Electron-electron dynamics in laser-induced nonsequential double ionization

For the description of nonsequential double ionization (NSDI) of rare-gas atoms by a strong linearly polarized laser field, the quantum-mechanical $S$-matrix diagram that incorporates rescattering impact ionization is evaluated in the strong-field approximation. We employ a uniform approximation, which is a generalization of the standard saddle-point approximation. We systematically analyze the manifestations of the electron-electron interaction in the momentum distributions of the ejected electrons: for the interaction, by which the returning electron frees the bound electron, we adopt either a (three-body) contact interaction or a Coulomb interaction, and we do or do not incorporate the mutual Coulomb repulsion of the two electrons in their final state. In particular, we investigate the correlation of the momentum components parallel to the laser-field polarization, with the transverse momentum components either restricted to certain finite ranges or entirely summed over. In the latter case, agreement with experimental data is best for the contact interaction and without final-state repulsion. In the former, if the transverse momenta are restricted to small values, comparison of theory with the data shows evidence of Coulomb effects. We propose that experimental data selecting events with small transverse momenta of both electrons are particularly promising in the elucidation of the dynamics of NSDI. Also, a classical approximation of the quantum-mechanical $S$ matrix is formulated and shown to work very well inside the classically allowed region.

Figure 1

Momentum correlation function (15) of the electron momenta parallel to the laser field for nonsequential double ionization computed with the uniform approximation using the contact interaction (21). The field frequency is $\omega =0.0551\text{a.u.}$ and the ponderomotive energy $U_P=1.2\text{a.u.}$, which corresponds to an intensity of $5.5\times {10}^{14}\mathrm{W}∕{\text{cm}}^2$. The first two ionization potentials are $\mid E_{01}\mid =0.79\text{a.u.}$ and $\mid E_{02}\mid =1.51\text{a.u.}$ corresponding to neon. Panel (a) shows the yield for the case where the transverse momenta ${\mathbf{p}}_{n\perp }\left(n=1,2\right)$ are completely integrated over, whereas in the remaining panels they are restricted to certain intervals. In panels (b) and (c), ${\mathbf{p}}_{2\perp }$ is integrated, while $0<p_{1\perp }∕{\left[U_p\right]}^{1∕2}<0.1$ and $0.4<p_{1\perp }∕{\left[U_p\right]}^{1∕2}<0.5$, respectively. In panels (d), (e), and (f), both transverse momenta are confined to the intervals $0<p_{n\perp }∕{\left[U_p\right]}^{1∕2}<0.5$, $0.5<p_{n\perp }∕{\left[U_p\right]}^{1∕2}<1$, and $1<p_{n\perp }∕{\left[U_p\right]}^{1∕2}<1.5$, respectively.

Figure 2

Same as Fig. 1, but calculated for the Coulomb interaction (22).

Figure 3

Same as Fig. 1, but including Coulomb repulsion of the two electrons in the final state. The transverse momenta of the two electrons are antiparallel, $\varphi =\pi$.

Figure 4

Same as Fig. 3, but the transverse momenta are at right angles, $\varphi =\pi ∕2$.

Figure 5

Same as Fig. 3, but the transverse momenta are parallel, $\varphi =0$.

Figure 6

Same as Fig. 3, but the relative orientation of the transverse momenta is integrated over.

Figure 7

Same as Fig. 1, but calculated for the Coulomb interaction (22) and final-state Coulomb repulsion. The transverse momenta of the final electrons are antiparallel, $\varphi =\pi$.

Figure 8

Same as Fig. 7, but the transverse momenta are perpendicular, $\varphi =\pi ∕2$.

Figure 9

Same as Fig. 7, but the transverse momenta are parallel, $\varphi =0$.

Figure 10

Same as Figs. 7, 8, 9, but the relative orientation of the transverse momenta is integrated over.

Figure 11

Momentum correlation function for the Coulomb form factor (23) in the absence of final-state repulsion for specific relative angles $\varphi$: $\varphi =\pi$ (a), $\varphi =\pi ∕2$ (b), and $\varphi =0$ (c). The other parameters are as in Fig. 1a; in particular, the transverse momenta are completely summed over.

Figure 12

Same as Fig. 1, but calculated from the classical model for the contact interaction. Equation (37) underlies panel (a), Eq. (38) panels (b) and (c), and Eq. (39) panel (d).

Figure 13

Same as Fig. 12, but including electron-electron repulsion in the final state. The transverse momenta are perpendicular, $\varphi =\pi ∕2$.