Attosecond-time-scale multielectron collisions in the Coulomb four-body problem: Traces in classical probability densities

In the triple ionization of the Li ground state by single photon absorption the three electrons escape to the continuum mainly through two collision sequences with individual collisions separated by time intervals on the attosecond scale. We investigate the traces of these two collision sequences in the classical probability densities. We show that each collision sequence has characteristic phase space properties which distinguish it from the other. Classical probability densities are the closest analog to quantum mechanical densities allowing our results to be directly compared to quantum mechanical results.

Figure 1

Averages over ensembles [left, (a) and (b)] $\alpha =\mathrm{I}$ and [right, (c) and (d)] $\alpha =\mathrm{II}$; see Eq. (9). The upper panels show the interelectronic repulsions, ${⟨1∕r_{12}⟩}_{\alpha }$ (solid), ${⟨1∕r_{13}⟩}_{\alpha }$ (dashed), and ${⟨1∕r_{23}⟩}_{\alpha }$ (dotted). The lower panels show in addition the three-body energies with thin lines ${⟨H_{12}⟩}_{\alpha }$ (solid), ${⟨H_{13}⟩}_{\alpha }$ (dashed), and ${⟨H_{23}⟩}_{\alpha }$ (dotted).

Figure 2

(Color online) Top panel: ${\mathcal{P}}_{\alpha }(x_i,t)$ for electrons $i=1,2,3$ (from left to right) for the $\alpha =\mathrm{I}$ ensemble; bottom panel: As for the top panel but for ${\mathcal{P}}_{\alpha }(y_i,t)$.

Figure 3

Single electron energy averages ${⟨H_i⟩}_{\alpha }$ of electrons 1 (solid), 2 (dashed), and 3 (dotted) for the ensemble (a) $\alpha =\mathrm{I}$ and (b) $\alpha =\mathrm{II}$.

Figure 4

(Color online) Momentum distributions ${\mathcal{P}}_{\alpha }(p_x,t)$ for electron 1 for the $\alpha =\mathrm{I}$ (top) and $\alpha =\mathrm{II}$ (bottom) ensemble. The left panels show the evolution for short times in greater detail.

Figure 5

(Color online) Same as Fig. 4 but for electron 2.

Figure 6

(Color online) Momentum distribution ${\mathcal{P}}_{\alpha }(p_x,t)$ for electron 3 for the $\alpha =\mathrm{I}$ (left) and $\alpha =\mathrm{II}$ (right) ensemble.

Figure 7

Averaged angles as a function of time over the $\alpha =\mathrm{I}$ ensemble. Solid: Angle $\varphi$ between (a) ${\mathbf{F}}_{21}$ and ${\mathbf{p}}_1$ and between (b) ${\mathbf{F}}_{31}$ and ${\mathbf{p}}_1$; Dashed: Angle $\varphi$ between (a) ${\mathbf{F}}_{21}$ and ${\mathbf{p}}_{x,1}$ and between (b) ${\mathbf{F}}_{31}$ and ${\mathbf{p}}_{x,1}$; Dash-dotted: Angle $\varphi$ between (a) ${\mathbf{F}}_{12}$ and ${\mathbf{p}}_2$ and between (b) ${\mathbf{F}}_{13}$ and ${\mathbf{p}}_3$; Dotted: Angle $\varphi$ between (a) ${\mathbf{F}}_{12}$ and ${\mathbf{p}}_{x,2}$ and between (b) ${\mathbf{F}}_{13}$ and ${\mathbf{p}}_{x,3}$.

Figure 8

(Color online) Same as Fig. 4 but for the probability density of the interelectronic angle ${\theta }_{12}$, ${\mathcal{P}}_{\alpha }({\theta }_{12},t)$. The arrows indicate the time of the collision $t_{ij}$.

Figure 9

(Color online) Same as in Fig. 8 but for the interelectronic angle ${\theta }_{13}$.

Figure 10

(Color online) Same as in Fig. 8 but for the interelectronic angle ${\theta }_{23}$.