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

TL;DR

This paper analyzes the mean-field dynamics of the Fröhlich Hamiltonian, showing that the many-body quantum state can be effectively approximated by a product state evolving according to Landau-Pekar and Bogoliubov equations, with improved convergence results.

Contribution

It introduces a norm approximation for the Fröhlich dynamics in the mean-field regime, extending previous results to a broader class of initial states with better convergence rates.

Findings

Effective dynamics approximates the many-body state in norm

Product state evolution follows Landau-Pekar equations

Includes Bogoliubov correction for improved accuracy

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.