A time-dependent momentum-resolved scattering approach to core-level spectroscopies

We thank the Referees for their insightful and rigorous assessment of our manuscript. We greatly value their attention to detail that has helped us to greatly improve the manuscript. We are also pleased that both consider it worthy of publication.

-We have added a clarifying sentence:
"After obtaining the ground state we connect the extended probe at time $t=0$ and measure the momentum distribution function $n_{b,d}(k,t)$ at the detector as a function of time. Since in our tDMRG simulations we use open boundary condition, the proper definition of momenta corresponds to particle in a box states $\sin{(k_jx)}$ with momenta $k_j=j\pi/(L+1)$ with $ (j=1,\cdots,L)$. However, as customary in DMRG calculations, we vary $k$ continuously."

-We have added an entire description or the Hubbard model with additional simulation details directly under "Results".

-We have added the explicit form of the projector in the sentence:

"This can be done by means of a projector:
\begin{eqnarray}
H_{source}= -|\kin\rangle \langle \kin|+\lambda \sum_{ij}n_{b,s,i}n_{b,s,j},
\end{eqnarray}
where $|\kin\rangle \langle \kin|=n_{b,s}(\kk)=\frac{1}{L}\sum_{mn}e^{i\kin(\mathbf{R}_m-\mathbf{R}_n)}b^\dagger_{s,m}b_{s,n}$"

"The full calculation proceeds as follows: The system is first initialized in the ground state of $H_0+H_{source}$...."

-We have added the following discussion in the summary:
"In our calculations accuracy is kept it under control by using a sufficiently large number of DMRG states (bond dimension). Notice that at time $t=0$ the system in in the ground state of $H_0+H_{source}$. The core orbitals are in a product state of double occupied states and do not contribute to the entanglement. As time evolves, one electron will be excited from the core-orbitals, and one photon will eventually be emitted when the core hole recombines. When this occurs, the core orbitals return to a product state. Moreover, there is no hopping for the core degrees of freedom. This means that any additional entanglement will stem from the perturbations left behind in the system (which is a gapped Mott insulator) and the single photon at the detector, which will contribute to the entanglement by a bounded amount $\mathcal{O}(1)$. As a consequence, the entanglement growth will be minimal and simulations can proceed to quite long times. While we have not done a detailed quantitative analysis, we believe that the entanglement growth will be comparable, if not lower, than typical tDMRG simulations of spectral functions, particularly in the case of single-site RIXS."

- We have corrected the notation for Eq.(8)

-We have fixed the notation:
"the system absorbs a photon with energy $\win$ and momentumm $\kin$ and emits another one with energy $\wout$, momentum $\kout$. We hereby focus on the so-called ``direct RIXS'' processes, see Fig.\ref{fig:fig2} and Fig.~1 in Ref.\cite{Kourtis2012-PhysRevB.85.064423}). As a consequence, the photon loses energy $\Delta \omega=|\wout-\win|=\win-\wout$ (from now one referred-to as simply $\omega$) and the electrons in the solid end up in an excited state with momentum $\kout-\kin$. In the following, we consider $\kin=0$ and refer to the momentum transferred simply as $\kk$."

- Modified the sentence: "By an appropriate choice of $\Gamma_s^{\sigma\sigma'}=\Gamma_d^{\tau\tau'}=\Gamma$, and all others set to zero, one evolves the system in time to obtain a wave-function $|\psi_{\sigma\sigma',\tau\tau'}(t)\rangle$. This allows us to resolve the different contributions to the spectrum that split into spin conserving and non-conserving ones:..."

- Modified legends in Fig. 4 and Fig. 5

- We have expanded our description of Fig.5 in the text.