  A distant-neighbor quantum-mechanical method is used to study the nonlinear
optical wave mixing in graphene nanoflakes (GNFs), including sum- and
difference-frequency generation, as well as four-wave mixing. Our analysis
shows that molecular-scale GNFs support quantum plasmons in the visible
spectrum region, and significant enhancement of nonlinear optical wave mixing
is achieved. Specifically, the second- and third-order wave-mixing
polarizabilities of GNFs are dramatically enhanced, provided that one (or more)
of the input or output frequencies coincide with a quantum plasmon resonance.
Moreover, by embedding a cavity into hexagonal GNFs, we show that one can break
the structural inversion symmetry and enable otherwise forbidden second-order
wave mixing, which is found to be enhanced by the quantum plasmon resonance
too. This study reveals that the molecular-sized graphene could be used in the
quantum regime for nanoscale nonlinear optical devices and ultrasensitive
molecular sensors.
