Nonsequential Double Recombination in Intense Laser Fields

A second plateau in the harmonic spectra of laser-driven two-electron atoms is observed both in the numerical solution of a low-dimensional model helium atom and using an extended strong field approximation. It is shown that the harmonics well beyond the usual cutoff are due to the simultaneous recombination of the two electrons, which were emitted during different, previous half-cycles. The new cutoff is explained in terms of classical trajectories. Classical predictions and the time-frequency analysis of the ab initio quantum results are in excellent agreement. The mechanism corresponds to the inverse single photon double ionization process in the presence of a (low frequency) laser field.

Figure 1

Return energies $E_r$ in units of $U_p$ vs the return time $t_r$ in laser cycles for a laser field $\mathit{A}(t)=\widehat{A}{\mathit{e}}_z\sin \omega t$. Electrons emitted during the first half cycle (1, black), the second (2, cyan), the third (3, red), etc., reach their maximum return energy $3.17U_p$ during the subsequent half cycle. At later returns only lower values are achieved. For two electrons returning at the same time but emitted at different times, the values indicated by “$2+1$” and “$3+1$” are obtained (the sum “$3+2$” equals “$2+1$” in height and is omitted in the plot).

Figure 2

HOHG spectra for a $n=6$-cycle laser pulse with $\omega =0.0584$ and $\widehat{A}=3.417$ ($I=1.4\times {10}^{15}\mathrm{W}{\mathrm{cm}}^{-2}$). The spectrum for the He model with both electrons active (drawn black) shows a second plateau. The corresponding single active electron (SAE) result and the spectrum obtained from the ${\mathrm{He}}^{+}$ ion are also included. The arrows indicate the expected cutoff positions (see text).

Figure 3

Time-frequency analysis of the HOHG spectrum in Fig. 2, showing the color-coded contour plot of ${log}_{10}|a(\Omega ,t){|}^2$. The relevant classical solutions for the 6-cycle ${sin}^2$-pulse are superimposed. The possible classical recombination energies of a single electron $\Omega =E_r+I_p^{(1)}$ are drawn white. The sums of the two highest classical return energies $E_{r1}$, $E_{r2}$ plus the two ionization potentials, $\Omega =\sum _i[E_{ri}+I_p^{(i)}]$, are superimposed in black for the recombination times $t_r$ of interest.

Figure 4

High-order harmonic spectra obtained by Fourier-transforming the SFA-NSDR expression for the dipole $d_z^{\mathrm{NSDR}}(t)$ (black). For comparison, the single active electron result for He (labeled “SAE,” blue) and ${\mathrm{He}}^{+}$ (labeled ${\mathrm{He}}^{+}$, red) are included: (a) same laser parameters as in Fig. 2; (b) $\omega =2\times 0.0584$, $\widehat{A}=2.0$. Classically expected cutoffs are indicated by arrows: (1) $|{\mathcal{E}}_1|+3.17U_p$, (2) $|{\mathcal{E}}_2|+3.17U_p$, (3) $|{\mathcal{E}}_1|+|{\mathcal{E}}_2|+4.70U_p$, (4) $|{\mathcal{E}}_1|+|{\mathcal{E}}_2|+5.55U_p$, and (5) $|{\mathcal{E}}_1|+|{\mathcal{E}}_2|+2\times 3.17U_p$.