Angular dependence of photoemission time delay in helium

Time delays of electrons emitted from an isotropic initial state with the absorption of a single photon and leaving behind an isotropic ion are angle independent. Using an interferometric method involving XUV attosecond pulse trains and an IR-probe field in combination with a detection scheme, which allows for full three-dimensional momentum resolution, we show that measured time delays between electrons liberated from the $1s^2$ spherically symmetric ground state of helium depend on the emission direction of the electrons relative to the common linear polarization axis of the ionizing XUV light and the IR-probing field. Such time delay anisotropy, for which we measure values as large as 60 as, is caused by the interplay between final quantum states with different symmetry and arises naturally whenever the photoionization process involves the exchange of more than one photon. With the support of accurate theoretical models, the angular dependence of the time delay is attributed to small phase differences that are induced in the laser-driven continuum transitions to the final states. Since most measurement techniques tracing attosecond electron dynamics involve the exchange of at least two photons, this is a general and significant effect that must be taken into account in all measurements of time delays involving photoionization processes.

Figure 1

Two-photon ionization pathways starting from ground-state helium. (a) Schematic defining the emission angle θ as the electron emission direction relative to the XUV and IR polarization axes and illustrating the different photoelectron partial waves of the corresponding final quantum states, which arise from the exchange of two photons. (b) Schematic illustrating the different quantum paths, which contribute to the final state of the liberated photoelectrons after the interaction with the XUV and IR fields. (c) XUV spectrum, which has been used to carry out the experiments.

Figure 2

Principle of the time delay extraction. (a–c) Examples of measured RABBITT spectrograms and oscillations of sideband (SB) 20 (marked by white dashed lines) for different ranges in emission angle. Note that the energy scale corresponds to the sum of photoelectron kinetic energy and the ionization potential of helium (ionization potential of helium: 24.5874 eV [34]). In (a) only the photoelectrons detected within a ${30}^{\circ }$ cone of emission [Fig. 1] are selected. Panel (c) comprises photoelectrons emitted within a hollow cone of emission between ${60}^{\circ }$ and ${65}^{\circ }$. Panel (b) shows an example of the intensity oscillations of SB 20 (red data points) obtained by integrating the counts within an energy window of 0.75 eV centered at the peaks of the SB oscillations (white dashed lines) together with their corresponding fits (blue solid lines). The time delay $\mathrm{\Delta }\tau \left(\theta \right)$ is clearly visible as a temporal shift between the two different SB oscillations.

Figure 3

Angular dependence of photoemission time delays in helium for different photoelectron kinetic energies. (a–d) For all photoelectron kinetic energies, referenced by the sidebands (SBs) of the harmonic spectrum of the attosecond XUV pulse train, the experimentally retrieved atomic delay (blue data points with error bars) is shown as a function of the emission angle $\theta$, following the procedure described in Fig. 2. For example, a delay at ${15}^{\circ }$ is understood as the delay between photoelectrons emitted at angles between ${10}^{\circ }$ and ${15}^{\circ }$ and photoelectrons emitted between $0^{\circ }$ and ${30}^{\circ }$ (reference). As a comparison, the corresponding theoretical predictions are also included in the graphs comprising an ab initio simulation (red dashed line with asterisks), where the time-dependent Schrödinger equation (TDSE) is solved using a nearly exact method [38] taking into account both electrons in He, a calculation solving the TDSE within the single-active electron (SAE) approximation (black dashed line with triangles) and lowest-order perturbation-theory (LOPT) calculation (green dashed line with diamonds). The different theories are in very good agreement and reproduce the experimental data well. The inset in (b) shows the typical behavior of the angle-dependent delay predicted by LOPT for an angular range up to ${90}^{\circ }$. As a consequence of the node in the $d$ wave, at large emission angles θ the delay changes significantly.

Figure 4

Spread of the data extracted from individual data sets and error estimation for SB 22. Spread of 13 individual data sets (blue circles) for each of the considered angular sectors. The means of the 13 individual data sets for each angular sector are shown as red diamonds. The according error bars correspond to the standard deviation of the 13 individual data sets at each angular sector.

Figure 5

Comparison between experiment and TDSE ab initio (two electrons) simulation. (a) Experimental data. (b) Results of the TDSE ab initio (two electrons) calculation. The two-dimensional (2D) plots show the delay-integrated photoelectron spectrum as a function of the emission angle θ, defined in Fig. 1. On the left- and right-hand side of (a,b), respectively, the angle-integrated projections for experiment and theory highlight the presence of four SBs comprising SB 18 to SB 24.

Figure 6

Effect of contamination in ab initio calculations on SB 18 in helium. Results of the angular anisotropy obtained from ab initio calculations using three distinct integration intervals $\mathrm{\Delta }E$. For the results represented by the red dashed line with circles $\mathrm{\Delta }E=1.6\mathrm{eV}$ was considered, while the calculations yielding the blue and cyan dashed lines with stars considered $\mathrm{\Delta }E=0.9\mathrm{eV}$ and $\mathrm{\Delta }E=0.6\mathrm{eV}$, respectively.

Figure 7

Comparison between TDSE ab initio two electrons and LOPT calculations in helium for SB 18 using idealized long pulses. The results obtained from TDSE ab initio two-electron (blue dashed line with asterisks) and LOPT (green dashed line with diamonds) calculations are in excellent agreement. This confirms that the discrepancy between the LOPT and the time-dependent simulations in Fig. 3 may be attributed to the effect of contamination.

Figure 8

Comparison between distinct LOPT calculations in helium for SB 18. The results of lowest-order perturbation theory (LOPT) calculations considering a single two-photon process ($\mathrm{XUV}+\mathrm{IR}$) and additionally its reversed time ordering ($\mathrm{XUV}+\mathrm{IR}$ and $\mathrm{IR}+\mathrm{XUV}$) are shown as dashed lines with violet filled and green open diamonds, respectively.