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.
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