  Optical conductivity measurements may provide access to distinct signatures
of Floquet electronic phases, which are described theoretically by their
quasienergy band structures. We characterize experimental observables of the
Floquet graphene antidot lattice (FGAL), which we introduced previously [Phys.
Rev. B 104, 174304 (2021)]. On the basis of Floquet linear response theory, the
real and imaginary parts of the longitudinal and Hall optical conductivity are
computed as a function of probe frequency. We find that the number and
positions of peaks in the response function are distinctive of the different
Floquet electronic phases, and identify multiple properties with no equilibrium
analog. First, for several intervals of probe frequencies, the real part of the
conductivity becomes negative. We argue this is indicative of a subversion of
the usual Joule heating mechanism: The Floquet drive causes the material to
amplify the power of the probe, resulting in gain. Additionally, while the Hall
response vanishes at equilibrium, the real and imaginary parts of the Floquet
Hall conductivity are non-zero and can be as large as the longitudinal
components. Lastly, driving-induced localization tends to reduce the overall
magnitude of and to flatten out the optical conductivity signal. From an
implementation standpoint, a major advantage of the FGAL is that the
above-bandwidth driving limit is reached with photon energies that are at least
twenty times lower than that required for the intrinsic material, allowing for
significant band renormalization at orders-of-magnitude smaller intensities.
Our work provides the necessary tools for experimentalists to map reflectance
data to particular Floquet phases for this novel material.
