Norm approximation for the Fr\"ohlich dynamics in the mean-field regime

Abstract

We study the time evolution of the Fr\"ohlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. Assuming that the particles are initially in a Bose-Einstein condensate and that the excitations of the phonon field are initially in a coherent state we provide an effective dynamics which approximates the time evolved many-body state in norm, provided that the number of particles is large. The approximation is given by a product state which evolves according to the Landau-Pekar equations and which is corrected by a Bogoliubov dynamics. In addition, we extend the results from [Arch. Ration. Mech. Anal. 240, 383-417 (2021)] about the approximation of the time evolved many-body state in trace-norm topology to a larger class of many-body initial states with an improved rate of convergence.