Nonlinear canonical transformations in the coupling between degenerate high and low energy excitations

Abstract

In microscopic many-body physics the coupling between the motion of fast particles (electrons) and slow particles (nuclei) is universal. The standard Born-Oppenheimer decoupling procedure breaks down, if the energy separation in the “fast” system is of the same order as the elementary excitation in the “slow” system. In this case “dynamical resonance” effects are to be expected. In the present investigation a model system of a coupling between a doubly degenerate high energy excitation and doubly degenerate low energy oscillator is handled by a non-linear canonical transformation which is shown to be quasi-exact in the sense that it diagonalizes the Hamiltonian in both extremal coupling cases. The transformation has some flexibility, so that the diagonalization regions can be enlarged. It is employed to calculate the “zero-phonon” optical response, which indeed displays aresonance effect. Likewise, another nonlinear transformation is devised, which only in the strong coupling limit yields diagonalization. This latter transformation in a natural way leads to the conventional semi-classical approaches to the dynamical Jahn-Teller problem. The results gotten with it are identical with those from our transformation in the strong coupling limit. On the basis of our results some remarks are made concerning the possible impact of the breakdown of the adiabatic approximation in other regions.