In this dissertation, we theoretically study the nonlinear response of ultrafast, ultrastrong and ultrashort optical pulse in 2D topological quantum materials, including graphene-like systems and Haldane model quantum dots. We investigate topological resonance, nonlinear optical absorbance and high harmonic generation (HHG) process. We study conditions under which topological resonance occurs in gapped graphene. For monolayer graphene, the topological resonance can occur only in the field of an elliptically polarized pulse while for graphene systems with many layers, the topological resonance can also be realized in a linearly polarized pulse.

We further study the Haldane model quantum dots, where a key tuning parameter is the phase accumulated by an electron hopping between next-nearest neighbors. The absorbance strongly depends on the frequency of the pulse. The nonlinear absorbance is shown to vary significantly with pulse frequency and amplitude. At low frequencies, much less than the bandgap, absorbance depends strongly on pulse amplitude and exhibits maxima at intermediate phase values. As the frequency increases and approaches the bandgap, this dependence weakens, and absorbance peaks at a phase of 90°, coinciding with the minimum bandgap. Nonlinear electron dynamics in such quantum dot systems changes from almost reversible one at small pulse frequencies to highly irreversible dynamics at large frequencies of the pulse.

With increasing the phase from its zero value to 90° in different size of quantum dots, the low-energy electron states in the conduction and valence bands become more localized near the edges of the quantum dot resulting in suppression of the band gap, enhancement of the dipole inter-band coupling, and strong suppression of the average low-energy electron density of the states. As a result, the nonlinear response of the electron system of a Haldane model quantum dot is the strongest at intermediate values of the phase, ≈ 30° − 40°. At these values of the phase, the electron dynamics is irreversible and a few first high-order harmonics have the largest intensities. When the phase approaches 90° value, the generation of high harmonics is strongly suppressed.