Real-time quantum trajectories for classically allowed dynamics in strong laser fields

TL;DR

This paper investigates the relevance of quantum features in classical trajectories of electrons in strong laser fields, demonstrating that classical solutions can incorporate quantum effects and linking imaginary trajectory components to initial wavepacket size.

Contribution

It introduces an exact classical Hamilton-Jacobi solution that accounts for quantum effects and clarifies the physical origin of complex trajectory components in strong field dynamics.

Findings

Classical equations can model quantum effects in electron dynamics.

Imaginary parts of trajectories relate to initial wavepacket size.

Purely classical trajectories are insufficient for localized wavepackets.

Abstract

Both the physical picture of the dynamics of atoms and molecules in intense infrared fields and its theoretical description use the concept of electron trajectories. Here we address a key question which arises in this context: Are distinctly quantum features of these trajectories, such as the complex-valued coordinates, physically relevant in the classically allowed region of phase space, and what is their origin? First, we argue that solutions of classical equations of motion can account for quantum effects. To this end, we construct an exact solution to the classical Hamilton-Jacobi equation which accounts for dynamics of the wave packet, and show that this solution is physically correct in the limit $\hbar \to 0$. Second, we show that imaginary components of classical trajectories are directly linked to the finite size of the initial wavepacket in momentum space. This way, if the electronic wavepacket produced by optical tunneling in strong infrared fiels is localised both in coordinate and momentum, its motion after tunneling {\em ipso facto\/} cannot be described with purely classical trajectories -- in contrast to popular models in the literature.