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I'm looking through the linked paper on x-ray scattering. The Section $5$ says that the expression $(67)$ (which contains an error but the editor mentions the correct variant at the beginning of the paper) describes the final state of the system after scattering. It has the form $$ \left| F \right\rangle = \left| {\Psi _0^{{N_{el}}}} \right\rangle {\widehat a^\dagger }_{{k_F},{\lambda _F}}\left| {{N_{EM}} - 1} \right\rangle, $$ where $\left| {\Psi _0^{{N_{el}}}} \right\rangle$ is the electronic ground state of the molecule with $N_{el}$ electrons,

$\left| {{N_{EM}} - 1} \right\rangle$ is the Fock state containing $N_{EM}-1$ photons in the mode (${\bf k_{in}}, \lambda_{in})$), and

${\widehat a^\dagger }_{{k_F},{\lambda _F}}$ creates a photon in the mode (${\bf k_{F}}, \lambda_{F})$)

I wonder why the field final state contains the creation operator? I thought that the final state should have been written like this $ \left| F \right\rangle = \left| {\Psi _0^{{N_{el}}}} \right\rangle \left| {{N_{EM}} - 1} \right\rangle$ because the field lost one photon. May it mean just that a field loses the photon in mode (${\bf k_{in}}, \lambda_{in})$) and creates a photon in mode $({\bf{k}_F},{\lambda _F})$ after scattering?

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I thought that the final state should have been written like this $ \left| F \right\rangle = \left| {\Psi _0^{{N_{el}}}} \right\rangle \left| {{N_{EM}} - 1} \right\rangle$ because the field lost one photon.

No, the field isn't losing one photon. One of the photons of the field is being scattered off of the atom. If the field were just losing one photon, that would be purely absorption and the atomic state would have to be in an excited state--not a ground state.

May it mean just that a field loses the photon in mode (${\bf k_{in}}, \lambda_{in})$ and creates a photon in mode $({\bf{k}_F},{\lambda _F})$ after scattering?

Yes, exactly. As the paper mentions, they are considering elastic scattering processes. So, the number of photons would remain conserved overall. The scattering would simply serve as a process in which the states of the scattered off photons are (potentially) changed because of their interaction with the atom.

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