Fig. 1. 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.

Fig. 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=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.

Fig. 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 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.

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.

Fig. 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.

Fig. 6. Diagrams for the nonlinear optical processes. (a) Double-photon emission and its conjugate; (b) difference frequency generation.

Fig. 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.

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.

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.