A new form of the redfield equation for dissipative systems related to the matrix of correlation functions

Abstract

In the Redfield theory framework, we consider the problem of the vibrational dynamics in dissipative systems. We decompose the Hamiltonian of interaction of the observed system with a thermal bath into terms that are products of system transition operators and bath transition operators. Using the decomposition, we construct a correlation function matrix containing all the information about the interaction of the system with the bath and obtain a new operator form of the Redfield equation. We consider the procedure for factoring the interaction operator and constructing the correlation function matrix. Using the diagonalization of the obtained matrix, we give correlated dissipation operators, whose introduction simplifies the form of the Redfield equations. We show that for several problems in which fundamental transition frequencies can be chosen, this procedure significantly reduces the dimensionality of the dissipative dynamics problem.

Exponentially damped operators for a harmonic oscillator linearly coupled to a heat bath

“…At the same time, analytic approaches, when possible, allow obtaining a clear physical picture of the phenomena under consideration and characterizing the dependence of the system behavior on the parameters of the model used. This paper continues previous papers aimed at analytically describing the dissipative dynamics in the harmonic approximation [12]. In [13], we obtained closed expressions for the Redfield tensor elements, which allowed finding the explicit time dependence of the mean oscillation energy.…”

“…Equation (12) implies that the spectrum of the relaxation exponents E n (t) can be obtained from the spectrum of E n−1 (t) by adding a single number, namely, k n = nJ(ω). Taking the values of k 0 and k 1 into account, we can easily prove by induction that the spectrum of relaxation exponents contains all the numbers of the form nJ(ω) and that the relaxation times are…”

“…It therefore seems better to seek general expressions for operators (9) in terms of temperature-independent operators. We now define one set of such operator using the diagonalization procedure for the matrix of correlation functions [12].…”

“…and we hence have a two-dimensional space [12] of system operators and the two transition frequencies ω and −ω. The heat bath operators corresponding to these frequencies are…”

“…Orthogonalizing the heat bath operators and selecting the resonance terms [12], we can rewrite the interaction energy operator as (see Appendix 2)…”