  We have established a rigorous theoretical formalism for Floquet engineering,
or investigating and eventually tailoring most crucial electronic properties of
tetragonal molybdenum disulfide (1T$^\prime$-MoS$_2$), by applying an external
high-frequency dressing field in the off-resonant regime. It was recently
demonstrated that monolayer semiconducting1T$^\prime$-MoS$_2$ may assume a
distorted tetragonal structure which exhibits tunable and gapped spin- and
valley-polarized tilted Dirac bandstructure. From the viewpoint of electronics,
1T$^\prime$-MoS$_2$ is one of the most technologically promising nanomaterials
and a novel representative of an already famous family of transition metal
dichalcogenides. The obtained dressed states strongly depend on the
polarization of the applied irradiation and reflect the full complexity of the
initial low-energy Hamiltonian of non-irradiated material. We have calculated
and analyzed the obtained electron dressed states for linear and circular types
of the polarization of the applied field focusing on their symmetrical
properties, anisotropy, tilting and bandgaps, as well as topological
signatures. Since a circularly polarized dressing field is also known to induce
a transition into a new state with broken time-reversal symmetry and a non-zero
Chern number, the combination of these topologically non-trivial phases and
transitions between them could reveal some truly unique and earlier unknown
phenomena.
