  Second order optical nonlinear processes involve the coherent mixing of two
electromagnetic waves to generate a new optical frequency, which plays a
central role in a variety of applications, such as ultrafast laser systems,
rectifiers, modulators, and optical imaging. However, progress is limited in
the mid-infrared (MIR) region due to the lack of suitable nonlinear materials.
It is desirable to develop a robust system with a strong, electrically tunable
second order optical nonlinearity. Here we demonstrate theoretically that
AB-stacked bilayer graphene (BLG) can exhibit a giant and tunable second order
nonlinear susceptibility \chi ^(2) once an in-plane electric field is applied.
\chi^(2) can be electrically tuned from 0 to ~ {10^5 pm/V}, three orders of
magnitude larger than the widely used nonlinear crystal AgGaSe2. We show that
the unusually large \chi^(2) arises from two different quantum enhanced
two-photon processes thanks to the unique electronic spectrum of BLG. The
tunable electronic bandgap of BLG adds additional tunability on the resonance
of \chi^(2), which corresponds to a tunable wavelength ranging from ~2.6 {\mu}m
to ~3.1 {\mu}m for the up-converted photon. Combined with the high electron
mobility and optical transparency of the atomically thin BLG, our scheme
suggests a new regime of nonlinear photonics based on BLG.
