Optical-response function of a dynamic Jahn-Teller system

The optical-absorption-band shape in the singlet-multiplet electronic transition of an impurity center interacting with the phonons in polar crystals, has been investigated, taking into account that the vibronic system is a Jahn-Teller system. In the model Hamiltonian considered here, the electron has a two-level (one of which is degenerate) structure, and the linear electron-phonon interaction is transient, because it takes place only when the impurity has been excited by the light absorption. The Fourier transform $R\mathrm{}\left(t\right)\mathrm{}$ of the response function has been calculated by resumming all the Feynman diagrams, generated by the linear electron-phonon interaction on the degenerate electronic state, and by considering the point symmetries of the electron. $R\mathrm{}\left(t\right)\mathrm{}$ is found to be an exponential whose argument is a quadratic expression of noncommuting matrices. The path-integration method for a Gaussian functional is then used. The time-dependent behavior of $R\mathrm{}\left(t\right)\mathrm{}$, as well as its contribution to the absorption coefficient, is related to the strength of the electron-phonon interaction. The semiclassical results for the Jahn-Teller splitting of the absorption band are obtained by integrating the Gaussian functional on the path in the short-time limit $t\to 0$. The "interaction modes" used in the molecular model of the electron-phonon interaction are then found to correspond to the short-time paths of different symmetries in the Gaussian functionals. The origin of the $t^2$ and $t^3$ terms and their effects on the line shape are commented on, as well as the structure coming from the behavior of $R\mathrm{}\left(t\right)\mathrm{}$ in the long-time limit.