Decoherence of Cooper pairs and subgap magnetoconductance of superconducting hybrids

We demonstrate that electron-electron interactions fundamentally restrict the penetration length of superconducting correlations into a diffusive normal metal (N) attached to a superconductor (S). We evaluate the subgap magnetoconductance $G$ of SN hybrids in the presence of electron-electron interactions and demonstrate that the effect of the magnetic field on $G$ is twofold: It includes ($i$) additional temperature independent dephasing of Cooper pairs and ($ii$) Zeeman splitting between the states with opposite spins. The dephasing length of Cooper pairs can be directly extracted from measurements of the subgap magnetoconductance in SN systems at low temperatures.