Materials with massless Dirac fermions can possess exceptionally strong and widely tunable optical nonlinearities. Experiments on graphene monolayer have indeed found very large third-order nonlinear responses, but the reported variation of the nonlinear optical coefficient by orders of magnitude is not yet understood. A large part of the difficulty is the lack of information on how doping or chemical potential affects the different nonlinear optical processes. Here we report the first experimental study, in corroboration with theory, on third harmonic generation (THG) and four-wave mixing (FWM) in graphene that has its chemical potential tuned by ion-gel gating. THG was seen to have enhanced by ~30 times when pristine graphene was heavily doped, while difference-frequency FWM appeared just the opposite. The latter was found to have a strong divergence toward degenerate FWM in undoped graphene, leading to a giant third-order nonlinearity. These truly amazing characteristics of graphene come from the possibility to gate-control the chemical potential, which selectively switches on and off one- and multi-photon resonant transitions that coherently contribute to the optical nonlinearity, and therefore can be utilized to develop graphene-based nonlinear optoelectronic devices.
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