  We investigate the effect of phenomenological relaxation parameters on the
third order optical nonlinearity of doped graphene by perturbatively solving
the semiconductor Bloch equation around the Dirac points. An analytic
expression for the nonlinear conductivity at zero temperature is obtained under
the linear dispersion approximation. With this analytic formula as starting
point, we construct the conductivity at finite temperature and study the
optical response to a laser pulse of finite duration. We illustrate the
dependence of several nonlinear optical effects, such as third harmonic
generation, Kerr effects and two photon absorption, parametric frequency
conversion, and two color coherent current injection, on the relaxation
parameters, temperature, and pulse duration. In the special case where one of
the electric fields is taken as a dc field, we investigate the dc-current and
dc-field induced second order nonlinearities, including dc-current induced
second harmonic generation and difference frequency generation.
