  According to the holographic principle, the description of a volume of space
can be thought of as encoded on its boundary. Holographic principle establishes
equivalence, or duality, between theoretical description of volume physics,
which involves gravity, and the gravity-free field theory, which describes
physics on its surface. While generally accepted as a theoretical framework, so
far there was no known experimental system which would exhibit explicit
holographic duality and be amenable to direct experimental testing. Here we
demonstrate that nonlinear optics of hyperbolic metamaterials admits such a
dual holographic description. Wave equation which describes propagation of
extraordinary light through the volume of metamaterial exhibits 2+1 dimensional
Lorentz symmetry. The role of time in the corresponding effective 3D Minkowski
spacetime is played by the spatial coordinate aligned with the optical axis of
the material. Nonlinear optical Kerr effect bends this spacetime resulting in
effective gravitational interaction between extraordinary photons. On the other
hand, a holographic dual theory may be formulated on the metamaterial surface,
which describes its nonlinear optics via interaction of cylindrical surface
plasmons possessing conserved charges proportional to their angular momenta.
Potential implications of this duality for superconductivity of hyperbolic
metamaterials are discussed.
