Strong-coupling theory of quantum-dot Josephson junctions: Role of a residual quasiparticle

We consider an interacting quantum dot strongly coupled to two superconducting leads in a Josephson junction geometry. By defining symmetry-adapted superpositions of states from the leads, we formulate an effective Hamiltonian for the strong-hybridization regime with a single orbital directly coupled to the dot and three additional indirectly coupled orbitals. This minimal basis set allows one to account for the quasiparticles in the vicinity of the dot as well as those further away in the leads and to describe how their role evolves as a function of coupling strength and phase bias. This formulation also reveals the changing nature of the spin-doublet state for the experimentally relevant coupling strengths. The binding of a nearly decoupled quasiparticle in the vicinity of the quantum dot explains the ``doublet chimney'' in the phase diagram for, in contrast to 0, where the residual quasiparticle escapes to infinity and plays no active role.