Dynamic interference in the photoionization of He by coherent intense high-frequency laser pulses: Direct propagation of the two-electron wave packets on large spatial grids

The direct ionization of the helium atom by intense coherent high-frequency short laser pulses is investigated theoretically from first principles. To this end, we solve numerically the time-dependent Schrödinger equation for the two-electron wave packet and its interaction with the linearly polarized pulse by the efficient time-dependent restricted-active-space configuration-interaction method (TD-RASCI). In particular, we consider photon energies which are nearly resonant for the $1s\to 2p$ excitation in the ${\mathrm{He}}^{+}$ ion. Thereby, we investigate the dynamic interference of the photoelectrons of the same kinetic energy emitted at different times along the pulse in the two-electron system. In order to enable observation of the dynamic interference in the computed spectrum, the electron wave packets were propagated on large spatial grids over long times. The computed photoionization spectra of He exhibit pronounced interference patterns the complexity of which increases with the decrease of the photon energy detuning and with the increase of the pulse intensity. Our numerical results pave the way for experimental verification of the dynamic interference effect at presently available high-frequency laser pulse sources.

Figure 1

Sketch of the presently studied process, in which absorption of a photon of energy $\omega$ from a high-frequency pulse promotes one of the two electrons of He into the photoionization (PI) continuum of energy ${\varepsilon }_{\mathrm{PI}}$. Subsequent absorption of another nearly resonant photon from the same pulse can either couple the $1s$ and $2p$ states of the second electron remaining in the ${\mathrm{He}}^{+}$ ion, or promote the photoelectron into continuum state of higher energy ${\varepsilon }_{\mathrm{ATI}}$ via the above threshold ionization (ATI) process. The two coupled ionization thresholds ${\mathrm{He}}^{+}\left(1s^1\right)$ and ${\mathrm{He}}^{+}\left(2p^1\right)$ repel each other and follow a time-dependent energy shifts $\mathrm{\Delta }\left(t\right)$ provided by the laser pulse. These shifts result in the dynamic interference in photoelectron spectrum of He.

Figure 2

Radial parts of the one-electron functions, $r{\varphi }_{\alpha }\left(r\right)$, used as the basis functions for the bound electron in the two-electron TD-RASCI ansatz Eq. (8). Shown are the $1s_{\mathrm{HF}}$ Hartree-Fock orbital of the He atom, $1s_{ion}$ and $2p_{ion}$ orbital of the ${\mathrm{He}}^{+}$ ion, and the correlation functions $\widetilde{2s}$ and $\widetilde{3p}$ defined via Eqs. (17) and (19), respectively.

Figure 3

The final photoelectron radial density (upper panel) and the final photoelectron spectrum (lower panel) after the intense linearly polarized Gaussian-shaped pulse has expired. The pulse duration $\tau$, carrier frequency $\omega$, and peak intensity $I_0$, used in the calculations, as well as assignments of the main structures in the spectrum are indicated in the figure. Note the logarithmic scales on the vertical axes. Here and below, the electron energy was corrected by $-0.264$ eV for the difference between the theoretical and experimental ionization potentials (see Sec. 2c for details).

Figure 4

Photoionization spectra of the helium atom computed for Gaussian-shaped pulses of 30 fs duration and peak intensity of $2.5\times {10}^{14}$ $W/{\mathrm{cm}}^2$ for different carrier frequencies (indicated near each spectrum) around the $1s\to 2p$ resonant excitation in the ${\mathrm{He}}^{+}$ ion. The photoelectron energies expected in the weak field regime are indicated by the vertical down arrows. For the carrier frequencies below 1.5 a.u., the shift of the $1s$ threshold is negative, which results in the positive shift in the corresponding major part of the electron spectrum. On the contrary, the shift of the $2p$ threshold is positive, and for the associated minor part of the spectrum it is thus negative. For the frequencies above 1.50 a.u., the shift of the $1s$ threshold is positive and of the $2p$ threshold is negative, which results in the negative shift in the corresponding major part of the spectrum and in the positive shift in its minor part. The complexity of the dynamic interference patterns in the spectrum increases with the decrease of the photon energy detuning. The resonant carrier frequency of 1.50 a.u. produces nearly symmetric Autler-Townes doublet structured by distinct dynamic interference patterns.

Figure 5

Photoionization spectra of the helium atom computed for Gaussian-shaped pulses of 30 fs duration, two carrier frequencies of 1.500 a.u. (left panel) and 1.528 a.u. (right panel), and different peak intensities (indicated near each spectrum). For further details, see caption of Fig. 4. The complexity of the dynamic interference patterns in the spectrum increases with the increase of the pulse intensity.

Figure 6

Comparison of the dynamic interference patterns in the major part of the photoionization spectrum, which corresponds to the ${\mathrm{He}}^{+}\left(1s^1\right)$ final state, computed numerically via the electron wave packet propagation and estimated as described in Sec. 3b for two Gaussian-shaped pulses. The computational parameters are indicated in the panels. Each estimated spectrum is normalized to the maximum of the corresponding numerical one.