Signatures of odd-frequency pairing in the Josephson junction current noise

Odd frequency (odd-$\omega$) electron pair correlations naturally appear at the interface between BCS superconductors and other materials. The detection of odd-$\omega$ pairs, which are necessarily non-local 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 non-local time correlations, are suitable for detecting these pairs. In this work, we study the signatures of odd-$\omega$ 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-$\omega$), odd-$\omega$ pair amplitudes, and electron-hole correlation functions. Odd-$\omega$ pairing, which is inter-lead (between electrons at different sides of the junction), provides a positive contribution to the noise, becoming maximal at a superconducting phase difference of $\pi$. In contrast, intra-lead even-$\omega$ pair amplitude tends to reduce the noise, except for a region close to $\pi$, controlled by the transmission of the junction.