Self-consistent RPA retarded polaron Green function for quantum kinetics

We compute numerically the time dependent retarded Green function of the polaron within the self-consistent RPA approximation. The results show an approximately Gaussian behaviour at t = 0 changing at later times its concavity to an exponential decay? as it has been predicted in the approximate form of an inverse hyperbolic cosine function. The result contrasts with the non-selfconsistent RPA: where the exponential decay is only a transitory behaviour and the asymptotics is rather oscillatory. Our conclusions are significant in the context of the quantum kinetics with LO-phonons, where the transition from an intitially coherent scattering kinetics to a Markov kinetics with energy conservation is controlled by the time behaviour of the retarded Green function.