## 2 Formal Analysis
### XUV One-Photon Absorption Cross-Section  
We make the electric dipole approximation and assume that the attosecond pulse and the dressing pulse are both linearly polarized along the \(z\)-axis. Following the logic presented in Ref. [41], we start by solving the time-dependent Schrodinger equation:  
\[i\frac{\partial}{\partial t}|\Psi,t\rangle=\hat{H}(t)|\Psi,t\rangle, \tag{1}\]  
\[\hat{H}(t)=\hat{H}_{0}-E_{0}-\varepsilon_{d}(t)\hat{Z}-\varepsilon_{x}(t)\hat {Z}, \tag{2}\]  
where \(H_{0}\) is the unperturbed Hamiltonian of the electronic system, \(E_{0}\) is the system ground-state energy, \(\hat{Z}\) is the \(z\)-component of the electric dipole operator, and \(\varepsilon_{d}(t)\) and \(\varepsilon_{x}(t)\) are the dressing and attosecond (XUV) electric fields, respectively. Assuming the effect of the attosecond pulse can be treated perturbatively, the solution of Equation (1) is:  
\[|\Psi,t\rangle=|\Psi_{d},t\rangle+|\Psi,t\rangle^{(1)}+..., \tag{3}\]  
where \(|\Psi_{d},t\rangle\) is the electronic state vector in the absence of the attosecond pulse and \(|\Psi,t\rangle^{(1)}\) is the first-order correction with respect to the attosecond pulse. We employ the time-evolution operator \(\hat{U}_{d}(t,t_{0})\) to determine the optically dressed state vector \(|\Psi_{d},t\rangle\):\[|\Psi_{d},t\rangle=\hat{U}_{d}(t,t_{0})|\Psi_{0},t_{0}\rangle. \tag{4}\]  
Here \(t_{0}\) is a time before the system is optically dressed and \(|\Psi_{0},t_{0}\rangle=|\Psi_{0}\rangle\) is the initial state. The dipole moment of the electronic system along the \(z\)-axis can be expressed as:  
\[D(t)=\langle\Psi,t|\hat{Z}|\Psi,t\rangle=D_{HG}(t)+D^{(1)}(t)+... \tag{5}\]  
The term \(D_{HG}(t)\) describes harmonic generation driven by the dressing pulse only, and \(D^{(1)}(t)\) is the dipole moment correction to first order with respect to \(\varepsilon_{x}(t)\):  
\[D_{HG}(t)=\langle\Psi_{d},t|\hat{Z}|\Psi_{d},t\rangle, \tag{6}\]  
\[D^{(1)}(t)=i\int\limits_{-\infty}^{t}dt^{\prime}\varepsilon_{x}(t^{\prime}) \langle\Psi_{d},t|\hat{Z}\hat{U}_{d}(t,t^{\prime})\hat{Z}|\Psi_{d},t^{\prime} \rangle+c.c. \tag{7}\]  
In the following, we will separate the \(t\) dependence in \(\langle\Psi_{d},t|\hat{Z}\hat{U}_{d}(t,t^{\prime})\hat{Z}|\Psi_{d},t^{\prime}\rangle\) from the \(t^{\prime}\) dependence. To this end, we employ the following identity:  
\[\hat{U}_{d}(t,t^{\prime})=\hat{U}_{d}(t,t_{0})\hat{U}_{d}^{-1}(t^{\prime},t_{ 0}), \tag{8}\]  
which is correct for any reference time \(t_{0}\). For reasons that will become clearer in Section 2.2, we write explicitly \(\hat{U}_{d}^{-1}(t^{\prime},t_{0})\) rather than \(\hat{U}_{d}^{t}(t^{\prime},t_{0})\). Assuming that the attosecond pulse is shorter than all relevant electronic time scales [41], we approximate it by a delta function \(\varepsilon_{x}(t)=A_{x}\delta(t-\tau)\), where \(\tau\) is the moment when the attosecond pulse interacts with the system. Hence, employing Equations (7) and (8), the dipole moment correction along the \(z\)-axis can be written as:  
\[ \begin{split} D^{(1)}(t)&=i\sum\limits_{f}\Big{[} \int\limits_{-\infty}^{t}dt^{\prime}\varepsilon_{x}(t^{\prime})F_{0f}(t)F_{f0} (t^{\prime})-\int\limits_{-\infty}^{t}dt^{\prime}\varepsilon_{x}(t^{\prime}) F_{0f}^{*}(t)F_{f0}^{*}(t^{\prime})\Big{]}\\ &=iA_{x}\sum\limits_{f}\Big{[}F_{0f}(t)F_{f0}(\tau)-F_{0f}^{*}( t)F_{f0}^{*}(\tau)\Big{]}\,\theta(t-\tau),\end{split}  \tag{9}\]  
where \(\theta(t-\tau)\) is the Heaviside function. The functions \(F_{mn}(t)\) in Equation (9) represent transition dipole matrix elements between optically dressed states \(|\Psi_{m}\rangle\) and \(|\Psi_{n}\rangle\):  
\[F_{mn}(t)=\langle\Psi_{m}|\hat{U}_{d}^{-1}(t,t_{0})\hat{Z}\hat{U}_{d}(t,t_{0})| \Psi_{n}\rangle. \tag{10}\]  
The XUV one-photon cross-section of the system is given by [41]:  
\[\sigma^{(1)}(\omega)=\frac{4\pi\omega}{c}\text{Im}\Big{[}\frac{\hat{D}^{(1)}( \omega)}{\varepsilon_{x}(\omega)}\Big{]}, \tag{11}\]  
where \(\hat{D}^{(1)}(\omega)\) is the Fourier transform of \(D^{(1)}(t)\).