Second-harmonic generation [SHG, $I\left(2\omega \right)$] of the three-dimensional topological insulators (3D TIs) provides significant insights into the nontrivial band topology through the second-order nonlinear susceptibility ${\chi }^{\left(2\right)}$ of the topological surface states (TSSs), along with the electric-field-induced SHG resulting from band bending (given by ${\chi }^{\left(3\right)}{E}_{\text{dc}}$). Notably, by carefully tuning the polarization of the incident laser field, second-harmonic intensity measurements make it possible to isolate and analyze the distinct contributions from the bulk $\left({\chi }^{\left(3\right)}{E}_{\text{dc}}\right)$ and the metallic surface states $\left({\chi }^{\left(2\right)}\right)$. Recently reported Sb-doped quaternary alloy ${\mathrm{Bi}}_{2-x}{\mathrm{Sb}}_x{\mathrm{Te}}_{3-y}{\mathrm{Se}}_y$, isostructural to conventional TIs, provides a suitable platform for studying the TSS due to its high bulk resistivity and the Dirac point lying within the bulk band gap. In this work, we report the enhancement of $I\left(2\omega \right)$ upon the application of circularly polarized light, attributed to the breaking of the time-reversal symmetry in the TSS of ${\mathrm{BiSbTe}}_{1.25}{\mathrm{Se}}_{1.75}$ crystal. Additionally, the differential change in SHG intensity $\left[\mathrm{\Delta }I\left({\tau }_{\text{pp}}\right)\right]$ as a function of pump-probe delay time $\left({\tau }_{\text{pp}}\right)$ for a linearly polarized probe exhibits two competing contributions to $E_{\text{dc}}$ from the surface and the bulk. The surface contribution is more pronounced than the bulk contribution, owing to the system's bulk insulating nature. Interestingly, by changing the probe polarization from linear to circular, we show that the second-harmonic response has an additional contribution from ${\chi }^{\left(2\right)}$, thereby extracting the time evolution of $\mathrm{\Delta }{\chi }^{\left(2\right)}\left({\tau }_{\text{pp}}\right)$ arising from the TSS after photoexcitation. Furthermore, we justify the time-dependent second-order response of the TSS by analytically solving the second-order conductivity $\left({\sigma }^{\left(2\right)}\right)$, including the band gap opening at the Dirac point, which depicts a sign change in $\mathrm{\Delta }I$ depending on the shape of the Fermi surface.

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• Nonlinear optical susceptibility
• Symmetry protected topological states
• Topological insulators
• Green's function methods
• Optical second-harmonic generation
• Ultrafast femtosecond pump probe