  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 electro-optics. It is shown that ultrastrong coupling is easily reached
with small number of pump photons at temperatures around 1K and capacitor areas
of the order of $1\mu{\textrm{m}}^2$. 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.
