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Owing to its peculiar energy dispersion, the quantum capacitance property of graphene can be exploited in a two- dimensional layered capacitor configuration. Using graphene and boron nitride, respectively, as the electrodes and the insulating dielectric, a strongly nonlinear behavior at zero bias and small voltages is obtained. When the temperature is sufficiently low, the strong nonlinear interaction emerging from the quantum capacitance exhibits a diverse range of phenomena. The proposed structure could take over the functionalities of nonlinear elements in many cryogenic quantum systems, and in particular, quantum electrooptics. It is shown that ultrastrong coupling is easily reached with small number of pump photons at temperatures around 1 K and capacitor areas of the order of 1 mu m(2). A measure of anharmonicity is defined and as potential applications, a qubit design as well as schemes for non- reciprocal devices such as an electromagnetic frequency circulator are discussed.