Bath correlation functions

By using special forms of the so-called spectral density J(w) it is possible to treat memory effects in QMEs. The spectral density J(w) contains information on the frequencies of the environmental modes and their coupling to the system. Tanimura and coworkers [18,20,26] were the first to do calculations along the lines described here using spectral densities of Drude shape. This spectral densities lead to bath correlation functions with purely exponential... 

To describe the effect of the environment one usually needs to determine the bath correlation function C(t). Let us start discussing this function for a bosonic bath where the subscript Ph indicates a bath of phonons. Using the numerical decomposition of the spectral density Eq. (2) together with the theorem of residues one obtains the complex bath correlation functions... 

The dynamics of the bath enters through the bath correlation functions, C (Z) =... 

Here the W are operators of the subsystem and the superscript dagger denote the Hermitian conjugate. The Redfield equation can be written in this form only when an additional symmetrization of the bath correlation functions is performed [48]. Note that this alternative equation also expresses the dissipative evolution of the density matrix in terms of N x N... 

Before going on to consider more complicated systems, we review here some of the basic behavior of a two-state quantum system in the presence of a fast stochastic bath. This highly simplified bath model is useful because it allows qualitatively meaningful results to be obtained from a density matrix calculation when bath correlation functions are not available in fact, the bath coupling to any given system operator is reduced to a scalar. In the case of the two-level system, analytic results for the density matrix dynamics are easily obtained, and these provide an important reference point for discussing more complicated systems, both because it is often possible to isolate important parts of more complicated systems as effective two-level systems and because many aspects of the dynamics of multilevel systems appear already at this level. An earlier discussion of the two-level system can be found in Ref. 80. The more... 

The bath correlation function that relates to the spectral density via the fluctuation-dissipation theorem [Eq. (2.11)] can now be obtained via the contour integration algorithm. We have (setting + 7 )... 

The Zusman equation (ZE)/ due mainly to its physically insightful picture on solvation dynamics, is (at least used to be) one of the most commonly used approaches in the study of quantum transfer processes. In this approach, the electronic system degrees of freedom are coupled to a collective bath coordinate that is assumed to be diffusive. The only approximation involved is the classical high temperature treatment of bath. To account for the dynamic Stokes shift, the standard ZE includes also the imaginary part of bath correlation function. This part does not depend on temperature and is therefore exact in the diffusion regime. 

The bath correlation function is evaluated via the fluctuation-dissipation theorem ... 

We turn now to the HEOM formalism. We will demonstrate that it not just recovers the ZE, but also leads to a formally simple but physically significant modification. To proceed we focus on the following extended singleexponential form of bath correlation function (see eqn (13.2) for the parameters) ... 

Here, 5R is related to the 5(t)-component of bath correlation function and, as mentioned earlier, is given by ... 

The resulting bath correlation function for the Drude dissipation (eqn (13.12)) does acquire the form of eqn (13.23), with the parameters of ... 

Simpler and more manageable expressions are obtained in the limit where the dynamics of the bath is much faster than that of the system. In this limit the functions M t) (i.e. the bath correlations functions C t) of Eq. (10.121) or Cmnit) of Eq. (10.124)) decay to zero as t oo much faster than any characteristic system timescale. One may be tempted to apply this limit by substituting r in the elements o mnit — t) in Eq. (10.141) by zero and take these terms out of the integral over r, however this would be wrong because, in addition to their relatively slow physical time evolution, non-diagonal elements of a contain a fast phase factor. Consider for example the integral in first term on the right-hand side of (10.141),... 

Within a permrbative approach, which is second order in the system-bath coupling, the entire bath dynamics enters into the system dynamics via the complex bath correlation function, defined by... 

---