The Nakajima-Zwanzig equation

It is actually simple to find a formal time evolution equation in P space. This formal simplicity stems from the fact that the fundamental equations of quantum dynamics, the time-dependent Schrodinger equation or the Liouville equation, are linear. Starting from the quantum Liouville equation (10.8) forthe overall system— system and bath. 

These equations look complicated, however, in form (as opposed to in physical contents) they are very simple. We need to remember that in any representation that uses a discrete basis set p is a vector and is a matrix. The projectors P and Q are also matrices that project on parts of the vector space (see Appendix 9A). For example, in the simplest situation 

Equations (10.90) and (10.91) are seen to be just the equivalent set of equations for pp and pq. One word of caution is needed in the face of possible confusion To avoid too many notations it has become customary to use Pp also to denote pp, PPP also to denote Ppp, etc., and to let the reader decide from the context what these structures mean. With this convention Eq. (10.94a) is written in the form d. . . . .  

We proceed by integrating Eq. (10.91) and inserting the result into (10.90). Again the procedure is simple inform. If Pp = x and Qp = y were two scalar variables and all other terms were scalar coefficients, this would be a set of two coupled first order differential equations 

We will be doing exactly the same thing with Eqs. (10.90) and (10 91) Integration of (10.91) yields 

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The Nakajima-Zwanzig equation containing an explicit time convolution can also be transformed into a time-convolutionless form, in which the equation of... 

The identity (10.100) is the Nakajima-Zwanzig equation. It describes the time evolution of the relevant part Pp f) of the density operator. This time evolution is determined by the three terms on the right. Let us try to understand their physical contents. In what follows we refer to the relevant and irrelevant parts of the overall system as system and bath respectively. 

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. 

Starting point for the derivation of the Nakajima-Zwanzig identity is the Lioville-von Neumann equation for the density operator a of the complete system, i.e. relevant system plus environment. As stated before h = 1 is used and thus the QME reads... 

So within a few steps it is possible to derive the Nakajima-Zwanzig identity starting from the Lioville-von Neumann equation. 

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. 

Nakajima-Zwanzig equation with memory, see, e.g., [Breuer 2002], Introducing the Nakajima-Zwanzig projectors... 

We have considered repeated projective measurements on an ancilla as a tool for manipulating the evolution of a dynamic quantum system of interest. Due to an interaction between the dynamic system and the ancilla, the nonunitary evolution of the ancilla extends equally to the dynamic system, but close to the Zeno-limit the coherence of the dynamic system may still be preserved. Of particular interest here are systems coupled with a nondemolition interaction, since they can be described in an essentially simplified manner. Depending on the dimension Na of the ancilla, individual elements of the reduced state of the dynamic part obey master equations that are iV order differential equations in time. Equivalently, the master equations can be written in the Nakajima-Zwanzig or time convolutionless form. 

Open systems are generally described within the density matrix formahsm [27-30,41,42]. The reduced density matrix Ps t) associated with the system is obtained by tracing over the bath coordinates. In the Nakajima-Zwanzig formalism [43], Ps t) is solution of a reduced equation containing a memory which depends on the whole history of the global system-bath... 

Due to complexity of the real world, all QDT descriptions involve practically certain approximations or models. As theoretical construction is concerned, the infiuence functional path integral formulation of QDT may by far be the best [4]. The main obstacle of path integral formulation is however its formidable numerical implementation to multilevel systems. Alternative approach to QDT formulation is the reduced Liouville equation for p t). The formally exact reduced Liouville equation can in principle be constructed via Nakajima-Zwanzig-Mori projection operator techniques [5-14], resulting in general two prescriptions. One is the so-called chronological ordering prescription (COP), characterized by a time-ordered memory dissipation superoperator 7(t, r) and read as... 

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