Redfield equation bath models

The fact that the lineshape (18.49) is Lorentzian is a direct consequence of the fact that our starting point, the Redfield equations (10.174) correspond to the limit were the thermal bath is fast relative to the system dynamics. A similar result was obtained in this limit from the stochastic approach that uses Eq. (10.171) as a starting point for the classical treatment of Section 7.5.4. In the latter case we were also able to consider the opposite limit of slow bath that was shown to yield, in the model considered, a Gaussian lineshape. 

There are three important issues to consider in the numerical solution of the Redfield equation. The first is the evaluation of the Redfield tensor matrix elements I ,To obtain these matrix elements, it is necessary to have a representation of the system-bath coupling operator and of the bath Hamiltonian. Two fundamental types of models are used. First, the system-bath coupling can be described using stochastic fluctuation operators, without reference to a microscopic model. In this case, the correlation functions appearing in the formulas for parame-... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

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