Asymmetry of Wigner's time delay in a small molecule

Ionization by an attosecond pulse launches an electron wave packet in the continuum which contains rich information about the pulse, the parent system, and the ionization dynamics. This emission process is not instantaneous in the sense that the electrons take a finite time to leave the potential. This time is closely related to the Wigner time. In this paper we introduce the stereo Wigner time delay, which measures the relative delay between electrons emitted to the left and right in an asymmetric system. We present a theoretical study of the delay in photoemission for a small asymmetric molecular system using the streaking technique. The stereo Wigner time delay shows advantages compared to previous schemes. Our numerical calculation shows that such a measurement removes the infrared laser-Coulomb coupling, which has been problematic in the interpretation of the measured delay in photoemission from atomic systems.

Figure 1

Dipole matrix elements. The amplitude (blue solid line) and phase (green dashed line) of the dipole $d\left(p\right)$ for 1D H by projection on (a) plane and (b) scattering waves. (c) and (d) The same as (a) and (b) but for a 1D CO molecular system. The inset graph depicts a zoom of the dipole amplitude, which demonstrates a small asymmetry in the case of scattering-wave calculation.

Figure 2

Stereo Wigner time with plane and scattering waves. (a) The left (green circles) and right (blue solid line) Wigner times and the SWTD (red dashed line) $\Delta t_{\mathrm{W}}^{\left(\mathrm{LR}\right)}$ in the H atom calculated by scattering waves. (b) and (c) Same as (a) but using plane and scattering waves for the asymmetric molecule CO, respectively. The minimum in the SWTD shifted in these two pictures. (d) The asymmetry of the dipole matrix element amplitude shows very small values in the molecular case.

Figure 3

Ionization by a single attosecond pulse. Electron densities as a function of the position $z$ and time $t$ for the ionization by an XUV attosecond pulse are shown for (a) the atomic and (b) molecular systems. The red horizontal lines are the symmetric positions on either side $z_d=\pm 30$ a.u., and the red dashed lines are the corresponding arrival TOFs. (c) Absolute values of the positions at the maxima of electron densities [green (light gray) and blue (dark gray) solid lines] and the average values [red (gray) and black circles with dashed lines] as a function of time. The horizontal dashed line indicates the target position 30 a.u., and the vertical ones mark the corresponding TOFs for both sides. (d) Comparison between the stereo delay from the TOF technique using (i) the position of maximal density [blue (dark gray) circles] and (ii) average position values [green (light gray) circles] and the results extracted from the exact dipole phase as a function of the photoelectron energy.

Figure 4

Streaking traces for measuring the stereo Wigner time delay. For H and CO, (a) and (b) show the streaking traces, which are the photoelectron momentum distributions as a function of time delay $\tau$ between the attosecond pulse and the IR laser field. White lines are the negative value of the vector potential, $-A_{\mathrm{L}}\left(\tau \right)$. The green squares and blue circles are the expectation values for the electrons with negative and positive momenta. These expectation values and the vector potential $-A_{\mathrm{L}}\left(\tau \right)$ for both systems are depicted in (c) and (d). The inset graph in (d) shows a clear time delay $\Delta t_S^{\left(\mathrm{LR}\right)}=-2.9$ a.u. between electrons emitted to the left and right. As expected, in the case of the H atom, this difference is zero; see the inset in (c).

Figure 5

Streaking measurement of the stereo Wigner time delay. The graph shows the comparison between the exact SWTD $\Delta t_{\mathrm{W}}^{\left(\mathrm{LR}\right)}$ (red line) and the time $\Delta t_S^{\left(\mathrm{LR}\right)}$ obtained by the streaking technique (black circles) as a function of the photoelectron energy.

Figure 6

Stereo Wigner time delay for the molecular case. (a) Stereo Wigner time delay as a function of XUV FWHM. The red squares depict the averages of the exact SWTD $\langle \Delta t_{\mathrm{W}}^{\left(\mathrm{LR}\right)}\rangle$ within the photoelectron energy bandwidth of the EWP. The black circles are the values measured by the streaking technique, $\Delta t_S^{\left(\mathrm{LR}\right)}$. (b) SSTD as a function of the IR peak electric field strength (black circles).