Restricted-space ab initio models for double ionization by strong laser pulses

A double-electron-ionization process occurs when an intense laser pulse interacts with atoms or molecules. Exact ab initio numerical simulation of such a situation is extremely computer resource demanding, thus often one is forced to apply reduced dimensionality models to get insight into the physics of the process. The performance of several algorithms for simulating double electron ionization by strong femtosecond laser pulses is studied. The obtained ionization yields and the momentum distributions of the released electrons are compared and the effects of the model dimensionality on the ionization dynamics are discussed.

Figure 1

Geometry of the ES model. (a) Saddle tracks forming the directions along which electrons are allowed to move. The polarization axis points along $z$. (b) Different regions of the configuration space used to define the state of the system: the neutral atom $A$, singly charged ions $S_i$, and doubly charged ions $D_i$. The parameters are $a=12.5$ a.u. and $b=6$ a.u.

Figure 2

(a) Geometry of the c.m. model. Here $Z_c$ and $z$, the directions of the center of mass and the relative electronic coordinate, respectively, are collinear with the electric-field polarization direction; $\rho$ is the cylindrical radial direction of relative motion in the $x-y$ plane. (b) Dimensions for evaluating the problem.

Figure 3

Single- and double-ionization yields (probability of a given process) as a function of laser field intensity obtained by quantum-mechanical simulations. Single- and double-ionization yields are marked by open and closed symbols, respectively; results for the ES model are marked by blue squares and for the c.m. model by red triangles. In both cases a four-cycle ${sin}^2$ pulse was assumed.

Figure 4

Single- and double-ionization yields as a function of the laser field intensity obtained by classical simulations. Single- and double-ionization yields are marked by open and closed symbols, respectively; results for the ES model are marked by blue squares, for the c.m. model by red triangles, for the Rochester model by black diamonds, and for the full 3D classical model by green circles. For all the simulations 800-nm four-cycle pulses of ${sin}^2$ shape were used.

Figure 5

Ratio of double- to single-ionization yields as a function of the laser field intensity for the c.m. (triangles) and the ES (squares) models obtained by quantum simulations. The solid and dash-dotted lines denote the ratio calculated with the rate equations with constants $C=0.012$ and $C=0.08$, respectively.

Figure 6

Single- and double-ionization yields as a function of the laser field intensity. The points correspond to quantum-mechanical simulations as described in Fig. 3. The lines without symbols correspond to yields obtained via the Ammosov-Delone-Krainov formula. The solid line denotes the single-ionization yield. Dash-dotted and dashed lines both correspond to double-ionization yields obtained with correct, $C=0.012$, and incorrect, $C=0.08$, coefficients, respectively.

Figure 7

Released electron momentum distribution computed with (a)–(c) quantum and (d)–(f) classical simulations for (a) and (d) the c.m., (b) and (e) the ES, and (c) and (f) the aligned electron models. Simulations were done for the 800-nm field of intensity $6.3\times {10}^{14}\text{W}/{\text{cm}}^2$. The parameters of the grid for quantum models are, for the ES and the aligned electron plot, $4096\times 4096$ points with coordinate step size 100/512 a.u., and for the c.m. model, $3372\times 1686\times 454$ points with coordinate step size 0.3 a.u.; the laser pulse consists of four cycles. The plots in (b) and (c) are smoothed with Gaussian functions 0.12 a.u. wide. The value of ${\epsilon }^2=0.58$ in (c) corresponds to the same energy of the ground state as in (b). Classical data were obtained with the use of (d) $4\times {10}^7$ sample trajectories for the c.m. model, (e) $4\times {10}^7$ sample trajectories for the ES model, and (f) $5\times {10}^6$ sample trajectories for the aligned electron model. For classical simulations the same spatial criterion of the double-ionization event was used and an eight-cycle laser pulse was applied.