Explanation of the anomalous redshift on nonlinear X-ray Compton scattering spectrum by a bound electron

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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The Compton effect is well known for proving the quantum hypothesis of light in 1923 experi-33 mentally [1]. Afterwards the impulse approximation (IA) approach to Compton scattering on treating the NCS could be shortly provided. Specifically, we will focus on the double differential 66 probability (DDP) for the NCS process of a bound electron in an X-ray laser field. Our calculation 67 will clearly demonstrate that in the DDP spectrum of the two-photon NCS, as the energy of the 68 scattered photon increases, a redshift peak will appear, which is in contrast to the results by  The frequency-domain theory is based on the nonperturbative quantum electrodynamic, where 73 the laser-matter system can be regarded as an isolated one, hence the total energy of the system 74 is conserved during the laser-matter interaction process and the formal scattering theory [35] 75 can be applied. In this theory, the incident laser field, as a part of the whole system, is regarded 76 as a quantized field, and all dynamic processes are treated as quantum transitions between two 77 states of the laser-matter system. In the following, we develop this theory to investigate the NCS 78 by a bound electron in intense laser fields. Natural units (ℏ = = 1) are used throughout unless 79 otherwise stated. The -matrix element between the initial state | and the final state | is where 0 is the non-interaction part of the Hamiltonian for the atom-radiation system, is the 81 atomic binding potential, and is the interaction operator between electron and photons. The 82 initial state is | = Φ (r) ⊗ | ⊗ |0 with energy = (− ) + ( + respectively. In this work, since the contribution of the two-step transition is much smaller than 92 that of the one-step transition under the present laser conditions at the scattered photon energy 93 around twice of the incident photon energy, the second term in Eq. (1) is dropped here and will 94 be investigated in the future. Hence, the -matrix element for NSC can be expressed as

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The matrix element of the LEM transition can be written as and 114 where Λ = 1 represents the half amplitude of the classical field in the limits of 1 → 0 115 and → ∞. In Eq.
(3)-(5), the two-photon NCS processes correspond to = 2. Furthermore, 116 the contribution of + can be ignored, since it is much smaller than that of the other two terms 117 under the present laser conditions.

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The expression of the DDP for a Compton scattering process can be written as [36] where Ω is the differential solid angle of vector k 2 . To analyze the results more clearly, we 120 may rewrite the DDP by three parts: where the three parts on the right-hand of the equation represent the contributions of EMM, 122 LEM and their cross-term (CT). In the following, we will see how these terms affect the peak 123 position of the DDP spectrum for two-photon NCS processes.  125 We now calculate the DDP for two-photon Compton scattering by a 1 electron of Be atom, 126 where the intensity of the laser field is 4 × 10 20 W/cm 2 and the photon energy is 9.25 keV. peak on the DDP spectrum is due to LEM transition, as shown in Fig. 2(b). To find the reason 143 of the shifts of these peaks, we may simplify Eqs.

The redshift
with 2 being the angle between * 2 and the electron momentum P . And the matrix element 149 of EMM transition can be approximated as with 1 being the angle between 1 and the electron momentum P . The DDP spectra by 151 Eq. (9) and (10) are shown by the triangles and circles respectively in Fig. 2(a)-(c), where they 152 agree with the corresponding numerical results. 153 We firstly consider the influence of the atomic wavefunction on the DDP spectra. By integrat-

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ing the modular square of the wavefunction over P , we obtain the electron density distribution 155 as a function of the scattered photon energy 2 and the scattering angle , as shown in Fig. 2(d).  Fig. 3(b) and (c), respectively. The dots in Fig. 3(b) and (c) show the peaks of the DDP spectra 186 and the solid lines show the prediction of the scattered photon energy by the free-electron model.

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On the one hand, we may find that the the peaks of the DDP spectra by EMM decrease with the 188 scattering angle and is always red shifted comparing with the value of the free-electron model, as shown in Fig. 3(b). In additionally, the DDP spectrum of EMM transition presents a dip at 190 about = 90 • , which is also confirmed in Fig. 2(b). This is because that the DDP of EMM 191 transition is proportional to | 1 · * 2 | 2 , which is zero at = 90 • due to the scattering geometry 192 of 2 ⊥ 1 since k 2 ⊥ k 1 and k 2 1 . On the other hand, it can be found that the peaks of

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We have extended the frequency-domain theory to investigate the NCS of two X-ray photons