  We study a mobile quantum impurity, possessing internal rotational degrees of
freedom, confined to a ring in the presence of a many-particle bosonic bath. By
considering the recently introduced rotating polaron problem, we define the
Hamiltonian and examine the energy spectrum. The weak-coupling regime is
studied by means of a variational ansatz in the truncated Fock space. The
corresponding spectrum indicates that there emerges a coupling between the
internal and orbital angular momenta of the impurity as a consequence of the
phonon exchange. We interpret the coupling as a phonon-mediated spin-orbit
coupling and quantify it by using a correlation function between the internal
and orbital angular momentum operators. The strong-coupling regime is
investigated within the Pekar approach and it is shown that the correlation
function of the ground state shows a kink at a critical coupling, that is
explained by a sharp transition from the non-interacting state to the states
that exhibit strong interaction with the surroundings. The results might find
applications in such fields as spintronics or topological insulators, where
spin-orbit coupling is of crucial importance.
