Calculation of the spectral dynamics of systems with a quadratic vibronic interaction

Progress in generating ultrashort laser pulses provides continuous development of the exploration methods of the transient emission and absorption spectra. The theoretical approaches are basically limited to linear electron-vibrational interaction. In this paper the influence of quadratic vibronic interaction on transient spectra is explored. The method of functional derivatives is applied for calculation of the generating function determining the optical spectrum of a two-level electron-vibration system. The analytical expression for the generating function for the two-level electron-vibration system with quadratic vibronic interaction is derived. It is found that the frequency change of the vibrational subsystem accompanying the electronic transition manifests itself in both the first and second cumulants of the optical spectrum. The second cumulant defines the frequency dispersion is not the same for stationary and non-stationary spectra. The evolution equation of the second cumulant is obtained. An analytical expression for the first cumulant determining the position of the spectrum gravity center is derived. The time dependent displacement of the spectrum gravity center is demonstrated to be accompanied by a change in its width. It is shown that in the absence of the frequency change of the vibrational subsystem the generating function is transformed to a form received earlier at the description of ultrafast dynamics of the reverse transition from a state produced by a short laser excitation of a donor-acceptor complex at the frequency of the charge transfer absorption band.