Explanation of the anomalous redshift on nonlinear X-ray Compton scattering spectrum by a bound electron
S. Shi, J. Chen, Y. J. Yang, Z. C. Yan, X. J. Liu, B. B. Wang
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Frequency-domain theory of Compton Scattering by a bound electron
Shang Shi , , Jing Chen , , Yu-Jun Yang , Zong-Chao Yan , , Xiao-Jun Liu and Bingbing Wang , ∗ Laboratory of Optical Physics, Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100871, China Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Insititute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China Department of Physics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China and University of Chinese Academy of Sciences, Beijing 100049, China (Dated: February 26, 2021)We have developed a frequency-domain formulation based on the nonperturbative quantum electrodynamicto study nonlinear Compton scattering of two photons o ff a bound electron inside an atom in a strong X-ray laserfield. In contrast to previous theoretical works, our results clearly reveal the existence of anomalous redshiftphenomenon observed experimentally by Fuchs et al. [Nat. Phys. , 964 (2015)] and suggest its origin as thebinding energy of the electron as well as the momentum transfer from incident photons to the electron duringthe scattering process. Our work builds a bridge between intense-laser atomic physics and Compton scatteringprocess that can be used to study atomic structure and dynamics at high laser intensities. With the first use of the Linac Coherent Light Source at theSLAC National Accelerator Laboratory in 2010 [1], the eraof exploring the nonlinear interaction of ultrafast and ultra-intense X-rays with matters has begun. By using X-ray freeelectron lasers [2–6], people have observed for the first timeextensive nonlinear phenomena at X-ray wavelengths, includ-ing the X-ray second harmonic generation in diamonds [7],two-photon absorption in the hard X-ray region [8, 9], elec-tron femtosecond response to an ultra-intense X-ray radia-tion [10], and nonlinear Compton scattering (NCS ) of X-rayphotons [11]. Among them, the NCS is a particularly inter-esting phenomenon because the observed anomalous redshiftof the scattered photon can be regarded as a breakdown ofthe widely-used free-electron approximation for bound elec-trons [12–22].As far as we know, there exist a few theoretical studies de-voted to the NCS processes involving bound electrons. Hop-ersky et al. [23] first studied the Compton scattering of twoX-ray photons by helium atom using second-order perturba-tion theory in dipole approximation. However, their predic-tion of the emission spectrum showed an oscillating structurethat has not been observed experimentally. Recently, Krebs et al. [24] developed a nonperturbative approach based on thetime-dependent Schr¨odinger equation to investigate linear andnonlinear Compton scatterings of X-ray photons by atoms.However, their results were consistent with the predictions ofthe free-electron model and do not support the existence ofthe redshift found in [11]. More recently, Venkatesh and Ro-bicheaux [25] even claimed that their theoretical results ex-hibit a blueshift compared with the scattered photon energypredicted by the free-electron model during an NCS process.Therefore, the origin of the anomalous redshift phenomenonobserved by Fuchs et al. [11] is still an open question. ∗ [email protected] Motivated by the theoretical gap of the NCS mechanism, inthis work we will apply the nonperturbative quantum electro-dynamic (QED) frequency domain theory to study the NCSprocess of bound electrons. This theory has previously beensuccessfully applied to recollision processes in strong laserfields [26–29]. The advantages of the QED method in treatingthe Compton scattering could be shortly provided. Specif-ically, we will focus on the double di ff erential probability(DDP) for the Compton scattering process of a bound electronin an X-ray laser field. Our calculation will clearly demon-strate that in the DDP spectrum of the two-photon NCS, asthe energy of the scattered photon increases, a redshift peakwill appear, which is in sharp contrast to the free-electron re-sults [30–32], and other theoretical predictions [24, 25]. Ourtheoretical results can be considered as the first qualitativeconfirmation of the measurement of Fuchs et al. [11].In the following, natural units ( ~ = c =
1) are used through-out unless otherwise stated. Since a Compton scattering in-volves incident and scattered photons of di ff erent frequencies,the quantum system consists of an atom and two linearly po-larized radiation fields with their corresponding vector poten-tials A ( r ) and A ( r ), where A ( r ) coincides with the inci-dent X-ray laser field of frequency ω and wave vector k ,and A ( r ) with the scattered photon field of frequency ω andwave vector k . The Hamiltonian of this atom-radiation sys-tem is thus H = H + U + V . (1)In the above, H = ( − i ∇ ) m + ω N a + ω N a (2)is the non-interaction part of the Hamiltonian, where N a = ( a † a + a a † ) / N a = ( a † a + a a † ) / a i ( a † i ) being the annihilation (creation) operatorfor i = , U is the atomic binding potential; and V is thetotal electron-photon interaction potential that can be writtenas V = V + V + V with V = e A ( r ) · ( − i ∇ ) m + e A ( r )2 m (3) V = e A ( r ) · A ( r ) m (4)and V = e A ( r ) · ( − i ∇ ) m . (5)Here, we have neglected the term e A ( r )2 m for its weak strength.The vector potentials of the incident and scattered pho-ton modes are A ( r ) = g ( ǫ a e i k · r + c . c . ) and A ( r ) = g ( ǫ a e i k · r + c . c . ), where g = (2 ω V γ ) − and g = (2 ω V ′ γ ) − with V γ and V ′ γ the normalization volumes of the photonmodes, and ǫ and ǫ are the corresponding polarization vec-tors. According to the experimental set-up in Ref. [11], thepolarization and wave vectors of the incident laser field arechosen along the x - and z -axis, respectively, while the direc-tion of k is characterized by the spherical coordinates ( θ, φ ).In the frequency-domain theory, since the atom-laser sys-tem is regarded as an isolated one, the total energy of the sys-tem is conserved throughout the transition process and hencethe formal scattering theory [33] can be applied. The S -matrixelement between the initial state | ψ i i and the final state | ψ f i is S fi = h ψ − f | ψ + i i , (6)where ψ ± j = ψ j + E j − H ± i ε V ψ ± j (7)with j taken to be i or f . Physically, ψ + i is the scattering state at t = ψ i in theremote past, whereas ψ − f is the scattering state at t = ψ f in the remote future.After some algebraic manipulation, the S -matrix element canbe recast into S fi = δ fi − π i δ ( E f − E i ) T fi , (8)where T fi = h ψ f | V | ψ i i + h ψ f | V E i − H − U − V + i ε V | ψ i i (9)is the T -matrix element. The first term in Eq. (9) correspondsto a one-step transition and the second term corresponds to atwo-step transition. The normal Compton scattering is a one-step transition. For an NCS, in principle, both transition pro-cesses should be taken into account. However, since the con-tribution of the two-step transition is much smaller than that of the one-step process under the present laser conditions, thesecond term in Eq. (9) can be dropped. Hence, the T -matrixelement for an NSC can be expressed as T fi = T AP + T AA − + T AA + , (10)where T AP = h ψ f | V | ψ i i , T AA − = h ψ f | V − | ψ i i , and T AA + = h ψ f | V + | ψ i i with V = em g e − i k · r a † ǫ ∗ · ( − i ∇ ), V − = e m g g ǫ ∗ · ǫ e i ( k − k ) · r a a † , and V + = e m g g ǫ ∗ · ǫ ∗ e − i ( k + k ) · r a † a † . Figure 1 illustrates the correspondingschematic diagrams of these three processes, in which we call T AP the laser-assisted electron-mode (LEM) transition shownin Fig. 1(a), and call T AA ± the electron-assisted mode-mode(EMM) transition shown in Fig. 1(b) and (c) respectively. TheLEM transition describes the process where the bound elec-tron is ionized after absorbing several photons from the laserfield, and at the same time, a photon of frequency ω is scat-tered, whereas the EMM transition describes a similar processexcept that a second photon of frequency ω is either absorbedor emitted.In Eq. (10), | ψ i i = Φ i ( r ) ⊗ | l i ⊗ | i is the initial stateof the scattering system, i.e., the eigenstate of the Hamilto-nian operator H + U ( r ) with the corresponding eigenvalue E i = ( − E B ) + ( l + ) ω + ω , where Φ i ( r ) is the ground-statewave function of the atomic electron with the binding energy E B >
0, and | l i and | i are the Fock states of the incident andscattered photons with photon number l and 0 respectively.Also in Eq. (10), | ψ f i = Ψ P f n f ⊗ | i is the final state of thesystem with energy E f = P f / (2 m ) + ( n f + + u p ) ω + ω ,where P f is the final momentum of the electron along the di-rection of ( θ f , φ f ), u p is the ponderomotive energy in unit ofthe incident photon energy, and Ψ P f n f is the Volkov state ofthe electron in the quantum laser field given by [26] Ψ P f n f = V − / e ∞ X j = − n f e i [ P f + ( u p − j ) k ] · r J j ( ζ, η ) ∗ | n f + j i . (11)In the above, J j ( ζ, η ) = ∞ X m = −∞ J − j − m ( ζ ) J m ( η ) , (12)is the generalized Bessel function, ζ = α √ u p P f · ǫ , η = u p / α ≈ /
137 is the fine-structure constant, and V e is thenormalization volume of the electron.The matrix element of the LEM transition T AP can be writ-ten as T AP = em V − / e g ǫ ∗ · [ P f + ( u p − q ) k ] J q ( ζ, η ) × Φ i ( P f + k + ( u p − q ) k ) , (13)where q = l − n f denotes the number of photons transferredfrom the incident laser field during the NCS process. For thecase of two-photon NCS, q =
2. The matrix elements of theEMM transitions T AA ± are given by T AA − = e m V − / e Λ g ǫ · ǫ ∗ J q − ( ζ, η ) × Φ i ( P f + k + ( u p − q ) k ) (14)and T AA + = e m V − / e Λ g ǫ ∗ · ǫ ∗ J q + ( ζ, η ) × Φ i ( P f + k + ( u p − q ) k ) , (15)where Λ = √ u p / ( α e ) represents the half amplitude of theclassical field in the limits of g → l → ∞ . Sincethe contribution of T AA + is much smaller than that of the othertwo terms, it is ignored under our laser conditions.The expression of the DDP for a Compton scattering pro-cess can be written as [34] dW i → f d ω d Ω = Z π | T fi | δ ( E i − E f ) V ′ γ (2 π ) V e (2 π ) ω d P f , (16)where d Ω is the di ff erential solid angle of vector k . We nowcalculate the DDP for the two-photon Compton scattering by a1 s electron of Be atom, where the intensity of the laser field is4 × W / cm and the photon energy is 9.25 keV. Figure 2presents the DDP of the NCS at the scattering angles of 95 ◦ (a)and 150 ◦ (b), where the wave vector of the scattered photon k is fixed in the polarization plane of the incident laser fielddefined by k and ǫ , i.e., the azimuthal angle φ = ◦ . InFig. 2(a)-(b), the vertical lines indicate the scattered photonenergy predicted by the free-electron model [35]: ω = q ω + q ω m (1 − cos θ ) . (17)It can be clearly seen from Fig. 2 that under these scatter-ing angles, the peak positions of the scattered spectra are al-ways red shifted with respect to the one given by the free-electron model, which confirms the experimental observationin [11]. The redshift value of the scattered photon energy isabout 500 eV for θ = ◦ and 580 eV for θ = ◦ , which aremuch larger than the binding energy of the 1 s electron in Beatom.From Eq. (10), the total DDP can be classified into twoparts: the contributions by LEM and EMM. In order to find FIG. 1. Schematic for the one-step transition of Compton scatteringby a bound electron. The single straight line represents the boundelectron state, the wavy line represents the Fock state of the inci-dent laser, the combination of wavy and double lines represents theVolkov state, the red and blue dashed lines represent the scatteredphotons of frequency ω and ω respectively, and the vertex denotesthe transition operator V in (a), V − in (b), and V + in (c). out the origin of the redshift, we calculate them separately ac-cording to Eqs. (13) and (14), and the results are indicatedby the blue dotted lines (LEM) and pink short-dashed lines(EMM) in Fig. 2(a)-(b). One can see that the distributionof the total DDP is dominated by LEM at θ = ◦ and isdominated by EMM at θ = ◦ . Furthermore, under thepresent laser conditions, we find that both parameters ζ and η are much smaller than 1 and hence the generalized Besselfunctions in Eqs. (13)-(14) can be replaced by J − ( η ) ≈ C and J ( ζ, η ) ≈ − ζ/ ≈ C P f · ǫ , where C and C are twoconstants. Thus, the matrix element of LEM transition can beapproximated as T AP ≈ C em V − / e g [ P f cos θ ǫ + ( u p − q ) k · ǫ ∗ ] × Φ i ( P f + k + ( u p − q ) k ) (18)with θ ǫ being the angle between ǫ ∗ and the electron momen-tum P f , and the matrix element of EMM transition can beapproximated as T AA − ≈ C e m V − / e Λ g ǫ · ǫ ∗ P f cos θ ǫ × Φ i ( P f + k + ( u p − q ) k ) (19)with θ ǫ being the angle between ǫ and the electron momen-tum P f . Substituting Eqs. (18) and (19) into Eq. (16) yieldsalmost the same results as the DDP with no approximation,as shown by the triangles and circles in Fig. 2(a)-(b). FromEqs. (18) and (19), one can find that, for a certain geometryamong the three vectors k , k , and P f , both equations dependon the value of the electron momentum and the wavefunctionof Be atom Φ i ( P f + k + ( u p − q ) k ). In order to investigatethe influence of this wavefunction on the distribution of DDP,we integrate its modulus squared over the final electron mo-mentum P f and obtain the angle-resolved energy spectrum, asshown in Fig. 2(c). It can be found that the peaks shown bydots in the spectrum of Fig. 2(c) are always redshifted relativeto the free-electron model shown by the solid line in Fig. 2(c),and these redshifts are smaller than 280 eV. It indicates thata redshift can be triggered by the bound electron in NCS, butis still much smaller than the redshifts shown in Fig. 2(a) and(b). Furthermore, by analyzing the terms in Eqs. (18) and(19), we find that the total DDP linearly depends on the valueof the ionized electron momentum P f . In particular, the valueof P f and thus the total DDP decrease as the scattered photonenergy increases due to the energy conservation. As a result,an obvious redshift of the DDP spectrum is formed, whichis mainly caused by the dependence of DDP on the ionizedelectron momentum and bound-state wavefunction.We now qualitatively compare our NCS spectra with theexperimental results observed in [11] by using a magnifica-tion factor δ , where the DDP of two-photon NCS is shown inFig. 3 at θ = ◦ for δ = . × (a) and θ = ◦ for δ = . × (b). In Fig. 3(a)-(b), the lines and geomet-ric symbols represent, respectively, the theoretical and experi-mental results under various intensities. As shown in Fig. 3(a)and (b), the peaks of our theoretical curves agree well withthe experimental results, which are the second-order e ff ects of FIG. 2. The DDP for two-photon NCS by atomic Be as a function of the scattered photon energy ω at the scattering angle of θ = ◦ (a), θ = ◦ (b). The black solid, blue dotted lines and pink short-dashed lines represent the total DDP, the DDP by LEM (Eq. (13)) and theDDP by EMM (Eq. (14)), respectively. The blue triangles are for the DDP by Eq. (18) and the pink circles the DDP by Eq. (19). The verticallines are ω predicted by Eq. (17). The scattered wave vector k is fixed in the polarization plane defined by k and ǫ . The integral of | Φ i ( P f + k + ( u p − q ) k ) | over P f as a function of ω and θ is shown in (c), where the dots represent the scattered photon energy correspondingto the peaks at di ff erent scattering angles and the solid line represents the prediction by Eq. (17). laser intensity.By integrating the DDP over the scattered photon energy,we obtain the corresponding single di ff erential probability(SDP) shown in Fig. 3 with δ = . × for the case of ω = .
84 keV (c) and ω = .
75 keV (d). It shows that theSDP increases as the scattering angle increases, which is inagreement with the experimental results displayed by the redstars in the graphs. Moreover, the total SDP also includes twoparts: the contributions from EMM and LEM. One may findthat the contribution from LEM transition is insensitive to thescattering angle, whereas the contribution from EMM transi-
FIG. 3. (a)-(b) Comparison of the DDP for two-photon NCS byBe between theory and experiment at θ = ◦ (a) and θ = ◦ (b). The lines and geometric figures represent, respectively, thetheoretical and experimental values under di ff erent laser intensities.The incident photon energy ω is 9.25 keV. (c)-(d) Comparison ofSDP between theory and experiment at di ff erent scattering anglesfor ω = .
84 keV (c) and ω = .
75 keV (d). The stars representthe experimental data points. Note: the theoretical values in the samegraph are magnified by the same multiple. tion increases with the scattering angle. Thus, as the scatteringangle increases, the EMM transition will become more impor-tant in SDP.In order to present the spectra distribution of the two-photon NCS more comprehensively, the angle-resolved en-ergy spectra of the scattered light are shown in Fig. 4 at φ = ◦ (a)-(c) and φ = ◦ (d)-(f), where the laser parameters are thesame as those in Fig. 2. The solid lines in Fig. 4(a) and (d)represent the values of the scattered photon energy, predictedby the free-electron model Eq. (17), and the dots representthe scattered photon energies corresponding to the maximumvalues of the DDP. It can be seen that these photon energiescalculated by our theory are always redshifted, relative to thevalues of the free-electron model for the whole range of the FIG. 4. The DDP for two-photon NCS by Be as a function of thescattering angle θ and the scattered photon energy ω for the case of φ = ◦ (a)-(c) and φ = ◦ (d)-(f), where the graphs in the left, mid-dle, and right column shows, respectively, the total DDP, the DDPdue to EMM, and the DDP due to LEM. The black lines represent ω determined by Eq. (17), and the red dots represent ω correspondingto the peaks at di ff erent scattering angles. scattering angle. Although the forward scattering is not givenin Fuchs et al. work [11], our results show that the redshift stillexists significantly in the forward scattering. In addition, withthe increase of the scattering angle, the peak energy of theNCS spectra decreases gradually, which is similar to the pre-diction of the free-electron model. It can be attributed to therelationship between the density distribution of the wavefunc-tion and the angle θ shown in Fig. 2(c). By analyzing the ar-gument of the wavefunction, we can understand directly fromthe inset of Fig. 4(a) that, with u p neglected, the peak of DDPoccurs at about the momentum transfer P f = q k − k . Sincethis momentum transfer increases with the scattering angle,according to the conservation of energy in the NCS process,the electron energy gained from the scattering must increase,resulting in a decrease in the energy of the scattered photon.The spectra shown in Fig. 4(b) and (e) are the DDP con-tributed by EMM transition, and that by the LEM transitionis shown in Fig. 4(c) and (f). One can see from Fig. 4(b) and(e) that the DDP of EMM transition obviously depends onthe azimuthal angle φ of the scattered wave vector; whereasthe DDP of LEM transition is almost independent of the az-imuthal angle, as shown in Fig. 4(c) and (f). Thus, for φ = ◦ ,the DDP has a minimum around θ = ◦ , which can be ex-plained as follows. Since in this case k ⊥ k and k k ǫ , wethen have ǫ ⊥ ǫ . As a result, the probability contributed byEMM transition becomes zero at θ = ◦ because it is propor-tional to ǫ · ǫ ∗ . Moreover, comparing the contributions fromEMM and LEM transitions under di ff erent azimuthal angles, the LEM transition dominates the two-photon NCS processas the polarization directions of the incident and the scatteredlights tend to be perpendicular with each other, i.e., ǫ ⊥ ǫ and k ⊥ k , whereas the EMM transition dominates the two-photon NCS process for all other scattering geometries.In conclusion, we have extended the frequency-domain the-ory based on the nonperturbative QED to investigate the NCSof two X-ray photons by an atom. Our theoretical resultsare in qualitative agreement with the experimental results ofRef. [11] and thus clarify the underlying physical mecha-nism for the nonlinear scattering process, i.e., it is a one-stepprocess that includes two contributions: the LEM transitionand the EMM transition. The observed anomalous redshifts,which cannot be explained by the free-electron model [35]and other theories [23–25], are attributed to the atomic bind-ing potential and the momentum transfer from the incidentphotons to the electron during the collision. Our results havedemonstrated that the redshift is a general phenomenon thatcan be observed in any scattering geometry. All these find-ings promote significantly the understanding of the nonlinearscattering processes of bound electrons in X-ray laser fields.We thank all the members of SFAMP club for helpful dis-cussions. SS thanks D. Krebs and M. Fuchs for helpful dis-cussions. This work was supported by the National NaturalScience Foundation of China under Grant Nos. 12074418,11774411 and 11834015. ZCY was supported by the NSERCof Canada. [1] P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek,A. Brachmann, P. Bucksbaum, R. Co ff ee, F.-J. Decker, et al. ,Nat. Photon. , 641 (2010).[2] B. W. McNeil and N. R. Thompson, Nat. 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