Theory of Andreev and shot noise spectroscopy for topological superconductors probed by $s$-wave superconducting tips

Abstract: Scanning tunneling microscopy (STM) and spectroscopy (STS) with $s$-wave superconducting tips has been widely applied to probe exotic superconductors, but its potential for investigating topological superconductors remains unclear. In junctions between an $s$-wave superconductor and a topological superconductor, the dominant tunneling process is Andreev reflection, in which Cooper pairs from the $s$-wave superconductor tunnel as particle--hole excitations into the surface state of the topological superconductor. In this work, we theoretically investigate the fundamental properties of Andreev and shot noise spectroscopy on topological superconductors, focusing on the $dI/dV$ characteristics and current noise. We develop a real-time description of an effective tunneling action incorporating Andreev reflection processes in the Keldysh formalism and derive analytical expressions for the Andreev reflection current and the associated current noise. Furthermore, we perform numerical simulations for representative topological superconductors and provide a catalog of $dI/dV$ spectra and the Fano factor. Our results establish guidelines for probing topological superconductivity using STM with $s$-wave superconducting tips, and provide theoretical benchmarks for future STS experiments.