Transfer dynamics of the two-site prototype polaron model

In earlier work on the temporal evolution of exciton-phonon systems usually a semiclassical approximation was employed, i.e., the phonon system was treated classically. However, it has been shown that this approach may lead to solutions which contradict the exact solution in relevant parameter ranges, even qualitatively. On the other hand, a fully quantum mechanical treatment is practically impossible, especially in extended exciton-phonon systems. One possibility for overcoming the huge computational difficulties is to subject the coupled system to a tractable unitary transformation which approximately diagonalizes the Hamiltonian. Then the temporal evolution can be calculated in the transformed space. This method is demonstrated for the example of the two-site model (1 exciton, 1 oscillator, 2 sites). A comparison of the temporal evolution obtained in this way with the exact numerical solution shows that the unitary transformation method yields semiquantitatively correct results even in parameter ranges where the semiclassical approximation becomes qualitatively false.