  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.
