Vector Cooper Pairs and Coherent-Population-Trapping-like States in Ensemble of Interacting Fermions

Using the standard Hamiltonian of the BCS theory, we show that in an ensemble of interacting fermions with the spin 1/2 there exist coherent states $|NC>$, which nullify the Hamiltonian of the interparticle interaction (scattering). These states have an analogy with the well-known in quantum optics the coherent population trapping (CPT) effect. The structure of these CPT-like states corresponds to Cooper pairs with the total spin $S$=1. The found states have a huge degree of degeneracy and carry a macroscopic magnetic moment, that allows us to construct a new model of the magnetism connected with the delocalized electrons in metals (conductors). A principal possibility to apply the obtained results to the superfluid $^3$He is also demonstrated.