Redfield equation approximations

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]

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

In most applications of the theory to date, the solution of the Redfield equation has required first the explicit calculation of the Redfield tensor elements [Eq. (11)] given these, Eq. (10) could be solved as an ordinary set of linear differential equations with constant coefficients, either by explicit time stepping [41, 42] or by diagonalization of the Redfield tensor [37,38]. Since there are such tensor elements for an A -state subsystem, the number of these quantities can become quite large. Because of this, until recently most applications of Redfield theory have been limited to small systems of two to four states, or else assumptions, such as the secular approximation, have been used to neglect large classes of tensor elements. 

An important feature of the reduced-density-matrix approach is that it allows the bath to be treated at different levels of approximation. In the Redfield equation, the bath enters only through the correlation functions of the coupled bath variables in Eq. (18). This means that a substantial part of the complexity of a realistic condensed-phase environment is... 

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

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). 

Following Cohen-Tannoudji and using the Markoffian approximation we have derived the equations of motion for the off-diagonal elements of the reduced density matrix, which determine the dephasing constant (7 ) and the optical lineshape. The following Redfield-like equations were obtained ... 

For simplicity, in equation (3c) S oc is assumed to be mainly along the director axis n, and n is aligned parallel tol. For Btoc Bo, the total 7, is equal to 7,2, whereas in the opposite case B,oc Bq one has 7, = 7,o- Evidently, the finite local field contribution complicates the control of the angle adjustment without an exact knowledge of, oc, and because of equation (3c) the inclinations of 90° become impossible by external field switches. Furthermore, the spectral densities for 7,0 are not discussed in the literature to the same extent as for 7,2, nor does there exist a critical experimental examination of the validity of the basic expression equation (3a). Approximate predictions about with the Redfield formalism give, for the completely isolated, i.e., uncoupled proton spin-pair (/ = 1) and high spin-temperature approach ... 

Despite the fact that the exact j or TZ can be formally expressed in terms of an infinite series expansion, its evaluation, however, amounts to solve the total composite system of infinite degrees of freedom. In practice, one often has to exploit weak system-bath interaction approximations and the resulting COP [Eq. (1.2)] and POP [Eq. (1.3)] of QDT become nonequivalent due to the different approximation schemes to the partial consideration of higher order contributions. It is further noticed that in many conventional used QDT, such as the generalized quantum master equation, Bloch-Redfield theory and Fokker-Planck equations, there involve not only... 

---