## 2 Gaussian Pulse  
### Time-Dependent Perturbation Theory  
In this subsection, let us do a simple analysis on how the relative importance of the resonant and nonresonant paths depends on pulse width, based on the second-order time-dependent perturbation theory within the common rotating wave approximation. The dynamic Stark effect is negligible for pulse parameters used in the present study. We consider the process where a laser pulse with a central frequency \(\omega\) and a pulse envelope \(f(t)\), linearly polarized in the \(z\) direction, promotes an atomic electron from an initial state \(|i\rangle\) to a final continuum state \(|f\rangle\) through two-photon absorption. The complex amplitude \(c_{f}\) of the final state after the pulse in the interaction picture can be written as,  
\[c_{f}=\sum_{m}\int_{-\infty}^{\infty}\mu_{fm}e^{i\Delta_{f}t}f(t)\left(\int_{- \infty}^{t}\mu_{mi}e^{i\Delta_{m}t^{\prime}}f(t^{\prime})dt^{\prime}\right)dt \tag{1}\]  
where \(\mu_{mn}\) denotes the dipole transition matrix element between states \(m\) and \(n\), \(\Delta_{m}=\omega_{m}-(\omega_{i}+\omega)\), \(\Delta_{f}=\omega_{f}-(\omega_{m}+\omega)\) with \(\omega_{m}\) being the energy eigen-value of state \(m\), and the sum runs over all the intermediate bound and continuum states \(m\). One can equivalently express Equation (1) by Equation (3) of [45] using the Fourier transform of \(f(t)\). Although a rectangular pulse is often assumed in previous work [2] and textbooks, we take, as a more realistic choice, a Gaussian profile \(f(t)=E_{0}e^{-t^{2}/2T^{2}}\), with \(E_{0}\) and \(T\) being the field amplitude and the pulse width, respectively. More precisely, \(T\) is related to the full-width-at-half-maximum (FWHM) pulse width \(T_{1/2}\) as \(T_{1/2}=2\sqrt{\ln 2}\,T\). For the case of \(\omega_{f}=\omega_{i}+2\omega\), in particular, one can perform the integrals in Equation (1) analytically to obtain a physically transparent expression:  
\[c_{f}=\pi E_{0}^{2}T^{2}\sum_{m}\,\mu_{fm}\mu_{mi}\left[e^{-\Delta_{m}^{2}T^{ 2}}-i\frac{2}{\sqrt{\pi}}\,F\left(\Delta_{m}T\right)\right] \tag{2}\]  
where \(F(x)=e^{-x^{2}}\int_{0}^{x}e^{t^{2}}\,dt\) denotes [PERSON]'s integral [60], which tends to \(x\) near the origin and \(1/2x\) for \(x\rightarrow\infty\). Only resonant states within the spectral width of the pulse contribute to the first term, corresponding to the resonant path. In particular, if a single intermediate level \(r\) is resonant with the pulse spectrum, Equation (2) can be rewritten as,  
\[c_{f}=\pi E_{0}^{2}T^{2}\left[\mu_{fr}\mu_{ri}e^{-\Delta_{r}^{2}T^{2}}-i\sum_ {m}\,\mu_{fm}\mu_{mi}\frac{2}{\sqrt{\pi}}\,F\left(\Delta_{m}T\right)\right] \tag{3}\]  
On the other hand, the asymptotic behavior of \(F(x)\) suggests that all the intermediate states except for the exact resonance (\(\Delta_{m}=0\)) participate in the second term, as expected for nonresonant paths. While either term dominates for a relatively long pulse (ps and ns), assuming that resonant excitation is not saturated, we can expect that the two terms are comparative for sufficiently short pulses and that their relative importance, which may be expressed as \(\arg c_{f}\), varies with \(T\). In such a situation, the amplitude ratio \(c_{S}/c_{D}\) between the final \(S\) and \(D\) continuum states is complex, since the branching ratio \(\mu_{Sm}/\mu_{Dm}\) of the transitions from the intermediate \(P\) states \(m\) to each state depends on \(m\). While the actual outgoing wave packets involve the contribution from the final states with \(\omega_{f}\
eq\omega_{i}+2\omega\), it is instructive to write \(\arg c_{S}/c_{D}\) using Equation (3) as follows:  
\[\frac{c_{S}}{c_{D}}=\frac{\mu_{Sr}}{\mu_{Dr}}\frac{\sqrt{\pi}Te^{-\Delta_{r}^ {2}T^{2}}-i[a_{S}+2F(\Delta_{r}T)T]}{\sqrt{\pi}Te^{-\Delta_{f}T^{2}}-i[a_{D}+ 2F(\Delta_{r}T)T]} \tag{4}\]  
with \(a_{f}(f=S,D)=(\mu_{fr}\mu_{ri})^{-1}\sum_{m(\
eq r)}\mu_{fm}\mu_{mi}/\Delta_{m}\). Here we have assumed that \(F(\Delta_{m}T)\approx(2\Delta_{m}T)^{-1}\) for all the other intermediate states.