Nonlinear Compton scattering is an inelastic scattering process where a photon is emitted due to the interaction between an electron and an intense laser field. With the development of X-ray free-electron lasers, the intensity of X-ray laser is greatly enhanced, and the signal from X-ray nonlinear Compton scattering is no longer weak. Although the nonlinear Compton scattering by an initially free electron has been thoroughly investigated, the mechanis of nonrelativistic nonlinear Compton scattering of X-ray photons by bound electrons is unclear yet. Here, we present a frequency-domain formulation based on the nonperturbative quantum electrodynamic to study nonlinear Compton scattering of two photons off a bound electron inside an atom in a strong X-ray laser field. In contrast to previous theoretical works, our results clearly reveal the existence of anomalous redshift phenomenon observed experimentally by Fuchs et al. (Nat. Phys. 11, 964 (2015)) and suggest its origin as the binding energy of the electron as well as the momentum transfer from incident photons to the electron during the scattering process. Our work builds a bridge between intense-laser atomic physics and Compton scattering process that can be used to study atomic structure and dynamics at high laser intensities.
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