Sequential ionization and energy gain of bound electrons in classical modeling of atoms in strong laser fields

Classical mechanics and classical ensembles have provided numerous insights into the dynamics of strong-field double ionization. In this paper, we show that in classical multidimensional modeling, the laser intensity at which sequential ionization begins to dominate depends on the softening of the interaction between the electron and nucleus. We show that an unsoftened interaction in two or three dimensions can lead to classical orbits in which an electron can start deep in the nuclear potential-energy well, gain energy from the oscillating laser field, and ionize over the barrier without any recollision. We discuss how this energy gain occurs, with the electron orbit favoring one side of the nucleus or the other, depending on which side corresponds with the rising potential-energy curve.

Figure 1

Double-ionization yield vs laser intensity for three-dimensional classical ensembles, with different values of the nuclear softening parameter $a$ after the first ionization (as described in the text). For $a=0.4$, the green dashed curve with diamonds is the total DI yield and the black long-dashed curve with inverted triangles is the sequential yield. For $a=0.01$, the red solid curve with circles is the total DI yield and the blue dotted curve with squares is the sequential yield. The onset for sequential ionization occurs at a lower intensity for a more exposed nucleus. Laser wavelength $\lambda =800$ nm, ensemble size 50 000 atoms, with a trapezoidal ($1+3+1$) pulse.

Figure 2

Electron energy vs time for a single electron with an oscillating laser field in a 2D nuclear well, with nuclear softening parameter $a=0.01$. The electron absorbs energy and escapes the well. The oscillatory curve (brown) neglects the laser interaction energy $+z{E}_0\sin \left(\omega t\right)$, whereas the smoother curve (black) includes it.

Figure 3

Top: Electron energy vs time, with (black smooth curve) and without (brown oscillatory curve) the laser interaction, for the trajectory of Fig. 2, zoomed in on $1.0<t<1.5$ cycles. Center: $z$ coordinate of that same electron. The curves in the top plot cross whenever $z=0$. Bottom: Spatial trajectory for a few oscillations before and after $t=1.25$ cycles. The curve changes from green to black (light to dark) at $t=1.25$ cycles (second quadrant, close to the origin). The electron switches its oscillation from one side of the nucleus to the other at about 1.27 cycles, when the laser force is strong in the $-z$ direction. That switch plays a key role in the electron's energy increase.

Figure 4

Effective energy plots from $t=1.0$ through 1.5 cycles for the same trajectory as the previous figure. The dashed horizontal line segment shows the lowest barrier height. The electron gains or loses energy depending on its value of $z$ as the laser raises the potential-energy curve on one side of the nucleus and reduces it on the other side. We plot the potential energy vs $z$ for the actual $x$ value of the electron; the parametric dependence on $x$ changes the shape of the well [18].

Figure 5

Position coordinates $x$ (dashed red or light curve) and $z$ (solid blue or dark curve) vs time, as well as angular momentum $L$ (solid green curve, transitioning from $-$ to $+$) vs time. The angular momentum of the orbit increases as the “jump” to the new orbit approaches, then passes through zero just after the jump.

Figure 6

Electron energy (excluding the laser interaction) at $t=2.75$ cycles vs electron energy at $t=0$ cycles for a single electron in a nuclear potential-energy well exposed to a sinusoidal laser force at $I=3\times {10}^{15}$ W/cm${}^2$ with $\lambda =800$ nm. The nuclear potential energy is $-2/\sqrt{r^2+a^2}$. In all plots, the vertical dashed line shows the energy of the maximally suppressed barrier. Top left panel: 1D, $a=0.4$. Top right panel: 1D, $a=0.01$. Bottom left panel: 2D, $a=0.4$. Bottom right panel: 2D, $a=0.01$. For all plots, the initial positions and velocities were systematically varied. The electron can “climb” out of the well only for the unsoftened potential in multiple dimensions.

Figure 7

Ionization probability vs initial energy for the two-dimensional system, with $a=0.01$, for intervals of 2.75 (lower curve) and 4.75 cycles (upper curve). The vertical dashed line shows the energy of the maximally suppressed barrier.

Figure 8

Top plot shows energy vs time for one sequential-ionization electron from our 3D ensemble, for $a=0.01$ and $I=3\times {10}^{15}$ W/cm${}^2$ over the interval $0.0<t<4.0$ cycles. As in Fig. 2, the highly oscillatory (brown) curve neglects the laser interaction energy. The middle plot shows the electron position for $3.16<t<3.24$ cycles. The curve changes from green to black (light to dark) at $t=3.19$ cycles. The bottom plot shows the magnitude of angular momentum, $\left|L\right|$, for the same time interval, and with color change at the same time. At about this time, the electron jumps from orbiting primarily on the $z>0$ side of the nucleus to oscillating primarily on the $z<0$ side, and the laser does considerable positive work.

Figure 9

Top panel: Position vs time for an electron in 1D; $I=3\times {10}^{15}$ W/cm${}^2$ with $\lambda$ $=$ 800 nm and $a=0.01$. Bottom panel: Energy vs time, with the laser interaction included. The energy of the maximally suppressed barrier is shown as a dashed (black) line. Dots show energies at the time of laser zeros (upper dots) and maxima (lower dots).

Figure 10

Effective energy plots of the electron of Fig. 9 over 1 laser cycle in intervals of 0.125 cycles. The horizontal line shows the energy of the maximally suppressed barrier. The electron never escapes.