Thermal Bath Correlation Functions (RWA)

The bath correlation functions simply are

$$C_{12}(t) \equiv {\rm Tr}_B \left[\tilde{B}_1(t){B}_2R_0\right] = {\rm Tr}_B \left[\sum_{QQ'}\gamma_Q\gamma_{Q'}a_Qe^{-i\omega_Q t} a^{\dagger}_{Q'}R_0\right]$$

$$= \sum_Q \gamma_Q^2 e^{-i\omega_Q t}(1+n_B(\omega_Q)) = \int_0^{\infty} d\omega \rho(\omega) e^{-i\omega t}(1+n_B(\omega))$$

$$C_{21}(t) \equiv {\rm Tr}_B \left[\tilde{B}_2(t){B}_1R_0\right] = {\rm Tr}_B \left[\sum_{QQ'}\gamma_Q\gamma_{Q'}a_Q^{\dagger}e^{i\omega_Q t} a_{Q'}R_0\right]$$

$$= \sum_Q \gamma_Q^2 e^{i\omega_Q t}n_B(\omega_Q) = \int_0^{\infty} d\omega \rho(\omega) e^{i\omega t}n_B(\omega)$$

$$C_{11}(t) = C_{22}(t)=0,$$ (36)

where all the information on the microscopic coupling to the bath in now comprised within one single function, the bath spectral density $\rho (\omega )$

$$\fbox{$ \begin{array}{rcl} \displaystyle \rho(\omega) &\equiv& \sum_Q \gamma_Q^2\delta(\omega_Q-\omega). \end{array}$\ }$$ (37)

Using $\tilde{S}_1(t)=a^{\dagger}e^{i\Omega t}$, $\tilde{S}_2(t)=ae^{-i\Omega t}$, we have

$$D_{1} \equiv \int_0^{\infty}d \tau C_{12}(\tau)\tilde{S}_2(-\tau)= \int_0^{\infty}d \tau C_{12}(\tau) ae^{i\Omega t} = \hat{C}_{12}(-i\Omega) a$$ (38)

$$D_{2} \equiv \int_0^{\infty}d \tau C_{21}(\tau)\tilde{S}_1(-\tau)= \int_0^{\in... ...\tau C_{21}(\tau) a^{\dagger}e^{-i\Omega t} = \hat{C}_{21}(i\Omega) a^{\dagger}$$

$$E_{1} \equiv \int_0^{\infty}d \tau C_{21}^*(\tau)\tilde{S}_2(-\tau)= \int_0^{\... ...\tau C_{21}^*(\tau) ae^{i\Omega t} = [\hat{C}_{21}(i\Omega)]^* a =D_2^{\dagger}$$

$$E_{2} \equiv \int_0^{\infty}d \tau C_{12}^*(\tau)\tilde{S}_1(-\tau)= \int_0^{\... ...^{\dagger}e^{-i\Omega t} = [\hat{C}_{12}(-i\Omega)]^* a^{\dagger}=D_1^{\dagger}$$

Here, we defined the Laplace transformation of a function $f(t)$,

$$\hat{f}(z) = \int_{0}^{\infty}dt e^{-zt} f(t).$$ (39)

The Master equation therefore is

$$\frac{d}{dt}\rho(t) = -i[\Omega a^{\dagger}a,{\rho}(t)]$$

$$- \sum\Big\{ {S}_k D_k {\rho}(t) - D_k {\rho}(t) {S}_k+ {\rho}(t)E_k {S}_k - {S}_k{\rho}(t) E_k \Big\}$$ (40)

$$= -i[\Omega a^{\dagger}a,{\rho}(t)]$$

$$- \Big\{ \left[\hat{C}_{12}(-i\Omega)a^{\dagger} a + \hat{C}_{21}(i... ...{21}(i\Omega)]^*a a^{\dagger} + [\hat{C}_{12}(-i\Omega)]^* a^{\dagger} a\right]$$

$$- \hat{C}_{12}(-i\Omega) a{\rho}(t)a^{\dagger} -\hat{C}_{21}(i\Omeg... ...\dagger}{\rho}(t) a - [\hat{C}_{12}(-i\Omega)]^*a {\rho}(t) a^{\dagger} \Big\}.$$