The multi-electron dynamics in atoms and molecules is one of the great challenges of current quantum physics, which is at the background of many problems ranging from chemical reactions to superconductivity. A classic example that has been studied for a long time is the so-called "non-sequential double ionization" (NSDI) of atoms by a strong laser field. In this case, "non-sequential" means that the ionization does not take place step by step, i. first the simple ionization of the atom and later the ionization of the singly charged ion, but immediately in a process that can not be decomposed into single steps. This is only possible if the two electrons are correlated. Although ionization seems to be a typical quantum mechanical process (which, according to some textbooks, proves the quantum nature of light), purely classical simulations have successfully described the available data, at least qualitatively. If this were generally true in multi-electron dynamics, then it would have far-reaching consequences, because classical simulations are far simpler than quantum mechanical ones. In fact, accurate quantum mechanical calculations for more than two particles have been virtually impossible. In a recently published paper we show limits for this classical approach. Namely, we attribute a known effect - a qualitative change in the electron-electron-momentum correlation with decreasing laser intensity - to a quantum mechanical cause, namely the interference of the contributions of different ionization scenarios.

# Does quantum mechanics play a role in laser-induced double ionization?

*Fig. 1 Illustration of RESI (rescattering excitation with subsequent ionization) mechanism. The interaction potential between the electron and the laser field is given by the red straight line (at a time when the field pulls the electron to the right, half a period later it is reversed, and the field moves to the left). The blue line represents the "effective potential". It is the sum of the interaction potential and the potential that binds the electron to the doubly positive charged ion. The first released electron, which is driven back through the field to the singly charged ion, is represented by the horizontal green arrow. It transports the second bound electron (vertical green arrows) into an excited state, from which it is freed by field ionization (horizontal broken lines).*

The usual notion is that non-sequential double ionization (NSDI) is a three-step process: first, an electron is released through a tunneling process. This electron then performs an oscillatory motion in the laser field where it can collide again with the ion. The second electron is released. If the kinetic energy of the first electron is sufficient, this can be done in a direct collision ("recollision impact ionization", RII). Otherwise, the colliding electron first transports the bound electron into an excited state, from which it then later tunnels out ("recollision excitation with subsequent ionization", RESI), cf. Fig. 1. The two scenarios lead to different momentum distributions of the two electrons. Namely, if, as in FIG. 2, the number of events is plotted against the longitudinal (parallel to the laser polarization) pulse components of the two electrons, then the RII mechanism results in a concentration in the first and third quadrants (ie, the two electrons tend to While the RESI Mechanimus does not produce such a preference (the electrons also like to travel in the same direction as in the opposite direction). Experimentally, however, it has been found that for argon at low intensities electron emission in the opposite direction is clearly preferred [Liu et al., Phys. Rev. Lett. 101, 053001 (2008)].

*Fig. 2 Examples of the electron-electron momentum distributions expected in different scenarios of double ionization. The two axes correspond to the momentum components of the two electrons parallel to the laser field and the density reflects the number of events with these pulses. The remaining pulse components are not measured or added up. If the two electrons are uncorrelated, one expects a distribution as in (a). The images in (b) and (c) are only possible if there is a correlation. Image (b) corresponds to the RII mechanism: the two electrons go "side by side" in the same direction. The distances of the centers of the two distributions from the origin are again the momentum which the two electrons acquire after their inelastic collision by acceleration in the field; the diameters correspond to the energy that the first electron introduces into the collision. Figure (c) illustrates a possible distribution from the RESI mechanism. Here, the second electron in an excited state can become free half a laser cycle later (or after an odd multiple of half cycles), so that the two electrons run away in the opposite direction.*

The theoretical description of the NSDI has essentially followed three different models: quantum mechanical calculations in the context of the so-called "strong-field approximation," solution of the completely classical two-electron equations of motion and semiclassical models, in which the first electron tunnels out quantum-mechanically, but subsequently the follows classical equations of motion In Rev. Mod. Phys. 84, 1011 (2012)]. Surprisingly, the purely classical and semiclassical models have so far been able to describe quite well all available data.

*Fig. 3 Distribution of longitudinal electron pulses (parallel to laser polarization) in the non-sequential double ionization of argon through a 800 nm laser field at different intensities: (a) and (b) 4 x 1013 W / cm2 (c) and (d) 7 x 1013W / CM22; (e) and (f) 9 x 1013 W / cm 2. In the upper row, the contributions of the different channels (cf the excited states in Fig. 1) are incoherent and in the lower coherent added considering the corresponding phases. The comparison illustrates the dramatic effect of quantum mechanical interference.*

In einer neueren Arbeit haben wir quantenmechanische Rechnungen vorgelegt, die berücksichtigen, dass das einfach geladene Argonion verschiedene angeregte Zustände wählen kann, die im RESI Mechanismus berücksichtigt werden müssen. Die Beiträge dieser verschiedenen Kanäle in denselben Endzustand müssen in der Ionisationsamplitude kohärent addiert werden. Die resultierende Interferenz führt dazu, dass die Symmetrie zwischen den verschiedenen Quadranten derart gebrochen wird, dass bei niedrigen Intensitäten die Emission in entgegengesetzter Richtung dominiert. Die Ergebnisse stimmen recht gut mit den Daten überein. Da sowohl die Existenz diskreter angeregter Zustände wie auch die Interferenz von deren Beiträgen keine klassischen Entsprechungen haben, scheint dies ein erster klarer Effekt der Quantenmechanik in der NSDI zu sein, der die unvermeidbaren experimentellen Beschränkungen wie focal averaging, Mittelung über unbeobachtete Impulskomponenten usw., überlebt.

### Original publication

Quantum effects in double ionization of Argon below the threshold intensity Physical Review Letters**112**(2014) 073002/1-5