Optimal control of the classical dynamics of an electron in the Morse-soft-Coulomb potential

We introduce the one-dimensional Morse-soft-Coulomb (MsC) potential to model the dynamics of an electron in an atomic potential. This potential has a single parameter that controls the softness of the repulsive barrier and the well depth. When this softening-depth parameter tends to zero, the MsC potential approaches the Coulomb potential with an infinite repulsive barrier, a known successful model for the hydrogen atom. We investigate the ionization under strong XUV laser fields by considering the classical chaotic dynamics in the MsC potential subjected to time-dependent external fields and comparing the results with the ones for the Coulomb potential. We show that the MsC potential reproduces the dynamics and the ionization probabilities of the Coulomb potential for sufficiently small values of the softening parameter. We also investigate the role of the softening parameter in the ionization probability and in the phase-space structures, showing that an increase in its value widens the chaotic sea and consequently elevates the ionization probability. Finally, we address the problem of controlling the dynamics of an electron in the MsC potential from the perspective of optimal control theory whose solution cannot be easily obtained in the case of the Coulomb potential due to the singularity at the origin. We analyze a particular analytical solution to the problem of transferring a given amount of energy to the atom at minimum cost, which has been previously termed the intrinsic solution. We find that the energy deposited in the atom essentially grows when the electron collides with the nucleus. At each collision, the optimal field follows the abrupt change of the electron momentum in the form of a single-cycle pulse. Thus, the optimal intrinsic solution is essentially composed of a sequence of single-cycle pulses separated by increasing time intervals as the electron climbs the anharmonic potential well. Our results show that the MsC potential is a useful simple model for investigating and furnishing insights into the physics of atomic excitation by external fields.