  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. 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. Here,
we adapt the concept of full counting statistics (FCS) to provide a very deep
insight into the spin-dependent transport in these hybrid systems. We
illustrate the power of FCS by analyzing different situations in which YSR
states show up including single-impurity junctions, as well as double-impurity
systems where one can probe the tunneling between individual YSR states. The
FCS concept allows us to unambiguously identify every tunneling process that
plays a role in these situations. Moreover, FCS provides all the relevant
transport properties, including current, shot noise and all the cumulants of
the current distribution. Our approach can reproduce the experimental results
recently reported on the shot noise of a single-impurity junction with a normal
STM tip. We also predict the signatures of resonant (and non-resonant) multiple
Andreev reflections in the shot noise of single-impurity junctions with two
superconducting electrodes. 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 a minimum value of 7/32, a new
fundamental 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, and it is also suitable to analyze multi-terminal
superconducting junctions, irradiated contacts, and many other basic
situations.
