Sequential and nonsequential double ionization of helium by intense XUV laser pulses: Revealing ac Stark shifts from joint energy spectra

Double ionization of a model helium system by intense extreme ultraviolet laser pulses is investigated by solving the time-dependent Schrödinger equation and extracting the photoelectron spectra. Considering a frequency regime where the nonsequential two-photon double ionization is expected to dominate the double ionization process according to the perturbation theory, we identify the sequential and nonsequential mechanisms from the joint energy spectra of two electrons in the strong-field regime. In the nonperturbative regime, we find that sequential multiphoton double ionization shows remarkable peaks in the joint energy spectra and these peaks spread their widths with increasing laser field strength, which is caused by the ac Stark effect. Thus in the strong-field regime, the ac Stark shifts of different ionic states can be observed from the sequential multiphoton features in the joint energy spectra and the signs of the ac Stark shifts can be controlled by the laser frequency. We also find that the nonsequential mechanism dominates double ionization only at low field strengths while the sequential multiphoton mechanism contributes substantially at high field strengths, in accordance with estimates from rate equations.

Figure 1

The laser frequency considered in this work, illustrated with some energy levels of He and ${\mathrm{He}}^{+}$. Nonsequential two-photon DI is allowed, while sequential two-photon DI is forbidden. Three of the three-photon DI mechanisms, i.e., the sequential three-photon DI via two different ionic channels and the nonsequential three-photon DI, are presented as examples.

Figure 2

(a) The DI JES $S_{\text{DI}}(E_1,E_2)$ for a laser pulse with $F_0=0.1,\omega =1.60,N_c=200$. (b) The SI PES in the lowest four ionic channels $S_{\text{SI},|1e\rangle },S_{\text{SI},|2o\rangle },S_{\text{SI},|3e\rangle },S_{\text{SI},|4o\rangle }$, for the same laser pulse. Based on energy conservation, the crossings of the black dotted lines along $E_1+E_2=3\omega -I_p^{\text{DI}}$ in (a) indicate some expected electron energy pairs $(E_1,E_2)$ in the sequential three-photon DI process while the black dotted lines in (b) indicate some expected PES peaks in SI. The maximum in the DI JES is $S_{\text{DI}}(E_1=0.801,E_2=1.097)=S_{\text{DI}}(E_1=1.097,E_2=0.801)=1.57$.

Figure 3

(a),(b) Same as Fig. 2, the DI JES and the channel-resolved SI PES, but for a field strength $F_0=0.5$. (c) The PES of the ground state (red solid line) and first-excited state (green dashed line) of ${\mathrm{He}}^{+}$, together with the DI JES integrated along $E_1,\int S_{\text{DI}}(E_1,E_2)d{E}_1$ (orange dotted line). (d)–(f) Same as (a)–(c), but for $F_0=1.0$. Arrows in (c) and (f) indicate the PES peak positions. For better visualization, zoom-ins of the selected peaks in (c) and (f) are presented in separate panels below (c) and (f). The maxima in the DI JES are (a) $S_{\text{DI}}(E_1=0.813,E_2=1.033)=S_{\text{DI}}(E_1=1.033,E_2=0.813)=3.96\times {10}^1$ and (d) $S_{\text{DI}}(E_1=0.691,E_2=1.319)=S_{\text{DI}}(E_1=1.319,E_2=0.691)=4.64\times {10}^2$, respectively.

Figure 4

The nonsequential two-photon DI probability $P_{\text{NS-2ph}}$ (multiplied by a factor of 100) and the ratio $P_{\text{NS-2ph}}/P_{\text{DI}}$ [see Eqs. (21) and (22)] vs field strength $F_0$, for fixed $\omega =1.60$ and $N_c=200$.

Figure 5

The nonsequential two-photon DI probability $N_{\text{DI-2ph}}/N_0$ (multiplied by a factor of 100) and the ratio $N_{\text{DI-2ph}}/(N_{\text{DI-2ph}}+N_{\text{DI-3ph}})$ based on the rate equations [see Eqs. (23, 24, 25, 26, 27)] vs field strength $F_0$, for fixed $\omega =1.60$ and $N_c=200$.

Figure 6

Similar to Fig. 3, for fixed $F_0=1.0$ and $N_c=200$. The laser frequencies are (a)–(c) $\omega =1.80$ and (d)–(f) $\omega =1.90$. Instead of the PES of the $|2o\rangle {\mathrm{He}}^{+}$ in Figs. 3 and 3, here the PES of the $|4o\rangle {\mathrm{He}}^{+}$ (blue dot-dot-dashed line) is plotted in (c) and (f). Insets in (c) and (f): zoom-ins of the PES peaks, for clear observation of the peak shifts. The DI JES integrated along $E_1,\int S_{\text{DI}}(E_1,E_2)d{E}_1$ (orange dotted line) are also plotted in the insets in (c) and (f) for comparison. The maxima in the DI JES are (a) $S_{\text{DI}}(E_1=0.899,E_2=1.603)=S_{\text{DI}}(E_1=1.603,E_2=0.899)=1.94\times {10}^3$ and (d) $S_{\text{DI}}(E_1=0.995,E_2=1.815)=S_{\text{DI}}(E_1=1.815,E_2=0.995)=5.65\times {10}^2$, respectively.