Scaling of subgap excitations in a superconductor-semiconductor nanowire quantum dot

A quantum dot coupled to a superconducting contact provides a tunable artificial analog of a magnetic atom in a superconductor, a paradigmatic quantum impurity problem. We realize such a system with an InAs semiconductor nanowire contacted by an Al-based superconducting electrode. We use an additional normal-type contact as a weakly coupled tunnel probe to perform tunneling spectroscopy measurements of the elementary subgap excitations, known as Andreev bound states or Yu-Shiba-Rusinov states. We demonstrate that the energy of these states zeta scales with the ratio between the Kondo temperature T-K and the superconducting gap Delta. zeta vanishes for T-K / Delta approximate to 0.6, denoting a quantum phase transition between the spin singlet and doublet ground states. By further leveraging the gate control over the quantum dot parameters, we determine the singlet-doublet phase boundary in the stability diagram of the system. Our experimental results show remarkable quantitative agreement with numerical renormalization group calculations.