  Electron dynamics of anatase TiO$_2$ under the influence of ultrashort and
intense laser field is studied using the real-time time-dependent density
functional theory (TDDFT). Our findings demonstrate the effectiveness of TDDFT
calculations in modeling the electron dynamics of solids during ultrashort
laser excitation, providing valuable insights for designing and optimizing
nonlinear photonic devices. We analyze the perturbative and non-perturbative
responses of TiO$_2$ to 30 fs laser pulses at 400 and 800 nm wavelengths,
elucidating the underlying mechanisms. At 400 nm, ionization via single photon
absorption dominates, even at very low intensities. At 800 nm, we observe
ionization through two-photon absorption within the intensity range of
$1\times10^{10}$ to $9\times10^{12}$ W/cm$^2$, with a transition from
multiphoton to tunneling ionization occurring at $9\times10^{12}$ W/cm$^2$. We
observe a sudden increase in energy and the number of excited electrons beyond
$1\times10^{13}$ W/cm$^2$, leading to their saturation and subsequent
laser-induced damage. We estimate the damage threshold of TiO$_2$ for 800 nm to
be 0.1 J/cm$^2$. In the perturbative regime, induced currents exhibit a phase
shift proportional to the peak intensity of the laser pulse. This phase shift
is attributed to the intensity-dependent changes in the number of free
carriers, indicative of the optical Kerr effect. Leveraging the linear
dependence of phase shift on peak intensities, we estimate the nonlinear
refractive index ($n_2$) of TiO$_2$ to be $3.54\times10^{-11}$ cm$^2$/W.
