Classical explanation for electrons above energy $2U_p$ in strong-field double ionization at 390 nm

A classical ensemble is employed to investigate double ionization of helium for laser wavelength 390 nm and intensity $1.1\text{PW}∕{\text{cm}}^2$, below the threshold for recollision impact ionization. Significant numbers of trajectories that lead to electron energy above $2U_p$ are shown to be present and to feature a time delay between recollision and final ionization. Processes through which nuclear and laser forces combine to give an electron final energy above $2U_p$ are identified.

Figure 1

(Color online) Final kinetic energies for all DI electrons from classical ensemble for laser wavelength 390 nm and intensity $1.1\text{PW}∕{\text{cm}}^2$. On the top the count densities of the recolliding and struck electrons are plotted as the blue dashed curve and the red solid curve, respectively. On the bottom is a density plot showing the final energy $E_2$ of the recolliding electron vs final energy $E_1$ of the struck electron. Pulse length was five cycles. For these laser parameters $U_p=0.58\text{au}$ and the black box is drawn at $2U_p$.

Figure 2

(Color online) Percent of the trajectories vs laser phase for recollision (top plot) and ionization (bottom plot) times. In each plot, the lower green band indicates high-energy trajectories and the upper red band the remaining trajectories. Ionizations peak as the field grows in strength. Results are shown for both ten- and five-cycle pulses.

Figure 3

(Color online) Percent of high-energy trajectories meeting specified conditions vs time since recollision for both ten- and five-cycle pulses. From top to bottom along the right: dotted magenta curve indicates both electrons are free; blue dashed curve indicates the high-energy electron is still bound but the other free; green curve with larger dashes has the low-energy electron bound but the other free. The solid red curve visible at shorter times indicates doubly excited atoms. The black dotted curve along the bottom denotes trajectories for which one electron is bound and both electrons finish with high energy.

Figure 4

(Color online) Trajectories of Figs. 5 (top) and 6. Curves begin same times as those plots, with successive dots indicating elapsed time of 0.02 cycles. Arrows show positions at zeros of the laser field. In each trajectory, the returning electron is incident from the right.

Figure 5

(Color online) Sequence for upper trajectory of Fig. 4. Curves show effective potential energies for each electron as functions of $z$ (the laser polarization axis). Upper arrows show laser force, and lower arrows show $v_z$. Vertical lengths of the “dumbbells” (more visible in Fig. 6) indicate kinetic energy for motion perpendicular to the $z$ axis. At $t=2.5c$, the struck electron (color coded in red with solid curve) remains bound and has just started traveling back in the positive $z$ direction. The laser force propels it over a suppressed barrier, and the electron finishes with energy greater than $2U_p$.

Figure 6

(Color online) Effective-energy collage for lower trajectory of Fig. 4. The electron that is free after recollision achieves energy greater than $2U_p$. In this example, the recolliding electron is recaptured, but reionizes when the field is strong.