  We consider the quantum dynamics of a large number $N$ of interacting bosons
coupled a tracer particle, i.e. a particle of another kind, on a torus. We
assume that in the initial state the bosons essentially form a homogeneous
Bose-Einstein condensate, with some excitations. With an appropriate mean-field
scaling of the interactions, we prove that the effective dynamics for $N\to
\infty$ is generated by the Bogoliubov-Fr\"ohlich Hamiltonian, which couples
the tracer particle linearly to the excitation field.
