We consider Dirac neutrinos interacting with background fermions in the frame of the standard model. We demonstrate that a time-dependent effective potential is quite possible in a protoneutron star (PNS) at certain stages of its evolution. For the first time, we formulate a nonperturbative treatment of neutrino processes in a matter with arbitrary time-dependent effective potential. Using linearly growing effective potential, we study the typical case of a slowly varying matter interaction potential. We calculate differential mean numbers of $ u bar{ u}$ pairs created from the vacuum by this potential and find that they crucially depend on the magnitude of masses of the lightest neutrino eigenstate. These distributions uniformly span up to $sim 10$ eV energies for muon and tau neutrinos created in PNS core due to the compression just before the hydrodynamic bounce and up to $sim 0.1$ eV energies for all three active neutrino flavors created in the neutronization. Considering different stages of the PNS evolution, we derive constraints on neutrino masses, $m_{ u}lesssim (10^{-8}-10^{-7})$ eV corresponding to the nonvanishing $ u bar{ u}$ pairs flux produced by this mechanism. We show that one can distinguish such coherent flux from chaotic fluxes of any other origin. Part of these neutrinos, depending on the flavor and helicity, are bounded in the PNS, while antineutrinos of any flavor escape the PNS. If the created pairs are $ u_{e}bar{ u}_{e}$, then a part of the corresponding neutrinos also escape the PNS. The detection of $ u $ and $bar{ u}$ with such low energies is beyond current experimental techniques.
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