Superconductor-ferromagnetic heterostructures have been suggested as one of the most promising alternatives of realizing odd-frequency superconductivity. In this work we consider the limit of shrinking the ferromagnetic region to the limit of a single impurity embedded in a conventional superconductor, which gives rise to localized Yu-Shiba-Rusinov (YSR) bound states with energies inside the superconducting gap. We demonstrate that all the sufficient ingredients for generating odd-frequency pairing are present at the vicinity of these impurities. We investigate the appearance of all possible pair amplitudes in accordance with the Berezinskii SP* OT * = -1 rule, with the symmetry under the exchange of spin, spatial, orbital (in our case O = +1), and time index, respectively. We study the spatial and frequency dependence of the possible pairing amplitudes, analyzing their evolution with impurity strength and identifying a reciprocity between different symmetries related through impurity scattering. We show that the odd-frequency spin-triplet pairing amplitude dominates at the critical impurity strength, where the YSR states merge at the middle of the gap, while the even components are quenched close to the impurity. We also show that the spin-polarized local density of states exhibits the same spatial and frequency behavior as the odd-co spin-triplet component at the critical impurity strength.

Odd-frequency (odd−ω) electron pair correlations naturally appear at the interface between BCS superconductors and other materials. The detection of odd−ω pairs, which are necessarily nonlocal in time, is still an open problem. The main reason is that they do not contribute to static measurements described by time-local correlation functions. Therefore, dynamical measurements, which depend on nonlocal time correlations, are suitable for detecting these pairs. In this work, we study the signatures of odd−ω pairs in the supercurrent noise through a weak link between two superconductors at different superconducting phases. We show that the finite-frequency current noise can be decomposed into three different contributions coming from even-frequency (even−ω), odd−ω pair amplitudes, and electron-hole correlation functions. Odd−ω pairing, which is interlead (between electrons at different sides of the junction), provides a positive contribution to the noise, becoming maximal at a superconducting phase difference of π. In contrast, intralead even−ω pair amplitude tends to reduce the noise, except for a region close to π controlled by the transmission of the junction.