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The result can be written in a closed form for $\phi^A_i(\vec{r},t)$, represented in real space, and   $\phi^B_i(\vec{k},t)$, in momentum space, with the following structure:  \begin{align}\label{eq:FMM_prop}  \left\{  \begin{array}{l}  \phi^A_i(\vec{r},t+\Delta t) =   \varphi^A_i(\vec{r},t+\Delta t) +\varphi^B_i(\vec{r},t+\Delta t)\\ t)\ ,\\  \phi^B_i(\vec{k},t+\Delta t) =  \vartheta^A_i(\vec{k},t+\Delta t)+\vartheta^B_i(\vec{k},t+\Delta t) t)\ ,  \end{array}  \right.\,,  \end{align}  andwith  the additional set of equations, \begin{align}\label{eq:FMM_prop_aux}  \left\{  \begin{array}{l}  \varphi^A_i(\vec{r},t+\Delta t) = M \hat{U}(\Delta t) \phi^A_i(\vec{r},t)\\ \phi^A_i(\vec{r},t)\ ,\\  \varphi^B_i(\vec{r},t+\Delta t) =  \frac{M}{(2\pi)^{\frac{3}{2}}}\int {\rm d}\vec{k} e^{\mathrm{i}\vec{k}\cdot\vec{r}} \hat{U}_{\rm v}(\Delta t)  \phi^B_i(\vec{k},t) \\ \ ,\\  \vartheta^A_i(\vec{k},t+\Delta t) =  \frac1{(2\pi)^{\frac{3}{2}}} \int {\rm d}\vec{r} e^{-\mathrm{i}\vec{k}\cdot\vec{r}} (1-M) \hat{U}(\Delta t)  \phi^A_i(\vec{r},t) \\ \ ,\\  \vartheta^B_i(\vec{k},t+\Delta t) =  \hat{U}_{\rm v}(\Delta t) \phi^B_i(\vec{k},t) -   \frac1{(2\pi)^{\frac{3}{2}}} \int {\rm d}\vec{r} e^{-\mathrm{i}\vec{k}\cdot\vec{r}}  \varphi^B_i(\vec{r},t+\Delta t) t)\ .  \end{array}  \right. .  \end{align}  The momentum-resolved photoelectron probability is then obtained directly from   the momentum components as~\cite{DeGiovannini_2012}