  Fluxons in a superconducting loop can be coherently coupled by quantum phase
slips occurring at a weak link such as a Josephson junction. If Cooper pair
tunneling at the junction occurs through a resonant level, $2\pi$ quantum phase
slips are suppressed, and fluxons are predominantly coupled by $4\pi$ quantum
phase slips. We analyze this scenario by computing the coupling between fluxons
as the level is brought into resonance with the superconducting condensate. The
results indicate that the $4\pi$-dominated regime can be observed directly in
the transition spectrum for circuit parameters typical of a fluxonium qubit. We
also show that, if the inductive energy of the loop is much smaller than the
plasma frequency of the junction, the low-energy Hamiltonian of the circuit is
dual to that of a topological superconducting island. These findings can inform
experiments on bifluxon qubits as well as the design of novel types of
protected qubits.
