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A quantum dot coupled to a superconducting contact provides a tunableartificial analogue of a magnetic atom in a superconductor, a paradigmaticquantum impurity problem. We realize such a system with an InAs semiconductornanowire contacted by an Al-based superconducting electrode. We use anadditional normal-type contact as weakly coupled tunnel probe to performtunneling spectroscopy measurements of the elementary sub-gap excitations,known as Andreev bound states or Yu-Shiba-Rusinov states. We demonstrate thatthe energy of these states, $\zeta$, scales with the ratio between the Kondotemperature, $T_K$, and the superconducting gap, $\Delta$. $\zeta$ vanishes for$T_K/\Delta \approx 0.6$, denoting a quantum phase transition between spinsinglet and doublet ground states. By further leveraging the gate control overthe quantum dot parameters, we determine the singlet-doublet phase boundary inthe stability diagram of the system. Our experimental results show remarkablequantitative agreement with numerical renormalization group calculations.

Scaling of sub-gap excitations in a superconductor-semiconductor nanowire quantum dot