  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-hybridisation regime with a single orbital
directly coupled to the dot and three additional indirectly coupled orbitals.
This minimal basis set allows 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 $\phi$.
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 QD explains the "doublet
chimney" in the phase diagram for $\phi \sim \pi$, in contrast to $\phi \sim 0$
where the residual quasiparticle escapes to infinity and plays no active role.
