Josephson tunneling through a Yu-Shiba-Rusinov state: Interplay of $\pi$-shifts in Josephson current and local superconducting order parameter
An impurity hosting a magnetic moment coupled to a conventional $s$-wave superconductor gives rise to so-called Yu-Shiba-Rusinov (YSR) states with energies inside the superconducting gap. Depending on the coupling between the impurity and the superconductor, the system can have two distinct quantum ground states separated by a quantum phase transition (QPT). We investigate the interplay of two effects observed at the QPT. First, the tunneling supercurrent through the impurity reverses its sign at the QPT, denoted as a $\pi$-shift in the current-phase relation. Secondly, the local superconducting order parameter at the impurity site is suppressed and becomes negative at the QPT, generally termed a $\pi$-shift in the local superconducting order parameter. We find that both these effects are governed by the presence of the YSR state, however, they do not significantly depend or influence each other. In particular, we establish that the $\pi$-shift in the superconducting order parameter does not induce a $\pi$-shift in the tunneling Josephson current, nor can the Josephson current and its spatial behavior be used to directly probe the impurity-induced changes in the local superconducting order parameter, which occur on a length scale substantially shorter than the superconducting coherence length.