Quantum interference in strong-field ionization by a linearly polarized laser pulse and its relevance to tunnel exit time and momentum

We investigate the liberation of an atomic electron by a linearly polarized single-cycle near-infrared laser pulse having a peak intensity that ensures tunneling. Based on phase space analysis and energy distribution in the instantaneous potential, we reveal the importance of quantum interference between tunneling and over-the-barrier pathways of escape. Tunneling is blurred both in space and time, and the contribution of tunneling at the mean energy is almost negligible. We suggest and justify improved initial conditions for a classical particle approximation of strong-field ionization, based on the quantum momentum function, and we show how to reconstruct them from the detected momentum of an escaped electron.

Figure 1

Contour plots of the reduced 1D Wigner function of Eq. (3) in logarithmic color scheme (upper panels) and plots of $V(z,0,t)$ (lower panels, in black) at $t=155$ (a), $t=160$ (b), and $t=165$ (c) during the liberation of the hydrogen atom's electron by the laser pulse of the ${\mathcal{E}}_z\left(t\right)$ plot in the insets. Phase space trajectories in gray with different styles represent classical motion in the instantaneous potential, corresponding to energy levels marked in the lower panels with respective styles. The trajectory with a thin gray solid line is the instantaneous separatrix between tunneling and OB ionization regimes. Black solid lines in the upper panels show the quantum momentum function of Eq. (4).

Figure 2

Energy probability density of the quantum state at $t=155$ (blue, $\blacksquare$), 160 (green, $⧫$), and 165 (red, $▾$) in the instantaneous potential. Thin solid vertical gridlines mark the $\langle E\rangle \left(t\right)\pm \mathrm{\Delta }E\left(t\right)$ values; dashed gridlines mark $V_{\mathrm{top}}\left(t\right)$, with respective colors and symbols. The inset zooms in on the region around the peaks; thick solid gridlines mark $\langle E\rangle \left(t\right)$.

Figure 3

Comparison of the probability currents $j(z,t)$ (black, $▾$), $j_{\mathrm{FT}}(z,t)$ (red, $\blacksquare$), $j_{\mathrm{OB}}(z,t)$ (blue, $⧫$), and $j_{\mathrm{ST}}(z,t)$ (magenta, $★$) in (a), and the corresponding quantum momentum functions in (b). We plot these curves at the peak of the laser pulse $t=165$ (solid lines) and somewhat earlier at $t=155$ (dashed lines). The vertical gridlines mark the momentary positions of the tunnel entrance, the $V_{\mathrm{top}}\left(t\right)$, and the tunnel exit, from left to right, with respective style.

Figure 4

Definition of the initial phase space point (shown for $t_{\mathrm{s}}=157$): there is only a single inflection point (along the dot-dashed line in red) among the inflection points (dashed line and dots in gray) of the stationary phase space trajectories (gray solid lines) of the instantaneous potential, which is on the instantaneous quantum momentum function (black solid line).

Figure 5

Classical phase space trajectories of the liberated electron (dashed lines) for the indicated starting time instants, plot from $t_{\mathrm{s}}$ to $t=NT$. Initial conditions correspond to Eq. (5); the inset zooms in to the phase space region around these start points. Directly escaping trajectories are represented and marked by $t_{\mathrm{s}}=149,153,\text{and}157$ (in purple, green, and red, respectively). The $t_{\mathrm{s}}=159$ (in dark blue) marks a limiting case having $\sim 0$ momentum at $t=NT$. The $t_{\mathrm{s}}=165$ (in light blue) marks a rescattering trajectory. Snapshots of the classical and the quantum propagation at $t=157,185,210,\text{and}230$, represented by the dots and by $q(z,t)$ (black solid line), respectively, show a good match for all of the different outcomes.

Figure 6

Comparison of classical and PMPE quantum dynamics of an escaping electron at $t=158$ (a) and at $t=180$ (b). Left panels: colored contour plots of $W_{\mathrm{PMPE}}$ and plot of $q(z,t)$ (solid black line). Solid and dashed thin gray trajectories are as in Fig. 1. Dashed lines in purple (marked by $★$), green ($\blacksquare$), red ($⧫$), and magenta ($▾$) show classical trajectories starting at $t_{\mathrm{s}}=149,153,157$, and 158, respectively, having different initial conditions according to Eq. (5). Dots on each of these dashed lines mark the momentary state of the classical dynamics. Right panels: momentum distribution of the PMPE wave packet (black solid line) and momentum values relevant to classical dynamics marked with gridlines in respective colors and symbols.