\textbf{Ultrafast Response of the Hubbard Model: Non-adiabatic TDDFT}$+$\textbf{DMFT versus Non-equilibrium DMFT Solution}*

We study the ultrafast response of electrons in the one-band Hubbard model to an external laser-pulse perturbation by using the Non-adiabatic~Time-Dependent Density Functional Theory $+$ Dynamical Mean-Field Theory (TDDFT$+$DMFT) approach.~The corresponding~exchange-correlation kernel (XC) is obtained from the DMFT charge susceptibility by using the~Quantum Monte Carlo~solver for the impurity problem. Detailed analysis of the time-dependent excited charge density, the Fermi distribution function, and the spatially nonhomogeneous response (metallic domain growth), is performed for different~values for the carrier density and local Coulomb repulsion. We compare the results with~the corresponding~non-equilibrium DMFT~solutions, and demonstrate that non-adiabaticity (frequency-dependence) of the XC kernel is important in order to~reproduce the non-equilibrium~DMFT solution. Also, from the numerical results for the charge susceptibility,~we obtain an approximate analytical expression for the XC kernel.~Using this kernel, we reveal possible types of "elementary" excitations and the dynamics of metallic domain growth in the case of the one-band Hubbard model. Possible generalization of the approach to the multi-orbital case is discussed.