Full counting statistics of Yu-Shiba-Rusinov bound states

With the help of scanning tunneling microscopy (STM) it has become possible to address single magnetic impurities on superconducting surfaces and to investigate the peculiar properties of the in-gap states known as Yu-Shiba-Rusinov (YSR) states. These systems are an ideal playground to investigate multiple aspects of superconducting bound states, such as the occurrence of quantum phase transitions or the interplay between Andreev transport physics and the spin degree of freedom, with profound implications for disparate topics like Majorana modes or Andreev spin qubits. However, until very recently YSR states were only investigated with conventional tunneling spectroscopy, missing the crucial information contained in other transport properties such as shot noise. In this paper we adapt the concept of full counting statistics (FCS) to provide the deepest insight thus far into the spin-dependent transport in these hybrid atomic-scale systems. We illustrate the power of FCS by analyzing different situations in which YSR states show up including single-impurity junctions with a normal and a superconducting STM tip, as well as double-impurity systems where one can probe the tunneling between individual YSR states [Nat. Phys. 16, 1227 (2020)]. The FCS concept allows us to unambiguously identify every tunneling process that plays a role in these situations and to classify them according to the charge transferred in them. Moreover, FCS provides all the relevant transport properties, including current, shot noise, and all the cumulants of the current distribution. In particular, our approach is able to reproduce the experimental results recently reported on the shot noise of a single-impurity junction with a normal STM tip [Phys. Rev. Lett. 128, 247001 (2022)]. We also predict the signatures of resonant (and nonresonant) multiple Andreev reflections in the shot noise and Fano factor of single-impurity junctions with two superconducting electrodes and show that the FCS approach allows us to understand conductance features that have been incorrectly interpreted in the literature. In the case of double-impurity junctions we show that the direct tunneling between YSR states is characterized by a strong reduction of the Fano factor that reaches aminimum value of 7/32, a significant result in quantum transport. The FCS approach presented here can be naturally extended to investigate the spin-dependent superconducting transport in a variety of situations, such as atomic spin chains on surfaces or superconductor-semiconductor nanowire junctions, and it is also suitable to analyze multiterminal superconducting junctions, irradiated contacts, and many other basic situations.