Projection operators master equation

We shall use the projection operator method to derive the Pauli master equation. With the Liouville equation, we separate the Liouville operator into two parts ... 

It seems that the conventional approach to the quantum mechanical master equation relies on the equilibrium correlation function. Thus the CTRW method used by the authors of Ref. 105, yielding time-convoluted forms of GME [96], can be made compatible with the GME derived from the adoption of the projection approach of Section III only when p > 2. The derivation of this form of GME, within the context of measurement processes, was discussed in Ref. 155. The authors of Ref. 155 studied the relaxation process of the measurement pointer itself, described by the 1/2-spin operator Ez. The pointer interacts with another 1/2-spin operator, called av, through the interaction Hamiltonian... 

The simplest and most elegant theoretical technique operating in line with this leading idea is the Nakajima-Zwanzig projection method. By using this approach we are naturally led to replace the standard master equations. 

We end this discussion with two comments. First, we note that the Nakajima-Zwanzig equation (10.100) is exact no approximations whatever were made in its derivation. Second, this identity can be used in many ways, depending on the choice of the projection operator P. The thermal projector (10.87) is a physically motivated choice. In what follows we present a detailed derivation of the quantum master equation using this projector and following steps similar to those taken above, however, we will sacrifice generality in order to get practical usable results. 

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). 

There are several ways of proceeding from here to arrive at the master equation for the reduced density matrix. The Zwanzig projection operator technique in Liouville space or the Kubo cumulant expansion may be used and both methods have recently been applied to study optical dephasing in solids. 

An approximate but efficient solution is achieved by assuming a time scale separation between the fast jiggering motions within the potential wells and the slow conformational jumps. Under this assumption, projection of the diffusion operator onto a set of site functions, in the same number of the potential minima, can be performed to convert the diffusion equation into a master equation for jumps between discrete sites ... 

Projection operators of the type given by Eq. (547) possess a global operational character in the sense that they operate on both the system of interest (system) and its surroundings (bath). An alternative approach to the problem of constructing contracted equations of motion is to use projection operators that operate only on the subspace spanned by the bath. Such bath projection operators have been used in conjimction with Zwanzig s master equation to consttuct equations of motion solely for the state vector l/of > for the system of interest. 

Making use of the projection operator defined by Eq. (585) and the generalized master equation given by Eq. (562), one can constmct the following generalized phenomenological equations for the case of spatially dependent thermodynamic coordinates ... 

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