Photoionization of helium by attosecond pulses: Extraction of spectra from correlated wave functions

We investigate the photoionization spectrum of helium by attosecond XUV pulses both in the spectral region of doubly excited resonances as well as above the double ionization threshold. In order to probe for convergence, we compare three techniques to extract photoelectron spectra from the wave packet resulting from the integration of the time-dependent Schrödinger equation in a finite-element discrete variable representation basis. These techniques are projection on products of hydrogenic bound and continuum states, projection onto multichannel scattering states computed in a $B$-spline close-coupling basis, and a technique based on exterior complex scaling implemented in the same basis used for the time propagation. These methods allow one to monitor the population of continuum states in wave packets created with ultrashort pulses in different regimes. Applications include photo cross sections and anisotropy parameters in the spectral region of doubly excited resonances, time-resolved photoexcitation of autoionizing resonances in an attosecond pump-probe setting, and the energy and angular distribution of correlated wave packets for two-photon double ionization.

Figure 1

Approximate regions of applicability of different extraction methods for the single-ionization continuum of helium as a function of photon energy and pulse duration for a single-photon transition. SI, single ionization; DE, double excitation; DI, double ionization. Green shaded: projection on scattering states (PSS); blue solid: projection onto asymptotic channel functions; red solid: Berkeley-ECS method. See text for details. The upper part shows the (logarithmic) ionization probability for single ionization and positions of excited bound states.

Figure 2

Photoelectron distribution in the $1s$ ${}^1{P}^o$ channel resulting from the photoionization of helium from the ground state with an XUV pulse with $\omega =2.4$ a.u., $I={10}^{12}$ W/cm${}^2$, and $T_{\mathrm{FWHM}}=200$ as. Projection onto scattering states (PSS), red solid line; Berkeley-ECS, blue dashed line; projection on the product of Coulomb functions at $t=1000$ as after the center of the XUV pulse, green dotted line.

Figure 3

Photoelectron spectrum as in Fig. 2 with (a) a close-up near the $s{p}_2^{+}$ resonance, and (b) the spectrum on a logarithmic scale to highlight small deviations in the tails of the direct ionization peak.

Figure 4

Photoelectron distribution in the $N=2$ ${}^1{P}^o$ channel resulting from the photoionization of helium from the ground state with an XUV pulse with $\omega =2.4$  a.u., $I={10}^{12}$ W/cm${}^2$, and $T_{\mathrm{FWHM}}=200$ as. PSS, red dashed line; ECS, blue solid line.

Figure 5

Photoionization cross section ${\sigma }_2\left(\omega \right)$ [Eq. (21)] and anisotropy parameter ${\beta }_2$ [Eq. (24)] for ionization of helium from the ground state to $N=2$ excited ionic states, in the region of autoionizing resonances between the $N=2$ and the $N=3$ thresholds. We compare our results obtained by the Berkeley-ECS and PSS methods with the theoretical values in velocity gauge by Moccia and Spizzo [67], earlier theoretical values by Sanchez et al. [82, 83], and experimental results by Menzel et al. [84], Zubek et al. [85], and Lindle et al. [86]. Our results agree perfectly with the theoretical values of Moccia et al. (and with results by Venuti et al. [74], not shown) and are in best agreement with the experimental data of Lindle et al.

Figure 6

Photoionization cross section and anisotropy parameter ${\beta }_2$ between the $N=2$ and the $N=3$ thresholds for ionization of helium from the metastable ${}^{1}S\left(1s2s\right)$ state. PSS, red solid line; ECS, blue dotted line.

Figure 7

Partial $N=2$ photoionization cross sections for the first ${}^1{P}^o$ autoionizing resonance below the $N=3$ threshold for ionization from the metastable ${}^{1}S$$\left(1s2s\right)$ state (a) and ionization from the ground state (c). In (b) the according ${\beta }_2$ parameters for $n=2$ are compared, where the energies are shifted such that the resonance appears at the same position. The lower energy axis corresponds to ionization from the ground state, while the upper energy axis corresponds to ionization from the metastable $\left(1s2s\right)$ state.

Figure 8

(a) Formation of the Fano profiles for a regular series of autoionizing Rydberg states according to the Fano model within the impulsive approximation. The energy distribution of the photoelectron is multiplied by a Gaussian spectral-shape function to simulate the effect of an attosecond pulse. This approach is justified as long as the duration of the pulse is much shorter than the lifetimes of the autoionizing states. See text for more details. (b) Total photoelectron spectrum of the wave packet created by the action of a subfemtosecond pulse on the ground state of the helium atom and computed by projecting the wave packet on products of bound ($Z=2$) and continuum ($Z=1$) Coulomb functions. Since such products are not eigenstates of the field-free Hamiltonian, the spectrum is only approximated and changes with time after the XUV pulse. As the doubly excited states populated by the pulse decay, characteristic Fano profiles build up. See text for more details.

Figure 9

Close-up of the evolution of the photoelectron spectrum determined by projection on products of Coulomb functions in comparison with the asymptotic Fano profile extracted from the same wave packet by projection onto scattering states immediately after the end of the external pulse.

Figure 10

(a) Time evolution of the photoelectron distribution $P(E,t)$ in the unperturbed continuum as a function of time after the impulsive excitation of a series of autoionizing resonances. (b) Asymptotic photoelectron distribution $\overline{P}(E,\tau )$ as a function of the time delay $\tau$ between the impulsive excitation and the impulsive depletion of the residual localized part of the autoionizing resonances. (c) shows the comparatively small differences that exist between the two spectra.

Figure 11

Singly differential photoelectron distribution after two-photon double ionization of helium by a subfemtosecond XUV pulse ($\omega =2.4$ a.u., $I={10}^{12}$ W/cm${}^2$, and $T_{\mathrm{FWHM}}=200$ as). Two different extraction methods are compared: projection on products of Coulomb functions with $Z=2$ performed 1 fs (red dashed line) and 7 fs (black solid line) after the peak of the XUV pulse and the Berkeley-ECS method (green dotted line) for a computational box with $R=244$ a.u. performed 1 fs after the peak of the XUV pulse.

Figure 12

Conditional angular distribution for one electron emitted in the laser polarization axis [gray arrow in (a)] and integrated over both electron energies after two-photon double ionization of helium by a subfemtosecond XUV pulse ($\omega =2.4$ a.u., $I={10}^{12}$ W/cm${}^2$, and $T_{\mathrm{FWHM}}=200$ as). Two different extraction methods are compared: (a) projection on products of Coulomb functions with $Z=2$ performed 7  fs (black solid line) after the peak of the XUV pulse and the Berkeley-ECS method (green dotted line) for an extraction radius of $R=244$ a.u. performed 1 fs after the peak of the XUV pulse. In (b) a close-up of the emission of the two electrons in the same direction is shown, illustrating the convergence of the Berkeley-ECS method with extraction radius (dotted lines) and of the Coulomb projection with propagation time (solid lines). The two techniques agree when the extraction radius of the Berkeley-ECS method approximately equals the position (of the relevant parts) of the wave packet at the time of projection on the Coulomb functions. The partial wave expansion includes angular momenta up to $l_1=l_2=15$.