  We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an
impurity particle on a three-dimensional box with periodic boundary condition.
For large Fermi momentum $k_\text{F}$, we prove that the effective dynamics is
generated by a Fr\"ohlich-type polaron Hamiltonian, which linearly couples the
impurity particle to an almost-bosonic excitation field. Moreover, we prove
that the effective dynamics can be approximated by an explicit coupled coherent
state. Our method is applicable to two relevant settings: first, an interaction
coupling $\lambda=1$ and masses of order 1 for time scales of order
$k_\text{F}^{-1}$; second to the case of $\lambda=k_\text{F}^{-1}$ and a heavy
Fermi gas with masses of order $k_\text{F}$ for time scales of order 1.
