## 2 Gaussian Pulse
### Hydrogen Atom
Let us assume that the spectral width of the pulse contains a sufficient number of levels that the sum in Equation (1) can be approximately replaced by an integral,  
\[c_{f}\approx\pi E_{0}^{2}T^{2}\int_{-\infty}^{\infty}\left[e^{-\Delta_{m}^{2 }T^{2}}-i\frac{2}{\sqrt{\pi}}\,F\left(\Delta_{m}T\right)\right]s(\Delta_{m})d \Delta_{m} \tag{12}\]  
where \(s(\Delta_{m})\) denotes the density of states multiplied with \(\mu_{fm}\mu_{mi}\). By noting that \(F(\Delta T)\approx(2\Delta T)^{-1}\), thus, \(F(\Delta T)\approx F(\Delta)/T\) is a good approximation in most region of \(\Delta T\), Equation (12) can be rewritten as,  
\[c_{f}\approx\pi E_{0}^{2}T\int_{-\infty}^{\infty}\left[e^{-\Delta^{2}}s( \Delta/T)-i\frac{2}{\sqrt{\pi}}\,F\left(\Delta\right)s(\Delta)\right]d\Delta \tag{13}\]  
Since \(s(\Delta/T)\) in the first term in the integrand can usually be approximated as \(s(0)\) for the central frequency, the ratio \(c_{S}/c_{D}\), and thus \(\delta\) and \(W\), becomes roughly independent of \(T\). Also, due to the continuity of the oscillator strength distribution across the ionization threshold [74], one can see that \(c_{S}/c_{D}\) also changes smoothly as a function of \(\hbar\omega\). It should be noticed that, for the case of two-photon above-threshold ionization [37] and in the long-pulse limit, the first (resonant path)and second (nonresonant path) terms of Equation (12) tend to, e.g., the second and first terms of Equation (3.2) of [37], respectively. Figure 4: Pulse-width dependence of the phase-shift difference \(\delta\) for H [(**a**)–(**d**)] and He [(**d**)–(**g**)] for different values of photon energy. Panel (**d**) is the close-up of the vertical-axis range between 2.5 and 3.0 radians of panel (**c**). The pulse is nonresonant for (**a**) and (**e**), resonant with a single level for (**b**) and (**f**), and resonant with multiple levels for (**c**), (**d**), and (**g**). The resonant excitation energies for He \(1s2p^{1}P\), \(1s3p^{1}P\), \(1s4p^{1}P\), \(1s5p^{1}P\) levels are 21.218, 23.087, 23.742, and 24.046 eV [48]. How dense should the levels be within the spectral width for the above discussion to be valid? One can estimate it by investigating how well the integral \(\int_{-\infty}^{\infty}e^{-x^{2}}dx\) can be approximated by the sum \(\sum_{n=-\infty}^{\infty}e^{-x_{n}^{2}}\Delta x\) of the values at discrete points, where \(x_{n}=(n+\delta n)\Delta x\), with \(\Delta x(>0)\) and \(\delta n\) being the increment of \(x\) and the offset of \(n\), respectively. In Figure 5 we plot the ratio,  
\[\frac{\sum_{n=-\infty}^{\infty}e^{-x_{n}^{2}}\Delta x}{\int_{-\infty}^{\infty }e^{-x^{2}}dx}=\theta_{3}(\delta n\,\pi,e^{-\frac{\pi^{2}}{\Delta x^{2}}}) \tag{14}\]  
as a function of \(\Delta x\) for \(\delta n=0,\frac{1}{4}\), and \(\frac{1}{2}\), where \(\theta_{a}(u,q)\) denotes the theta function [60]. The approximation is surprisingly good even with \(\Delta x=1\), though the FWHM of \(e^{-x^{2}}\) is \(2\ln 2\,(=1.66511\cdots)\). This explains why \(\Delta\) varies smoothly even in the range of \(\hbar\omega\) where the level spacing is not negligibly small compared with the spectral width in Figure 1.  
\[i\frac{\partial\Phi(\mathbf{r},t)}{\partial t}=\left[-\frac{1}{2}\
abla^{2}- \frac{1}{r}-zf(t)\sin\omega t\right]\Phi(\mathbf{r},t) \tag{10}\]0 _Cont._  
\[i\frac{\partial\Phi(\mathbf{r},t)}{\partial t}=\left[-\frac{1}{2}\
abla^{2}- \frac{1}{r}-zf(t)\sin\omega t\right]\Phi(\mathbf{r},t) \tag{10}\]1 Ratio given by Equation (14) as a function of \(\Delta x\) for different offset values \(\delta n\).