  There are a variety of nonlinear optical effects including higher harmonic
generations, photovoltaic effects, and nonlinear Kerr rotations. They are
realized by the strong light irradiation to materials that results in nonlinear
polarizations in the electric field. These are of great importance in studying
the physics of excited states of the system as well as for applications to
optical devices and solar cells. Nonlinear properties of materials are usually
described by the nonlinear susceptibilities $\chi$'s, which have complex
expressions including many matrix elements and energy denominators. On the
other hand, a nonequilibrium steady state under a electric field periodic in
time has a concise description in terms of the Floquet bands of electrons
dressed by photons. Here, we theoretically show by using the Floquet formalism
that various nonlinear optical effects, such as the shift current in
noncentrosymmetric materials, photovoltaic Hall response, and photo-induced
change of order parameters under the continuous irradiation of monochromatic
light, can be described in a unified fashion by topological quantities
involving the Berry connection and Berry curvature. It is found that vector
fields defined with the Berry connections in the space of momentum and/or
parameters govern the nonlinear responses. This topological view offers a new
route to design the nonlinear optical materials.
