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We study the critical Josephson current flowing through a double quantum dotweakly coupled to two superconducting leads. We use analytical as well asnumerical methods to investigate this setup in the limit of small and largebandwidth leads in all possible charging states, where we account for on-siteinteractions exactly. Our results provide clear signatures of nonlocalspin-entangled pairs, which support interpretations of recent experiments[Deacon, R. S. et al., Nat. Commun. 6, 7446 (2015)]. In addition, we find thatthe ground state with one electron on each quantum dot can undergo a tunablesinglet-triplet phase transition in the regime where the superconducting gap inthe leads is not too large, which gives rise to an additional new signature ofnonlocal Cooper pair transport.

Signatures of nonlocal Cooper-pair transport and of a singlet-triplet transition in the critical current of a double-quantum-dot Josephson junction