We analyze the attosecond electron dynamics in hydrogen molecular ion driven by an external intense laser field using ab-initio numerical simulations of the corresponding time-dependent Schr{{o}}dinger equation and Bohmian trajectories. To this end, we employ a one-dimensional model of the molecular ion in which the motion of the protons is frozen. The results of the Bohmian trajectory calculations do agree well with those of the ab-initio simulations and clearly visualize the electron transfer between the two protons in the field. In particular, the Bohmian trajectory calculations confirm the recently predicted attosecond transient localization of the electron at one of the protons and the related multiple bunches of the ionization current within a half cycle of the laser field. Further analysis based on the quantum trajectories shows that the electron dynamics in the molecular ion can be understood via the phase difference accumulated between the Coulomb wells at the two protons. Modeling of the dynamics using a simple two-state system leads us to an explanation for the sometimes counter-intuitive dynamics of an electron opposing the classical force of the electric field on the electron.
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