Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas

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 Fröhlich-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 a range of interaction couplings, in particular including interaction couplings of order 1 and time scales of the order $k_{\text{F}}^{-1}$.