  We present a comprehensive analysis of the the nonlinear optical Kerr effect
in graphene. We directly solve the S-matrix element to calculate the absorption
rate, utilizing the Volkov-Keldysh- type crystal wave functions. We then
convert to the nonlinear refractive index coefficients through the
Kramers-Kronig relation. In this formalism, the source of Kerr nonlinearity is
the interplay of optical fields that cooperatively drive the transition from
valence to conduction band. This formalism makes it possible to identify and
compute the rates of distinct nonlinear processes that contribute to the Kerr
nonlinear refractive index coefficient. The four identified mechanisms are two
photon absorption, Raman transition, self coupling, and quadratic AC Stark
effect. We also present a comparison of our theory with recent experimental and
theoretical results.
