  Intense efforts have been made in recent years to realize nonlinear optical
interactions at the single-photon level. Much of this work has focused on
achieving strong third-order nonlinearities, such as by using single atoms or
other quantum emitters while the possibility of achieving strong second-order
nonlinearities remains unexplored. Here, we describe a novel technique to
realize such nonlinearities using graphene, exploiting the strong per-photon
fields associated with tightly confined graphene plasmons in combination with
spatially nonlocal nonlinear optical interactions. We show that in properly
designed graphene nanostructures, these conditions enable extremely strong
internal down-conversion between a single quantized plasmon and an entangled
plasmon pair, or the reverse process of second harmonic generation. A separate
issue is how such strong internal nonlinearities can be observed, given the
nominally weak coupling between these plasmon resonances and free-space
radiative fields. On one hand, by using the collective coupling to radiation of
nanostructure arrays, we show that the internal nonlinearities can manifest
themselves as efficient frequency conversion of radiative fields at extremely
low input powers. On the other hand, the development of techniques to
efficiently couple to single nanostructures would allow these nonlinear
processes to occur at the level of single input photons.
