  Floquet states can be realized in quantum systems driven by continuous
time-periodic perturbations. It is known that a state known as the Floquet Weyl
semimetal can be realized when free Dirac fermions are placed in a rotating
electric field. What will happen if strong interaction is introduced to this
system? Will the interaction wash out the characteristic features of Weyl
semimetals such as the Hall response? Is there a steady state and what is its
thermodynamic behavior? We answer these questions using AdS/CFT correspondence
in the $\mathcal{N}=2$ supersymmetric massless QCD in a rotating electric field
in the large $N_c$ limit realizing the first example of a "holographic Floquet
state". In this limit, gluons not only mediate interaction, but also act as an
energy reservoir and stabilize the nonequilibrium steady state (NESS). We
obtain the electric current induced by a rotating electric field: In the high
frequency region, the Ohm's law is satisfied, while we recover the DC nonlinear
conductivity at low frequency, which was obtained holographically in a previous
work. The thermodynamic properties of the NESS, e.g., fluctuation-dissipation
relation, is characterized by the effective Hawking temperature that is defined
from the effective horizon giving a holographic meaning to the "periodic
thermodynamic" concept. In addition to the strong (pump) rotating electric
field, we apply an additional weak (probe) electric field in the spirit of the
pump-probe experiments done in condensed matter experiments. Weak DC and AC
probe analysis in the background rotating electric field shows Hall currents as
a linear response, therefore the Hall response of Floquet Weyl semimetals
survives at the strong coupling limit. We also find frequency mixed response
currents, i.e., a heterodyning effect, characteristic to periodically driven
Floquet systems.
