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We study the subgap spectrum of the interacting single-level quantum dotcoupled between two superconducting reservoirs, forming the Josephson-typecircuit, and additionally hybridized with a metallic normal lead. This systemallows for the phase-tunable interplay between the correlation effects and theproximity-induced electron pairing resulting in the singlet-doublet (0-$\pi$)crossover and the phase-dependent Kondo effect. We investigate the spectralfunction, induced local pairing, Josephson supercurrent, and Andreevconductance in a wide range of system parameters by the numerically exactNumerical Renormalization Group and Quantum Monte Carlo calculations along withperturbative treatments in terms of the Coulomb repulsion and the hybridizationterm. Our results address especially the correlation effects reflected independencies of various quantities on the local Coulomb interaction strength aswell as on the coupling to the normal lead. We quantitatively establish thephase-dependent Kondo temperature $\log T_{K}(\phi)\propto\cos^{2}(\phi/2)$ andshow that it can be read off from the half-width of the zero-bias enhancementin the Andreev conductance in the doublet phase, which can be experimentallymeasured by the tunneling spectroscopy.

Josephson-phase-controlled interplay between correlation effects and electron pairing in a three-terminal nanostructure