  In transition metal dichalcogenides (TMDCs), charge carriers have spin,
pseudospin, and valley degrees of freedom associated with magnetic moments. The
monolayers and bilayers of the TMDCs, in particular, MoS$_2$, lead strong
couplings between the spin and pseudospin effects. This feature have drawn
attention to TMDCs for their potential use in advanced tech devices. Meanwhile,
high-order harmonic generation (HHG) has recently been applied to the
characterization of the electronic structure of solids, such as energy
dispersion, Berry-curvature, and topological properties. Here, we show
theoretical results obtained with the `philosophy' of using HHG to investigate
the structural effects of the monolayer and bilayers of MoS$_2$ on nonlinear
optical emission. We use a simple model for MoS$_2$ in the 3R AB staking. We
find that the pseudospin and valley indexes (the Berry curvature and the dipole
transition matrix element) in TMDC driven by circularly polarized laser (CPL)
can encode in the high energy photon emissions. This theoretical investigation
is expected to pave the way for the ultrafast manipulation of valleytronics and
lead to new questions concerning the spin-obit-coupling (SOC) effects on TMDC
materials, Weyl Semimetals, and topological phases and transitions in
topological insulators.
