  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.
