## 2 Methods
The coefficients are propagated in time within the diagonalized Hilbert space by multiplication with a pure phase factor, \(\mathrm{e}^{-\mathrm{i}E_{i}\Delta t/\hbar}\), containing the eigenvalues \(E_{i}\left[i=1,\;2,\;\left(3\mathrm{only}\;\mathrm{when}\;c_{c}\;\mathrm{is }\;\mathrm{included}\right)\right]\) of the diagonalized Hamiltonian. The complex time-dependent dipole moment \(\mathrm{d}^{+}(t)\) of the system rotating with positive frequencies (containing terms \(\propto\mathrm{e}^{\mathrm{i}\omega t}\)) is given by:  
\[\mathrm{d}^{+}(t)=d_{ge}\;c_{g}(t)c_{e}^{*}(t)+d_{gc}\;c_{g}(t)\;c_{c}^{*}(t), \tag{2}\]  
with the previously defined dipole matrix elements \(d_{ge}\) and \(d_{gc}\). The second term in Equation (2) does not contribute for the two-level system. The optical density (OD) of transient absorption is calculated with the frequency spectrum of the time-dependent dipole moment \(\mathrm{d}(\omega)=\mathrm{FT}\!\left\{\mathrm{d}^{+}(t)\right\}\) obtained after Fourier transformation and is given by taking the negative decadic logarithm of the ratio between transmitted (numerator) and incoming (denominator) spectral intensity:  
\[\mathrm{OD}(\omega)=-\log_{10}\!\left(\frac{\left|E(\omega)+i\,\eta\!\cdot\! \mathrm{d}(\omega)\right|^{2}}{\left|E(\omega)\right|^{2}}\right). \tag{3}\]  
here, \(E(\omega)\) is the frequency spectrum of the incoming laser pulse and \(\eta=10^{-4}\) is a constant connected to the macroscopic particle density of the target medium. Its numerical value is arbitrary and here chosen small enough to avoid macroscopic propagation effects. Equation (3) contains the term \(\left|E(\omega)+i\,\eta\!\cdot\!\mathrm{d}(\omega)\right|^{2}\), which is an interferometric superposition of the polarization response (\(P=\eta\!\cdot\!\mathrm{d}\)[22]) and the driving field following [PERSON]'s equations, which is at the heart of transient absorption spectroscopy (see Figure 1a). The resulting optical density (Figure 1d,e) is, thus, directly sensitive to the relative phase between the driving field and the dipole response [8] manifesting in an absorption profile, which can be parametrized by a generalized Fano line shape. To quantify the asymmetry of the resonant absorption lines we hence fit the OD with a Fano profile:  
\[\mathrm{OD}(\omega)=A\frac{\left(\varepsilon(\omega)+q\right)^{2}}{\left(1+ \varepsilon^{2}(\omega)\right)}+B,\;\varepsilon(\omega)=\frac{\hbar\omega-E_{ r}}{\Gamma_{r}/2}, \tag{4}\]  
where \(q\) is the Fano asymmetry parameter, \(\varepsilon(\omega)\) is the relative photon energy depending on the resonance position \(E_{r}\), its spectral width \(\Gamma_{r}\) and the photon energy \(\hbar\omega\). A and B are fitting parameters for the amplitude and offset, respectively. In the case of the weak-field two-level system, the absorption profile is unperturbed and assumes a Lorentzian shape (Figure 1d), which is the limiting case of a Fano profile for \(q\rightarrow\pm\infty\). When the third quasi-continuum state for the 2s2p resonance in helium is included in the weak-field case, the \(q\) parameter equals \(-2.75\)[23]. With strong laser fields and in the \"impulsive limit\" [8; 24; 25], i.e., when the pulse duration (here: 3 fs) is short compared to the resonance lifetime (here: \(\tau=1/\Gamma_{r}=17\) fs), the absorption profile asymmetry, which is captured by the \(q\) parameter, can be modified. For the transmission-geometry transient absorption, this change is related to a phase shift \(\Delta\varphi_{D}\) of the time-dependent dipole moment with respect to the weak-field response of the system [8] and is given by:  
\[q\left(\Delta\varphi_{D}\right)=-\cot\!\left(\frac{\Delta\varphi_{D}}{2} \right)\Leftrightarrow\;\Delta\varphi_{D}=2\;\mathrm{arg}(q-i). \tag{5}\]