Quantum Entanglement and Correlation Lengths of a S-wave Superconductors in the Presence of a Weak Constant External Potential

By considering a s-wave Bardeen-Cooper-Schrieffer superconductor, as a many body system, subject to a weak constant external potential, U, using perturbed linearized Gorkov equations at zero temperature and calculating perturbed Green's functions up to the first approximation, we obtain the two-particle space-spin density matrix of the system. Then, we investigate the effect of the potential on bipartite entanglement (via concurrence) of electron spins of a Cooper pair and also quantum discord in terms of the potential and the relative distance of electrons of a Cooper pair, r. At some fixed values of r, concurrence is zero and does not change until U increases and receives to a special value. Specially, quantum entanglement length and quantum correlation length (in which quantum discord becomes zero) with respect to the potential are derived. We result that by increasing the potential, these lengths are increased. At higher values of U, quantum correlation length is not very sensitive to changes in U. Finally, the relation between these lengths is given.