Here we report on an elastic and inelastic neutron scattering study addressing the effect of electric fields on quantum paraelectric SrTiO₃. Our elastic scattering results find small changes as a function of field in a superlattice reflection that sample the octahedral rotations, which is indicative of only weak coupling of octahedral rotation and electric polarization. Using inelastic neutron scattering, we measured the dispersion of the transverse acoustic and transverse optic phonon in SrTiO₃ as a function of applied electric fields. By collecting not only the change in gap, but also of the dispersion, we can better quantify the changes in the lattice dynamics. The findings are put in context to recent DFT calculations predicting the E-field effect on the atomic motions of the lowest lying transverse optical (soft) mode. We find hints of nonlinear coupling to the acoustic mode and to the phonon with polarization perpendicular to the E field, which shows the nonlinearity in the chemical potential that is also relevant when SrTiO₃ is excited by strong E-field (THz) pulses.