Universal features in sequential and nonsequential two-photon double ionization of helium

We analyze two-photon double ionization of helium in both the nonsequential ($\hslash \omega <I_2\approx 54.4\hspace{0.5em}\mathrm{eV}$) and sequential ($\hslash \omega >I_2$) regime. We show that the energy spacing $\Delta E=E_1-E_2$ between the two emitted electrons provides the key parameter that controls both the energy and the angular distribution and reveals the universal features present in both the nonsequential and sequential regime. This universality, i.e., independence of $\hslash \omega$, is a manifestation of the continuity across the threshold for sequential double ionization. For all photon energies considered, the energy distribution can be described by a universal shape function that contains only the spectral and temporal information entering second-order time-dependent perturbation theory. Angular correlations and distributions are found to be more sensitive to the value of $\hslash \omega$. In particular, shake-up interferences have a large effect on the angular distribution. Energy spectra, angular distributions parametrized by the anisotropy parameters ${\beta }_j$, and total cross sections presented in this paper are obtained by fully correlated time-dependent ab initio calculations.

Figure 1

(a) Singly differential energy distribution $P^{\text{DI}}(\Delta E)$ as a function of the energy difference $\Delta E=E_1-E_2$, normalized to the yield at $\Delta E=0$ for photon energies between $42$ and $80\hspace{0.5em}\mathrm{eV}$. The pulses had a duration $T=4.5\hspace{0.5em}\mathrm{fs}$ and an intensity $I_0={10}^{12}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$. The gray lines show the expected positions of the peaks for the sequential process for different intermediate states (i.e., with and without shake up). Here, ${\mathcal{E}}_n$ is the excitation energy from the ionic $n=1$ state to an excited state $n$ (${\mathcal{E}}_2\approx 40.8\hspace{0.5em}\mathrm{eV}$). The black line shows the energy distribution based on second-order perturbation theory $P_{\mathcal{G}}^{\text{DI}}(\Delta E)$, calculated for the same pulse parameters as in the numerical simulations (sin${}^2$ pulse with $T=4.5\hspace{0.5em}\mathrm{fs}$). (b) and (c) show the two-electron energy distribution $P^{\text{DI}}(E_1,E_2)$ on the log scale for (b) $\hslash \omega =48\hspace{0.5em}\mathrm{eV}$ and (c) $\hslash \omega =80\hspace{0.5em}\mathrm{eV}$.

Figure 2

(a) Anisotropy parameters $\beta (\omega ,\Delta E)$ and $\gamma (\omega ,\Delta E)$ [see Eq. (9)] for photon energies between $42$ and $80\hspace{0.5em}\mathrm{eV}$ as a function of the energy difference $\Delta E=E_1-E_2$ in the interval $[-E_{\text{tot}},E_{\text{tot}}]$, with $E_{\text{tot}}=2\hslash \omega +E_0$ and $E_0\approx -79\hspace{0.5em}\mathrm{eV}$. The pulse parameters are the same as in Fig. 1. The upper group of lines contains the $\beta$ parameters, while the lower group contains the values for $\gamma$. Note that the angular distribution of the electron with $E_1$ is shown; thus $\Delta E>0$ characterizes the faster electron and $\Delta E<0$ the slower one. (b) Angular distribution at equal energy sharing $P^{\text{DI}}(\Delta E=0,\theta )$ for $\hslash \omega =42,48$, and $70\hspace{0.5em}\mathrm{eV}$ [from inside to outside, same color code as in (a)], normalized to a value of 1 for $\theta =0^{°}$. The laser polarization axis is indicated by the arrow. The black solid line shows a $\cos {}^2(\theta )$ distribution.

Figure 3

Two-dimensional energy-angular differential distribution $P^{\text{DI}}(\Delta E,{\theta }_1)$ for an xuv pulse with $T=4.5\hspace{0.5em}\mathrm{fs}$ and $\hslash \omega =80\hspace{0.5em}\mathrm{eV}$ as a function of the energy difference $\Delta E=E_1-E_2$ between the two electrons, and emission angle ${\theta }_1$, relative to the polarization axis. The vertical dashed white lines show the expected positions of the peaks for the sequential process; see Fig. 1. The horizontal nodal line ${\theta }_1=\pi /2$ is visible except near shake-up resonances.

Figure 4

(a) Anisotropy parameters $\beta (E_1)$ and $\gamma (E_1)$ as well as $\gamma {}^{\prime }(E_1)$ (dashed line), which according to Eq. (10) enforces vanishing probability of electron ejection at ${90}^{°}$ to the polarization axis. (b)–(f) show angle-resolved one-electron distributions at different electron energies, normalized to a value of $1$ for $\theta =0^{°}$: (b) equal energy sharing, (c) asymmetric energy sharing, (d) main sequential peak (intermediate state: $|{\mathrm{He}}^{+}1s\rangle$), (e) second sequential peak (intermediate state: $|{\mathrm{He}}^{+}2l\rangle )$, and (f) third sequential peak (intermediate state: $|{\mathrm{He}}^{+}3l\rangle$). Parameters: central frequency $\hslash \omega =80\hspace{0.5em}\mathrm{eV}$ and duration $T=4.5\hspace{0.5em}\mathrm{fs}$. The thin black line in (b)–(f) shows a $\cos {}^2$ distribution.

Figure 5

Anisotropy parameters $\beta$ (upper group of lines and squares) and $\gamma$ (lower group of lines and circles) for TPDI in the sequential regime with an xuv pulse with (a) $\hslash \omega =90\hspace{0.5em}\mathrm{eV}$ in the long pulse limit and $T=4.5\hspace{0.5em}\mathrm{fs}$ (FWHM–$\sin {}^2$ pulse envelope), in comparison with recent time-independent results by Ivanov et al. [19], and (b) $\hslash \omega =91.6\hspace{0.5em}\mathrm{eV}$ for ultrashort pulse durations $T=150$ and 450 as (FWHM–Gaussian pulse envelope), in comparison with the calculations of Barna et al. [18].

Figure 6

Angular correlation characterized by the expectation value of the angle between the emitted electrons, $\langle \cos {\theta }_{12}\rangle$ as a function of $(\Delta E)$ for photon energies between $42$ and $80\hspace{0.5em}\mathrm{eV}$ (the other field parameters are the same as in Figs. 1 and 2).

Figure 7

Energy distribution divided by the pulse duration $P^{\text{DI}}(\Delta E)/T$, anisotropy parameters $\beta$, $\gamma$, and $\langle \cos {\theta }_{12}\rangle$ as a function of $\Delta E$ for two different pulse durations: $T=4.5\hspace{0.5em}\mathrm{fs}$ (blue dashed line) and $T=10\hspace{0.5em}\mathrm{fs}$ (red solid line) and $\hslash \omega =80\hspace{0.5em}\mathrm{eV}$.

Figure 8

Total TPDI cross section obtained using $\sin {}^2$ pulses with different durations $T$. Spectral distributions of the pulse are displayed for one data point for each $T$, showing the area over which the cross section obtained is effectively integrated. The intensity is ${10}^{12}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$ for all pulses. Angular momenta up to $L_{\max }=3$ for the total angular momentum and $l_{1,\max }=l_{2,\max }=7$ for the single angular momenta are included. The radial boxes had extensions up to $r_{\max }=1400\hspace{0.5em}\mathrm{a}.\mathrm{u}.$ for the $10\hspace{0.5em}\mathrm{fs}$ pulses.