Definition

We first re-call the definition of the bath correlation function,

$$C(t) \equiv {\rm Tr}_B \left[\tilde{B}(t){B}R_0\right] = {\rm Tr}_B \left[\su... ...-i\omega_Q t}+a^{\dagger}_Qe^{i\omega_Q t}) (a_{Q'}+a^{\dagger}_{Q'})R_0\right]$$

$$= \sum_Q \gamma_Q^2 \left[e^{-i\omega_Q t}(1+n_B(\omega_Q)) + e^{i\omega_Q t} n_B(\omega_Q)\right]$$

$$= \int_0^{\infty} d\omega \rho(\omega) \left[e^{-i\omega t}(1+n_B(\omega)) + e^{i\omega t} n_B(\omega)\right] =C^*(-t).$$ (53)

Furthermore, $n_B(\omega) \equiv {1}/[{e^{\beta\omega}-1}]$ is the Bose function.