# Attosecond Time Delay in Photoionization of Noble-Gas and Halogen Atoms
## 3 Results and Discussion
### Phase Shift and Time Delay in the Vicinity of Giant Resonance and Cooper Minimum  
In this section, we discuss the general features of the Wigner time delay for noble-gas and halogen atoms. A simple observation of the bottom row of Figures 1-3 suggests that one can divide the time delay into three parts based on the photon energy. Within about 10 eV above the photoionization threshold, all Wigner time delays show large positive values and decrease dramatically due to the dominant Coulomb phase shift explained in the previous section. For photon energy larger than 200 eV in Figures 1 and 2, or 70 eV in Figure 3, the phase shift (third row in all figures) becomes constant and structureless, thus the time delay goes to zero. The photon energy range from tens of eV to a few hundred eV is where the giant resonance and Cooper minimum happens, and we will discuss how these cross section features affect the time delay. Let us first investigate Figure 1 for Xe and I atoms. The calculations for these two atoms look very similar. The following discussion applies to both of them and we will not distinguish them unless it is necessary. A resonance, including the Cooper minimum, usually means a phase change of \(\pi\). The broad \(4d\) giant resonance Figure 1a,b is dominated by the \(4d\rightarrow\varepsilon f\) channel and stretches roughly from 60-160 eV. Within the same photon energy range, the phase shift of \(4d\rightarrow\varepsilon f\) channel increases approximately at a constant slope from \(\pi\) to \(2\pi\), as shown in Figure 1e,f. Since the Wigner time delay is the energy derivative of the phase shift, we can estimate the time delay in the giant resonance regime using the average phase change slope  
\[\tau_{\rm W}=\frac{\rm phase\ change\ \pi}{\rm energy\ change\ of\ 100\ eV}= \frac{\pi}{3.675\ a.u.}=0.855\ a.u.=21\ as. \tag{22}\]  
The calculated time delay Figure 1g,h decreases slowly across the giant resonance regime and is about 30 \(as\) at the resonance peak 100 eV. The estimated \(21\ as\) is therefore not too far off from the calculation. In Figure 1a,b, the ionization cross sections of \(5p\) and \(5s\) subshells also have giant resonances due the interchannel interaction with \(4d\) subshell. Across the giant resonance energy range from 70 to 110 eV, Wigner time delays of \(5p\) and \(5s\) subshells in Figure 1g,h shows plateaus of roughly 10 \(as\), corresponding to a monotonic phase change across the giant resonance regime. Besides giant resonance, Cooper minimum is also accompanied by a \(\pi\) phase change. In Figure 1e,f, there are two Cooper minima at 50 and 150 eV for the \(5p\rightarrow\varepsilon d\) channel, and one Cooper minimum at 170 eV for the \(4d\rightarrow\varepsilon f\) channel. The width of these minima is much narrower than the giant resonance but still has a \(\pi\) phase change. One would expect large value (negative or positive) Wigner time delay. But these dramatic phase changes does not contribute much to the time delay, because the decreased angular momentum channels \(5p\rightarrow\varepsilon s\) and \(4d\rightarrow\varepsilon p\) dominate the ionization cross section and have a monotonically-decreased phase shift (not shown in the figure), which will generate moderate-value negative time delays. Our calculated time delays in Figure 1g,h support the above analysis, showing negative sinks at the Cooper minimum regime. At both Cooper minima, 50 eV for the \(5p\rightarrow\varepsilon d\) channel and 170 eV for the \(4d\rightarrow\varepsilon f\) channel, the phase shifts quickly increase by \(\pi\), indicating large positive time delay, but the actual time delays are about \(-50\ as\) due to the dominance of \(l_{0}\to l_{0}-1\) channels. In the vicinity of Cooper minimum, while the sign of time delay is determined by the \(l_{0}\to l_{0}-1\) channels, the width of the negative time delay sinks reflect the width of the Cooper minimum.