Implications of the Redfield equation

Next we consider the physical implications of Eq. (10.155). We assume, as discussed above, that the terms involving V are included with the system Hamiltonian to produce renormalized energies so that the frequency differences Mab correspond to the spacings between these renormalized levels. This implies that the second term on the right of Eq. (10.155), the term involving V, does not exist. We also introduce the following notation for the real and imaginary parts of the super-matrix R-. 

We will see below that it is the real part of R that dominates the physics of the relaxation process. 

To get a feeling for the physical content of Eq. (10.155) let us consider first the time evolution of the diagonal elements of fr in the case where the nondiagonal elements vanish. In this case Eq. (10.155) (without the term involving E) 

This is a set of kinetic equations that describe transfer of populations within a group of levels. The transition rates between any two levels a and c are given by 

Using Eqs (10.121), (10.138) and (10.151) we find that these rates are given by 

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The Redfield equation describes the time evolution of the reduced density matrix of a system coupled to an equilibrium bath. The effect of the bath enters via the average coupling V = and the relaxation operator, the last sum on the right of Eq. (10.155). The physical implications of this term will be discussed below. 

Like the exact QDT counterpart [cf. Eq. (4.6)], the POP-CS-QDT preserves both the reduced Gaussian dynamics and the effective local field pictinre for the DBO system. Its TZg [Eq. (4.11a)] has the same dissipation superoperator terms as those in ]Zf [Eq. (4.6b)]. The first and the last terms in the right-hand-side of Eq. (4.11a) for TZg or Eq. (4.6b) for are mainly responsible for the energy renormalization (or self-energy) contribution [38] and their dynamics implications are often neglected in phenomenological quantum master equations such as the optical Bloch-Redfield theory [36]. Note that the bath response function relates to the spectral density as [cf. Eq. (2.8)]... 

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