Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

We consider the Fr\"ohlich Hamiltonian with large coupling constant $\alpha$. For initial data of Pekar product form coherent phonon field and electron minimizing corresponding energy, we provide a norm approximation evolution, valid up to times order $\alpha^2$. The is given in terms state, evolved through Landau-Pekar equations, corrected by Bogoliubov dynamics taking quantum fluctuations into account. This allows us show that equations approximately describe evolution electron- one-phonon reduced density matrices under