Charge Kondo effect in negative- U quantum dots with superconducting electrodes

Recent experimental realization of superconducting quantum dot devices with intradot attraction $U$ [Nature (London) 521, 196 (2015); Phys. Rev. X 6, 041042 (2016)] offers unique opportunities to study the charge Kondo effect in a superconducting environment. In such devices pseudospin flips are caused by two tunneling processes. One is the cotunneling of normal electrons which generates near-gap Kondo resonances in the single-electron spectral density. This negative-$U$ charge Kondo effect is more robust than the conventional spin Kondo effect against the suppression by the superconductivity. The other tunneling is the mean-field Cooper-pair tunneling which produces a zero-energy bound state in the pair spectral density. Interesting crossover physics from the strongly-correlated Kondo screening to the mean-field polarization of local pseudospin is demonstrated. Due to the interplay of these two tunnelings, the supercurrent is suppressed for intermediate couplings, but it can increase to the unitary limits both in the strong and weak coupling regimes. We obtain the magnetic field-dependent supercurrent which is consistent with the key experimental findings.