• Photonics Research
  • Vol. 13, Issue 6, 1510 (2025)
Leshi Zhao1,†, Linfeng Zhang1,2,†, Haitan Xu2,3,6,*, and Zheng Li1,4,5,7,*
Author Affiliations
  • 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
  • 6e-mail: haitanxu@nju.edu.cn
  • 7e-mail: zheng.li@pku.edu.cn
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    DOI: 10.1364/PRJ.539785 Cite this Article Set citation alerts
    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) Copy Citation Text show less

    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 [16]. 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 Si1χGeχ mixtures that take advantage of the different lattice constants of Si and Ge [7,912], 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 200  GeV electrons due to an ultrastrong effective magnetic field of 200 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.

    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

    The 3D free-electron lattice in the CU. (a) Transversal structure of the Si1−χGeχ 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 24d, where d≈0.543 nm is the crystal constant. The cross sections of the channels denoted by golden rectangles form a 2D Bravais lattice with primitive vectors a and b. (b) The premicrobunched free electrons channeling in the CU (green spheres) form a 3D free-electron lattice, where L is the interval between adjacent microbunches in the lab frame.

    Figure 1.The 3D free-electron lattice in the CU. (a) Transversal structure of the Si1χGeχ 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 24d, where d0.543  nm is the crystal constant. The cross sections of the channels denoted by golden rectangles form a 2D Bravais lattice with primitive vectors a and b. (b) The premicrobunched free electrons channeling in the CU (green spheres) form a 3D free-electron lattice, where L 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 xlab1=L0sin(kuxlab3) in the channels, and ku=2π/λu (see Fig. 2). We use the relativistic four-vector notation, i.e., we use x0,x1,x2,x3 to represent t,x,y,z. The corresponding pseudo-potential reads Alabμ=a(0,cos(kuxlab3),0,0), where a=B0/ku, B0=2πmK1/eλu is the effective magnetic field amplitude, and K1=ea/m is the effective CU parameter. Here we use the natural units =c=ϵ0=1.

    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 xlab1=L0 sin(kuxlab3) with period λu and amplitude L0, where ku=2π/λu. 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 θ1 and θ2.

    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 xlab1=L0sin(kuxlab3) with period λu and amplitude L0, where ku=2π/λu. 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 θ1 and θ2.

    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].

    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 ω1≃ωfd/3. The two axes correspond to the emission angles θ2 and A2 of photon 2, where A2 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.

    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 ω1ωfd/3. The two axes correspond to the emission angles θ2 and A2 of photon 2, where A2 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 Δl=λu,preγ1+K12/22+K12/2, where the incident electron bunch has a zero initial average velocity in the EF before entering the undulator, λu,pre is the period of the undulator for premicrobunching, γ is the Lorentz factor of the electron, and K1 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 Ne and Lb, respectively. The longitudinal coherence of emissions from different free electrons in each channel is given by the coherence length lcoh, and the electrons can be treated as coherent electrons within the coherence length lcohΔl/ρP, where ρP represents the Pierce parameter of the electron bunches in premicrobunching. The dimensionless Pierce parameter ρP was defined in Ref. [1], which is about 1/166 in our case. The coherence length lcoh encompasses 1ρP microbunches (in our case, lcoh166ΔlΔl) where the electrons are coherent. The number of the single-channel coherent electrons (SCCEs) is denoted by NSCCE. The phase difference of the double-photon emission from electrons within a coherence length is given by ϕe,l=Δk·re,l, where Δk=k1+k2k, k1 and k2 are the momenta of the two emitted photons, k is the momentum of quasi-electromagnetic wave, and re,l is the wave-packet center position of the lth electron with l=1,2,,NSCCE. If the lth electron lies in the j(l)th microbunch, we obtain re,l=j(l)Δlx^3+Δxl1x^1+Δxl2x^2, where Δxl1 and Δxl2 are the transversal positions of the lth electron relative to the center of the channel, and x^1,2,3 are the three spatial unit vectors, and j(l)=1,2,,Ncmb, where Ncmblcoh/Δl is the number of microbunches within the coherence length. The phase factor of the NSCCE electrons is given by ESCCE=l=1NSCCEeiϕe,l.

    As Δllcohλu and L0λu [8], all the electrons within the coherence length are approximately at the same transversal position, i.e., |Δxl1Δxl1|<2πL0Δl/λuΔl for any l,l=1,2,,NSCCE. For the emitted photon wave vectors considered in this work, Δk1·Δxl11 and Δk2·Δxl21. Thus, the cumulative phase factor for the enhancement can be approximated by ESCCEl=1NSCCEeiΔk3·j(l)Δl=j=1NcmbNjeiΔk3·jΔl,where Ncmb is the number of microbunches within the coherence length, and Nj is the number of electrons within the jth microbunch. The summed phase factor by all the electrons in a bunch and a single channel is given by |ESC|2|ESCCE|2Lblcoh.

    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 Nch channels, where NchAetr/Actr reflects the ratio between the transversal area of the electron bunch Aetr and the transversal area of the channel Actr. The cross section of the crystalline channels forms a 2D Bravais lattice. We index the different channels by (na,nb), which indicates the channels in the two directions of Bravais primitive vectors a, b. For the channel (na,nb), the corresponding phase factor is ϕc,l=Δk·(naa+nbb). Thus, the summed phase factor of the Nch channels is Emc=l=1Ncheiϕc,l=na=1Nch,anb=1Nch,beiΔk·(naa+nbb),where Nch,a and Nch,b are the number of channels along the two directions of a and b, respectively, which satisfy Nch,aNch,b=Nch. Combining the enhancement factors, the entangled photon pair emission from the 3D free-electron lattice is (dNddt)lattice=(dNddt)single·|ESC|2·|Emc|2,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. Nd is the photon pair number of double-photon emission, and (dNd/dt)single is the entangled photon pair emission from a single electron [see Appendix D for details of (dNd/dt)single].

    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 C(ρf)=max(0,λ1λ2λ3λ4), where λ1,2,3,4 are the four eigenvalues of the matrix Q=ρf(σyσy)ρf*(σyσy) in descending order, ρf is the density matrix of entangled photon pair, and σy 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 Re=0.5  s1 and concurrence C=0.86 of the emitted entangled photon pairs for certain angular parameters of γtanθ1=1.17,A1=3π/8,γtanθ2=0.725,A2=11π/8. θ1,A1 and θ2,A2 are chosen to satisfy the phase-matching condition, i.e., (k11+k21k1)dch,x2mxπ and (k12+k22k2)dch,y2myπ, where mx,my are integers, dch,x is the x component of the spacing between channels, dch,y is the y component of the spacing between channels, k1 and k2 depend on the emission zenith angles Z1,Z2 and the polar angles A1,A2 in the EF, and Z1,Z2 are related to off-axis angles θ1 and θ2 in the lab frame by the Lorentz transformation. The energies of the emitted photons are chosen to be around ω1,labωfd/3 and ω2,lab0.54ωfd with an energy range of ΔE=0.0045  MeV. 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.

    (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.

    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 y0 due to the different channeling transverse energies Ey, 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 z 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 C=0.1 (see Appendix F for details of C), 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] g˜z+iωmbg˜(z;ξ,Ey)(1vz(z))=D0[Ey(Eyg˜Ey)+1Ee2g˜ξ2],where g˜(z;ξ,Ey)=f(t,z;ξ,Ey)eiωmbc(zct), f(t,z;ξ,Ey) is the electron distribution function periodically modulated in the longitudinal direction, ωmb=4πγ2c/(1+K12/2)λu,pre is the frequency of modulation, vz is the electron velocity projection in the longitudinal direction, Ey is the transverse energy of the electrons, ξ is the angle between the particle trajectory and longitudinal direction, and D0 is the scattering parameter associated with the dechanneling [2527].

    We use the Pöschl–Teller potential to calculate the motion equation of electron channeling oscillation y(t) [8] (see Appendix F for details). Meanwhile, the electrons travel at an angle ξ away from the plane formed by the channeling oscillation direction of the y axis and the longitudinal direction of the z axis, and the motion in the x-axis direction is x=ztanξ. Thus, the electron velocity component in the longitudinal direction is vz(z)=c11γ2(z˙/x˙2+y˙2+z˙2)=c11γ21+sin2ξ+bPT2(2πcTy)2sinh2(y0bPT)sin2(2πzcTy)1+sinh2(y0bPT)cos2(2πzcTy),where bPT=0.145d for the case of Si(110) planar channel [8], y0 is the channeling oscillation amplitude, and Ty is the time period of channeling oscillating motion of the electrons (see Appendix F for details).

    Note that ξ1 and bPT2(2πcTy)21; we expand Eq. (5) to the second order and average it as vz(z)=c[112γ212bPT2(2πcTy)2I],where I=0Ty1Tysinh2(y0bPT)sin2(2πtTy)1+sinh2(y0bPT)cos2(2πtTy)dtrefers to the average over time period Ty.

    By applying Eq. (6) to Eq. (4), and using separation of variables as g˜(z;ξ,Ey)=Z(z)Ξ(ξ)E(Ey), we obtain D0E(Ey)Ey[EyE(Ey)Ey]iωmbcbPT22(2πcTy)2I=CEy,D0Ee1Ξd2Ξ(ξ)dξ2iωmbcξ22=Cξ,1ZdZ(z)dz+iωmb2cγ2=Cz,where CEy, Cξ, and Cz, satisfying Cz=CEy+Cξ, are parameters to be determined with boundary conditions. As the electron distribution at Ey=Umax is zero, where Umax is the potential well depth, using the parameters given in Table 1 and choosing λu,pre/λu=400 for larger demodulation length, we get from Eq. (8) that CEy(3.9×10250i)m1. 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] Cξ,n=(1+i)ωmbD0cEe(n+12),n=0,1,2,.

    For large z, g˜(z;ξ,Ey) is dominated by the term with n=0, and Cξ,0=1.6×102(1+i)m1. According to Cz=CEy+Cξ, we obtain the demodulation length Ldm=1Cz=1.8  mm, and the number of demodulation periods Ndm=Ldm/λu12. We show in Fig. 5(b) more results of Ndm for different undulator parameters K1 and centrifugal parameters C under the conditions λu,pre/λu=400,L0=0.2  nm.

    Dependence of (a) Ndc and (b) Ndm on the undulator parameter K1 and the centrifugal parameter C under the conditions λu,pre/λu=400,L0=0.2 nm.

    Figure 5.Dependence of (a) Ndc and (b) Ndm on the undulator parameter K1 and the centrifugal parameter C under the conditions λu,pre/λu=400,L0=0.2  nm.

    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 [3035]. The dechanneling length Ldc 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 C [8]. Under the parameters given in Table 1, we get the number of dechanneling periods Ndc20. We show in Fig. 5(a) more results of Ndc for different undulator parameters K1 and centrifugal parameters C under the condition L0=0.2  nm (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].

    Diagrams for the nonlinear optical processes. (a) Double-photon emission and its conjugate; (b) difference frequency generation.

    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 md2rdt2=e[Eu(r,t)+E2(r,t)]ev×[Bu(r,t)+B2(r,t)],where v is the electron velocity in the EF, Eu,Bu and E2,B2 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 v of the electrons will change in the EF.

    In the EF, as the order of magnitude of K1 is 1, the electron velocity in the EF is much less than c, i.e., βEF=|v|/c1, and we can expand Eq. (12) to the first order [37], md2rdt2=e[Eu(t)+E2(t)]ev×[Bu(t)+B2(t)]e(r·)[Eu(t)+E2(t)],where the last term is also of first order, as |(r·)E|βEF|E||v×B|.

    We apply a perturbation technique with Fourier transformation to solve Eq. (13). The leading orders of r read r(0)(ω)=emω2[Eu(ω)+E2(ω)],r(1)(ω)=(emω2)2{dω(ωωω)[Eu(ωω)+E2(ωω)]×{×[Eu(ω)+E2(ω)]}+dω{[Eu(ωω)+E2(ωω)]·}[Eu(ω)+E2(ω)]}.

    We consider the electric field of the quasi-EM wave Eu to be monochromatic with frequency ωu and an amplitude Eu0 much larger than that of the vacuum field E20. Thus, the dominant linear component of r at ωu reads r(0)(t)=emωu2Eucos(ωut+ϕ0),where ϕ0 is a phase factor. And for a specific emission frequency ω1, the DFG process is contributed only by the component of the vacuum field at ω2=ωuω1. We obtain the corresponding component of r so that |rDFG(1)(ω1)|e2m2ω13cEu0E20(ω2),where E20(ω2) is the amplitude of E2 at the frequency ω2. In this section, we use the SI units and write ,c,ϵ0 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 p(t)=p0cos(ωt), the emission power per solid angle reads dPdΩω4p0232π2ϵ0c3,where the detailed angular dependence is neglected for an order-of-magnitude evaluation. For a multifrequency oscillation p(t), the total emission energy per solid angle is dtdPdΩdωω4|p˜(ω)|232π2ϵ0c3.

    Thus, from Eq. (16), we obtain the differential single-photon emission rate of CUR as dNsdtdΩe4Eu0232π2ϵ0c3m2ωu,where dNs/dtdΩ reflects the number of emitted photons in unit solid angle per unit time.

    Via Lorentz transformation and the relationship Eu0=2πγmc2eλuK1 in the EF, we obtain the differential single-photon emission rate in the lab frame as (dNsdtdΩ)labγ2e2K1216πϵ0λu.

    By Fourier transformation we can get the distribution function of differential single-photon emission rate in frequency and spatial domains so that dNsdΩdEe4|E˜u0(ω)|232π2ϵ0c3m22ω.

    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 (dPdΩk1)one modeω1432π2ϵ0c3(e3Eu0ω2ϵ0Vm2ω13c)2,where only one mode of the vacuum field E2 at the frequency ω2 is present. We further sum over the vacuum modes with frequency around ω2, dPdΩk1e6Eu0232π2ϵ0c5m4ω12ω2ϵ0V|k2|2d|k2|dΩk28π3V.

    Thus, the differential double-photon emission rate is given by dNddtdΩk1dΩk2dE2=dPdΩk1dΩk2dE2ω1e6Eu02ω23256π5ϵ02c8m4ω13.

    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 (dNddtdΩk1dΩk2dE2)labe4γ4K1264π3ϵ02c4m2λu2.

    By Fourier transformation we can get the distribution function of differential double-photon emission rate in frequency and spatial domains as dNddΩk1dΩk2dE1dE2e6|E˜u0(ω)|22562π5ϵ02c8m4.

    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 1018  s1 to 1020  s1 (at different angles) and the nonlinear optics evaluation gives 1020  s1. For the double-photon emission, the numerical result of the nonperturbative QED method yields differential emission rate ranging from 1032  J1s1 to 1034  J1s1 and the nonlinear optics model gives 1032  J1s1. 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 (dNsdΩdE)totale4|E˜u0(ω)|232π2ϵ0c3m22ω|ESC,s|2|Emc,s|2,and (dNddΩk1dΩk2dE1dE2)totale6|E˜u0(ω)|22562π5ϵ02c8m4|ESC|2|Emc|2,where Ns is the photon number of single-photon emission, E˜u0(ω) is Fourier component of the undulator electric field, Ω and E are solid angle and energy of single-photon emission, Ωk1,2 and E1,2 are solid angles and energies of emitted photon pairs, |ESC,s| and |Emc,s| are the enhancement factors of single-photon emission with Δk=kk, and k 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 ωfd has a broadened linewidth. Therefore, the single-photon emission may have a contribution at the frequencies ω1,lab and ω2,lab 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 ΔE and ΔΩ, and the ranges of energies and solid angles of double-photon emission are denoted as ΔE1, ΔE2, ΔΩ1, and ΔΩ2.

    In the lab frame, the four-pseudo-potential provided by the CU reads Alabμ=(0,Alab). Via Lorentz transformation, we obtain the four-potential in the EF as AEFμ(xEF)=(0,AEF), where AEF=Alab(xlab(xEF)), as we consider the case of Alab·x^3=0. After a four-dimensional Fourier transformation, we obtain A˜EF(k)=dxEF0dxEF1dxEF2dxEF3AEF(xEF)eik·xEF=dxEF0dxEF1dxEF2dxEF3eik·xEF×Alab(xEF1,xEF2,γ(xEF3+βxEF0)),where Alab is dependent only on the spatial coordinates but not on time in the lab frame. With a substitution of integral variable u=γ(xEF3+βxEF0) and β1, we obtain A˜EF(k)=dxEF0ei(k0+k3)xEF0dxEF1dxEF2duγei(k1xEF1+k2xEF2+k3γu)Alab(xEF1,xEF2,u),u=1γ2πδ(k0+k3)A˜lab(k1,k2,k3γ),where k0,1,2,3 are the components of four-vector k. For the electric field in the EF EEF(xEF)=cAEF/xEF0, we have E˜EF(k)=ick0A˜EF(k)=ick0γ2πδ(k0+k3)A˜lab(k1,k2,k3γ).

    In Eq. (32), the δ function ensures the survival of only the electric field components with k3=k0=ω/c.

    The longitudinal coherence length of emissions from different electrons in each channel is approximately lcohΔlρP, where ρP 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 ρce(xEF)=ρce(xEF1,xEF2,xEF3+βEFxEF0) within the coherent length where about NSCCENch electrons are present. The corresponding distribution in the wave-vector space can be obtained via Fourier transformation as ρ˜ce(k)=d4x·ρce(xEF1,xEF2,xEF3+βEFxEF0)eik·xEF=2πδ(k0+βEFk3)ρ˜ce(k1,k2,k3),where ρ˜ce(k1,k2,k3)ρ˜ce(k)=d3xρce(x)eik·x.

    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 N1,lab(k,ΔE,ΔΩ)γLblcohΔΩΔEe4|d4xEFeik·xEFEEF(xEF)ρce(xEF)|232π2ϵ0c6m22k0γLblcohΔΩΔEe4|d4kE˜EF(k)ρ˜ce(kk)|232π2ϵ0c6m22k0LblcohΔΩΔE·e4(ks0)28γϵ0c4m22k0|dk1dk2A˜lab(k1,k2,ks0γ)×ρ˜ce(k1k1,k2k2,k3+ks0)|2,where ks0=(k0+βEFk3)/(1βEF). And for double-photon emission, we have N2,lab(k1,ΔE1,ΔΩ1;k2,ΔE2,ΔΩ2)γ2LblcohΔE2ΔΩ1ΔΩ2×e6(k20)3|d4xEFei(k1+k2)·xEFEEF(xEF)ρce(xEF)|22562π5ϵ02c10m4(k10)3γ2Lblcoh(ΔΩΔE)2e6(k20)3|d4kE˜EF(k)ρ˜ce(k1+k2k)|22562π5ϵ02c10m4(k10)3Lblcoh(ΔΩΔE)2·e6(k20)3(kd0)2642π3ϵ02c8m4(k10)3·|dk1dk2A˜lab(k1,k2,kd0γ)×ρ˜ce(k11+k21k1,k12+k22k2,k13+k23+kd0)|2,where kd0=[k10+k20+βEF(k13+k23)]/(1βEF).

    The transversal broadening of the electrons in the x1,x2 directions is much smaller than the scale of the undulator field in the x1,x2 directions, so it can be approximated by considering the integrals of A˜lab(k1,k2,k30γ) only in the vicinity of k1=0 and k2=0. Thus, we obtain N1,lab(k,ΔE,ΔΩ)N1,lab(k)·|A˜lab(0,0,ks0/γ)|2,where N1,lab(k,ΔE,ΔΩ)LblcohΔΩΔEe4(ks0)28γϵ0c4m22k0|ρ˜ce(k1,k2,k3+ks0)|2,and N2,lab(k1,ΔE1,ΔΩ1;k2,ΔE2,ΔΩ2)N2,lab(k1,k2)·|A˜lab(0,0,kd0/γ)|2,where N2,lab(k1,ΔE1,ΔΩ1;k2,ΔE2,ΔΩ2)LblcohΔE2ΔΩ1ΔΩ2·e6(k20)3(kd0)2642π3ϵ02c8m4(k10)3×|ρ˜ce(k11+k21,k12+k22,k13+k23+kd0)|2.

    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 Alab(x1,x2,x3)=al=112g(l)h(x3lλux03)x^1, where the tapering factor g(l) is chosen to be g(l)={exp[(llt)2/σt2],l=1,2,3,4,51,l=6,7exp{[l(13lt)]2/σt2},l=8,9,10,11,12.h(y)=sin(kuy)Θ(y+λu)Θ(λuy), Θ is the step function, x03 is the coordinate of the injection point of the undulator, and lt and σt 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 Ns(k,ΔE,ΔΩ) within ΔE around k and the photon pair number of double-photon emission Nd(k1,ΔE1,ΔΩ1;k2,ΔE2,ΔΩ2) within ΔE1 around k1 and ΔE2 around k2 in the lab frame as follows: Ns(k,ΔE,ΔΩ)N1(k,ΔE,ΔΩ)·|A˜lab(0,0,ks0/γ)|2,and Nd(k1,ΔE1,ΔΩ1;k2,ΔE2,ΔΩ2)N2(k1,ΔE1,ΔΩ1;k2,ΔE2,ΔΩ2×|A˜lab(0,0,kd0/γ)|2,where A˜lab(k1,k2,k3/γ) is the Fourier component of the tapered undulator field, ks0=(k0+βEFk3)/(1βEF), kd0=[k10+k20+βEF(k13+k23)]/(1βEF), βEF=|v|/c1, v is the average velocity of electrons in the EF, and N1(k,ΔE,ΔΩ) and N2(k1,ΔE1,ΔΩ1;k2,ΔE2,Δω2) are given in Eqs. (38) and (40).

    We show the reduction of the single-photon emission background at ω2,lab in Fig. 7, where Ft is the ratio between the single-photon emission rate Ns,taper with the tapered undulator field and the single-photon emission rate Ns,step with untapered undulator field [i.e., g(l)1]. With lt=5.45 and σt=2.97, the tapered undulator field can reduce the single-photon emission background by a factor of 106. In our case, the emission rate of the entangled photon pair is about 1  s1 within the given solid angle range and energy interval. The single-photon emission rate is on the order of 1  s1 without tapering. While considering tapering suppression factor, the single-photon emission rate is suppressed to the order of 106  s1, which is much lower than the entangled photon pair emission rate. The effect of tapering suppression is very obvious.

    Ratio Ft between Ns,taper and Ns,step at the energy of ω2,lab as a function of σt for various lt. When Ft<1, the tapered undulator field leads to a reduction of the single-photon emission background.

    Figure 7.Ratio Ft between Ns,taper and Ns,step at the energy of ω2,lab as a function of σt for various lt. When Ft<1, 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) [3943], where the background noise of bremsstrahlung may be neglected.

    There are many methods to fabricate periodic CU, including molecular beam epitaxy (MBE) [4446], 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 (λu102  μm) [10]. In this work, our calculation is based mainly on Si1χGeχ 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 λu and the undulator oscillation amplitude L0 are determined by the average germanium concentration χ¯, 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 λu has an uncertainty on the order of nanometers (0.001  μm), and the undulator oscillation amplitude L0 has an uncertainty on the order of picometers (0.001  nm) in Ref. [53]. X-ray radiation experiments based on Si1χGeχ 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 γtanθ1=1.17,A1=3π/8,γtanθ2=0.725,A2=11π/8, the emission rate of entangled photon pair can reach 0.5  s1, 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 106 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 nth harmonic frequency of CUR is given by ωn=2γ2ω0n1+γ2θ2+K22,where K=2πγL0/λu is the undulator parameter of the CU, γ is the average Lorentz factor of the incident electrons, θ is the off-axis angle of the emitted photons of single-photon emission CUR in the lab frame, and ω0=2πc/λu. 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 xlab1(xlab3)=L0sin(kuxlab3), and ku=2π/λu. In this article, we use the relativistic four-vector notation; that is, we use x0,x1,x2,x3 to represent t,x,y,z. We can realize such a trajectory in a planar magnetic undulator, where the magnetic field is B=(0,By,0) with By=B0sin(kuxlab3). 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 =c=ϵ0=1. For a planar magnetic undulator, the four-potential is Alabμ=a(0,cos(kuxlab3),0,0), where a=B0/ku.

    By Lorentz transformation, we obtain the four-potential of the quasi-EM wave in the EF as AEFμ=a(0,cos[γku(βtEF+xEF3)],0,0)=a(0,cos(k·xEF),0,0),where xEF is the four-vector of position and β is the average velocity of the electron bunch. k=γku(β,0,0,1) 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 β 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 (peAm)Ψ=0, where p=γμpμ and γμ 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, ΨpV(x)=(1eAk2k·p)exp{i[p·x+0ϕ(ep·A(ϕ)k·pe2A2(ϕ)2k·p)dϕ]}up,where ϕ=k·x=k0x0k·x, and up is the free-electron spinor. Substituting the magnetic pseudo-potential into the expression of the Volkov state, it can be obtained that ΨpV(x)=(1ea1k2k·pcosϕ)exp[i(p+e2a24k·pk)·xi(α1sinϕ+ζ1sin2ϕ)]up,where a1=a(0,1,0,0), α1=ea1p/kp, and ζ1=e2a28k·p. For an electron of initial four-momentum p, the corresponding Volkov state has a time-varying four-momentum whose average is the quasi-momentum, qμ=pμ+e2a24k·pkμ,satisfying |q|2=m2+e2a2/2=m*2.

    In the Volkov state of the electron, the four-quasi-momentum of the linearly polarized magnetic undulator case qμ=pμ+e2a24k·pkμ satisfies q·q=m2+e2a2/2m*2. Here we use the on-shell condition k·k=0. In the EF, the emission process satisfies the four-dimensional energy momentum conservation condition, qf+k1=qi+nk,where qf is the four-quasi-momentum of the final state of electron, qi is the four-quasi-momentum of the initial state of electron, k1 is the four-dimensional momentum of the emitted single photon, k 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 n represents the number of absorbed photons. We set the momentum of the initial electron as pi=m(1,0,0,0), and the momentum of the incident photon as k=k0(1,0,0,1).

    Square both sides of Eq. (A6), and use the on-shell condition: qi·qi=m*2, qf·qf=m*2, k·k=0, k1·k1=0, qf·k1=qi·nk,and then consider qf=qi+nkk1 and k1·k1=0, it can be obtained that (qi+nk)·k1=qi·nk.

    Let k1=k10(1,ek1), for a specific direction ek1, the harmonic frequency is k10=qi·nk(qi+nk)·(1,ek1).

    Under typical initial conditions in EF qi=(m+e2a24m,0,0,e2a24m), and (1,ek1)=(1,sinZ,0,cosZ), where Z refers to the off-axis angle between the three-momentum k1 of the emitted photon and the z axis, we have k10=nk0·11+(e2a24m2+nk0m)(1+cosZ).

    In the lab frame, the corresponding four-momentum k1,lab could be obtained by Lorentz transformation. We introduce the off-axis angle θ, which refers to an off-axis angle between the three-momentum k1 of the emitted photon and the z axis in the lab frame, and obtain the relation of the zenith angle Z in the EF and off-axis angle θ in the lab frame as tanZtanθ=k1,lab3k13=γ(β+cosZ)cosZ,k1,lab0k10=γ(1+βcosZ).

    Recalling that k0=γku, and defining an effective undulator parameter K1=eam of the pseudo-potential, the harmonic frequency reads k1,lab0=γ2nku·1+βcosZ1+(K124+nk0m)(1+cosZ).

    As a=B0/ku and B0=2πmK1/eλu, numerically K1=K, where K=2πγL0/λu. As β1, we obtain that (γtanθ)2=21+cosZ1 from tanZtanθ=k1,lab3k13=γ(β+cosZ)cosZ. Then, k1,lab0=2γ2nku·11+(γtanθ)2+K122+2nγkum.

    We consider a set of parameters for the numerical analysis: L0=0.2  nm, λu=156.25  μm, and Ee=196.97  GeV. We determine the strength of the magnetic pseudo-potential to be B0=212.49T; then K1=3.1001. In Eq. (A14), 2nγkum changes to 2nγkumc in the SI units, i.e., it is reduced by the light speed c. For n=1, we have 2γkumc=0.012, and 2nγkumK1.

    Meanwhile, we assume the emission angle of the photon in the lab frame to be small due to the beaming effect [1], i.e., θ1, and we have tanθθ. In summary, we obtain the frequency of single-photon emission in the case of pseudo-potential, k1,lab0=2γ2nku·11+(γθ)2+K122.

    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 (n=1), ωfd=2γ2(2πλu)1+K122+(γtanθ)2.

    For the Volkov state of electrons, we define Gj(n˜,α1,ζ1)=12πππeiα1sinϕiζ1sin2ϕ+in˜ϕcosjϕdϕ,for j=0,1,2,, and Floquet expansion of the Volkov state reads ΨpV(x)=n˜=+[G0(n˜,α1,ζ1)ea1k2k·pG1(n˜,α1,ζ1)]×ei(q·x+n˜k·x)up.

    In the Floquet picture, the Volkov state corresponds to the superposition state of the electron dressed by n˜ Floquet photons, i.e., ΨpV(x)=n˜=+Fn˜exp[i(q·x+n˜k·x)]up. 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 k1 and wave vector of incident quasi-EM wave k are set as follows: k1=ω1(1,sinZ,0,cosZ),k=ω(1,0,0,1),where ω1 is the angular frequency of the emitted photon, Z refers to the off-axis angle between the three-momentum k1 of the emitted photon and the z axis in the EF.

    In the single-photon emission process, we set the four-momentum of the initial electron as pi, the four-quasi-momentum as qi, the four-spin as si, the four-momentum of the final state electron as pf, the four-quasi-momentum as qf, the four-spin as sf, the four-wave vector of the emitted photon as k1, and the four-polarization as ϵ. Under such conventions, the scattering matrix element from any specific initial state (labeled as i) to any specific final state (labeled as f) could be expressed following the Feynman rules as SfiS=d4xΨ¯f(x)(ieA1)Ψi(x),where A1(x)=12k10Vϵ*μeik1·x, u¯=uγ0 refers to the Dirac conjugate of a spinor u, and γ0 is the Dirac matrix. For convenience, we define [GG]n,r=G0(n,α1r,ζ1r)ea1k2k·pG1(n,α1r,ζ1r), where r is used to label the four-momentum of the different states. The integration of Sfi in Eq. (B3) gives SfiS=12k0Vn,n=+{u¯(pf,sf)[GG]¯n,f(ieϵ*)[GG]n,i×u(pi,si)·(2π)4δ(4)(qf+nk+k1qink)},where n,n=+[GG]¯n,f(ieϵ*)[GG]n,i can be further simplified. By definition, G0(n,α1,ζ1)=12πππeiα1sinϕiζ1sin2ϕ+inϕdϕ,and eiα1sinϕiζ1sin2ϕ=n=+G0(n,α1,ζ1)einϕ.

    For Bessel functions of the first kind Jn(x), eiα1sinϕ=n1=+Jn1(α1)ein1ϕ,eiζ1sin2ϕ=n2=+Jn2(α1)ei2n2ϕ.

    So, there is G0(n,α1,ζ1)=n2=+Jn2n2(α1)Jn2(ζ1).

    By inserting the complex form of the cosine function into Eq. (A18), we find the recurrence relation of the G function, Gj(n,α1,ζ1)=12[Gj1(n+1,α1,ζ1)+Gj1(n1,α1,ζ1)].

    Thus, we can calculate the value of the G functions from the recurrence relation and the Bessel function of the first kind. For the [GG]n,r function, we use the above results, n=+[GG]n,reinϕ=(1ea1k2k·prcosϕ)×eiα1,rsinϕiζ1,rsin2ϕ,n=+[GG]¯n,reinϕ=(1eka12k·prcosϕ)×eiα1rsinϕiζ1rsin2ϕ.

    Then n=+[GG]¯n,reinϕϵ*n=+[GG]n,reinϕ=(1eka12k·prcosϕ)ϵ*(1ea1k2k·prcosϕ)×ei[(α1rα1r)sinϕ+(ζ1rζ1r)sin2ϕ].

    From commutators of the Dirac matrices, there is fg+gf=2f·g; then ka1ϵ*a1k=a1kϵ*ka1=a1(2k·ϵ*)ka1a1ϵ*(kk)a1=(2k·ϵ*)ka1a10=a2(2k·ϵ*)k.

    So, we get SfiS=ie2k0Vn=+[u¯(pf,sf)Γ^Su(pi,si)×(2π)4δ(4)(qf+nk+k1qink)],where Γ^S is Γ^S=ϵ*G0(n,α1if,ζ1if)(eka1ϵ*2k·pf+eϵ*a1k2k·pi)G1(n,α1if,ζ1if)+e2a2(2k·ϵ*)(2k·pf)(2k·pi)kG2(n,α1if,ζ1if),where n=ss is the change of the number of Floquet photons in the initial and final states of the quasi-EM field, α1if=α1iα1f and ζ1if=ζ1iζ1f.

    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 k1=ω1(1,sinZ1cosA1,sinZ1sinA1,cosZ1),k2=ω2(1,sinZ2cosA2,sinZ2sinA2,cosZ2),where ω1,ω2 are the angular frequencies of the two photons, Z1,Z2 and A1,A2 refer to the zenith and azimuth angles of the three-momenta k1,k2 of the two photons relative to the z 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 Sfi,(a)D=d4x1d4x2Ψ¯f(x2)(ieA2(x2))(iGV(x2,x1))×(ieA1(x1))Ψi(x1),where GV 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 k1 and k2 are exchanged in Fig. 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.

    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 Sfi,(a)D=ie2V12k202k10d4pm(2π)4×n˜1,n˜2,n˜3,n˜4=+[u¯(pf,sf)Γ^Du(pi,si)]×(2π)4δ(4)[qf+k2qm(n˜3n˜2)k]×(2π)4δ(4)[qm+k1qi(n˜1n˜4)k],and Γ^D=[ϵ2*G0(n2,α1mf,ζ1mf)(eka1ϵ2*2k·pf+eϵ2*a1k2k·pm)G1(n2,α1mf,ζ1mf)+e2a2(2k·ϵ2*)(2k·pf)(2k·pm)kG2(n2,α1mf,ζ1mf)]×pm+mpm2m2+iδ[ϵ1*G0(n1,α1im,ζ1im)(eka1ϵ1*2k·pm+eϵ1*a1k2k·pi)G1(n1,α1im,ζ1im)+e2a2(2k·ϵ1*)(2k·pm)(2k·pi)kG2(n1,α1im,ζ1im)],where nj is the number of Floquet photons absorbed from the quasi-EM wave at the jth vertex for j=1,2 that satisfies n1=n˜1n˜4, n2=n˜3n˜2. We define n=n1+n2, and using the Cauchy–Schwarz inequality, we find that the sum over n actually starts at n=1, and SfiD=ie2V12k202k10n=1+[n1=+M(n,n1)|MC(n1)×(2π)4δ(4)(qf+k1+k2qink)],where M(n,n1)|MC(n1)=M(a)(n,n1)|MC((a),n1)+M(b)(n,n1)|MC((b),n1), and |MC((a),n1),|MC((b),n1) refer to the constraint of qm+k1qin1k=0 and qm+k2qin1k=0, respectively. Since the photons are bosons, M(b)(n,n1) and MC((b),n1) can be obtained from M(a)(n,n1) and MC((a),n1) by the exchanges of photon external lines that k1k2, ϵ1ϵ2, and mm.

    APPENDIX D: CUR EMISSION RATE FROM A SINGLE ELECTRON

    We first calculate the single-photon emission rate in CUR that [24] dNsdt=Vd3qf(2π)3Vd3k1(2π)3|SfiS|2VTV,where |SfiS|2 refers to the probability of the single-photon emission in CUR from a specified initial state i to the final state f by emission of a photon with the wave vector of k1, and V, T refer to the volume and temporal scale of the interaction zone.

    We can obtain the differential emission rate as dNsdtdΩk1=e2m2(2π)2qi0n=1+(k0)2|MRS(n)|22qi·nk|SE(n),where SE(n) means that the quasi-momentum conservation condition qf+k1qink=0 and the on-shell condition of final quasi-momentum (qf)2m*2=0 are satisfied simultaneously. MRS(n)=u¯R(pf,sf)Γ^SuR(pi,si), where uR(p) is the renormalized free-electron spinor that satisfies uR(p)uR(p)=p0m.uR(p) is calculated from u(p), which satisfies u(p)u(p)=p0q0V.dΩk1 is the differential solid angle of k1.

    Similarly, the double-photon emission rate in CUR reads [24] dNddt=Vd3qf(2π)3Vd3k1(2π)3Vd3k2(2π)3|SfiD|2VTV,where |SfiD|2 refers to the probability of the double-photon emission in CUR from a specified initial state i to the final state f by emission of two photons with wave vectors of k1 and k2.

    We further obtain the differential emission rate, dNddtdk10dΩk1dΩk2=e4(2π)5m22qi0n=1+k10(k20)2|n1=+MR(n,n1)|MC(n1)|22|(qi+nkk1)·k2|×Θ(qi0+nk0k10k20)|DE(n),where Θ is the step function, and MR(n,n1)|MC(n1) can be calculated from M(n,n1)|MC(n1) by replacing u(p) with uR(p). DE(n) means that qf+k1+k2qink=0 and qf2m*2=0 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 s 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] ρi=si=12|si,0si,0|.

    Via scattering, the final density matrix can be expressed as ρf=RρiR, where R is the evolution operator of ρi. Since the two-photon polarization basis is four-dimensional, the final density matrix is a 4×4 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] sf,j1|ρf|sf,j2=sf,si=12sf,j1|R|si,0si,0|R|sf,j2,where j1,j2=1,2,3,4.

    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 SfiD can be used to represent the final density matrix with an appropriate normalization constant Nn so that [24] ρf,j1j2=12sf,si=12NnSfi,j1DSfi,j2D*,where j1,j2=1,2,3,4.

    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 C. For the case of a sinusoidal periodically bent crystal [16,58], C=4π2EeL0Umaxλu21,where Umax is the maximal gradient of potential energy, and C=1 corresponds to the case of a straight crystal.

    The Pöschl–Teller potential [59], U(ρ)=aPTtanh2(ρbPT),is used to calculate the planar channeling oscillation motion of electrons in bent crystals, where ρ is the distance of the electrons to the channel center, and aPT and bPT are parameters determined by the potential well depth ΔUmax and the maximal gradient of potential energy Umax. For the case of Si(110) planar channel, aPT=23  eV and bPT=0.145d [8].

    Using the Pöschl–Teller potential, the equation of motion for the channeling electrons is given by [8] y(t)=bPTarcsinh[sinh(y0bPT)cos(2πtTy(y0))],where y0 is the channeling oscillation amplitude and Ty(y0)=2πbPTγm2aPTcosh(y0bPT)is the channeling oscillation period.

    The dechanneling length of an ultra-relativistic electron in the straight channel is [60] Ldc,0αΔUmaxEeπm2c4Lr,where α is the fine-structure constant, Lr is the radiation length, and Lr=9.47  cm for amorphous silicon.

    In scattering theory, the dechanneling length of an ultra-relativistic electron in bent crystals is proportional to the effective well depth ΔUC [61]. The Pöschl–Teller potential is used to calculate ΔUC/ΔUmax, and the result is approximately (1C)2, so the dechanneling length for the bent channels is [8] Ldc(C)=(1C)2Ldc,0.

    The corresponding number of dechanneling periods is Ndc=(1C)2Ldc,0λu.

    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 K of the CU is used. In fact, K is equal to K1 numerically. According to the definition of the undulator parameters K, γλu=K2πL0.

    Applying Eq. (G1) and Eq. (F1) to Eq. (F7), we obtain the expression of the number of dechanneling periods as Ndcαπm2c4ΔUmaxEe,0Lr2πL0·K·(12πEe,0UmaxλuK)2,where Ee,0 is the electron rest energy.

    The electrons are initially at rest in the EF before entering the undulator. However, a reverse velocity βEF will be induced when the electron interacts with the undulator field. The average velocity along the x3 axis of electrons in the laboratory frame is β0=λuλu+λfd=11+1+K2/22γ2,where λfd=(1+K2/2)λu/2γ2 is the fundamental wavelength. Then, by using the velocity-addition formula, we obtain the average electron velocity along the x3 axis in the EF as βEF=β0β1β0β=2γ21+2γ2+K2/211γ212γ21+2γ2+K2/211γ2.

    Under the condition L0=0.2  nm, we calculate the dependence of the undulator period length λu, the electron velocity βEF in the EF, the electron Lorentz factor γ, and the electron energy Ee on the undulator parameter K and the centrifugal parameter C, shown in Fig. 9.

    Dependence of the undulator period length λu, the electron average velocity βEF inside the undulator in the EF, the electron Lorentz factor γ, and the electron energy Ee on the undulator parameter K and the centrifugal parameter C under the condition L0=0.2 nm.

    Figure 9.Dependence of the undulator period length λu, the electron average velocity βEF inside the undulator in the EF, the electron Lorentz factor γ, and the electron energy Ee on the undulator parameter K and the centrifugal parameter C under the condition L0=0.2  nm.

    APPENDIX H: EXPLANATION OF THE PARAMETERS IN TABLE?1

    The undulator oscillation amplitude L0=0.2  nm is controlled by the average germanium concentration χ¯, which is in the large-amplitude regime, i.e., dip<L0λu, and dip is the interplanar distance determined by the material and dip=0.192  nm in our case of Si(110) channeling [8]. Meanwhile, L0=0.2  nm 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 K1=3.1 and the centrifugal parameter C=0.1, which correspond to the electron energy Ee=197  GeV and the undulator period length λu=156  μm. K1=3.1 corresponds to the number of demodulation periods Ndm12 and the number of dechanneling periods Ndc20. The undulator period number N should satisfy NNdm and NNdc, so we choose the undulator period number N=12. According to the definition of the pseudo-potential magnetic field amplitude B0=2πmK1/eλu, we get B0=212  T. The fundamental frequency ωfd is calculated by Eq. (A1). Since it is the fundamental frequency, we take n=1 and θ=0 (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.).

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    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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