Tunneling of fluxons via a Josephson resonant level

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.