  We demonstrate the photonic Floquet topological insulator (PFTI) in an atomic
vapor with nonlinear susceptibilities. The interference of three coupling
fields splits the energy levels periodically to form a periodic refractive
index structure with honeycomb symmetry that can be adjusted by the choice of
frequency detunings and intensities of the coupling fields, which all affect
the appearance of Dirac cones in the momentum space. When the honeycomb lattice
sites are helically ordered along the propagation direction, we obtain a PFTI
in the atomic vapor in which an obliquely incident beam moves along the zigzag
edge without scattering energy into the PFTI, due to the confinement of the
edge states. The appearance of Dirac cones and the formation of PFTI is
strongly affected by the nonlinear susceptibilities; i.e. the PFTI can be shut
off by the third-order nonlinear susceptibility and re-opened up by the
fifth-order one.
