We study a strongly interacting quantum dot connected to two superconducting leads using a slave-spin representation of the dot. At the mean-field level, the problem maps to a resonant level model with superconducting leads, coupled to an auxiliary spin-1/2 variable accounting for the parity of the dot. We obtain the mean-field phase diagram, showing a transition between a Kondo (singlet) and a local moment (doublet) regime, corresponding to the $0\text{−}\pi$ transition of the junction. The mean-field theory qualitatively captures the Kondo singlet phase and its competition with superconductivity for weak values of the Bardeen-Cooper-Schrieffer (BCS) gap, including the nontrivial dependence of the Andreev bound states on the interaction, but fails in the doublet regime where it predicts a dot decoupled from the bath. Using diagrammatic techniques and a random-phase approximation, we include fluctuations on top of the mean-field theory to describe finite-frequency dynamics of the effective spin variable. This leads to the formation of high-energy Hubbard bands in the spectral function and a coherent Kondo peak with a BCS gap at low energies. We compute the Josephson current and the induced superconducting correlations on the dot. Finally, we evaluate the microwave response in the strongly interacting Kondo regime.

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• Josephson effect
• Kondo effect
• Mesoscopics
• Josephson junctions
• Quantum dots
• Anderson impurity model