  In this work we use Floquet-Bloch theory to study the influence of circularly
and linearly polarized light on two-dimensional band structures with semi-Dirac
band touching points, taking the anisotropic nearest neighbor hopping model on
the honeycomb lattice as an example. We find circularly polarized light opens a
gap and induces a band inversion to create a finite Chern number in the
two-band model. By contrast, linearly polarized light can either open up a gap
(polarized in the quadratically dispersing direction) or split the semi-Dirac
band touching point into two Dirac points (polarized in the linearly dispersing
direction) by an amount that depends on the amplitude of the light. Motivated
by recent pump-probe experiments, we investigated the non-equilibrium spectral
properties and momentum-dependent spin-texture of our model in the Floquet
state following a quench in absence of phonons, and in the presence of phonon
dissipation that leads to a steady-state independent of the pump protocol.
Finally, we make connections to optical measurements by computing the frequency
dependence of the longitudinal and transverse optical conductivity for this
two-band model. We analyze the various contributions from inter-band
transitions and different Floquet modes. Our results suggest strategies for
optically controlling band structures and experimentally measuring topological
Floquet systems.
