  We study low-energy transport through a quantum dot coupled to one normal and
two superconducting (SC) leads in a junction of Y-shape. In this geometry a
crossover between Kondo dominated and Cooper-pairing dominated states occurs by
tuning the parameters such as the quantized energy level of the dot and the
Josephson phase, which induces a supercurrent flowing between the two SC leads
through the dot. Furthermore, Andreev scattering takes place at the interface
between the dot and normal lead. The low-lying energy states of this system can
be described by a local Fermi-liquid theory for interacting Bogoliubov
particles. In a description based on an Anderson impurity model we calculate
transport coefficients, renormalized parameters and spectral function, using
Wilson's numerical renormalization group (NRG) approach, in the limit of large
SC gap. Our results demonstrate how the Andreev resonance level approaches the
Fermi level in the crossover region between Cooper-pairing singlet state and
strong coupling situation as the impurity level or Josephson phase are varied.
The strong coupling situation shows a Kondo effect with a significantly
renormalized resonance width. The crossover is smeared when the coupling
between the dot and normal lead is large. Furthermore, asymmetry in the
Josephson junction suppresses the cancellations of the SC proximity for finite
Josephson phase, and it favors the SC singlet state rather than the Kondo
singlet.
