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Efficient frequency shifting and beam splitting are important for a wide range of applications, including atomic physics^,, microwave photonics^{, , –}, optical communication^, and photonic quantum computing^{, , , , –}. However, realizing gigahertz-scale frequency shifts with high efficiency, low loss and tunability—in particular using a miniature and scalable device—is challenging because it requires efficient and controllable nonlinear processes. Existing approaches based on acousto-optics^{,, –}, all-optical wave mixing^{,,, , , –} and electro-optics^{, , , –} are either limited to low efficiencies or frequencies, or are bulky. Furthermore, most approaches are not bi-directional, which renders them unsuitable for frequency beam splitters. Here we demonstrate electro-optic frequency shifters that are controlled using only continuous and single-tone microwaves. This is accomplished by engineering the density of states of, and coupling …