The electro-optic effect, where the refractive index of a medium is modified by an electric field, is of central importance in nonlinear 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 could have profound implications for precision electrometry and nonlinear optics at the single-photon level. Here we propose and demonstrate a giant d.c. electro-optic effect on the basis of polarizable (Rydberg) dark states. When a medium is prepared in a dark state consisting of a superposition of ground and Rydberg energy levels, it becomes transparent and acquires a refractive index that is dependent on the energy of the highly polarizable Rydberg state. We demonstrate phase modulation of the light field in the Rydberg-dark-state medium and measure an electro-optic coefficient that is more than six orders of magnitude larger than in usual Kerr media
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