  The quantum kinetic framework provides a versatile method for investigating
the dynamical optical and transport currents of crystalline solids. In this
paper, starting from the density-matrix equations of motion, we present a
general theoretical path to obtain nonlinear optical responses elegantly and
transparently. We devise a kinetic theory applicable to materials with
arbitrary band structures and captures intraband and interband coherence
effects, finite Fermi surfaces, and disorder effects. We present a
classification of nonlinear optical currents arising from the interference of
interband and intraband components of the density matrix with distinct symmetry
and quantum geometrical origin for each contribution. In this context, we
report four findings. (i) The Fermi Golden Rule approach is insufficient to
derive the correct expression for the injection current, a shortcoming that we
remedy in our theory while associating the injection current with the
intraband-interband contribution to the second-order density matrix. (ii) The
interband-intraband contribution yields a resonant current that survives
irrespective of any symmetry constraint in addition to the well-known anomalous
nonlinear current (non-resonant), which requires time-reversal symmetry. (iii)
Quite generally, the nonlinear current is significantly enhanced by
contributions from the finite Fermi surface. (iv) The finite Fermi surface and
Fermi sea additionally lead to sizable novel nonlinear effects via
contributions we term double resonant and higher-order pole. As an
illustration, we compute the nonlinear response of topological antiferromagnet
CuMnAs and thin film tilted Weyl semimetals dominated by interband coherence
effects. We find that the nonlinear response of CuMnAs is responsive to the
direction of finite magnetization field and choice of the polarization angle of
the beam, while Weyl semimetal only to tilt.
