  The effect of a time-periodic perturbation, such as radiation, on a system
otherwise at equilibrium has been studied in the context of Floquet theory with
stationary states replaced by Floquet states and the energy replaced by
quasienergy. These quasienergy bands in general differ from the energy bands in
their dispersion and, especially in the presence of spin-orbit coupling, in
their states. This may, in some cases, alter the topology when the quasienergy
bands exhibit different topological invariants than their stationary
counterparts. In this work, motivated by advances in pump-probe techniques, we
consider the optical response of driven topological systems when the drive is
not purely periodic but is instead multiplied by a pulse shape/envelope
function. We use real time-evolved states to calculate the optical conductivity
and compare it to the response calculated using Floquet theory. We find that
the conductivity bears a memory of the initial equilibrium state even when the
pump is turned on slowly and the measurement is taken well after the ramp. The
response of the time-evolved system is interpreted as coming from Floquet bands
whose population has been determined by their overlap with the initial
equilibrium state. In particular, at band inversion points in the Brillouin
zone the population of the Floquet bands is inverted as well.
