Spin noise and magnetic screening of impurities in a BCS superconductor

The coupling of a localized impurity to a BCS superconductor (SC) leads to the formation of impurity Cooper-pairs via the proximity effect, generating two bound states within the SC energy gap, the so-called Yu-Rusinov-Shiba (YSR) states. They are similar to the Andreev Bound States that originate from Andreev reflection, e.g. when the impurity is hosted in a Josephson junction, and are known to produce sharp sub-gap resonances in charge noise [de Sousa et al., PRB 2009], providing a natural explanation for the observation of microresonators in superconducting devices [Simmonds et al., PRL 2004]. Here we present a theory for the spin noise generated by magnetic impurities in a SC, and discuss the impact of the Shiba states on models of flux noise in superconducting qubits. We use a combination of analytical methods and the numerical renormalization group technique to calculate the spin noise of an Anderson impurity in a SC, unveiling the competition between the proximity effect and Kondo correlations. Both mechanisms produce magnetic screening and a corresponding reduction in spin noise, giving rise to new insights on the kinds of impurities that are responsible for the observed $1/f^{\alpha}$ flux noise in superconducting circuits.