Quantum master equation

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

The best one can hope is that there is an approximate equation of type (3.6), called Redfield equation 510 - or in the present context the quantum master equation . The approximation requires an expansion parameter the obvious choice is the parameter a. To prepare for this expansion we transform pT to its interaction representation [Pg.437]

This is an equation for the system S by itself. It is not yet quite of the type (3.6) inasmuch as the superoperator [ ] still depends on the time elapsed since that special time at which (3.4) had been postulated. Suppose, however, that there exists a finite time interval tc such that the integrand practically vanishes for t > tc. Then the integral may be extended to infinity as soon as t > xc and one finds the quantum master equation ... 

The Schrodinger-Langevin equation and the quantum master equation... 

What follows is a heuristic derivation of the quantum master equation (5.6) suggested by a formal resemblance between the Schrodinger equation and the... 

Thus we have found the general form of the quantum master equation that corresponds to the Schrodinger-Langevin equation (5.3). 

We have arrived at the quantum master equation (5.6) heuristically via the Schrodinger-Langevin equation (5.3). It turns out, however, that (5.6) has a firmer foundation it can be proved mathematically on the basis of the following three general conditions concerning the evolution of p. 

A serious difficulty now appears. The quantum master equation (3.14), obtained by eliminating the bath, does not have the required form (5.6) and therefore results in a violation of the positivity of ps(/). Only by the additional approximation rc Tm was it possible to arrive at (3.19), which does have that form (see the Exercise). The origin of the difficulty is that (3.14) is based on our assumed initial state (3.4), which expresses that system and bath are initially uncorrelated. This cannot be true at later times because the interaction inevitably builds up correlations between them. Hence it is unjustified to use the same derivation for arriving at a differential equation in time without invoking a repeated randomness assumption, such as embodied in tc rm. ) At any rate it is physically absurd to think that the study of the behavior of a Brownian particle requires the knowledge of an initial state. 

This is the quantum master equation describing internal fluctuations. The energy e enters only as a parameter and a stands for the entire set a, b,. 

The main difference with the second edition is that the contrived application of the quantum master equation in section 6 of chapter XVII has been replaced with a satisfactory treatment of quantum fluctuations. Apart from that, throughout the text corrections have been made and a number of references to later developments have been included. Of the more recent textbooks, the following are the most relevant. 

The equation of motion for the reduced density operator (quantum master equation) takes the form [40]... 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

Time-Local Quantum Master Equations and their Applications to Dissipative Dynamics and Molecular Wires... 

Open quantum systems have attracted much attention over the last decades. While most of the studies dealt with systems coupled to bosonic heat baths, recently systems coupled to fermionic reservoirs describing for example molecular wires have been in the focus of many investigations. This chapter will not try to give a concise overview of the available literature but will focus on a particular approach time-local (TL) quantum master equations (QMEs) and in particular their combination with specific forms of the spectral density. 

Fig. 3 Forward rate coefficient kAB(t) as a function of time for f3 = 1.0. The upper (blue) curve is the adiabatic rate, the purple curve is the result obtained by Tully s surface-hopping algorithm, the middle (black) curve is the quantum master equation result, the green curve is the QCL result, and the lowest dashed line (grey) is the result using mean-field dynamics.

Q. Shi and E. Geva (2004) A semiclassical generalized quantum master equation for an arbitrary system-bath coupling. J. Chem. Phys. 120, p. 10647... 

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. 

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