Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential

We present a novel numerical method and algorithm for the solution of the 3D axially symmetric time-dependent Schr\"odinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the split-operator approximation of the exponential operator with the implicit equations of second order cylindrical 2D Crank-Nicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full Crank-Nicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the split-operator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a few-cycle laser pulse with linear polarization

Quantum entanglement in strong-field ionization

We investigate the time-evolution of quantum entanglement between an electron, liberated by a strong few-cycle laser pulse, and its parent ion-core. Since the standard procedure is numerically prohibitive in this case, we propose a novel way to quantify the quantum correlation in such a system: we use the reduced density matrices of the directional subspaces along the polarization of the laser pulse and along the transverse directions as building blocks for an approximate entanglement entropy. We present our results, based on accurate numerical simulations, in terms of several of these entropies, for selected values of the peak electric field strength and the carrier-envelope phase difference of the laser pulse. The time evolution of the mutual entropy of the electron and the ion-core motion along the direction of the laser polarization is similar to our earlier results based on a simple one-dimensional model. However, taking into account also the dynamics perpendicular to the laser polarization reveals a surprisingly different entanglement dynamics above the laser intensity range corresponding to pure tunneling: the quantum entanglement decreases with time in the over-the-barrier ionization regime

Density-based one-dimensional model potentials for strong-field simulations in He\text{He}, H2+\text{H}_{2}^{+} and H2\text{H}_{2}

We present results on the accurate one-dimensional (1D) modeling of simple atomic and molecular systems excited by strong laser fields. We use atomic model potentials that we derive from the corrections proposed earlier using the reduced ground state density of a three-dimensional (3D) single-active electron atom. The correction involves a change of the asymptotics of the 1D Coulomb model potentials while maintaining the correct ground state energy. We present three different applications of this method: we construct correct 1D models of the hydrogen molecular ion, the helium atom and the hydrogen molecule using improved parameters of existing soft-core Coulomb potential forms. We test these 1D models by comparing the corresponding numerical simulation results with their 3D counterparts in typical strong-field physics scenarios with near- and mid-infrared laser pulses, having peak intensities in the $10^{14}-10^{15}\,\mathrm{W/cm}^2$ range, and we find an impressively increased accuracy in the dynamics of the most important atomic quantities on the time scale of the excitation. We also present the high-order harmonic spectra of the He atom, computed using our 1D atomic model potentials. They show a very good match with the structure and phase obtained from the 3D simulations in an experimentally important range of excitation amplitudes

Oscillations in Quantum Entanglement During Rescattering

We study the time evolution of quantum entanglement between an electron and its parent ion during the rescattering due to a strong few-cycle laser pulse. Based on a simple one-dimensional model, we compute the Neumann entropy during the process for several values of the carrier-envelope phase. The local maxima of the oscillations in the Neumann entropy coincide with the zero crossings of the electric field of the laser pulse. We employ the Wigner function to qualitatively explain the quantum dynamics of rescattering in the phase space.Comment: 2 page