  The electro-optic effect, where the refractive index of a medium is modified
by an electric field, is of central importance in non-linear optics, laser
technology, quantum optics and optical communications. In general,
electro-optic coefficients are very weak and a medium with a giant
electro-optic coefficient would have profound implications for non-linear
optics, especially at the single photon level, enabling single photon
entanglement and switching. Here we propose and demonstrate a giant
electro-optic effect based on polarizable dark states. We demonstrate phase
modulation of the light field in the dark state medium and measure an
electro-optic coefficient that is more than 12 orders of magnitude larger than
in other gases. This enormous Kerr non-linearity also creates the potential for
precision electrometry and photon entanglement.
