1State Key Laboratory for Mesoscopic Physics and Collaborative Innovation Center of Quantum Matter, School of Physics, Peking University, Beijing 100871, China
2School of Materials Science and Intelligent Engineering, Nanjing University, Suzhou 215163, China
3Shishan Laboratory, Nanjing University, Suzhou 215163, China
4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
5Peking University Yangtze Delta Institute of Optoelectronics, Nantong 226010, China
Leshi Zhao, Linfeng Zhang, Haitan Xu, Zheng Li, "Quantumness of gamma-ray and hard X-ray photon emission from 3D free-electron lattices," Photonics Res. 13, 1510 (2025)
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【AIGC One Sentence Reading】:Crystalline undulator radiation generates entangled 100 MeV photons, enhanced by 3D free-electron lattices, with effects of demodulation and dechanneling explored.
【AIGC Short Abstract】:This study reveals that crystalline undulator radiation can generate entangled high-energy photons, and a 3D free-electron lattice enhances this emission through phase matching. It also explores the impact of demodulation and dechanneling on the process, highlighting the dependence on undulator parameters.
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Abstract
Crystalline undulator radiation (CUR) is emitted by charged particles channeling through a periodically bent crystal. We show that entangled high-energy photons of the order of 100 MeV can be generated from CUR and obtain the quantum entanglement properties of the double-photon emission of CUR with a nonperturbative quantum field theory. We demonstrate that the crystalline undulator (CU) can induce a 3D free-electron lattice with premicrobunched electrons, and the resulting free-electron lattice can enhance the entangled high-energy photon emission for certain angles by phase matching. We also examine the effects of demodulation and dechanneling during the electron beam channeling process, and show the dependence of the dechanneling and demodulation lengths on the undulator parameters.
1. INTRODUCTION
High-quality radiation from free electrons has become an important tool in various scientific areas, which covers the frequency ranging from microwave up to hard X-rays [1–6]. In the past four decades, crystalline undulators (CUs) based on periodically bent crystals have been proposed and demonstrated in experiments, which provide compact light sources to achieve hard X-ray radiation [7,8]. Methods commonly used to prepare bent crystals of the CUs include growing mixtures that take advantage of the different lattice constants of Si and Ge [7,9–12], or fabricating trenches on crystal surfaces with diamond blades [13,14] or lasers [15] to achieve internal deformations. With periodic deformation, one can obtain periodically bent crystals for the CU. When electrons are injected into a crystal with periodic bending, the electron motion acquires an oscillating mode with a frequency determined by the deformation periodicity, which leads to crystalline undulator radiation (CUR) [8].
In this work, we study the entangled high-energy photon pair emission from a 3D free-electron lattice by injecting premicrobunched electrons into the CU and analyze the quantum entanglement of the emitted photon pairs. We present a nonperturbative quantum electrodynamical (QED) theory for the entangled high-energy photon pair emission and also elaborate a nonlinear optics model that qualitatively reveals the underlying mechanism. Such a 3D free-electron lattice can produce entangled high-energy photon pairs with electrons due to an ultrastrong effective magnetic field of T. We also propose an undulator field-tapering scheme that can reduce the single-photon emission around the frequencies of the entangled photon pairs.
Before diving into the theoretical details, we first give an intuitive picture about the creation and enhancement of entangled CUR emission. A CU provides a periodic pseudo-potential magnetic field. In the lab frame, the injected relativistic electrons move in the undulator with a periodic magnetic field, while in the electron frame (EF) where the initial velocity of the electrons is 0 before entering the undulator, the periodic magnetic field of the CU is seen by the electrons as an electromagnetic wave, which is scattered by the electrons.
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The scattering process includes not only single-photon emission, but also double-photon emission and other higher-order multiphoton emissions that can be calculated with Feynman diagrams using nonperturbative QED theory. The CU also plays an important role in enhancing the entangled photon pair emission. As the CU has a natural polycrystalline cell structure, a free-electron lattice can be formed. The regular transverse periodic structure in the CU allows the 3D phase-matching condition that enhances the emission rate.
2. QUANTUM EMISSION FROM A SINGLE FREE ELECTRON IN A CU
Figure 1.The 3D free-electron lattice in the CU. (a) Transversal structure of the CU [9,16]. The free electrons channel along the (110) direction. The Si (Ge) atoms denoted by red spheres are in the same transversal plane, while the blue ones are in a parallel plane separated by a distance of , where is the crystal constant. The cross sections of the channels denoted by golden rectangles form a 2D Bravais lattice with primitive vectors and . (b) The premicrobunched free electrons channeling in the CU (green spheres) form a 3D free-electron lattice, where is the interval between adjacent microbunches in the lab frame.
The radiation from the free electrons in a CU is fully determined by the electron kinematics [8]. By introducing a pseudo-potential that retains the electron kinematics [18], we can obtain the radiation from the CU via quantum electrodynamics (QED) theory (see Appendix A for the pseudo-potential model) [19]. For an ideal CU, the free electrons move along the sinusoidal trajectory in the channels, and (see Fig. 2). We use the relativistic four-vector notation, i.e., we use to represent . The corresponding pseudo-potential reads , where , is the effective magnetic field amplitude, and is the effective CU parameter. Here we use the natural units .
Figure 2.Schematic of entangled double-photon emission from CU (lower panel) and the microscopic mechanism (upper panel). The circles mark the atoms in the periodically bent crystal. The electron trajectory in the lab frame represented by the orange curve is with period and amplitude , where . The dashed curve represents the magnetic pseudo-potential of the CU. The electrons are premicrobunched before being injected into the CU [20,21], which coherently enhances the double-photon emission. The entangled photons are emitted along the angles of and .
The differential single-emission rate for a single free electron in CU can be calculated based on the Weizsäcker–Williams formalism; the result is shown in Fig. 3(a) (see Appendix B for details of calculation). The schematic of the double-photon emission in the CU is shown in Fig. 2. The double-photon emission process of CUR can also be calculated using nonperturbative QED theory, as shown in Figs. 3(b) and 3(d) (see Appendix C for details of the calculation), where photon pairs are emitted during the scattering between the free electrons and the quasi-EM wave of CU [22].
Figure 3.The normalized differential emission rates of CUR. (a) The angular distribution of the differential single-photon emission rate of CUR in the lab frame. (b), (d) The double-angular distribution of the differential double-photon emission rate of CUR for different emission directions of photon 1 in the lab frame. The red point represents the directions of photon 1, and we choose . The two axes correspond to the emission angles and of photon 2, where is the polar angle of photon 2 in the lab frame. (c), (e) The double-angular distribution of the concurrence of CUR for different emission directions of photon 1 in the lab frame. The configuration corresponds to (b) and (d), respectively.
3. QUANTUM EMISSION FROM A 3D FREE-ELECTRON LATTICE
Now we study the collective double-photon emission from a 3D free-electron lattice, which is formed by premicrobunched electrons traveling in the CU. The electron microbunches are centered along the channels, and the distance between adjacent microbunches in the EF is , where the incident electron bunch has a zero initial average velocity in the EF before entering the undulator, is the period of the undulator for premicrobunching, is the Lorentz factor of the electron, and is the CU parameter, which is the same as that of the premicrobunching undulator. The total number and the length of the electron bunch are denoted by and , respectively. The longitudinal coherence of emissions from different free electrons in each channel is given by the coherence length , and the electrons can be treated as coherent electrons within the coherence length , where represents the Pierce parameter of the electron bunches in premicrobunching. The dimensionless Pierce parameter was defined in Ref. [1], which is about 1/166 in our case. The coherence length encompasses microbunches (in our case, ) where the electrons are coherent. The number of the single-channel coherent electrons (SCCEs) is denoted by . The phase difference of the double-photon emission from electrons within a coherence length is given by , where , and are the momenta of the two emitted photons, is the momentum of quasi-electromagnetic wave, and is the wave-packet center position of the th electron with . If the th electron lies in the th microbunch, we obtain , where and are the transversal positions of the th electron relative to the center of the channel, and are the three spatial unit vectors, and , where is the number of microbunches within the coherence length. The phase factor of the electrons is given by .
As and [8], all the electrons within the coherence length are approximately at the same transversal position, i.e., for any . For the emitted photon wave vectors considered in this work, and . Thus, the cumulative phase factor for the enhancement can be approximated by where is the number of microbunches within the coherence length, and is the number of electrons within the th microbunch. The summed phase factor by all the electrons in a bunch and a single channel is given by .
For a 3D free-electron lattice, the transversal beam size of the electron bunch is much larger than the transversal area of a crystal channel, and the resulting interchannel coherence provides the 3D phase-matching condition of the entangled photon modes and the enhancement of double-photon emission rate in CUR. We assume a random distribution of electrons for the channels, where reflects the ratio between the transversal area of the electron bunch and the transversal area of the channel . The cross section of the crystalline channels forms a 2D Bravais lattice. We index the different channels by , which indicates the channels in the two directions of Bravais primitive vectors , . For the channel , the corresponding phase factor is . Thus, the summed phase factor of the channels is where and are the number of channels along the two directions of and , respectively, which satisfy . Combining the enhancement factors, the entangled photon pair emission from the 3D free-electron lattice is which results in strong enhancement for entanglement quality and emission rate of the two emitted photons due to 3D phase-matched photon modes and coherence. is the photon pair number of double-photon emission, and is the entangled photon pair emission from a single electron [see Appendix D for details of ].
We use concurrence to quantify the entanglement degree of the emitted photon pairs [23]. Each photon pair can be treated as a two-qubit system [24] (see Appendix E for details), and its concurrence is defined as , where are the four eigenvalues of the matrix in descending order, is the density matrix of entangled photon pair, and is a Pauli matrix. Figures 3(c) and 3(e) show the results for the double-angular distribution of the concurrence of CUR for given emission directions of photon 1 (represented by red points) (see Appendices A, C, and E for details of calculation). Using Figs. 3(c) and 3(e), we can tell the entanglement degree of the emitted photon pairs for certain emission angles of photons 1 and 2.
By numerical calculation, we can obtain the emission rate and concurrence of the emitted entangled photon pairs for certain angular parameters of . and are chosen to satisfy the phase-matching condition, i.e., and , where are integers, is the component of the spacing between channels, is the component of the spacing between channels, and depend on the emission zenith angles and the polar angles in the EF, and are related to off-axis angles and in the lab frame by the Lorentz transformation. The energies of the emitted photons are chosen to be around and with an energy range of . The frequencies of the two photons of an entangled photon pair satisfy the four-dimensional energy momentum conservation condition. In addition, the frequencies of the emitted entangled photon pairs are chosen to avoid the integer multiples of the fundamental frequency in order to reduce the interference effect of the fundamental frequency on the entangled emission. Meanwhile, the frequencies of the two photons in an entangled photon pair also need to be different from each other for demonstration purposes. We calculate the density matrix of the emitted photon pairs with the basis of the helicity eigenstates . The density matrix of the emitted photon pairs is shown in Fig. 4.
Figure 4.(a) Absolute values of the density matrix elements of the emitted photon pairs. (b) Arguments of the density matrix elements of the emitted photon pairs.
4. EFFECTS OF DEMODULATION AND DECHANNELING ON THE FREE ELECTRONS IN CU
For the multi-electron channeling process, different electrons have different channeling oscillation amplitudes due to the different channeling transverse energies , and the electrons also have emittance in a plane perpendicular to the channeling oscillation plane [10,25,26]. Although the electrons have almost the same energy, their velocity projections in the longitudinal direction axis are different, resulting in the electron beam gradually losing its longitudinal modulation during channeling, and thus the weakening of the coherent enhancement of the high-energy photon pair emission.
When the bending amplitude of the crystalline channel is small, e.g., when the centrifugal parameter (see Appendix F for details of ), we can estimate the demodulation length using a straight channel as an approximation. In the random scattering theory, the diffusion equations for both electrons and positrons have an identical structure, that is [25,26] where , is the electron distribution function periodically modulated in the longitudinal direction, is the frequency of modulation, is the electron velocity projection in the longitudinal direction, is the transverse energy of the electrons, is the angle between the particle trajectory and longitudinal direction, and is the scattering parameter associated with the dechanneling [25–27].
We use the Pöschl–Teller potential to calculate the motion equation of electron channeling oscillation [8] (see Appendix F for details). Meanwhile, the electrons travel at an angle away from the plane formed by the channeling oscillation direction of the axis and the longitudinal direction of the axis, and the motion in the -axis direction is . Thus, the electron velocity component in the longitudinal direction is where for the case of Si(110) planar channel [8], is the channeling oscillation amplitude, and is the time period of channeling oscillating motion of the electrons (see Appendix F for details).
Note that and ; we expand Eq. (5) to the second order and average it as where refers to the average over time period .
By applying Eq. (6) to Eq. (4), and using separation of variables as , we obtain where , , and , satisfying , are parameters to be determined with boundary conditions. As the electron distribution at is zero, where is the potential well depth, using the parameters given in Table 1 and choosing for larger demodulation length, we get from Eq. (8) that . Note that Eq. (8) is different from that for the positrons. Only Eqs. (9) and (10) have identical forms to those for the positron case [25,26]. From Eq. (9), we have [25,26]
For large , is dominated by the term with , and . According to , we obtain the demodulation length , and the number of demodulation periods . We show in Fig. 5(b) more results of for different undulator parameters and centrifugal parameters under the conditions .
Figure 5.Dependence of (a) and (b) on the undulator parameter and the centrifugal parameter under the conditions .
When free electrons travel through the channels, they are subject to random scattering by atoms in the crystal, resulting in the dechanneling phenomenon [28,29]. The free electrons within the CU undergo multiple scatterings with the atoms in the crystal, changing the electrons’ transverse motion. When the scattering elevates the energy of the electron’s transverse motion beyond the interplanar potential well, the electrons may exit the channel, a phenomenon known as dechanneling. Meanwhile, scattering with atoms can also reduce the energy of the electron’s transverse motion, allowing the electrons to be recaptured by the channel, known as rechanneling. The possibility of rechanneling is normally lower than that of dechanneling, resulting in a reduction in the number of free electrons within the channel [30–35]. The dechanneling length refers to the average distance over which an electron travels before it undergoes dechanneling. For straight crystals, the dechanneling length is influenced by both the crystal characteristics and the electron’s energy, while for bent crystals, the dechanneling length is also affected by the centrifugal parameter [8]. Under the parameters given in Table 1, we get the number of dechanneling periods . We show in Fig. 5(a) more results of for different undulator parameters and centrifugal parameters under the condition (see Appendix G for more details).
5. NONLINEAR OPTICS MODEL OF ENTANGLED CUR
From a nonlinear optical perspective, the double-photon emission of CUR in the EF can be treated as a downconversion with a single electron. Such downconversion is conjugate to a difference frequency generation (DFG) process with one of the incident EM fields being a vacuum field (see Fig. 6), i.e., the nonlinear optical susceptibility tensors of the two processes are symmetric [36].
Figure 6.Diagrams for the nonlinear optical processes. (a) Double-photon emission and its conjugate; (b) difference frequency generation.
Thus, we consider the equation of motion of a single electron in the quasi-EM field of the magnetic pseudo-potential and a vacuum field, given by where is the electron velocity in the EF, and are the EM field components of the quasi-EM field of the magnetic pseudopotential and the vacuum field, respectively. The EF is defined by the initial velocity of the electrons, i.e., the electrons are initially at rest in the EF before entering the undulator. When the electrons interact with the undulator field, the velocity of the electrons will change in the EF.
In the EF, as the order of magnitude of is 1, the electron velocity in the EF is much less than , i.e., , and we can expand Eq. (12) to the first order [37], where the last term is also of first order, as .
We apply a perturbation technique with Fourier transformation to solve Eq. (13). The leading orders of read
We consider the electric field of the quasi-EM wave to be monochromatic with frequency and an amplitude much larger than that of the vacuum field . Thus, the dominant linear component of at reads where is a phase factor. And for a specific emission frequency , the DFG process is contributed only by the component of the vacuum field at . We obtain the corresponding component of so that where is the amplitude of at the frequency . In this section, we use the SI units and write explicitly.
A. Single-Photon Emission Rate
We first calculate the differential single-photon emission rate of CUR using the classical theory of electromagnetism for convenience. For an electric dipole oscillating as , the emission power per solid angle reads where the detailed angular dependence is neglected for an order-of-magnitude evaluation. For a multifrequency oscillation , the total emission energy per solid angle is
Thus, from Eq. (16), we obtain the differential single-photon emission rate of CUR as where reflects the number of emitted photons in unit solid angle per unit time.
Via Lorentz transformation and the relationship in the EF, we obtain the differential single-photon emission rate in the lab frame as
By Fourier transformation we can get the distribution function of differential single-photon emission rate in frequency and spatial domains so that
B. Double-Photon Emission Rate
The double-photon emission rate of CUR is comparable to that of the DFG process according to the conjugation between them. By substituting Eq. (17) into Eq. (18), we obtain where only one mode of the vacuum field at the frequency is present. We further sum over the vacuum modes with frequency around ,
Thus, the differential double-photon emission rate is given by
In the lab frame, we focus on the cases where the two emitted photons are at frequencies of similar orders of magnitude. Via Lorentz transformation, we obtain
By Fourier transformation we can get the distribution function of differential double-photon emission rate in frequency and spatial domains as
Here we compare the results of differential single-photon and double-photon emission rates obtained from the nonlinear optics model [based on Eqs. (21) and (26)] with the results from the nonperturbative QED method. For the single-photon emission, the numerical result of the nonperturbative QED method yields the differential emission rate ranging from to (at different angles) and the nonlinear optics evaluation gives . For the double-photon emission, the numerical result of the nonperturbative QED method yields differential emission rate ranging from to and the nonlinear optics model gives . We can see that the numerical results obtained using the nonperturbative QED method are consistent with the estimations obtained using the nonlinear optics model in orders of magnitude.
For the enhanced single-photon and double-photon emission, we thus have and where is the photon number of single-photon emission, is Fourier component of the undulator electric field, and are solid angle and energy of single-photon emission, and are solid angles and energies of emitted photon pairs, and are the enhancement factors of single-photon emission with , and is the momentum of the emitted photon of single-photon emission.
6. REDUCTION OF SINGLE-PHOTON EMISSION BACKGROUND BY A TAPERED CU
A practical CU interacts with the incident electrons in a finite volume, and the single-photon emission spectrum of CUR that peaks at the fundamental frequency has a broadened linewidth. Therefore, the single-photon emission may have a contribution at the frequencies and of the double-photon emission. Here we propose a scheme of a tapered CU which can reduce the single-photon emission background around the frequencies of the entangled photon pairs.
With the help of the nonlinear optics model presented in Section 5, we can analyze the effect of the tapered undulator field on the single-photon and double-photon emission in CUR. We assume that frequency and spatial filters are placed after the CU, which allow photons emitted within only certain range of energy and solid angle to pass. The ranges of energy and solid angle of single-photon emission are denoted as and , and the ranges of energies and solid angles of double-photon emission are denoted as , , , and .
In the lab frame, the four-pseudo-potential provided by the CU reads . Via Lorentz transformation, we obtain the four-potential in the EF as , where , as we consider the case of . After a four-dimensional Fourier transformation, we obtain where is dependent only on the spatial coordinates but not on time in the lab frame. With a substitution of integral variable and , we obtain where are the components of four-vector . For the electric field in the EF , we have
In Eq. (32), the function ensures the survival of only the electric field components with .
The longitudinal coherence length of emissions from different electrons in each channel is approximately , where refers to the Pierce parameter in premicrobunching [1,38]. To estimate the collective effects of the electron bunch, we introduce a formal expression of the spatial electron distribution in the EF as within the coherent length where about electrons are present. The corresponding distribution in the wave-vector space can be obtained via Fourier transformation as where
Consequently, we can evaluate the differential single-photon and double-photon emission rates by applying Eq. (32) and Eq. (33) to Eq. (22) and Eq. (27). For single-photon emission, we have where . And for double-photon emission, we have where .
The transversal broadening of the electrons in the directions is much smaller than the scale of the undulator field in the directions, so it can be approximated by considering the integrals of only in the vicinity of and . Thus, we obtain where and where
The envelope of the pseudo-magnetic potential of the CU can be tapered by adjusting the internal deformation of the crystal. As an example, we consider a 12-period undulator field of , where the tapering factor is chosen to be , is the step function, is the coordinate of the injection point of the undulator, and and are taper parameters. The tapered undulator field can reduce the broadening of the single-photon emission spectrum in the frequency domain. For the tapered undulator, we get the photon number of single-photon emission within around and the photon pair number of double-photon emission within around and around in the lab frame as follows: and where is the Fourier component of the tapered undulator field, , , , is the average velocity of electrons in the EF, and and are given in Eqs. (38) and (40).
We show the reduction of the single-photon emission background at in Fig. 7, where is the ratio between the single-photon emission rate with the tapered undulator field and the single-photon emission rate with untapered undulator field [i.e., ]. With and , the tapered undulator field can reduce the single-photon emission background by a factor of . In our case, the emission rate of the entangled photon pair is about within the given solid angle range and energy interval. The single-photon emission rate is on the order of without tapering. While considering tapering suppression factor, the single-photon emission rate is suppressed to the order of , which is much lower than the entangled photon pair emission rate. The effect of tapering suppression is very obvious.
Figure 7.Ratio between and at the energy of as a function of for various . When , the tapered undulator field leads to a reduction of the single-photon emission background.
Compared to the 1D free-electron laser undulator, the 3D CU has a higher double-photon emission rate and a much smaller system size. Nonetheless, the double-photon emission from CUR may still be contaminated by the bremsstrahlung of electrons in the crystal. A possible solution is a 3D free-electron lattice generated via other electron beam modulation techniques, such as emittance exchange (EEX) [39–43], where the background noise of bremsstrahlung may be neglected.
There are many methods to fabricate periodic CU, including molecular beam epitaxy (MBE) [44–46], ion implantation [47], mechanical scratching [13], grooving [48,49], laser ablation [15], sandblasting [50], and tensile/compressive strips deposition [48,51,52]. These methods are feasible for the preparation of large period length CUs (μ) [10]. In this work, our calculation is based mainly on produced with MBE (note that our theory is general and can be extended to CUs produced by other methods as well). The undulator period length and the undulator oscillation amplitude are determined by the average germanium concentration , which can be controlled with high precision using the molecular beam epitaxy technique [53]. The germanium concentration has an uncertainty of 0.1%, the undulator period length has an uncertainty on the order of nanometers (μ), and the undulator oscillation amplitude has an uncertainty on the order of picometers () in Ref. [53]. X-ray radiation experiments based on CUs have been performed on Mainz Microtron MAMI [54]. We expect the small fabrication uncertainty of the mature technology would not affect our results qualitatively.
7. CONCLUSION
We have theoretically investigated and proposed the entangled high energy emission from a 3D free-electron lattice by injecting premicrobunched electrons into a periodically bent CU. We carry out a nonperturbative QED analysis of the double-photon emission rate of entangled high-energy photons in CUR and the entanglement property of the emitted photon pairs, which can be enhanced by the collective effect of the 3D free-electron lattice. Under the angular parameters of , the emission rate of entangled photon pair can reach , the concurrence of entangled photon pairs is 0.86, and the energy of the emitted photon can reach the order of 100 MeV. We have also proposed a 12-period undulator tapering scheme to reduce the single-emission background by a factor of at the energies of the entangled photons. Note that when electrons travel in the CU, there are dechanneling and demodulation effects, and we have analyzed the dependence of the dechanneling and demodulation lengths on the undulator parameters. The 3D free-electron lattice in this work can also be extended to synchrotron and free-electron laser systems.
Acknowledgment
Acknowledgment. We thank Zunqi Li, Andrei Benediktovitch, Nina Rohringer, Zhirong Huang, and Zhaoheng Guo for discussions.
Code Availability.The codes for calculation of density matrix and emission rate are available from the corresponding authors upon reasonable request.
APPENDIX A: QUANTUM ELECTRODYNAMICAL TREATMENT OF CUR WITH MAGNETIC PSEUDO-POTENTIAL
From a classical perspective of radiation theory [8], the th harmonic frequency of CUR is given by where is the undulator parameter of the CU, is the average Lorentz factor of the incident electrons, is the off-axis angle of the emitted photons of single-photon emission CUR in the lab frame, and . We construct a pseudo-potential such that an electron inside it moves along the given trajectory, and the radiation from the electron scattered by the pseudo-potential is the same as that from an electron moving along the given trajectory [18]. Specifically, in a periodically bent crystal, we can construct a pseudo-potential like an undulator potential, as similarly treated in Ref. [18], and use this pseudo-potential to calculate the properties of CU radiation via QED scattering theory [19].
We adopt an ideal CU model [8], in which the electrons move along the sinusoidal trajectory , and . In this article, we use the relativistic four-vector notation; that is, we use to represent . We can realize such a trajectory in a planar magnetic undulator, where the magnetic field is with . The relativistic electrons travel through the CU, and interact with the magnetic pseudo-potential inside the crystal (see Fig. 2) [18]. We use the EF for calculation, where the initial average velocity of the incident electrons is zero. We adopt the frame transformation formalism of the Weizsäcker–Williams method [55], and use the natural units . For a planar magnetic undulator, the four-potential is , where .
By Lorentz transformation, we obtain the four-potential of the quasi-EM wave in the EF as where is the four-vector of position and is the average velocity of the electron bunch. is the wave vector of the quasi-EM wave of magnetic pseudo-potential in the EF, which is close to the photon mass shell for approaching the light speed. For an electron whose initial state is almost stationary in the EF, it satisfies the Dirac equation in the external field , where and are the Dirac matrices.
The quantum formalism using Volkov states is applicable for the emission close to saturation, which is consistent with the case of premicrobunched electrons. In quasi-EM wave from Lorentz transformation of the magnetic pseudo-potential, the electron is in a Volkov state, where , and is the free-electron spinor. Substituting the magnetic pseudo-potential into the expression of the Volkov state, it can be obtained that where , , and . For an electron of initial four-momentum , the corresponding Volkov state has a time-varying four-momentum whose average is the quasi-momentum, satisfying .
In the Volkov state of the electron, the four-quasi-momentum of the linearly polarized magnetic undulator case satisfies . Here we use the on-shell condition . In the EF, the emission process satisfies the four-dimensional energy momentum conservation condition, where is the four-quasi-momentum of the final state of electron, is the four-quasi-momentum of the initial state of electron, is the four-dimensional momentum of the emitted single photon, is the momentum of the incident photon, as the quasi-EM field of the pseudo-potential is equivalent to the photon in our model, and represents the number of absorbed photons. We set the momentum of the initial electron as , and the momentum of the incident photon as .
Square both sides of Eq. (A6), and use the on-shell condition: , , , , and then consider and , it can be obtained that
Let , for a specific direction , the harmonic frequency is
Under typical initial conditions in EF , and , where refers to the off-axis angle between the three-momentum of the emitted photon and the axis, we have
In the lab frame, the corresponding four-momentum could be obtained by Lorentz transformation. We introduce the off-axis angle , which refers to an off-axis angle between the three-momentum of the emitted photon and the axis in the lab frame, and obtain the relation of the zenith angle in the EF and off-axis angle in the lab frame as
Recalling that , and defining an effective undulator parameter of the pseudo-potential, the harmonic frequency reads
As and , numerically , where . As , we obtain that from . Then,
We consider a set of parameters for the numerical analysis: , μ, and . We determine the strength of the magnetic pseudo-potential to be ; then . In Eq. (A14), changes to in the SI units, i.e., it is reduced by the light speed . For , we have , and
Meanwhile, we assume the emission angle of the photon in the lab frame to be small due to the beaming effect [1], i.e., , and we have . In summary, we obtain the frequency of single-photon emission in the case of pseudo-potential,
The CUR emission can be viewed as quasi-EM wave being scattered by the electron bunch in the EF, and for the fundamental frequency emission (),
For the Volkov state of electrons, we define for , and Floquet expansion of the Volkov state reads
In the Floquet picture, the Volkov state corresponds to the superposition state of the electron dressed by Floquet photons, i.e., . With the change of quasi-momentum, the initial Volkov state of the electron can evolve into another final Volkov state, and photons are emitted in this process, which can be regarded as a superposition of scattering amplitudes of emission processes with different photon numbers.
APPENDIX B: SINGLE-PHOTON EMISSION IN CUR
In the EF, the quasi-EM wave formed by the pseudo-potential magnetic field scatters with the electron, and induces photon emissions [56]. The wave vector of the emitted photon and wave vector of incident quasi-EM wave are set as follows: where is the angular frequency of the emitted photon, refers to the off-axis angle between the three-momentum of the emitted photon and the axis in the EF.
In the single-photon emission process, we set the four-momentum of the initial electron as , the four-quasi-momentum as , the four-spin as , the four-momentum of the final state electron as , the four-quasi-momentum as , the four-spin as , the four-wave vector of the emitted photon as , and the four-polarization as . Under such conventions, the scattering matrix element from any specific initial state (labeled as ) to any specific final state (labeled as ) could be expressed following the Feynman rules as where , refers to the Dirac conjugate of a spinor , and is the Dirac matrix. For convenience, we define , where is used to label the four-momentum of the different states. The integration of in Eq. (B3) gives where can be further simplified. By definition, and
For Bessel functions of the first kind ,
So, there is
By inserting the complex form of the cosine function into Eq. (A18), we find the recurrence relation of the function,
Thus, we can calculate the value of the functions from the recurrence relation and the Bessel function of the first kind. For the function, we use the above results,
Then
From commutators of the Dirac matrices, there is ; then
So, we get where is where is the change of the number of Floquet photons in the initial and final states of the quasi-EM field, and .
APPENDIX C: DOUBLE-PHOTON EMISSION IN CUR
The double-photon emission process of CUR can be similarly treated in the EF via the Weizsäcker–Williams formalism, where the photon pairs are emitted upon scattering of incident quasi-EM wave [22]. The wave vector of the photon pair is set as where are the angular frequencies of the two photons, and refer to the zenith and azimuth angles of the three-momenta of the two photons relative to the axis in the EF, respectively.
Similar to the analysis of the single-photon emission process, we begin with the calculation of the scattering matrix of the double-photon emission. In the double-photon emission in CUR, two photons are emitted as the electron in the Volkov state decays spontaneously during propagation. The S-matrix of the double-photon emission can be written as where is the propagator of Volkov electron [57] and the subscript (a) refers to the Feynman diagram (a) in Fig. 8 and (b) refers to the Feynman diagram in which and are exchanged in Fig. 8.
Figure 8.The Feynman diagrams of the single electron process in the CU. The black lines represent the Volkov states of electron, and the orange lines represent the photons. The first and second diagrams of the second-order process refer to the double-photon emission CUR of high-energy photons.
Following the Feynman rules, we obtain and where is the number of Floquet photons absorbed from the quasi-EM wave at the th vertex for that satisfies , . We define , and using the Cauchy–Schwarz inequality, we find that the sum over actually starts at , and where , and refer to the constraint of and , respectively. Since the photons are bosons, and can be obtained from and by the exchanges of photon external lines that , , and .
APPENDIX D: CUR EMISSION RATE FROM A SINGLE ELECTRON
We first calculate the single-photon emission rate in CUR that [24] where refers to the probability of the single-photon emission in CUR from a specified initial state to the final state by emission of a photon with the wave vector of , and , refer to the volume and temporal scale of the interaction zone.
We can obtain the differential emission rate as where means that the quasi-momentum conservation condition and the on-shell condition of final quasi-momentum are satisfied simultaneously. , where is the renormalized free-electron spinor that satisfies is calculated from , which satisfies is the differential solid angle of .
Similarly, the double-photon emission rate in CUR reads [24] where refers to the probability of the double-photon emission in CUR from a specified initial state to the final state by emission of two photons with wave vectors of and .
We further obtain the differential emission rate, where is the step function, and can be calculated from by replacing with . means that and are satisfied simultaneously.
APPENDIX E: DENSITY MATRIX OF THE PHOTON PAIRS FROM DOUBLE-PHOTON EMISSION IN CUR
In order to calculate the entanglement degree of the emitted photon pair from double-photon emission in CUR, we construct the corresponding density matrix from the S-matrix elements [24]. For the double-photon emission in CUR, the corresponding quantum system consists of an electron and two photons, and we use to label the spin state of the electron. For the photon, we decompose their quantum states with the circular polarization basis that contains four states of the two photons: , , , and , and for convenience, we index these basis as 1, 2, 3, and 4. For the initial state, there is only one electron and no photon, so the initial density matrix is [24]
Via scattering, the final density matrix can be expressed as , where is the evolution operator of . Since the two-photon polarization basis is four-dimensional, the final density matrix is a matrix. The electron–photon entanglement in the density matrix is eliminated by summing the spin of the electron; we get the matrix elements of the two-photon density matrix so that [24] where .
In fact, each transition matrix element from the initial state to the final state is proportional to the corresponding scattering matrix element, so the scattering matrix can be used to represent the final density matrix with an appropriate normalization constant so that [24] where .
With the density matrix, we can calculate the emission rate of double-photon emission and the entanglement of two photons. For potential applications, we need entangled photon pairs with an emission rate and an entanglement degree as high as possible.
APPENDIX F: DECHANNELING AND DEMODULATION
When traveling through the CU, the particle can attain a state of stable channeling within periodically bent crystals as long as the electromagnetic field force can supply the necessary centripetal force characterized by the centrifugal parameter . For the case of a sinusoidal periodically bent crystal [16,58], where is the maximal gradient of potential energy, and corresponds to the case of a straight crystal.
The Pöschl–Teller potential [59], is used to calculate the planar channeling oscillation motion of electrons in bent crystals, where is the distance of the electrons to the channel center, and and are parameters determined by the potential well depth and the maximal gradient of potential energy . For the case of Si(110) planar channel, and [8].
Using the Pöschl–Teller potential, the equation of motion for the channeling electrons is given by [8] where is the channeling oscillation amplitude and is the channeling oscillation period.
The dechanneling length of an ultra-relativistic electron in the straight channel is [60] where is the fine-structure constant, is the radiation length, and for amorphous silicon.
In scattering theory, the dechanneling length of an ultra-relativistic electron in bent crystals is proportional to the effective well depth [61]. The Pöschl–Teller potential is used to calculate , and the result is approximately , so the dechanneling length for the bent channels is [8]
The corresponding number of dechanneling periods is
APPENDIX G: THE DEPENDENCE OF Ndc, λu, βEF, Ee ON K AND C
Here we discuss the dechanneling process in a CU, the electron velocity in the EF, and the electron energy, so the undulator parameter of the CU is used. In fact, is equal to numerically. According to the definition of the undulator parameters ,
Applying Eq. (G1) and Eq. (F1) to Eq. (F7), we obtain the expression of the number of dechanneling periods as where is the electron rest energy.
The electrons are initially at rest in the EF before entering the undulator. However, a reverse velocity will be induced when the electron interacts with the undulator field. The average velocity along the axis of electrons in the laboratory frame is where is the fundamental wavelength. Then, by using the velocity-addition formula, we obtain the average electron velocity along the axis in the EF as
Under the condition , we calculate the dependence of the undulator period length , the electron velocity in the EF, the electron Lorentz factor , and the electron energy on the undulator parameter and the centrifugal parameter , shown in Fig. 9.
Figure 9.Dependence of the undulator period length , the electron average velocity inside the undulator in the EF, the electron Lorentz factor , and the electron energy on the undulator parameter and the centrifugal parameter under the condition .
APPENDIX H: EXPLANATION OF THE PARAMETERS IN TABLE?1
The undulator oscillation amplitude is controlled by the average germanium concentration , which is in the large-amplitude regime, i.e., , and is the interplanar distance determined by the material and in our case of Si(110) channeling [8]. Meanwhile, is very close to the parameters of the previous experimental works [46]. In order to improve the numbers of dechanneling periods and demodulation periods and to avoid too high electron energy for practical reasons, according to the dependencies we deduced in Section 4 and Appendix G, we choose the appropriate undulator parameter and the centrifugal parameter , which correspond to the electron energy and the undulator period length μ. corresponds to the number of demodulation periods and the number of dechanneling periods . The undulator period number should satisfy and , so we choose the undulator period number . According to the definition of the pseudo-potential magnetic field amplitude , we get . The fundamental frequency is calculated by Eq. (A1). Since it is the fundamental frequency, we take and (forward emission). The repetition rate, the charge per pulse, and the transverse width of the electron bunch, which determine the number of channels, are chosen for demonstration purposes based on the latest technology in this field (such as LCLS-II, FLASH, CEPC, etc.).
References
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