Radiation-damage-free ghost diffraction with atomic resolution

The X-ray free electron lasers (XFEL) can enable diffractive structural determination of protein crystals or single molecules that are too radiation-sensitive for conventional X-ray analysis. However the electronic form factor could have been modified during the ultrashort X-ray pulse due to photoionization and electron cascade caused by the intense X-ray pulse. For general X-ray imaging techniques, to minimize radiation damage effect is of major concern to ensure faithful reconstruction of the structure. Here we show that a radiation-damage-free diffraction can be achieved with an atomic spatial resolution, by using X-ray parametric down-conversion (XPDC), and two-color two-photon ghost diffraction. We illustrate that the formation of the diffraction patterns satisfies a condition analogous to the Bragg equation, with a resolution that could be as fine as the lattice length scale of several Angstrom. Because the samples are illuminated by the optical photons of low energy, they can be free of radiation damage.


Introduction
The advent of femtosecond X-ray free electron lasers (XFEL) has enabled diffractive structural determination of protein crystals or single molecules that are too small and radiation-sensitive for conventional X-ray analysis [1][2][3] . Using X-ray pulses of ∼ 10fs, sufficient diffraction signals could be collected before significant changes occur in the sample 1 . Nevertheless, the electronic form factor could be modified due to photoionization and electron cascades caused by the intense X-ray pulse 4,5 . For general X-ray imaging techniques, minimizing the effects of radiation damage is of major concern to guarantee a reliable reconstruction of the structure. Here we show that a radiation damage free diffraction is achievable with an atomic spatial resolution by using X-ray parametric down-conversion (XPDC) [6][7][8] and ghost diffraction with entangled photons of X-ray and optical frequencies. An intense hard X-ray pulse generated by XFEL is used to pump a nonlinear medium, and is down-converted to two-photon pairs that consist of an X-ray and an optical photon with wavelengths λ X and λ o , respectively. The optical photons are sent to illuminate the sample crystals or molecules, and the reflected photons are collected with a bucket detector D A . The X-ray photons entangled with the optical photons travel a certain distance, and are captured by a pixel photon counting detector D B . Based on the basic principle of ghost imaging [9][10][11][12][13][14] , the output pulses of the bucket detector D A and the pixel detector D B are sent to a coinicidence circuit with certain time gate for counting the joint-detection of the two-photon pairs.
We show here that the X-ray photons can form diffraction patterns on pixel detectors. We illustrate that the formation of the diffraction patterns satisfies a condition analogous to the Bragg equation with a resolution that could be as fine as the lattice length scale of severalångstrom. Because the samples are illuminated with the optical photons of low energy, they can be free of radiation damage. The ultrabright intensity of XFEL could be crucial for realization of the proposed scheme to ensure sufficient two-photon flux and signal strength. Since the diffraction pattern formation is based on photon counting, the requirement for signal intensity is benign, we also show that the analogous Bragg condition can be satisfied with feasible experimental parameters.

Results
An optical beam and an X-ray beam from XPDC are directed to illuminate a crystal in the optical arm and form diffraction patterns in the X-ray arm (Fig. 1). Without loss of generality, we assume that the signal photon has an optical wavelength of λ s = λ o , and the idler photon has an X-ray wavelength of λ i = λ X . From the detailed derivation presented in Section S1 of Supplementary Information (Eq. S1-S24) 15 , as the optical photons scatter off lattice planes of inter-plane distance d hkl = d, the condition for the X-ray photons to form peaks in the diffraction pattern is for an integer n. θ is the reflection angle of the optical photon from the lattice planes of a Miller index [hkl] (Fig. 2), and ρ B is a vector on the plane of the pixel detector. The magnification factorm is found to be (Eq. S23) 15 where α is defined as d s D i = α λ i λ s .m guarantees that the Bragg condition (Eq. 1) can be satisfied in the case d λ s . The two-photon coincidence counting rate of the bucket detector A and the pixel detector B can be written as 10,12 , where S(t B ,t A ) is the coincidence time function that vanishes unless 0 ≤ t B − t A ≤ T , and describes the finite time gate of a joint-detection of two-photon pair 10 . ρ A is on the plane of the bucket detector A with an area σ A , and 2/8 σ B is the area of the pixel detector B. ρ is the density matrix of the two-photon state on the output plane of the nonlinear crystal. E (+) j , j = A, B is the positive frequency part of the photon field, and E denotes the trace, i.e. coherent summation over ρ A . R c T gives the pixelwise number of counted photons in one joint measurement. The photon fields at the plane of the bucket detector A and the pixel detector B,Ê where z s = d s + L s and z i = D i are the full optical path lengths for the signal and the idler photon, κ is the transverse momentum of the photon field with k = √ k 2 − κ 2ê z + κ .â † p ( k) andâ p ( k) are creation and annihilation operators of the signal and the idler photon field in a specific mode at the output plane of the nonlinear crystal, with the commutator relation, â p ( k),â † q ( k ) = δ p,q δ ω,ω δ κ , κ . g( κ , ω, ρ , z) is the Green's function for a specific mode of the photon field. Assume two atoms in two lattice planes of Miller index [hkl] with distance d = d hkl are in a plane a, ρ a is a vector in this plane, and the photon-atom scattering amplitude is t( ρ a ), the photon fieldsÊ B at the planes of detectors can be expressed as, Using first-order perturbation theory, the two-photon amplitude from an XPDC process is shown to be 15 where |Ψ is the two-photon state vector, γ = is the second-order susceptibility of the nonlinear crystal, U j is the group velocity of the signal and the idler photon inside the nonlinear crystal of length L, T j are their transmission coefficients, and E p is the strength of the pump field. Ω s and Ω i are the frequencies of the signal and the idler photon that satisfy the phase matching condition, ω p n p (ω p ) − Ω s n s (Ω s ) − Ω i n i (Ω i ) = 0. Provided that the Bragg equation (Eq. 1) and the phase matching condition are satisfied, diffraction patterns can be formed on the X-ray photon arm from the coincidence photon count in a joint measurement of the bucket detector D A and the pixel detector D B such hat the reflection angle θ is measured with the detector D A , and the corresponding intensity is measured with the detector D B from coincidence counts on pixels of a given radius | ρ B |. For the schematic configuration in Fig. 1, the Bragg peaks are manifested as modulation of the coincident counting rate at ρ B in the plane of the pixel detector with a background 15 where γ and D si are parameters of the XPDC process 15 . ψ 0 is the scattering amplitude of an atom in the lattice with the optical photons. And R s = λ i λ s D i + d s is the effective optical path length from the sample to the pixel detector B. We assume the phase matching condition to be Ω s n s (Ω s ) + Ω i n i (Ω i ) = ck p , ignoring G for simplicity, which can be restored for a given experimental configuration. The frequency of the signal field is ω s = Ω s + ν s , with ν s characterizing the deviation from the central frequency Ω s of the phase matching condition, and ν s = −ν i .
For realistic experimental consideration, we assume an X-ray beam of 3.1 keV (λ i = 4Å) and an optical beam of 3.1 eV (λ s = 4000Å). Nonlinear X-ray process with energy difference of the two-photon on such scale was  experimentally demonstrated 6,16 . Assume the sample is placed at a distance of 1 cm from the XPDC source, the reflection angle of optical photon is measured on detector D A , and the x-ray photon in the two-photon pair is measured by the pixels with radius of 1 m on the pixel detector D B placed at a distance of 10 m. In the XFEL nanocrystal diffraction experiment, virtual powder diffraction patterns are formed for various reflection angles. The singles photons from the XPDC process have a certain angular spread 17 . It is practically advantageous to determine the actual reflection angle θ by a pixel detector D A , although in an idealized setup the reflection angle can be uniquely determined by the geometrical configuration of the photon source and a single crystal, for which a bucket detector D A is sufficient. The Bragg equation gives the ghost diffraction pattern in Fig. 3. The resolution requirement can be satisfied with the state-of-the-art pixel pnCCD detector that can achieve 75µm pixel pitch 1 . The requirement to the spectral and angular resolution of the diffractometer determined by ∆d d + cot θ ∆θ = ∆λ s λ s , which is on the same scale of the conventional Laue diffraction. For nanocrystals with the size L c , the analogous Scherrer equation is found to be (see Section S3 of Supplementary Information , Eq. S40-S43) 15 which determines the width B of Bragg peaks. As shown Fig. 3, the width of Bragg peaks is on the same scale of Laue diffraction and is resolvable by the state-of-the-art diffractometer. The optical photon energy of 3.1 eV may fall below band gap of the crystal, thus significant absorption can be avoided. The proposed two-color entangled ghost diffraction approach could eventually achieve atomic resolution without radiation damage.

Discussion
The physical mechanism of the two-color two-photon ghost diffraction can be understood from the particle nature of the optical and X-ray photons and their position-momentum entanglement. The optical photon of frequency ω scatters with the atoms in a nanocrystal or a molecule and experiences momentum transfer Q , with frequency independent Thomson scattering cross section dσ ( Q ) dΩ th = r 2 (1 + cos 2 θ ) r 2 e | f (ω) + i f (ω)| 2 is equivalent to the dispersive corrections of form factor in the X-ray regime (see explanation in Section S5 of Supplementary Information ) 15 , which can be used for the phase problem in crystallography 18 . For photons as particles, the probability of reflection from an atom is in fact on the similar orders of magnitude for the optical and X-ray photons. Without considering the molecular form factor of f ( Q ) , we could model the form factor of a crystal as periodic distributions of point scatterers as , which is proportional to the incident optical photon energy, is too small to form interference patterns at non-imaginary reflection angle because Q G hkl . Thus, Laue diffraction requires X-ray photons of wavelength λ < 2d.
The momentum transfer can be effectively magnified in the two-photon ghost diffraction from Q tom Q . The mechanism of momentum transfer magnification can be understood by a simple quantum model of the "unfolded" two-photon ghost diffraction 20 (see Fig. 4). In this simplified model, we do not consider the transverse momenta of the down converted photons as in the exact treatment presented in the previous section. The photon fields on detectors D A and D B can be written in this case aŝ whereâ l p , l = 1, 2, p = s, i are the annihilation operators of signal and idler photons at position 1, 2 in Fig. 4. The two-photon state |Ψ can be expressed as where ε is the two-photon amplitude, and we can assume φ 1 = φ 2 since the pump beam from an XFEL or a synchrotron must be transversely coherent at position 1 and 2. The two-photon interference is formed due to the uncertainty in the birth place of the two-photon pair (either at position 1 or 2 in Fig. 4), and the property of the two-photon product state that allows us to have the uncertainty relation ∆( k s + k i ) = 0 and ∆( ρ s − ρ i ) = 0 at the same time 19 , which guarantees the operation to concatenate the optical paths of the X-ray and optical photons exactly at the point of their birth position. It is straightforward to find the second-order coherence function G AB ∝ ε 2 cos 2 π λ s (r A1 − r A2 ) + π λ i (r B1 − r B2 ) , and Bragg peak is formed under condition In Eq. 11, the Bragg condition can be satisfied despite d λ s , because the optical path length difference can be compensated by that of the arm of idler photons magnified by a factor λ s λ i of ∼ 10 3 . As a result, we have a qualitative explanation to the origin of the magnification factorm that appears in Eq. 1, and equivalently to the effective form factor f eff ( Q ) = −r e δ (m Q − 2π G hkl ) , which is able to produce diffraction patterns for reciprocal vectors on the 1/Å scale. It is important to notice that the photon behaves as point particle in the elastic scattering, and its wavelength is irrelevant to the effective size of the photon, thus the optical photon does not smear out the crystal structure with period that is much shorter than its wavelength. In fact the wavelength and the size of a quantum object reflect its wave-like and particle-like natures respectively, and should be distinguished from each other. It is known that the intensity of Bragg peaks should be reduced due to the dynamic structural factor S( Q , ω s ) 21 by a Figure 5. Sketch of the seeded Kapitza-Dirac-like XPDC process using single electron as the nonlinear medium. An X-ray photon of ω X3 is down converted to two photons of frequencies ω o1 and ω X2 , as the electron is deflected with Bragg angle θ B in coincidence. For the use of ghost diffraction, the down converted photon pair can be then spatially separated by a metal foil that reflects the optical photon. The collinear geometry of k X3 and (k o1 ,k X2 ) can be loosened for more convenient experimental setup, provided the phase matching condition is satisfied.
Because the frequency of the optical photon is much lower than the X-ray photon, ω o ω X , the experiment would suffer from strong absorption from photon-bound electron scattering. Thus the optical photons in the vacuum ultraviolet (VUV) regime is not favorable for the proposed ghost diffraction. The entanlged ghost diffraction scheme using optical photons to interact with the crystal could also potentially suffer from skin effect when the samples are good conductors like copper, which has ∼3 nm skin depth for optical photons in the visible and ultraviolet regime, and optical photons can only penetrate several uppermost lattice planes. In this sense, the proposed ghost diffraction scheme is suitable to diffractive structural determination of insulators, some semiconductors and proteins, which should have poor conductivity. For good conductors, our method is limited to the cases of thin and homogeneous samples, or the studies of surfaces.
In order to achieve sufficient photon counting rate, it is crucially important to have a two-photon source with high flux using X-ray quantum optical techniques 22 . Because the XPDC and the sum-frequency generation (SFG) processes 6 are subject to the same nonlinear susceptibility, we can expect the feasibility of two-photon production with similar λ s λ i ratio. Consider the XPDC process ω X3 → ω X2 + ω o1 , which is physically equivalent to the Thomson scattering of X-rays by an atom illuminated with an optical field, Doppler-shifted sideband is induced by this process 23 . As elaborated in Section S4.1 of Supplementary Information , the XPDC cross section from nonlinear crystal scales as dσ dΩ (2) ∝ 1/ω o1 and especially favors conversion to low energy optical photons. Using the semiclassical treatment 15, 23, 24 , we estimated the cross section of the instance in this work to be ∼ 1.9 × 10 3 fm 2 (see Section S4.1), which can be comparable to that of the Thomson scattering. We also show in Section S4.1 of Supplementary Information that the quasi-degenerate XPDC to pairs of two X-ray photons has cross section that is four orders of magnitude lower than the XPDC to pairs of X-ray and optical photons. Moreover the strong absorption of X-ray photons by the diamond crystal 8 and radiation damage caused deterioration of phase matching condition can significantly suppress the XPDC efficiency.
To overcome this problem, we could use radiation-hard multilayer photonic crystal to enhance quantum efficiency of XPDC, or to use beam of single electrons as nonlinear medium for XPDC through the three-color Kapitza-Diraclike mechanism (Fig. 5) 25 . In Section S4.2 of Supplementary Information 15 , we used perturbation theory 25-27 to find the probability of the desired XPDC process under the phase matching condition k X3 = k o1 + k X2 as In the three-color Kapitza-Dirac-like process, the electrons are scattered 2) ensures the energy conservation of the electrons, and v z is the initial velocity of the electrons along the z axis. With electric field intensity I ∼ 10 18 W/cm 2 well below the QED critical intensity and moderate electron velocity in non-relativistic regime, the XPDC probability could reach 10 −5 .

6/8
In conclusion, we have theoretically described a mechanism to realize X-ray diffraction with an atomic resolution by two-color entanlged ghost diffraction. Because the sample is irradiated by photons of optical wavelength, the proposed scheme can be free of radiation damage by X-ray photons. In principle the proposed scheme could also be applied for single molecules to determine the molecular structure using phase retrieval techniques of coherent diffractive imaging. Moreover, achieving resolution on a much smaller length scale than the wavelength of the illuminating photons using quantum product state of fundamental particle entanglement opens the possibility for future development of quantum optics based imaging techniques, such as using entangled photon-electron, electron-electron pair or electron-anti-neutrino pair from β -decay.
difference of optical path lengths θ reflection angle of the optical photon from the crystal ψ 0 photon-atom scattering amplitudẽ m magnification factor L (θ ) line shape of Bragg peak with reflection angle θ L c length of a nanocrystal S1/S15  Figure S1. Proposed layout for two-color two-photon ghost diffraction of X-ray and optical photon pair. ρ A , ρ B , ρ s , ρ a are vectors on the planes of the bucket detector A, the pixel detector B, the X-ray PDC source and atoms in the sample. The path of the optical photon in the arm A is labeled by blue lines, with the distance d s from the XPDC plane to the crystal plane and the distance from L s to the bucket detector A. The path of the X-ray photon in the arm B is labeled by green line, with the distance D i from the XPDC plane to the pixel detector B. k p , k s and k i are the momentum vectors of the pump, the signal (optical) and the idler (X-ray) photons.

S1 Bragg equation for two-color two-photon diffraction
We assume paraxial approximation for the wave vectors of modes | k of the photon field, k = √ k 2 − κ 2ê z + κ , with k = ω c κ and κ = (k x , k y , 0). The two-photon coincidence counting rate of the bucket detector A and the pixel detector can be written as 1,2,3 , where S(t B ,t A ) is the coincidence time function that vanishes unless 0 ≤ t B − t A ≤ T and can be approximated as a rectangular function, ρ A and ρ B are vectors on the plane of the bucket detector A with an area σ A and the plane of the pixel detector B with an area σ B respectively.ρ is the density matrix of the two-photon state on the output plane of the nonlinear crystal.Ê where z s = d s + L s and z i = D i are the full optical path lengths for the signal and the idler photon.â † p ( k) andâ p ( k) are operators of signal and idler photon fields in a specific mode at the output plane of the nonlinear crystal, with the commutator relation, g( κ , ω, ρ , z) is the Green's function for a specific mode of photon field. Assume two atoms in two lattice planes of Miller index [hkl] with the distance d = d hkl are in a plane a, ρ a is a vector in this plane (Fig. S1), and the photon-atom scattering amplitude is t( ρ a ), we shows thatÊ × −ik s 2πL s e ik s L s e i ks 2Ls | ρ A − ρ a | 2 e −iω s t Aâ s ( k s ) Denote G(| α |, β ) = e i β 2 | α | 2 , its two-dimensional Fourier transformation is The photon fieldsÊ B can be then written aŝ With Eq. S6, we can obtain explicitly the second-order coherence function in Eq.S1, where |Ψ is the two-photon state vector, and U AB is the joint photon detection amplitude. Without loss of generality, we ignore here the reciprocal lattice vector G of the nonlinear crystal, which can be added for a given experimental configuration. Suppose the phase matching condition is satisfied for ω p n p (ω p ) − Ω s n s (Ω s ) − Ω i n i (Ω i ) = 0, with K j = Ω j c , and deviation from the central frequency ω j = Ω j + ν j , j = s, i. Frequency and spatial filtering guarantees ν j Ω j and κ j K j , thus we have Taken Eqs. S6 and S7, we have Using a first-order perturbation theory, the two-photon amplitude from a PDC process is shown to be 2,4 , where γ = and U i are the group velocity of the signal and the idler photons inside the nonlinear crystal of length L. χ (2) is the second-order susceptibility of the nonlinear crystal, U j is the group velocity of the signal and the idler photons inside the nonlinear crystal of length L, T j are their transmission coefficients, and E p is the strength of the pump field. Using the relation where V Q is the quantization volume. We can write U in Eq. S9 as Ls ρ a · ρ A e iK s z s , (S13) , and r i are the full lengths of the optical paths. Define the effective sample-to-pixel detector path length R s = d s + λ i λ s D i , the integral I κ s over transverse momentum κ s in Eq. S13 can be rewritten as The amplitude for joint photon detection at the bucket detector A and the pixel detector B is then where V ( ρ A , ρ B ) is the interference kernel of the joint photon detection amplitude U AB that characterizes the formation of the two-photon diffraction pattern. We now consider the scattering of the optical photon with the atoms in a lattice by integrating over ρ a , and the collection of optical photons at the bucket detector A by integrating over ρ A in the detector plane, because the bucket detector collects photons without distinguishing their actual position. We expect that diffraction patterns can be formed on the image plane of the pixel detector B, with intensity I( ρ B ) and joint photon counting rate R c ( ρ B ) at ρ B (S16) To obtain the analogous Bragg equation for two-color two-photon ghost diffraction, we consider two optical paths A 1 and A 2 of the optical photons that scatter with two atoms in two lattice planes with distance d ≡ d hkl (Fig. S1). For optical path A 1 , we have and for optical path where δ = d sin θ . Thus the effective two-photon joint detection amplitude consists of contributions from the two optical paths, (S19) Take far-field approximation δ R s that is similar to the Laue diffraction, we have Analogous to the procedure in Sec. S2, we obtain the Bragg equation for two-color two-photon ghost diffraction from Eq. S21 by requiring the δ -dependent phase to vanish, i.e.
with an integer n. Define the magnification factorm as the Bragg equation can be then written as 2md sin θ = nλ s .
Provided the Bragg condition is satisfied, and define ρ B = (x, 0) and d = (d, 0) for simplicity, we find which forms a broad and flat background. Using Eqs. S1, S7, S15 and S25, we reach the final equation for the counting rate of joint photon detection Eq. S26 can be simplified using the relation S5/S15 and the counting rate of joint-detection can be written as

S2 Kinematic description of Laue diffraction
In Section S1, the Bragg condition for the two-color two-photon ghost diffraction is obtained from a kinematic scenario. Here we show the conventional Bragg equation can be obtained from similar procedure for the Laue diffraction with monochromatic X-ray beam. According to Fourier optics, a light wave that passes the plane z = 0, and arrives at the plane z = ∆ can be described by the Huygens principle as E(x, y, ∆) = dk x dk y e i(k x x+k y y) e ik z ∆ F [E(x, y, 0)] = dk x dk y e i(k x x+k y y) e i∆ √ where P(x, y, ∆) is The physical scenario of Eq. S29 reflects a typical statement of Huygens principle, that (a) the Fourier transformation on the z = 0 plane makes a map to the momentum space E(x, y, 0) →Ȇ(k x , k y , 0), (b) each Fourier modeȆ(k x , k y , 0) corresponds to a sub-source that travels as a plane wave e i(k x x+k y y) e ik z ∆ to the z = ∆ plane, and (c) an inverse Fourier transformation on the z = ∆ plane gives the image E(x, y, ∆). For Laue diffraction, we follow a standard treatment by considering two optical paths 1 and 2 for photons that scatter off atoms in two lattice planes ( Fig. S2(a)). Assume the photon scatters off the two atoms O and O with amplitude t (1) = ψ 0 δ (x − d, y, 0) and t (2) = ψ 0 δ (x, y, 0), and denote δ = d sin θ , we have Thus the intensity of diffraction pattern I(x) at the detector plane is given by ∆+2δ e −i 2π λ 2δ + c.c. For far-field diffraction ∆ 2δ , we have using the relation 1 + cos α = 2 cos 2 α 2 . The Bragg condition is obtained by requiring the δ -dependent phase in Eq. S33 to vanish, 2δ = 2d sin θ = nλ . (S34) We obtain the diffraction pattern on the detector at the distance ∆ from the sample, with a period a = 2λ ∆ d . Meanwhile, we show that the kinematic description is consistent with the description of Laue diffraction in momentum space. The form factor f ( Q ) is the Fourier transformation of the charge distribution. For simplicity, we model the atoms as point charges, thus and the intensity of diffraction pattern is From Fig. S2(b), we can find that sin θ x 2∆ and Taken Eqs. S37 and S38, we obtain the intensity of diffraction pattern with a period a = 2λ ∆ d , which is consistent with Eq. S35 from the kinematic description of Laue diffraction.

S3 Broadening of Bragg peaks
Analogous to the Scherrer equation, we calculate the width of the Bragg peaks of the two-color two-photon ghost diffraction, which determines the resolution of the proposed scheme. It is especially important for the application of XFEL to the diffraction of nanocrystals with a finite size. Supposing a nanocrystal of length L c that contains N lattice planes with inter-plane distance d, such that Nd = L c , the accumulated δ -dependent phase for the diffraction from N atoms can be calculated using Eq. S21 as Thus the line shape L (θ ) of a Bragg peak at a reflection angle θ is The width of the Bragg peak can be simply found through zeros of the line shape function as Defining the width of the Bragg peak as B = 2∆θ , we find The factor λ s m in the width of Bragg peaks guarantees that the two-color two-photon ghost diffraction will have a similar resolution to the Laue diffraction, i.e. on the atomic scale.

S4 X-ray parametric down conversion through nonlinear crystal and single electron
In this section, we present details of the two proposed schemes for X-ray parametric down conversion (XPDC) using conventional nonlinear crystals or single electron as the nonlinear medium. We also present an estimation of the efficiency from experimental consideration for the XPDC of an X-ray photon of frequency ω X3 to a pair of X-ray and optical photons of frequencies ω X2 and ω o1 . Both schemes rely dominantly on the figure eight motion of electrons turned by Lorentz force. For the nonlinear crystal based XPDC, the phase matching condition relies critically on the well-defined lattice vectors of reciprocal space, thus the structural damage to the crystal inevitably causes deterioration of the XPDC process. Moreover the conventionally used crystals, like diamond, can strongly absorb the photons and substantially suppress the XPDC efficiency 3 . In contrast, the XPDC using single electron, which is Kapitza-Dirac-like scattering, could have the advantage of being free of damage of the nonlinear crystal by the intense X-ray light and strong absorption of the photons inside of the crystal medium.

S4.1 XPDC using nonlinear crystal
To obtain an order-of-magnitude estimation of the XPDC cross section by nonlinear crystalline medium, we use a semiclassical formalism to calculate nonlinear response functions for X-rays 5 . For electrons in the atom with density ρ and velocity v, we apply the equation of motion and continuity condition ∂ v ∂t Under perturbative expansion ρ = ρ (0) + ρ (1) + ρ (2) + · · · , v = v (1) + v (2) + · · · , and J = J (1) + J (2) + · · · , we obtain the second order nonlinear current for a general XPDC process ω 3 → ω 1 + ω 2 where the second term vanishes in the case We now consider the XPDC of a hard X-ray photon to a pair of X-ray and optical photons ω X3 → ω o1 + ω X2 . Physically this process is equivalent to the Thomson scattering of X-rays by an atom illuminated with an optical field, which induces Doppler-shifted sideband 6 . Of the several terms that occur in the second order current (Eq. S45) for the nonlinear response of an atom to applied electromagnetic fields of frequencies ω o1 , ω X2 , ω X3 , only one is of importance to the present instance 6,7 , where ω X3 = ω o1 +ω X2 , and ρ is the electron density of the crystal. Eq. S46 describes the Doppler-shifted reflection of E 2 from electron that is driven by the optical field E 1 and moving with velocity v 1 . For simplicity, we denote thereafter ω o1 = ω 1 , ω X2 = ω 2 , ω X3 = ω 3 , the same notation applies for the momentum k and electromagnetic fields E and B. We expand the electron density in terms of reciprocal lattice vector G m that ρ( r ) = ∑ m ρ m e i G m · r . The dominant nonlinear current is where e i is the polarization vector of the electric field E i . In the general case, the relation between the nonlinear response functions R (n) and the corresponding nonlinear susceptibilities χ (n) is 5 χ (n) (ω; ω 1 , ω 2 , · · · , ω n ) = i 1−n c n ω 1 ω 2 ω n R (n) (ω; ω 1 , ω 2 , · · · , ω n ) .
For the quadratic process, the nonlinear current is Since A(ω) ∼ cE ω , by inspection the nonlinear response function is The nonlinear susceptibility is thus in atomic units, where α is the fine structure constant and r e is the classical electron radius. The estimation of χ (2) in Eq. S51 is consistent with the earlier work (see comment 28 in Ref. 8). Using Fermi's golden rule, we obtain the differential cross section for XPDC as 5 dσ dΩ (2) = ω 3 ω 3 2 ω 3 1 288π 3 c 7 χ (2) 2 .
Eqs. S51 and S53 give the scaling of cross section as ∝ 1/ω 1 , which especially favors the low energy of the optical photon. For the instance given in the main text of XPDC ω 3 (= 3.1 keV) → ω 2 (= 3096.9 eV) + ω 1 (= 3.1 eV) with S9/S15 diamond crystal, we obtain χ (2) ∼ 9.1 × 10 −12 cm 2 StC −1 , where StC is the Gaussian unit of StatCoulomb. The XPDC cross section is dσ dΩ (2) ∼ 1.9 × 10 3 fm 2 = 19 b , which could be comparable to the cross section of Thomson scattering. Nevertheless, the low XPDC efficiency in actual experiments could be attributed to the radiation damage caused deterioration of phase matching condition and strong absorption of photons inside the crystal.
We also show here that the (quasi-)degenerate XPDC ω X → ω X /2 + ω X /2 (Ref. 3) has cross section that is almost four orders of magnitude lower than the XPDC to X-ray and optical photons. In this case ω 3 = ω and ω 1 = ω 2 = ω/2 are both in the X-ray regime, the dominant nonlinear current is Similarly we obtain the nonlinear response function for (quasi)-degenerate XPDC and the nonlinear susceptibility is in this case Taken the experimental parameters in the earlier work 3 ,ω 3 (= 18 keV) → ω 1 (= 9 keV) + ω 2 (= 9 keV) with diamond crystal, we obtain χ (2) ∼ 7.7 × 10 −20 cm 2 StC −1 , and the XPDC cross section is dσ dΩ which is lower than the cross section in Eq. S53 by four orders of magnitude.

S4.2 XPDC using single electron
As depicted in Fig. S3, the incident free electron plays the role of the nonlinear medium. A linearly polarized X-ray field (ω X3 ,k X3 ) propagates from the right along the x axis, while two seeding fields (ω o1 ,k o1 ) and (ω X2 ,k X2 ) propagate from the left along the x axis, and ω o1 + ω X2 = ω X3 , k o1 + k X2 = k X3 . The electric fields E o1 , E X2 , E X3 are polarized along the z axis. The figure eight motion of the electron in the fields of E o1 and E X2 created polarization along the x axis P (2) x (ω X3 ) = χ (2) xzz (ω X3 ; ω o1 , ω X2 )E o1,z E X2,z . Meanwhile the X-ray field of ω X3 induces polarization P (1) x (ω X3 ) = χ (1) xz (ω X3 ; ω X3 )E X3 . The two wave mixing of P (2) (ω X3 ) and P (2) (ω X3 ) induces the effective grating for diffractive scattering of the electron with Bragg angle sin θ B = k X3 p i (Fig. S3), where p i is the initial momentum of the electron 9 . From the perspective of nonlinear optics, this three color Kapitza-Dirac-like scattering is an XPDC process, in which the electron absorbs an X-ray photon of ω X3 , converts it to a pair of photons with ω o1 , ω X2 and changes the momentum by 2k X3 for momentum conservation. Since we use the fields of ω o1 and ω X2 as seeding fields, we could eventually identify the entangled photon pair from XPDC by coincidence with the deflected electron.
(S59) S10/S15 Figure S3. Sketch of the seeded Kapitza-Dirac-like XPDC process using single electron as the nonlinear medium. An X-ray photon of ω X3 is down converted to two photons of frequencies ω o1 and ω X2 , as the electron is deflected with Bragg angle θ B in coincidence. For the use of ghost diffraction, the down converted photon pair can then be spatially separated by a metal foil that reflects the optical photon. The collinear geometry of k X3 and (k o1 ,k X2 ) can be loosened for more convenient experimental setup, provided the phase matching condition is satisfied.