Double ionization (DI) is a fundamental process that despite its apparent simplicity provides rich opportunities for probing and controlling the electronic motion. Even for the simplest multielectron atom, helium, new DI mechanisms are still being found. To first order in the field strength, a strong external field doubly ionizes the electrons in helium such that they are ejected into the same direction (front-to-back motion). The ejection into opposite directions (back-to-back motion) cannot be described to first order, making it a challenging target for control. Here, we address this challenge and optimize the field with the objective of back-to-back double ionization using a (1 + 1)-dimensional model. The optimization is performed using four different control procedures: (1) short-time control, (2) derivative-free optimization of basis expansions of the field, (3) the Krotov method, and (4) control of the classical equations of motion. All four procedures lead to fields with dominant back-to-back motion. All the fields obtained exploit essentially the same two-step mechanism leading to back-to-back motion: first, the electrons are displaced by the field into the same direction. Second, after the field turns off, the nuclear attraction and the electron–electron repulsion combine to generate the final motion into opposite directions for each electron. By performing quasi-classical calculations, we confirm that this mechanism is essentially classical.
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Throughout, we use atomic units (me = e = 4πϵ0 = a0 = 2|ERyd| = 1) unless indicated otherwise.
It may also happen that the larger fraction of the doubly ionized wavepacket is in D1, but then, typically its main component is not along the u = 0 line, indicating not a true back-to-back motion.
To simplify notation, we drop the hats in the coordinate operators and . It will be clear from the context whether operators are meant or not.
Here, global optimization means not an optimization to a global minimum but that the field is optimized globally.
Note that the field in the length gauge would have components with both positive and negative values. Note further that due to symmetry of the potential in u, the overall sign of the field does not matter.
The larger momentum components for v = 0 are due to the form of the potential and as such are “ground state quantities” and not very interesting for this study.