移其次 $|u\rangle \rightarrow|e\rangle$ 衰变比过程中的所有其他时间尺度短 得多，特别是下面讨论的浴的记忆时间。可以通过对最初末占用的状态进行类似操作来 执行相同的过程 $|g\rangle$.
(b) 对系统作用微弱的连续侧量，在任何给定时间仅导致部分状态减少。这种测量不应 被误解为由极短的间隔分隔的连续投影的极限， $\tau \rightarrow 0 ：$ 这种无限频繁的投射是非物理 的，对应于无限的能量传播 $\hbar / \tau$. 持续监测状态时，连续测量是可行的，但它们仍然需 要有限的时间 $\tau$ 完成观察 (即，他们有一个有限的有效率 $1 / \tau_d$ ).

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