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Abstract
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 mechanism 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 electrodynamics to study nonlinear Compton scattering of two photons by an atom in a strong X-ray laser field. In contrast to previous theoretical works, our results clearly reveal the existence of a 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 processes that can be used to study atomic structure and dynamics at high laser intensities.
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1. Introduction
The Compton effect is well known for proving the quantum hypothesis of light in 1923 experimentally [1]. Afterwards the impulse approximation (IA) approach to Compton scattering on bound electrons, which considers the initial electron to be free with the momentum distribution of the bound state, was put forward by DuMond in his work on the scattering of photons from solids [2,3]. Since the Compton profile is directly related to the electron momentum distribution, the electronic structure of atoms, molecules and solids can be probed by X- and $\gamma$-ray linear Compton scattering (LCS) [4–7] and analyzed by IA. For LCS process, Eisenberger and Platzman verified the validity of nonrelativistic IA for doubly differential cross sections [8], and later Ribberfors extended IA to relativistic region [9]. Modification of IA and reexamination of the validity of IA have been the main interests of recent studies [10–12].
With the first use of the Linac Coherent Light Source at the SLAC National Accelerator Laboratory in 2009 [13], the era of exploring the nonlinear interaction of ultrafast and ultra-intense X-rays with matters has begun. By using X-ray free-electron lasers [14–17], people have observed for the first time extensive nonlinear phenomena at X-ray wavelengths, including the X-ray second harmonic generation in diamonds [18], two-photon absorption in the hard X-ray region [19,20], electron femtosecond response to an ultra-intense X-ray radiation [21], and nonlinear Compton scattering (NCS) of X-ray photons [22]. Among them, the NCS is a particularly interesting phenomenon because the observed anomalous redshift of the scattered photon can be regarded as a breakdown of the widely-used IA theory for bound electrons.
There exist a few theoretical studies [23–25] devoted to the NCS processes involving bound electrons in recent years, but no theoretical work demonstrated the experimental observations by Fuchs et al. [22]. For example, Krebs et al. [24] developed a nonperturbative approach based on the time-dependent Schrödinger equation to investigate linear and nonlinear Compton scatterings of X-ray photons by atoms. However, their results were consistent with the predictions of the free-electron model and do not support the existence of the redshift found in Ref. [22]. More recently, Venkatesh and Robicheaux [25] claimed that their theoretical results exhibit a blueshift compared with the scattered photon energy predicted by the free-electron model during an NCS process. Therefore, the origin of the anomalous redshift phenomenon observed by Fuchs et al. [22] is still an open question.
Motivated by the theoretical gap of the NCS mechanism, in this work we will apply the frequency-domain theory based on the nonperturbative quantum electrodynamics (QED) to study the NCS processes of bound electrons. This theory has previously been successfully applied in investigating recollision processes in strong laser fields [26–29]. The advantages of the QED method in treating the NCS could be shortly provided. Specifically, we will focus on the double differential probability (DDP) of a NCS process by a bound electron in an X-ray laser field. Our calculation will clearly demonstrate that in the DDP spectrum of the two-photon NCS, as the energy of the scattered photon increases, a redshift peak will appear, which agrees with the observation by Fuchs et al. [22] qualitatively.
2. Frequency-domain theory of the NCS by bound electrons
In the frequency-domain theory based on the QED, the whole system consists of an atom and radiation fields, where the radiation fields are linearly polarized and quantized, involving incident and scattered photons in Compton scattering processes. In the following, natural units $(\hbar =m=c=1)$ are used throughout unless otherwise stated, where $m$ is the electron rest mass and $c$ the speed of light. The Hamiltonian of this atom-radiation system is
where is the non-interaction part of the Hamiltonian, with $N_{a_1}=(a_1^{\dagger } a_1+a_1a_1^{\dagger })/2$ and $N_{a_2}=(a_2^{\dagger } a_2+a_2a_2^{\dagger })/2$ being the number operators of the incident and the scattered photons, respectively, with $a_i(a_i^{\dagger })$ being the annihilation (creation) operator for $i=1,2$, and $\omega _1$ is the incident laser frequency and $\omega _2$ the scattered photon frequency. $U$ is the atomic binding potential, and $V$ is the total electron-photon interaction potential that can be written as $V=V_1+V_{21}+V_{22}$ with and where $e$ is the elementary charge. Here, we have neglected the term $\frac {e^{2}\textbf {A}_2^{2}(\textbf {r})}{2}$ for its weak strength compared with the above three terms $V_1$, $V_{21}$ and $V_{22}$. The vector potentials of the incident and scattered photon modes are $\textbf {A}_1(\textbf {r})=g_1(\boldsymbol {\epsilon }_1a_1e^{i\textbf {k}_1\cdot \textbf {r}}+\textrm {c.c.})$ and $\textbf {A}_2(\textbf {r})=g_2(\boldsymbol {\epsilon }_2a_2e^{i\textbf {k}_2\cdot \textbf {r}}+\textrm {c.c.})$, where $\textbf {r}$ is the spatial coordinate of the electron. $g_1=(2\omega _1V_{\gamma })^{\frac {-1}{2}}$ and $g_2=(2\omega _{2}V^{\prime }_{\gamma })^{\frac {-1}{2}}$ with $V_{\gamma }$ and $V^{\prime }_{\gamma }$ the normalization volumes of the photon modes, $\textbf {k}_1$ ($\textbf {k}_2$) and $\boldsymbol {\epsilon }_1$ ($\boldsymbol {\epsilon }_2$) are the wave vector and polarization vector of the incident laser field (scattered photon mode), respectively. By substituting the expressions of $\textbf {A}_1(\textbf {r})$ and $\textbf {A}_2(\textbf {r})$ into Eq. (4) and Eq. (5), we obtain the explicit expressions of $V_{21}$ and $V_{22}$, where $V_{21}$ can be split into two termsHere, the terms containing the annihilation operator $a_2$, which make no contribution to NCS processes, are not included in Eq. (6)-(8).
Since the atom-radiation system can be regarded as an isolated one, the total energy of the system is conserved during the atom-radiation interaction process and the formal scattering theory [30] can be applied. The scattering matrix ($S$-matrix) element [27,28] is expressed as
where with $j$ taken to be $i$ or $f$ and $\varepsilon$ to be a small positive real number. Physically, $\psi _i^{+}$ is the scattering state at $t=0$ which has developed from a pre-collision state $\psi _i$ in the remote past, whereas $\psi _f^{-}$ is the scattering state at $t=0$ which will develop to a post-collision state $\psi _f$ in the remote future. After some algebraic manipulation, the $S$-matrix element can be recast into where $T_{fi}$ is the exact transition matrix ($T$-matrix) element [31,32] between the initial state $|\psi _i\rangle$ and the final state $|\psi _f\rangle$, and can be written asSubstituting Eq. (10) into Eq. (12), we have
Here, the initial state $|\psi _i\rangle =\Phi _i(\textbf {r})\otimes |l\rangle \otimes |0\rangle$ is the eigenstate of the operator $H_0+U$ with the eigenenergy $E_i=(-E_B)+(l+\frac {1}{2})\omega _1+\frac {1}{2}\omega _2$. $\Phi _i(\textbf {r})$ is the ground-state wave function of the electron in an atom with the binding energy $E_B>0$, and $|l\rangle$ and $|0\rangle$ are the Fock states of the incident and scattered photons with photon number $l$ and 0, respectively. The final state $|\psi _f\rangle =\Psi _{\textbf {P}_fn_f}\otimes |1\rangle$ is the eigenstate of the operator $H_0+V$ with the eigenenergy $E_f=\textbf {P}_f^{2}/2+(n_f+\frac {1}{2}+u_p)\omega _1+\frac {3}{2}\omega _2$, where $\textbf {P}_f$ is the final momentum of the electron, $n_f$ is the laser photon number of the final state and $u_p$ the ponderomotive energy in unit of the incident photon energy. $\Psi _{\textbf {P}_fn_f}$ is the Volkov state of the electron in the laser field [26]
In the above, $V_e$ is the normalization volume of the final momentum of the electron, $\mathcal {J}_{n}(\zeta,\eta )=\sum _{s=-\infty }^{\infty } J_{-n-2s}(\zeta )J_s(\eta )$ is the generalized Bessel function with the arguments $\zeta =2\sqrt {\frac {u_p}{\omega _1}}\textbf {P}_f\cdot \boldsymbol {\epsilon }_1$ and $\eta =u_p/2$, the subscript $n$ represents the number of transferred photons and $s$ is an integer.
The first and second terms in Eq. (13) correspond to one-step and two-step transitions, respectively. In order to compare with the experimental results [22], we focus on peak positions of the DDP spectra in this work, where there is no contribution from the two-step transition under the laser condition of laser intensity being ${4 \times 10^{20}\,\textrm{W}/{\textrm{cm}}}^{2}$ with photon energy $\omega _1$=9.25 keV, hence the second term in Eq. (13) is dropped and will be investigated in the future. Therefore, the $T$-matrix element for NCS can be expressed as
where the matrix elements of the EMM transitions $T_{AA\pm }$ are given byIn Eq. (16)–(18), the two-photon NCS processes correspond to $q=2$.
Figure 1 illustrates the corresponding schematic diagrams of the three terms in Eq. (15), where we name $T_{AP}$ the laser-assisted electron-mode (LEM) transition shown in Fig. 1(a), and name $T_{AA-}$ and $T_{AA+}$ the electron-assisted mode-mode (EMM) transitions shown in Fig. 1(b) and (c) respectively. The LEM transition describes the process where the bound electron is ionized after absorbing several photons from the laser field, and at the same time, a photon of frequency $\omega _2$ is scattered, whereas the EMM transition describes a similar process except that a second photon of frequency $\omega _1$ is either absorbed ($T_{AA-}$) or emitted ($T_{AA+}$).
Fig. 1. Schematic for the one-step transition of the NCS by a bound electron. The single straight line represents the bound electron state, the wavy line represents the Fock state of the incident laser, the combination of wavy and double lines represents the Volkov state, the red and blue dashed lines represent the scattered photons of frequency $\omega _1$ and $\omega _2$ respectively, and the vertex denotes the transition operator $V_{22}$ in (a), $V_{21-}$ in (b), and $V_{21+}$ in (c).
The expression of the DDP for a Compton scattering process can be written as
3. Redshift
According to the experimental set-up in Ref. [22], the polarization and wave vectors of the incident laser field are chosen along the $x$- and $z$-axis, respectively, while the direction of $\textbf {k}_2$ is characterized by the spherical coordinates ($\theta,\phi$), as shown in Fig. 2. Since the polarization direction of the scattered light is not fixed in Ref. [22], we integrate all the polarization directions of the scattered light in our calculation.
Fig. 2. Geometry of the NCS for initially linearly polarized light. The polarization and wave vectors of the incoming photons define the $xz$ plane, while the direction of the scattered photons is characterized by the spherical coordinates ($\theta,\phi$).
We now calculate the DDP for Compton scattering of two X-ray laser photons by a 1$s$ electron of Be atom, where the intensity of the incident laser field is $4 \times 10^{20}~\textrm{W/cm}^{2}$ with the photon energy of 9.25 keV. In order to compare the DDP spectra with those of the experiment by Fuchs et al., the wave vector of the scattered photon $\textbf {k}_2$ is fixed in the polarization plane of the incident laser field defined by $\textbf {k}_1$ ($z$-axis) and $\boldsymbol {\epsilon }_1$ ($x$-axis), i.e., the azimuthal angle $\phi =0^{\circ }$. The DDP of the NCS at the scattering angles of $\theta =89^{\circ }$ (a), $103^{\circ }$ (b), $117^{\circ }$ (c) and $131^{\circ }$ (d) are displayed by solid lines in Fig. 3. The vertical lines indicate the scattered photon energy predicted by the free-electron model [24,25,35]:
It can be seen from Fig. 3 that the peak positions in the scattered spectra for the four angles are all red shifted to the scattered photon energy predicted by the free-electron model, which qualitatively agrees with the experimental observations [22].
Fig. 3. The DDP as a function of the scattered photon energy $\omega _2$ at the scattering angle of $\theta =89^{\circ }$ (a), $\theta =103^{\circ }$ (b) , $\theta =117^{\circ }$ (c) and $\theta =131^{\circ }$ (d). The solid lines represent the total DDP. The short-dotted lines, dash-dotted lines, and short-dashed lines represent the DDP determined by $T$-matrix elements $T_{AP}$, $T_{AA-}$ and $T_{AA+}$. The dashed lines show the distribution of the cross term (CT). The vertical lines are $\omega _2$ predicted by Eq. (19). The scattered wave vector $\textbf {k}_2$ is fixed in the polarization plane defined by $\textbf {k}_1$ and $\boldsymbol {\epsilon }_1$. The incident X-ray laser intensity is $4 \times 10^{20}~\textrm{W/cm}^{2}$ with photon energy of 9.25 keV.
In order to explain the results shown in Fig. 3, we present separately the contributions to the total DDP by tranistions $T_{AP}$ (short-dotted lines), $T_{AA-}$ (dash-dotted lines) and $T_{AA+}$ (short-dashed lines) as well as the cross term (dashed lines) in Fig. 3(a)-(d). One can see that the contribution of $T_{AA+}$ can be ignored, hence the EMM transition mentioned below refers to the $T_{AA-}$ transition. For $\theta =89^{\circ }$, although the total DDP is dominated by the LEM transition which is blue shifted to the energy predicted by the free-electron model, the contribution from the cross term plays a crucial role in the DDP spectrum, where it increases the probability in lower energy region and decreases the probability in higher energy region. Thus, a bimodal structure with higher peak in lower energy of the spectrum appears. For $\theta =103^{\circ }, 117^{\circ }$ and $131^{\circ }$, since the total DDP is dominated by EMM transition, the peak positions of total DDP spectra are all redshifted to the energy predicted by the free-electron model.
Furthermore, Fig. 3 shows that the peak positions of the DDP spectrum by EMM transition are always red shifted to the energy given by the free-electron model, while the peak positions of that by LEM transition can be either blue shifted or red shifted. To find the reason of these calculation results, we simplify Eq. (16) and Eq. (18) by replacing the generalized Bessel functions $\mathcal {J}_{1}(\zeta,\eta )$ with $J_{-1}(\zeta )\approx C_1\textbf {P}_f\cdot \boldsymbol {\epsilon }_1$ and $\mathcal {J}_{2}(\zeta,\eta )$ with $J_{-1}(\eta )+J_{-2}(\zeta )\approx C_2$, as the values of $\zeta$ and $\eta$ are much smaller than 1 under the present laser conditions ($|\zeta |<0.034,~\eta =3.6\times 10^{-5}$). Here the parameters $C_1$ and $C_2$ are constants determined by the laser conditions. Therefore, the matrix element of EMM transition can be approximated as
Fig. 4. The DDP by the $T$-matrix element $T_{AA-}$ (red short-dashed lines) and the DDP by Eq. (22) (olive short-dotted lines), and the DDP by the $T$-matrix element $T_{AP}$ (blue dash-dotted lines) and the DDP by Eq. (23) (magenta dotted lines) at $\theta =89^{\circ }$ (a) and $\theta =117^{\circ }$ (b). The laser parameters and azimuthal angle of the scattered wave vector are the same as Fig. 3.
We firstly consider the influence of the atomic wavefunction on the DDP spectra, which is contained in Eq. (22)-(23). By integrating the modular square of the wavefunction over $\textbf {P}_f$, we obtain the electron density distribution as a function of the scattered photon energy $\omega _2$ and the scattering angle $\theta$, as shown in Fig. 5. It shows that the peak positions presented by dots on the density distribution decrease with the scattering angle. This can be explained as follows: By analyzing the argument of the wavefunction, it can be found that the peak of the electron density distribution occurs at the momentum transfer $\textbf {P}_f=q\textbf {k}_1-\textbf {k}_2$, as illustrated by the inset of Fig. 5. Since this momentum transfer increases with the scattering angle, the energy gained by electron from the scattering process increase inevitably, leading to a decrease of the scattered photon energy. Moreover, one may find that the peaks of the density distribution are always redshifted to the prediction of the free-electron model shown by the solid line, where the value of the redshift, in a range between 127 eV and 153 eV, is close to the binding energy of the 1s state of Be atom. This indicates that a bound electron can provide a redshift comparable to its binding energy.
Fig. 5. The integral of $|\Phi _i(\textbf {P}_f+\textbf {k}_2+(u_p-q)\textbf {k}_1)|^{2}$ over $\textbf {P}_f$ as a function of $\omega _2$ and $\theta$. The dots denote the scattered photon energy corresponding to the peaks at different scattering angles and the solid line represents the prediction by Eq. (21).
Next, by analyzing other terms in Eq. (22)-(23), we find that the shifts of the peak positions of the DDP spectra can also be affected by the value of the final electron momentum besides the binding energy of the electron. Specifically, for EMM transition, the $T$-matrix element depends linearly on the value of the electron momentum $P_f$. According to the energy conservation, $P_f$ decreases with the increase of the scattered photon energy, such that the peak positions of the DDP determined by EMM transition are always red shifted to the prediction of the free-electron model. On the contrary, for LEM transition, with the variation of the scattering angle, the dependence of the $T$-matrix element on the value of the electron momentum $P_f$ is nonmonotonic, resulting in the non-unidirectional shift of the peak position (may red shift or blue shift).
To illustrate the DDP spectra more comprehensively, the total DDP, the DDP due to EEM and LEM transitions as a function of the scattering angle and the scattered photon energy at different azimuthal angles are calculated and shown at the top, middle and bottom of Fig. 6, respectively. The dots show the peak positions of the total DDP spectra and the solid lines predict the scattered photon energy by the free-electron model. Firstly, for $\phi =0^{\circ }$, one may find that, the peak positions are below the energy predicted by the free-electron model at $\theta <90^{\circ }$, while the peak position jumps above the prediction of the free-electron model at $\theta =90^{\circ }$, and keeps blue shifted until $\theta >94^{\circ }$. These results can be explained as follows: (1) At $\theta =90^{\circ }$, the total DDP is determined by LEM transition, since EMM transition and the cross term are both zero due to $\boldsymbol {\epsilon }_1\cdot \boldsymbol {\epsilon }_2^{*}=0$, where $\textbf {k}_2 \perp \textbf {k}_1$ and $\textbf {k}_2\parallel \boldsymbol {\epsilon }_1$ causes $\boldsymbol {\epsilon }_2\perp \boldsymbol {\epsilon }_1$. Therefore, the DDP spectrum changing from redshift to blueshift is due to the contribution of the LEM transition which is blue shifted at $\theta =90^{\circ }$; (2) For $\theta \in 90^{\circ }\pm 4^{\circ }$ except $90^{\circ }$, the cross term begins to play a role in the contribution to DDP, as shown in Fig. 3(a) and (b), hence the peak position of the DDP spectrum changes to redshift for $\theta <90^{\circ }$ and to blueshift for $\theta >90^{\circ }$; (3) For the angles $\theta <86^{\circ }$ or $\theta >94^{\circ }$, EMM transition dominates the contribution to the total DDP, hence the peak position of the total DDP is redshifted to the energy predicted by the free-electron model.
Fig. 6. Angle-resolved energy spectra of two-photon NCS by Be. The DDP is shown as a function of the scattering angle $\theta$ and the scattered photon energy $\omega _2$ at different azimuthal angles $\phi =0^{\circ }$ (left column), $\phi =45^{\circ }$ (middle column) and $\phi =90^{\circ }$ (right column). The total DDP, the DDP due to EMM, and the DDP due to LEM are shown in (a)-(c), (d)-(f) and (g)-(i), respectively. The solid lines represent $\omega _2$ determined by the free-electron model Eq. (21) and the dots denote the peak positions of the total DDP. The laser parameters are the same as Fig. 3.
Secondly, for $\phi =45^{\circ }$ and $90^{\circ }$, it shows that the peak positions of total DDP spectra are always redshifted to the energy predicted by the free-electron model. By comparing the DDP spectra contributed by EMM (Fig. 6(d)-(f)) with those contributed by LEM (Fig. 6(g)-(i)), one can find that the total DDP is dominated by the contribution from EMM transition. Owing to the fact that the peak positions of the DDP of EMM transition are always redshifted, the peak positions of total DDP will also red shift to the prediction of the free-electron model when EMM transition is dominant.
We now qualitatively compare our NCS spectra with the experimental results [22] by using a magnification factor $\delta$. The DDP of two-photon NCS is shown in Fig. 7 at $\theta =89^{\circ }$ with $\delta =2\times 10^{16}$ (a), $\theta =103^{\circ }$ with $\delta =3.5\times 10^{14}$ (b), $\theta =117^{\circ }$ with $\delta =1.1\times 10^{14}$ (c) and $\theta =131^{\circ }$ with $\delta =7.3\times 10^{13}$ . In Fig. 7, the curves and geometric symbols represent, respectively, the theoretical and experimental results under various laser intensities. The consistency between the theoretical and experimental results is reflected in the following two points: First, the peak positions in the experiment and our theory are both red shifted to the predicted value of free-electron model. Second, the dependence of the DDP spectra on the X-ray laser intensity in our theory is the same as that of the experiment. Especially, in our method, the second-order nonlinear effects of the laser intensity can be found in the expression of the $T$-matrix elements of LEM and EMM, i.e., the term $\mathcal {J}_q(\zeta,\eta )$ in $T_{AP}$ and the term $\Lambda \mathcal {J}_{q-1}(\zeta,\eta )$ in $T_{AA-}$ are proportional to the laser intensity. For the discrepancies between theoretical and experimental results, one may notice that the experimental spectra are wider than the theoretical spectra, this is probably because the laser in the experiment cannot be monochromatic and have a certain spectral width, but the spectral width is not considered in our theoretical calculation. On the other hand, the 1-mm-thick Be target is used in the experiments, while we only consider the interaction between the X-ray laser and a Be atom in our calculation, this may also induce some differences.
Fig. 7. Comparison between the DDP of theory and experiment. The DDP for two-photon NCS by Be at $\theta =89^{\circ }$ (a), $\theta =103^{\circ }$ (b), $\theta =117^{\circ }$ (c) and $\theta =131^{\circ }$ (d). The curves and geometric figures represent, respectively, the theoretical and experimental values under different laser intensities. The incident photon energy $\omega _1$=9.25 keV. The azimuthal angle of the wave vector of scattered light is same to Fig. 3. Note: the theoretical values in the same graph are magnified by the same magnification factor.
4. Conclusion
We have extended the frequency-domain theory to investigate the NCS of two X-ray photons by an atom. Our theoretical results are in qualitative agreement with the experimental results of Ref. [22]. By analyzing the EMM and LEM transition matrix elements, we find that the redshift in the DDP spectra of the NCS process can be attributed to the atomic binding potential and the momentum transfer from the incident photons to the electron during the collision. Our results have demonstrated that the redshift can be observed in both forward and backward directions. All these findings promote significantly the understanding of the nonlinear scattering processes of bound electrons in X-ray laser fields.
Funding
National Natural Science Foundation of China (11774411, 11834015, 12074418).
Acknowledgments
We thank all the members of SFAMP club for helpful discussions. S.S thanks D. Krebs and M. Fuchs for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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