Recombination of an atomic system in one, two, and three dimensions in the presence of an ultrastrong attosecond laser pulse: A comparison of results obtained using a Coulomb and a smoothed Coulomb potential

The dynamics of recombination in an ultrastrong laser field is studied by numerical simulations performed for one-dimensional (1D), two-dimensional, and three-dimensional atomic systems modeled by the hydrogen atom Coulomb potential as well as by a model potential with a smoothed core. A nonmonotonic behavior of the total bound-states' final population is studied as a function of the laser field amplitude. The dependence of the results on the used atomic potential is demonstrated. It is shown that the recombination probabilities calculated in these two cases may differ even qualitatively. Even eigenstates of the 1D hydrogen atom are shown to play a significant role in the time evolution.

Figure 1

The final level of the recombination (sum of the populations of a few lowest bound states) as a function of the electric field intensity in the case of the real hydrogen atom (upper plot) and the 3D smoothed model potential (lower plot). The populations were calculated after exactly one, two, and three optical cycles except for the insets which correspond to time instants at which the sum of the populations exhibits a local maximum. Also shown are the initial total populations of the bound states, the values of which are not negligible due to packet being initially not very far from the nucleus. The abbreviation a.u., used in this figure as well in the next ones, stands for atomic units.

Figure 2

The sum of the final population levels of the KH states depending on electric field intensity in the case of the 3D smoothed model potential. The populations were calculated at the same time instants as in the previous figure.

Figure 3

The population of a few lowest bound states for ${\epsilon }_0=5\text{a}\text{.u}\text{.}$ in both studied 3D cases (singular potential, upper plot; smoothed potential, lower plot) ordered according to growing energy.

Figure 4

The final level of the recombination as in Fig. 1 but averaged over the packet initial positions $\left[z_0\left({\epsilon }_0\right)-1,z_0\left({\epsilon }_0\right)+1\right]$ for all values of ${\epsilon }_0$; singular potential, upper plot; smoothed potential, lower plot.

Figure 5

The wave function in the $z$ and $s$ coordinates (transformed back to the standard cylindrical coordinates, i.e., divided by $\sqrt{s}$) for $t=3T$ and for various laser field intensities. The spatial range of the images is $\left(-30,15\right)\text{a}\text{.u}\text{.}$ in the $z$ direction and (0,17) a.u. in the $s$ direction. The blackness of the plot is proportional to the square of wave-function modulus. The case of the singular hydrogen potential is presented in the left-hand side, and that of the smoothed model potential is in the right-hand side.

Figure 6

The final expectation value of the position $z_f$ of the 3D wave packet as a function of the laser intensity ${\epsilon }_0$. The position of the wave packet after an odd number of half cycles should be close to zero if the slow drift were absent; the position after a full number of the cycles should be close to $z_0=-2{\epsilon }_0/{\omega }^2$ in such a case. Any deviations from that pattern reveal the presence of the slow drift. The inset of the upper plot shows, for the potential with the singularity, the series of plots of $⟨x⟩$ after full cycles for various grid steps and for the same space range (from $1024\times 512$ up to $8192\times 4096$). Except for the largest grid step the differences are almost indistinguishable.

Figure 7

The final level of the recombination as a function of the electric field intensity in the case of the 2D hydrogen atom singular potential (upper plot) and the 2D smoothed model potential (lower plot). The populations were calculated at initial moment as well as after exactly one, two, and three optical cycles, except for the inset in the lower plot which is prepared for time instants at which the sum of the populations exhibits a local maximum. The inset in the upper plot is prepared exactly for $1.5T$ and $2.5T$.

Figure 8

The population of a few lowest bound states for ${\epsilon }_0=5\text{a}\text{.u}\text{.}$ in both studied 2D cases (singular potential, upper plot; smoothed potential, lower plot) ordered according to growing energy. Higher states are less populated thus the truncation to a few lowest states is justified.

Figure 9

The dynamics of the sum of the populations of a few lowest bound states in the singular hydrogen atom (upper) and smoothed model atom (lower). A more complicated structure of the recombination level in the former case is caused by a better developed double-ring structure of the wave function.

Figure 10

The evolution of the 2D wave function for ${\epsilon }_0=5\text{a}\text{.u}\text{.}$ presented in three snapshots calculated for $t_f=1T$, $2T$, and $3T$. The spatial range of the images is $\left(-15,15\right)\text{a}\text{.u}\text{.}$ in both directions. The blackness of the plot is proportional to the square of wave-function modulus. The case of the singular hydrogen potential is presented in the left-hand side, and that of the smoothed model potential is in the right-hand side.

Figure 11

The final expectation value of the position $z_f$ of the wave packet as in Fig. 6 but for 2D; singular potential, upper plot; smoothed potential, lower plot.

Figure 12

The sum of the final populations of the 2D KH well ten lowest bound states as a function of the electric field intensity, cf. Fig. 2.

Figure 13

The final level of the recombination as in Fig. 7 but for 1D; singular potential, upper plot; smoothed potential, lower plot.

Figure 14

The populations of the bound states as in Fig. 8 but for 1D. The sum of populations of the odd and even states is drawn for the singular hydrogen atom (upper plot). The inset of this plot presents the populations of only even states.

Figure 15

The final mean positions of the wave packet as in Fig. 11 but for 1D; singular potential, upper plot; smoothed potential, lower plot.

Figure 16

The probability of finding the electron in the vicinity of the nucleus as a function of the electric field intensity; the singular potential, upper plot; the smoothed potential, lower plot. In the inset: the sum of the final populations of the 1D KH well ten lowest bound states as a function of the electric field intensity.