  Graphene irradiated by a circularly polarized laser has been predicted to be
a Floquet topological insulator showing a laser-induced quantum Hall effect. A
circularly polarized laser also drives the system out of equilibrium resulting
in non-thermal electron distribution functions that strongly affect transport
properties. Results are presented for the Hall conductance for two different
cases. One is for a closed system such as a cold-atomic gas where transverse
drift due to non-zero Berry curvature can be measured in time of flight
measurements. For this case the effect of a circularly polarized laser that has
been switched on suddenly is studied. The second is for an open system coupled
to an external reservoir of phonons. While for the former, the Hall conductance
is far from the quantized limit, for the latter, coupling to a sufficiently low
temperature reservoir of phonons is found to produce effective cooling, and
thus an approach to the quantum limit, provided the frequency of the laser is
large as compared to the band-width. For laser frequencies comparable to the
band-width, strong deviations from the quantum limit of conductance is found
even for a very low temperature reservoir, with the precise value of the Hall
conductance determined by a competition between reservoir induced cooling and
the excitation of photo-carriers by the laser. For the closed system, the
electron distribution function is determined by the overlap between the initial
wavefunction and the Floquet states which can result in a Hall conductance
which is opposite in sign to that of the open system.
