Second-order perturbation evolution

The simplest close-form description of the relaxation enhancement is achieved if the second order perturbation theoty of Bloch-Wangsness-Redfield (BWR) " is applicable. The starting equation for the BWR description is the second order perturbation formula for the evolution of the spin density matrix ... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

In the weak field limit, the time evolution of wave packets in pump-probe experiments can be evaluated by second-order time-dependent perturbation theory. The second-order solution of Eq. (24) is expressed as... 

By the definition of the steady-state condition, the first terms on the right-hand side are zero and provided the first-order partial derivative terms do not all vanish, we can ignore the additional terms which are second-order in the perturbations. Thus, we obtain a pair of linear equations for the evolution of these perturbations in the vicinity of the steady-state point... 

The algebraic steps presented above can now be applied in the development of a perturbation expansion of the evolution operator in a similar way used by Bulaevskii . We consider only terms of the second order in the perturbation... 

The evolution of a t) is usually slow since the perturbation interaction H t) is small and fluctuates rapidly to satisfy the motional narrowing limit. Starting from d-(O), the time evolution of a may be calculated in the interaction representation through a second-order expansion. Formal integration of Eq. (5.4) gives... 

Owing to the coherence, we need to consider the macroscopic evolution of the field in a medium that shows a macroscopic polarization induced by the field-matter interaction. This will be done in three steps. First, the polarization induced by an arbitrary field will be calculated and expanded in power series in the field, the coefficients of the expansion being the material susceptibilities (frequency domain) or response function (time domain) of wth-order. Nonlinear Raman effects appear at third order in this expansion. Second, the perturbation theory derivation of the third-order nonlinear susceptibility in terms of molecular eigenstates and transition moments will be outlined, leading to a connection with the spontaneous Raman scattering tensor components. Last, the interaction of the initial field distribution with the created polarization will be evaluated and the signal expression obtained for the relevant techniques of Table 1. 

On less formal grounds a Markovian type of equation was also obtained by Redfield [167]. Redheld used perturbation theory up to second order and derived a Master equation for the evolution of the density matrix Oaa y where a, a denote quantum states of the molecule with energy defined by ha. The simple equation for the relaxation of the density matrix was obtained by neglecting the ensemble-averaged first-order terms. Thus we have [167]... 

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