We explore the potential of a static electric field to induce Anderson localization of light in a large three-dimensional (3D) cloud of randomly distributed, immobile atoms with a degenerate ground state (total angular momentum $J_g = 0$) and a three-fold degenerate excited state ($J_e = 1$). We study both the spatial structure of quasimodes of the atomic cloud and the scaling of the Thouless number with the size of the cloud. Our results indicate that unlike the static magnetic field, the electric field does not induce Anderson localization of light by atoms. We explain this conclusion by the incomplete removal of degeneracy of the excited atomic state by the field and the relatively strong residual dipole-dipole coupling between atoms which is weaker than in the absence of external fields but stronger than in the presence of a static magnetic field. A joint analysis of these results together with our previous results concerning Anderson localization of scalar waves and light suggests the existence of a critical strength of dipole-dipole interactions that should not be surpassed for Anderson localization to be possible in 3D.
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