Numerical attoclock on atomic and molecular hydrogen

Numerical attoclock is a theoretical model of attosecond angular streaking driven by a very short, nearly a single oscillation, circularly polarized laser pulse. The reading of such an attoclock is readily obtained from a numerical solution of the time-dependent Schrödinger equation as well as a semiclassical trajectory simulation. By comparing the two approaches, we highlight the essential physics behind the attoclock measurements. In addition, we analyze the predictions of the Keldysh-Rutherford model of the attoclock [A. W. Bray et al., Phys. Rev. Lett. 121, 123201 (2018)]. In molecular hydrogen, we highlight a strong dependence of the width of the attoclock angular peak on the molecular orientation and attribute it to the two-center electron interference. This effect is further exemplified in the weakly bound neon dimer.

Figure 1

Photoelectron momentum distribution $P(k_x,k_y,k_z=0)$ of atomic hydrogen in the polarization plane at the driving field intensity $I=8\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$. The attoclock offset angle ${\theta }_A$ and the angular width $\mathcal{W}$ are marked. The coloration ranges from zero (red) to the maximum (black) linearly. Results of the TDSE (top) and SPM (bottom) calculations are displayed. The $c$ and $f$ labels on the bottom panel are used to mark the center and fringes of the PMD.

Figure 2

Saddle-point solutions corresponding to the fringes $f$ (blue or red) and center $c$ (black) of the PMD as marked on the bottom panel of Fig. 1. The direction of each of the chosen momenta $\widehat{\mathit{k}}$ are given by dashed lines of the matching color. Top: solutions of Eq. (6) for atomic hydrogen with $k_z=0.0$ a.u. (filled circles) and $k_z=1.0$ a.u. (open circles). Bottom: solutions of Eq. (9) for ${\mathrm{H}}_2$ with $j=1,2$ given by filled and open symbols, respectively. The perpendicular orientation $\widehat{R}\parallel {\mathit{e}}_z$ is given with diamonds of dark hue, the major axis orientation $\widehat{R}\parallel {\mathit{e}}_x$ with circles of light hue, and the minor axis $\widehat{R}\parallel {\mathit{e}}_y$ orientation with squares of standard hue. The insets highlight the solutions about each fringe.

Figure 3

The attoclock offset angle ${\theta }_A$ plotted as a function of the laser field intensity. The TDSE1 (red squares) and TDSE2 (blue triangles) calculations and the experimental points (with error bars) visualize the data of Sainadh et al. [11] obtained with multicycle pulses. The TDSE calculation (red open circles) and the KR model prediction (red solid line) from Bray et al. [8] refer to the single-oscillation pulses. Similar calculation of Torlina et al. [5] (blue asterisks) is also shown. The KR model is extended beyond the range of its validity by Hofmann et al. [32] and displayed with the thick dotted line. The thin dotted line fits the data of [11] with the characteristic field intensity dependence $I^{-0.5}$ predicted by the KR model.

Figure 4

Top: Angular width $\mathcal{W}\left(I\right)$ of the PMD of atomic hydrogen in the polarization plane as a function of the field intensity $I$. Bottom: The width parameter $\mathcal{W}\left(k_z\right)$ as a function of the transverse photoelectron momentum outside the polarization plane at the field intensity of $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 5

The attoclock offset angle ${\theta }_A$ (top) and the angular width $\mathcal{W}$ (bottom) of the ${\mathrm{H}}_2$ molecule as functions of the field intensity. The molecule is aligned perpendicular to the polarization plane (red), parallel to the major polarization axis ${\widehat{\mathit{e}}}_y$ (black), and parallel to the minor polarization axis ${\widehat{\mathit{e}}}_x$ (green). The top panel shows the offset angle ${\theta }_A$ of the H atom scaled to the ionization potential of ${\mathrm{H}}_2$.

Figure 6

Radially integrated PMD $P\left(\theta \right)$ in the polarization plane for the three orientations of the Yukawa molecule with $R=2$ a.u. (top) and 4 a.u. (bottom).

Figure 7

Photoelectron momentum distribution in ${\mathrm{Ne}}_2$ in the polarization plane of the circularly polarized continuous laser field. Top: atomic-like SPM calculation using analytic expressions of [33]. Middle and bottom: the same calculation modulated by the interference factor (10) with the phase difference $\mathrm{\Delta }\varphi =\pi$ (gerade) and 0 (ungerade), respectively.