## 6 Spontaneous Four-Wave Mixing (SFWM)  
The SFWM process is mediated by a \(\chi^{(3)}\) nonlinear susceptibility, wherein two pump photons are annihilated by the medium, and then create two different photons, called signal and idler. MNFs can be used as quantum light sources generating entangled photon pairs via SFWM process. The small mode area of the MNF strongly enhances nonlinear coefficient which is roughly 100 times larger than that of the standard single-mode fiber. The SFWM process must satisfy both energy conservation and phase-matching condition which are given by  
\[2\omega_{p} = \omega_{s}+\omega_{i},\] \[2\beta_{p} = \beta_{s}+\beta_{i}+2\gamma P_{p}, \tag{19}\]  
Figure 12: Group velocity of the fundamental modes (**a**) as a function of diameter at \(\lambda=780\) nm and \(\lambda=1.5\) μm, (**b**) as a function of wavelength at different diameters. Figure 13: Group velocity dispersion of the fundamental modes (**a**) as a function of diameter at \(\lambda=780\) nm and \(\lambda=1.5\) μm, (**b**) as a function of wavelength at different diameters. where \(\beta_{j}\) and \(\omega_{j}\) are propagation constants and angular frequencies, respectively. The last term, \(2\gamma P_{p}\), refers to the contribution of the self-phase modulation. The nonlinear coefficient, \(\gamma\), can be given as \(2\pi n_{2}/(\lambda_{p}A_{\mathrm{eff}})\), where \(n_{2}=2\times 10^{-20}\) m\({}^{2}\)/W is the second-order nonlinear index of silica, \(P_{p}\) is the peak power of the pump laser, and \(A_{\mathrm{eff}}\) is the effective mode area. When we consider the reasonable pump power, i.e., 0.5 mW-average power, the self-phase modulation term leads the phase-matching condition to have two different regions in the normal dispersion regime. Figure 14a shows numerically calculated phase-matching curves of the signal and idler photons for various MNF diameters as a function of the pump wavelength. As a diameter increases, overall phase-matching curves are shifted toward the long wavelength. In the region where the phase-matched wavelengths are far-detuned from the pump wavelength, the wavelengths of created photon pair are weakly power-dependent. On the contrary, the region which is close to the pump wavelength has strong dependence of the pump power. As the pump power increases, the wavelength difference between signal and idler photons becomes large as shown in Figure 14b. Here, in order to consider pump power dependences in numerical calculations, we assume that a femto-second pulsed laser is operated at 78 MHz repetition rate. When a spectral filter with the full width at half-maximum of 1 nm is used, for the sech\({}^{2}\)-shape pulse, the pulse duration and peak power are obtained. From the conditions of Figure 14a,b, we can design from visible to NIR photon-pair sources working in the silicon-based single photon detector (SPD) [43]. In Figure 14c, one of the entangled photon-pair is generated at a telecom-band wavelength which is compatible with the existing fiber-based system, while the other photon has a NIR wavelength which can be detected in high efficiency by using the silicon-based SPD. Moreover, due to the large detuning between the pump and signal (idler) photons, Raman noise can be significantly suppressed at the room temperature [42]. Therefore, by the proper diameter control and the pump wavelength selection, photon-pair can be created at one's desired wavelengths. We now present a theoretical study of the two-photon state generated by SFWM with degenerated pump photons. The interaction Hamiltonian of SFWM process can be written as  
\[H_{int}(t)=e_{0}\chi^{(3)}\int_{-L/2}^{L/2}\hat{E}_{p_{1}}^{(+)}\hat{E}_{p_{2} }^{(+)}\hat{E}_{s}^{(-)}E_{i}^{(-)}dz+\mathrm{H.C.}, \tag{20}\]  
Figure 14: (**a**) Phase-matching curves of the signal and idler as a function of the pump wavelength at 610 nm (red), 615 nm (blue), and 630 nm (green) in diameter. Average pump power is 0.5 mW. (**b**) Power dependence of the phase-matching curves at 1.0 mW (brown), 5.0 mW (black), and 10 mW (gray) in average pump power.