In Chapter 7 we found that a time dependent Hamiltonian (Eq. 7.2) is the source of infrared lineshapes. It also causes energy transfer between eigenstates that will alter the intensity of cross- and diagonal peaks or create completely new peaks. As for lineshapes, the time dependence of the Hamiltonian results from the fluctuations of the solvent and molecular structure. Three types of relaxation occur, all of which have been observed experimentally: population relaxation, population transfer, and coherence transfer. Figure 8.1 shows an example of a Feynman diagram for each of these cases. In contrast to the Feynman diagrams that we have seen so far, these diagrams time-propagate with more than just the usual oscillatory terms. They flip quantum states (indicated by the dotted lines and the arrows in Fig. 8.1). Population relaxation, population transfer, and coherence transfer correspond to flipping an excited population state |1〈〉1| back into the ground state |0〈〉0| (Fig. 8.1a), flipping an excited population state |1〈〉1| into another excited state |1'〈〉1'| (Fig. 8.1b), and flipping a coherence state |0〈〉1| into another coherence state |0〈〉1'| (Fig. 8.1c), respectively. A rigorous treatment of relaxation processes has become quite demanding, and we outline here only the simplest cases. For a more comprehensive discussion, we refer readers to Ref. [106].

Population transfer

As the simplest case, consider a dimer of coupled oscillators that is sitting in a fluctuating bath.