Effective Dynamics for the Bose Polaron in the Large-Volume Mean-Field Limit

We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of $N$ bosons evolving in $\mathbb{R}^3$ in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density $\rho$ and large volume $\Lambda$ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint $\Lambda^3 \ll \rho$, the effective description by the translation-invariant Bogoliubov-Fr\"ohlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.