  A circularly polarized a.c. pump field illuminated near resonance on
two-dimensional transition metal dichalcogenides (TMDs) produces an anomalous
Hall effect in response to a d.c. bias field. In this work, we develop a theory
for this photo-induced anomalous Hall effect in undoped TMDs irradiated by a
strong coherent laser field. The strong field renormalizes the equilibrium
bands and opens up a dynamical energy gap where single-photon resonance occurs.
The resulting photon dressed states, or Floquet states, are treated within the
rotating wave approximation. A quantum kinetic equation approach is developed
to study the non-equilibrium density matrix and time-averaged transport
currents under the simultaneous influence of the strong a.c. pump field and the
weak d.c. probe field. Dissipative effects are taken into account in the
kinetic equation that captures relaxation and dephasing. The photo-induced
longitudinal and Hall conductivities display notable resonant signatures when
the pump field frequency reaches the spin-split interband transition energies.
Rather than valley polarization, we find that the anomalous Hall current is
mainly driven by the intraband response of photon-dressed electron populations
near the dynamical gap at both valleys, accompanied by a smaller contribution
due to interband coherences. These findings highlight the importance of
photon-dressed bands and non-equilibrium distribution functions in achieving a
proper understanding of photo-induced anomalous Hall effect in a strong pump
field.
