Attosecond dynamics of light-induced resonant hole transfer in high-order-harmonic generation

We present a study of high-order-harmonic generation (HHG) assisted by extreme ultraviolet (XUV) attosecond pulses, which can lead to the excitation of inner-shell electrons and the generation of a second HHG plateau. With the treatment of a one-dimensional model of krypton, based on time-dependent configuration interaction singles (TDCIS) of an effective two-electron system, we show that the XUV-assisted HHG spectrum reveals the duration of the semiclassical electron trajectories. The results are interpreted by the strong-field approximation (SFA) and the importance of the hole transfer during the tunneling process is emphasized. Finally, coherent population transfer between the inner and outer holes with attosecond pulse trains is discussed.

Figure 1

(a) Comparison of the XUV-assisted HHG spectra in logarithmic scale using different four-cycle driving IR fields and a single XUV resonant pulse with different time delay $\tau$: cos-like IR field, $\tau =0$ (red solid curve); cos-like IR field, $\tau =T_{\text{IR}}/4$ (green dashed curve); sin-like IR field, $\tau =0$ (blue dot-dash curve). (b) Comparison of the HHG spectrum using the same cos-like IR field but different XUV intensity at the fixed time delay $\tau =T_{\text{IR}}/4$: $I_{\text{XUV}}={10}^{12}\text{W}/{\text{cm}}^2$ (green dashed curve, as the same as the upper panel); $I_{\text{XUV}}={10}^{14}\text{W}/{\text{cm}}^2$ (red solid curve); $I_{\text{XUV}}={10}^{16}\text{W}/{\text{cm}}^2$ (blue dot-dash curve).

Figure 2

The upper panels are power spectra in the second plateau region of HHG for (a) cos-like and (b) sin-like short IR laser field with single XUV pulse. The lower panels are the return energy (in terms of the harmonic order with recombination to the inner shell) plotted as a function of the ionization time (red filled circles) and the recombination time (green open circles). The blue curves in the lower panels are the corresponding IR field: cos-like (left) and sin-like (right). The arrows in (c) and (d) are the classical trajectories mainly allowed and labeled with I-V.

Figure 3

The 2D plot of HHG spectra in the second plateau region for the cos-like four-cycle IR laser field with single XUV pulse as a function of time delay: (a) SFA with SPA for $p$ using Eq. (23); (b) SFA with SPA for both $p$ and $t_1$ using slowly varying approximation in $a_{\mathrm{xuv}}$; (c) TDCIS in the independent particle approximation with the use of short-range potential supporting only two bound states; (d) TDCIS in the independent particle approximation with the use of short-range potential supporting more than two bound states.

Figure 4

The comparison of HHG spectrum at $\tau =-0.4T_{\text{IR}}$ for Figs. 3–3 and 2.

Figure 5

Enlarged region of Fig. 3 for delay times of roughly $-0.4$ optical cycles. In this region, the stripe structure shows a positive slope. The dotted curve shows emitted photon energy in the second plateau as a function of ionization time. Long trajectories, with ionization times earlier than the cutoff trajectory (trajectory with maximal return energy) marked in blue, and short trajectories marked in black. The green dashed lines mark the minima of the interference pattern predicted by Eq. (44) for $\tau \approx t_i$.

Figure 6

Electric field of the IR field (red dash-dot line) and the XUV pulse train (blue solid line). If the XUV pulse train is generated via HHG in atomic gases, then $\mathrm{\Delta }{\varphi }_c=\pi$ and the repetition period is equal to half cycles of the IR field, $T_r=T_{\text{IR}}/2$.

Figure 7

The HHG in the second plateau region with two different kinds of XUV pulse train, $\mathrm{\Delta }{\varphi }_c$ is $\pi$ (left) and 0 (right), as a function of delay phase $\delta$. (a) and (b) HHG power spectra in the second plateau region with acceleration form. (c) and (d) Inner-shell population ${\rho }_1\left(t\right)$ as the function of delay phase $\delta$. The white lines on $\delta =-0.15\pi$ and $\delta =0.15\pi$ indicate the local maximum and minimal area of the HHG spectrum.

Figure 8

$|a_{\text{xuv}}(t,t_i,\tau ){|}^2$ curve (red lines) from $t=t_i$ to $t_r$ (blue dash line) for XUV pulse trains (green curves) with $\mathrm{\Delta }{\varphi }_c=\pi$ (left) and $\mathrm{\Delta }{\varphi }_c=0$ (right). Here $\delta$ is chosen $-0.15\pi$ and $0.15\pi$ (white lines in Fig. 7). When $\delta =-0.15\pi$, the parent ion interacts with the XUV-PT once and both of these two cases show the same curve in (a) and (c). When $\delta =0.15\pi$, the parent ion interacts with the XUV-PT twice and different CEO values show different features on the curves in (b) and (d).