Spatial and temporal interference during the ionization of H by few-cycle XUV laser pulses

We have investigated the ionization of H atoms by few-cycle XUV laser pulses, in particular the interference between electronic wave packets emitted at different time moments (temporal interference), and the interference between wave packets emitted at the same time, but following different paths (spatial interference). During the spatial interference, under appropriate conditions, the holographic mapping (HM) of the target atom's state is achieved, which has the potential to become a powerful tool to study atomic structure. First, by using classical trajectory Monte Carlo calculations, we have confirmed the existence of two distinct types of electron trajectories, which contribute to the formation of the HM interference pattern. Then, by using ab initio quantum-mechanical calculations, we have studied how the shape of the HM pattern is influenced by the laser pulse parameters. Finally, we have identified the optimal laser pulse parameters for the observation of the HM interference in the case of atomic species with an ionization potential close to 0.5 a.u.

Figure 1

The shape of the two-cycle, cosinelike laser pulses used in our calculations, where $T=2\pi /\omega$ is the period of the carrier wave.

Figure 2

The direct and scattered electron trajectories in the two-path model.

Figure 3

The two-path interference pattern for $z_0=5$ a.u.

Figure 4

Momentum distribution of the continuum electrons as a function of parallel and perpendicular momentum components at different time moments indicated by the arrows on the laser pulse. Laser pulse parameters: $\omega =0.4445$ a.u., $E_0=1$ a.u., and $\tau =28.26$ a.u.

Figure 5

The total ionization probability as a function of time. Laser pulse parameters: $\omega =0.4445$ a.u., $E_0=1$ a.u., and $\tau =28.26$ a.u.

Figure 6

Solid line: $t_2>t_1$ values for which the center of the interference circles is located at $k_{z_0}=-1.898$ a.u. Dashed line: $t_2=t_1$ line.

Figure 7

SFA interference rings (white circles) on top of the TDCC ionization probability density expressed as a function of parallel and perpendicular momentum components. Laser pulse parameters: $\omega =0.4445$ a.u., $E_0=1$ a.u., and $\tau =28.26$ a.u. On the upper figure, the “up arrows” indicate the $\{15.25,18.59\}$ pairs of $\{t_1,t_2\}$ values, while the “down arrows” indicate the $\{17.00,20.26\}$ pairs of $\{t_1,t_2\}$ values.

Figure 8

Ionization probability densities as a function of parallel and perpendicular momentum components obtained in the framework of the CTMC and TDCC models. Laser pulse parameters: $\omega =0.4445$ a.u., $E_0=1$ a.u., and $\tau =28.26$ a.u.

Figure 9

The distribution of the $r_{\text{min}}$ values for (a) all scattered CTMC trajectories, and (b) scattered CTMC trajectories with $0.5<p_{\text{per}}<0.7$, and $0.7<p_{\text{par}}<0.9$. The distributions were normalized to 1.

Figure 10

Ionization probability densities as a function of parallel and perpendicular momentum components at different field intensities, and at fixed $\omega =0.4445$ a.u.; (a) $E_0=1$ a.u.; (b) $E_0=0.75$ a.u.; (c) $E_0=0.50$ a.u.; (d) $E_0=0.25$ a.u.; (e) $E_0=0.10$ a.u.

Figure 11

Ionization probability densities as a function of parallel and perpendicular momentum components at different field parameters: (a) $\omega =0.66675$ a.u., $\tau =18.84$ a.u., $E_0=1.5$ a.u.; (b) $\omega =0.4445$ a.u., $\tau =28.26$ a.u., $E_0=1$ a.u.; (c) $\omega =0.22225$ a.u., $\tau =56.51$ a.u., $E_0=0.50$ a.u.

Figure 12

Ionization probability densities as function of electron ejection angle for fixed $p$ electron momentum values and for different laser pulse parameters: $\omega =0.4446$ a.u., $\tau =28.26$ a.u. (a) $E_0=1.0$ a.u., (b) $E_0=0.75$ a.u., (c) $E_0=0.5$ a.u., (d) $E_0=0.25$ a.u. The thick line denotes the TDCC, while the thin line denotes the two-path model results. The $\theta$ electron ejection angle is measured from the laser polarization (i.e., $Oz$) axis.