Quantum indirect damping classical limits

Recall that in the presence of quantum direct damping, this ACF must be multiplied by an exponential decay e 7° as it appears by comparison of Eqs. (124) and (125). Also recall that the ACF (123) that is at the origin of Eq. (135), is an explicit expression of the formal one given by Eq. (114). We need to know what is the classical limit of this ACF (135) involving quantum indirect damping. 

The different situations illustrated in Fig. 6 correspond more and more to approximate models when passing up to down. The top of the figure given the case described by Eq. (126) and corresponds to the reference quantum indirect damping. Below, is depicted the situation described by Eq. (146), where the Dyson time ordering operator is ignored in the quantum model. Further below is given the behavior that corresponds to both Eqs. (152) and (174), that is, to weak approximations on the classical limit of the quantum model [Eq. (152)] and to the semiclassical model [Eq. (174)]. At last, at the bottom we visualize the semiclassical model of Robertson and Yarwood [described by Eq. (185)]. 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

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