物理代写|热力学代写thermodynamics代考|Markovian Limit of the Master Equation

物理代写|热力学代写thermodynamics代考|Markovian Limit of the Master Equation

Here we consider the case of time-independent $H_{\mathrm{S}}$ and $S$, so that (11.46) holds, and assume sufficiently long times, $t \gg t_{\mathrm{c}}$, where the bath correlation (response) time $t_{\mathrm{c}}$ is the characteristic decay or memory time of $\Phi_T(t)$ (Sec. 11.3). Then the integral in (11.45) tends to its $t \rightarrow \infty$ limit, and (11.45) becomes a Markovian ME (MME),
$$\dot{\rho}=-i\left[H_{\mathrm{S}}, \rho\right]+\int_0^{\infty} d \tau\left{\Phi_T(\tau)[\tilde{S}(-\tau) \rho, S]+\text { H.c. }\right} .$$
We next perform the secular simplification, as follows. We write $H_{\mathrm{S}}$ in the form
$$H_{\mathrm{S}}=\sum_E E \Pi(E)$$
where ${E}$ is the complete set of those Hamiltonian eigenvalues, which differ from each other (i.e., if some $E$ is degenerate, it appears in ${E}$ only once), $\Pi(E)$ is an orthogonal projector onto the eigenspace corresponding to $E$, such that $\sum_E \Pi(E)=I_{\mathrm{S}}$, and $I_{\mathrm{S}}$ is the identity operator for the system. Then we can decompose the system-interaction operator,
$$S=\sum_{E, E^{\prime}} \Pi(E) S \Pi\left(E^{\prime}\right)=\sum_\alpha S_\alpha$$
The $S_\alpha$ are dubbed the “jump” operators. They are related to $S$ by the equality
$$S_\alpha=\sum_{E^{\prime}-E=\omega_\alpha} \Pi(E) S \Pi\left(E^{\prime}\right)$$
where the sum is over all $E^{\prime}$ and $E$ with a given difference $\omega_\alpha$.
The operator (11.57) in the interaction picture results from (11.47) in the form
$$\tilde{S}(t)=\sum_\alpha e^{-i \omega_\alpha t} S_\alpha$$
Upon inserting this expression and its $\tau=0$ value (11.57) (rewritten in the form $\left.S=\sum_{\alpha^{\prime}} S_{\alpha^{\prime}}^{\dagger}\right)$ into (11.55) and using (11.41), the MME acquires the form
$$\dot{\rho}=-i\left[H_{\mathrm{S}}, \rho\right]+\sum_{\alpha, \alpha^{\prime}}\left{\left(\gamma_\alpha / 2+i \Delta_\alpha\right)\left[S_\alpha \rho, S_{\alpha^{\prime}}^{\dagger}\right]+\text { H.c. }\right}$$
where the meaning of $\gamma_\alpha$ and $\Delta_\alpha$ is explained below.

物理代写|热力学代写thermodynamics代考|Non-Markovian Master Equation for Periodically Modulated TLS

In this section, we restrict the non-Markovian ME derived above to the case of a TLS that is coupled to a thermal bath, under periodic modulations of the TLS frequency. We dwell on unconventional squeezing effects incurred by a modulation that is fast enough to violate the RWA.

In the joint system-plus-bath Hamiltonian (11.2), now the modulated TLS Hamiltonian is given by
$$H_{\mathrm{S}}(t)=\frac{1}{2} \sigma_z\left[\omega_{\mathrm{a}}+\delta_{\mathrm{a}}(t)\right]$$
where $\delta_{\mathrm{a}}(t)$ is the frequency-modulation function. We take in (11.26) the dipolar system-interaction operator that couples it to the bath to be
$$S(t)=\tilde{\epsilon}(t) \sigma_x,$$
where $\tilde{\epsilon}(t)$ is the real amplitude describing the interaction-strength modulation. The operator (11.30) is given by
$$\tilde{S}(\tau, t)=\tilde{\epsilon}(\tau)\left[\varepsilon^(t) \varepsilon(\tau) e^{-i \omega_{\mathrm{a}}(t-\tau)} \sigma_{+}+\varepsilon(t) \varepsilon^(\tau) e^{i \omega_{\mathrm{a}}(t-\tau)} \sigma_{-}\right],$$
where the time-dependent phase factor follows from (11.64) to be
$$\varepsilon(t)=e^{i \int_0^t d t^{\prime} \delta_{\mathrm{a}}\left(t^{\prime}\right)} .$$

热力学代考

物理代写|热力学代写thermodynamics代考|Markovian Limit of the Master Equation

$t \gg t_{\mathrm{c}}$ ，其中沐浴相关 (响应) 时间 $t_{\mathrm{c}}$ 是特征衰减或记忆时间 $\Phi_T(t)$ (第 11.3 节)。 那么 (11.45) 中的积分趋于它的 $t \rightarrow \infty$ 极限，并且 (11.45) 成为马尔可夫 ME (MME)，

$$H_{\mathrm{S}}=\sum_E E \Pi(E)$$

$$S=\sum_{E, E^{\prime}} \Pi(E) S \Pi\left(E^{\prime}\right)=\sum_\alpha S_\alpha$$

$$S_\alpha=\sum_{E^{\prime}-E=\omega_a} \Pi(E) S \Pi\left(E^{\prime}\right)$$

$$\tilde{S}(t)=\sum_\alpha e^{-i \omega_\omega t} S_\alpha$$

物理代写|热力学代写thermodynamics代考|Non-Markovian Master Equation for Periodically Modulated TLS

$$H_{\mathrm{S}}(t)=\frac{1}{2} \sigma_z\left[\omega_{\mathrm{a}}+\delta_{\mathrm{a}}(t)\right]$$

$$S(t)=\tilde{\epsilon}(t) \sigma_x$$

$$\left.\left.\tilde{S}(\tau, t)=\tilde{\epsilon}(\tau)\left[\varepsilon^{(} t\right) \varepsilon(\tau) e^{-i \omega_{\mathrm{a}}(t-\tau)} \sigma_{+}+\varepsilon(t) \varepsilon^{(} \tau\right) e^{i \omega_{\mathrm{a}}(t-\tau)} \sigma_{-}\right],$$

$$\varepsilon(t)=e^{i \int_0^t d t^{\prime} \delta_a\left(t^{\prime}\right)}$$

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