  The dynamics of a single impurity interacting with a many particle background
is one of the central problems of condensed matter physics. Recent progress in
ultracold atom experiments makes it possible to control this dynamics by
coupling an artificial gauge field specifically to the impurity. In this paper,
we consider a narrow toroidal trap in which a Fermi gas is interacting with a
single atom. We show that an external magnetic field coupled to the impurity is
a versatile tool to probe the impurity dynamics. Using Bethe Ansatz (BA) we
calculate the eigenstates and corresponding energies exactly as a function of
the flux through the trap. Adiabatic change of flux connects the ground state
to excited states due to flux quantization. For repulsive interactions, the
impurity disturbs the Fermi sea by dragging the fermions whose momentum matches
the flux. This drag transfers momentum from the impurity to the background and
increases the effective mass. The effective mass saturates to the total mass of
the system for infinitely repulsive interactions. For attractive interactions,
the drag again increases the effective mass which quickly saturates to twice
the mass of a single particle as a polaron of the impurity and one fermion is
formed. For excited states with momentum comparable to number of particles,
effective mass shows a resonant behavior. We argue that standard tools in cold
atom experiments can be used to test these predictions.
