Exact solution for the dynamical decoupling of a qubit with telegraph noise

As a high-resistivity normal-metal barrier for superconducting Josephson junctions, metal-silicon alloys appear to be a good replacement for noble metals and have been applied to the development of quantum voltage standard circuits. We observed that the electrical properties of Nb-based junctions made with amorphous NbxSi1-x barriers were slowly evolving over time when stored at room temperature. After systematically investigating both junctions and thin films, we have concluded that the changes in junction electrical parameters are due to changes in the amorphous NbxSi1-x and not due to barrier-electrode interface effects. The resistivity of amorphous NbxSi1-x increases after heat treatment at temperatures as low as 80 degrees C when the Nb concentration is less than 33%, that of NbSi2. Furthermore, we found that annealed barriers behave similarly to barriers with the Nb concentration intentionally reduced to obtain smaller critical current and larger normal resistance with the same barrier thickness, as explained by our I-c-R-n relation developed based on the dirty-limit superconductor-normal-metal-superconductor Josephson junction theory. To explain these effects, we adopt and corroborate a microscopic picture based on alloy phase stability that was previously demonstrated by others. We also successfully demonstrate a method to stabilize junctions made with NbxSi1-x barriers by intentionally annealing wafers during fabrication.