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We theoretically study the superconducting proximity effect in a quantum dotcoupled to two superconducting leads when the intradot interaction betweenelectrons is made attractive. Because of the superconducting proximity effect,the electronic states for the embedded quantum dot are either spin-polarizedstates with an odd occupation number or BCS-like states with an even occupationnumber. We show that in the presence of an external magnetic field, the systemcan exhibit quantum phase transitions of fermion parity associated with theoccupation number. In this work, we adopt a self-consistent theoretical methodto extend our considerations beyond the so-called superconducting atomic limitin which the superconducting gap for the leads is assumed to be the largestenergy scale. The method enables us to numerically investigate the electronicstructure of the dot as results of the attractive interaction. For energy phasediagrams in the regime away from the atomic limit, we find a reentrant behaviorwhere a BCS-like phase of the dot exists in an intermediate range of thehybridization strength between the quantum dot and the leads. We also considerJosephson current phase relations and identify a number of examples showing$0-\pi$ phase transitions that may offer important switching effects.

Quantum phase transitions in superconductor--quantum-dot--superconductor Josephson structures with attractive intradot interaction