Kondo effect in asymmetric Josephson couplings through a quantum dot

Asymmetry in the Josephson couplings between two superconductors through a quantum dot is studied based on a single impurity Anderson model using the numerical renormalization group (NRG). Specifically, we examine how the difference between the couplings Gamma(L) and Gamma(R) affects the ground state, which is known to show a quantum phase transition between a nonmagnetic singlet and a magnetic doublet depending on the various parameters; the Coulomb interaction U, onsitepotential epsilon(d), level width Gamma(nu) caused by the hybridization, and superconducting (SC) gap Delta(nu) = vertical bar Delta vertical bar e(i theta nu) for the leads on the left and right (nu = L, R). Our results show that whether the local moment is fully screened or not depends substantially on the asymmetry in the couplings Gamma L not equal Gamma(R). It tends to make the singlet ground state stable, while the amplitude vertical bar Delta(nu)vertical bar and phase difference phi = theta(R) - theta(L) of the SC gaps tend to suppress the screening. We also discuss some general symmetry properties of the system and their relation to the current conservation.