# 物理代写|热力学代写thermodynamics代考|CL3010

## 物理代写|热力学代写thermodynamics代考|Dynamical “Filter Function” Control

In what follows, we consider several generic phase-modulation spectra that may act as spectral “filter functions,” which modify decay or relaxation rates. All modulations are taken to be quasiperiodic, namely, the quantity $\epsilon(t)=\varepsilon(t)$ in (11.141) [with $\tilde{\epsilon}(t)=1$ ] has the form,
$$\epsilon(t)=e^{i \phi(t)}=\sum_q \epsilon_q e^{i v_q t},$$
$v_q(q=0, \pm 1, \ldots)$ being arbitrary discrete frequencies such that
$$\left|v_q-v_{q^{\prime}}\right| \geq \Omega_0 \quad \forall q \neq q^{\prime},$$
with the minimal spectral interval $\Omega_0$. The complex coefficients $\epsilon_q$ satisfy $\sum_q\left|\epsilon_q\right|^2=1$.

Since it is not necessarily periodic, (11.204) is more general than (11.76). Nevertheless, the derivation leading to (11.104) still holds. Thus, the relaxation rates are dynamically modified at long times as
$$\gamma_{e(g)}=2 \pi \sum_q\left|\epsilon_q\right|^2 G_T\left(\pm \omega_q\right) .$$
These long-time limits are approached when
$$\Omega_0 t \gg 1, \quad t \gg t_{\mathrm{c}} \equiv \max q\left{1 / \xi_q\right} .$$ Here, the bath-memory (correlation) time $t{\mathrm{c}}$ is defined as the inverse of the narrowest spectral interval $\xi_q$ over which $G_T(\omega)$ changes appreciably near the relevant frequencies $\omega_q$ [Fig. 11.5(a)]. Thus, the correlation time $t_{\mathrm{c}}$ depends on the modulation frequencies $\omega_q$. The quasiperiodic modulation (11.204) is thus a filter function that shifts or splits the TLS resonant response at frequency $\omega_{\mathrm{a}}$ or 0 for $\mathrm{AN}$ or PN, respectively, converting it into peaks at the frequencies $\omega_q$ : The relaxation rate (11.206) is “filtered” by the modulation spectrum that assigns the desired weight $\left|\epsilon_q\right|^2$ to the coupling spectrum value at the frequency $\omega_q$ (Fig. 11.5).

## 物理代写|热力学代写thermodynamics代考|Phase Modulation (PM) of the Coupling for AN Control

A monochromatic (CW) modulation control of AN yields a constant frequency shift $\delta_{\mathrm{a}}$, so that
$$\epsilon(t)=e^{-i\left(\omega_\alpha+\delta_\alpha\right) t},$$
resulting in the upper state decay rate
$$\gamma_e=2 \pi G_T\left(\omega_{\mathrm{a}}+\delta_{\mathrm{a}}\right)$$
Such a shift is induced by the AC Stark effect of the control field (for atoms) or by the Zeeman effect (for spins). This shift may either enhance or suppress the Golden-Rule decay rate,
$$\gamma_{\mathrm{GR}}=2 \pi G_T\left(\omega_{\mathrm{a}}\right)$$
Equation (11.212) yields the maximal change of the decay rate achievable by external control, since it does not involve any smoothing of the bath response (coupling spectrum) $G_T(\omega)$ incurred by the width of the filter function $F_I(\omega)$. The modified $\gamma$ can vanish, if the shifted frequency $\omega_{\mathrm{a}}+\delta_{\mathrm{a}}$ is beyond the cutoff frequency of the bath coupling spectrum, where $G_T(\omega)=0$ [Fig. 11.5(d)]. Conversely, the increase of $\gamma$ due to a shift $\delta_{\mathrm{a}}$ can exceed that achievable by repeated measurements, that is, the anti-Zeno effect (AZE) (Ch. 10). However, AC Stark shifts are usually small for CW perturbations. Typically, only pulsed perturbations may result in multiple shifted frequencies $\omega_q$ of the coupling spectrum as per (11.206).

## 物理代写|热力学代写thermodynamics代考|Dynamical “Filter Function” Control

$$\epsilon(t)=e^{i \phi(t)}=\sum_q \epsilon_q e^{i v_q t}$$
$v_q(q=0, \pm 1, \ldots)$ 是任意离散频率，使得
$$\left|v_q-v_{q^{\prime}}\right| \geq \Omega_0 \quad \forall q \neq q^{\prime},$$

$$\gamma_{e(g)}=2 \pi \sum_q\left|\epsilon_q\right|^2 G_T\left(\pm \omega_q\right) .$$

## 物理代写|热力学代写thermodynamics代考|Phase Modulation (PM) of the Coupling for AN Control

$\mathrm{AN}$ 的单色 $(\mathrm{CW})$ 调制控制产生恒定的频移 $\delta_{\mathrm{a}}$ ， 以便
$$\epsilon(t)=e^{-i\left(\omega_\alpha+\delta_\alpha\right) t},$$

$$\gamma_e=2 \pi G_T\left(\omega_{\mathrm{a}}+\delta_{\mathrm{a}}\right)$$

$$\gamma_{\mathrm{GR}}=2 \pi G_T\left(\omega_{\mathrm{a}}\right)$$

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