Highly efficient polarization-entangled photon-pair generation in lithium niobate waveguides based on bound states in the continuum

Integrated optics provides a platform for the experimental implementation of highly complex and compact circuits for practical applications as well as for advances in the fundamental science of quantum optics. The lithium niobate (LN) waveguide is an important candidate for the construction of integrated optical circuits. Based on the bound state in the continuum (BIC) in a LN waveguide, we propose an efficient way to produce polarization-entangled photon pairs. The implementation of this method is simple and does not require the polarization process needed for periodically poled LN. The generation rate of the entangled photon pairs increases linearly with the length of the waveguide. For visible light, the generation efficiency can be improved by more than five orders of magnitude with waveguides having the length of only a few millimeters, compared with the corresponding case without BICs. The phenomena can appear in a very wide spectrum range from the visible to THz regions. This study is of great significance for the development of active integrated quantum chips in various wavelength ranges.

Recently, the approach of integrated optics is regarded to be essential for practical applications as well as for advances in the fundamental science of quantum optics [21][22][23]. The integration of various optical components on the same chip allows the realization of on-chip optical gate operations [24], multi-photon quantum interference [25], quantum Boson sampling [26], and the simulation of quantum walks [27]. An essential aspect for implementing a fully integrated platform for quantum information processing requires the realization of quantum light sources, in particular, entangled photon sources, on a chip. Although some schemes have been proposed [24], there is still a lack of optimal designs for on-chip generation of entangled photons.
In this work, we propose a new way to efficiently produce polarization-entangled photon pairs by designing bound states in continuum (BICs). Analogous to localized electrons with energies larger than their potential barriers, light BICs, which are also known as embedded trapped modes and correspond to discrete eigenvalues which coexist with the extended modes of a continuous spectrum, have been realized in recent years [47][48][49][50][51][52][53][54][55][56][57][58]. They have been shown to exist in dielectric gratings, waveguide structures, object surfaces, photonic crystal slabs, and some open subwavelength nanostructures [59][60][61][62][63][64][65][66][67][68]. Recently, photonic integrated circuits with BICs have been designed using the LN platform [69,70]. Here, we show theoretically the improvement of generation rate for the polarization-entangled photon pair by several orders of magnitude in LN waveguides with BICs, compared with the corresponding case without BICs.
This design does not require the dedicate polarization process necessary for periodically poled LN waveguides. The phenomenon can appear in a wide spectral range from the visible to the THz regions.

System and theory
We consider a waveguide structure composed of a z-cut LN layer, which optical axis along the z axis [69], sandwiched between a low-refractive index organic polymer layer and a silica (SiO2) substrate as shown in Fig. 1(a) and Fig. 1(b). In Refs. [69,70], it was proposed that such a structure supports a transverse-magnetic (TM) bound mode which lies within the continuous spectrum of the transverse-electric (TE) modes when the parameters are properly selected. The effective refractive indices of different waveguide modes in the structure can be calculated by the planar waveguide analysis method. The propagation of electromagnetic field in the waveguide structure satisfies the wave equation. Using the boundary condition of field continuity in the structure, we can calculate wave propagation constants. Then, the effective refractive indices of different waveguide modes can be obtained. Because the LN is an anisotropic material, we usually need to calculate the effective indices along different directions simultaneously. However, for a z-cut LN layer, the optical axis is along the z axis. It can be demonstrated that non-zero terms are only diagonal elements [77]. In such a case, the 4 effective indices along different directions can be calculated separately. The details of the calculation method are given in Refs. [69,70].
In Fig. 1(c) It is easy to find that the potential distribution for photons is inversely related to the refractive index distribution by comparing the Schrödinger equation with the wave equation [69,70]. Thus, we can calculate the photonic potential distributions for the above waveguide structure after obtaining the effective refractive indices. In fact, the low-refractive index material is patterned in the lateral direction (y direction) and forms a high-effective index channel for the transverse confinement of the TE and TM modes. This makes the TM mode bound within the LN waveguide and the TE mode extended.
The spatial confinement of the TE and TM modes can also be analyzed by the transmission loss spectrum and the electric field distribution in the structure. According to the analysis in Ref. [69], the transmission loss can be described by       exp , 0   We now discuss the generation of coherent photon pairs in the above structure. The system is divided into the three layers of air-LN-air along the direction of propagation. The nonlinear interaction in the structure is described by the Hamiltonian

 
Ht [71]: V is the volume of the LN waveguide, and    r is the LN second-order nonlinear coefficient.
The last integral depends on the waveguide volume . V We assume that in the waveguide the solution can be written as: where c is the speed of light, Hamiltonian can be expressed as: Here we assume that only the intense pump wave is fed into the sample and the initial state is the vacuum state .
We are interested only in the second term on the right side of Eq. (8) The explicit expressions of We are now ready to use the power of the signal field s P to measure the conversion efficiency of correlated photon pairs. In the one-dimensional state space, the relationship between the change of signal photon number and the change of energy is [72]: The same holds for the idler light. Thus, the density of state (DOS) is given by [72]: The above formula can be extended to three-dimensional space by using Fermi Golden Rule [72]: The differential of the down-converted signal optical power can be expressed as: Then, the power at the different signal wavelength is obtained: Using Eqs. (17) and (22), the mean number of photon pairs and the conversion efficiency can be obtained easily by numerical calculations.

Numerical results and discussions
In the following, we present numerical results for the generation of polarization-entangled photon pairs based on the LN waveguide. The technique for numerical calculations is not difficult, which can be realized by using MATLAB or any computation language such as Fortran and so on. We first consider the structure parameters as shown in Fig. 1 , and 8.6 mm . In addition, we can see that the maximum conversion efficiencies of each peaks increase with the increase of L because of the "quasi-phase matching" between the bound TM mode and the TE modes. The present quasi-phase matching is different from the usual cases. To distinguish this, here we have added quotation marks. There are usually two ways to achieve the quasi-phase matching [76]. One is to control the directions of wave propagation in anisotropic nonlinear crystals.
Another is to use periodically poled materials. The present "quasi-phase matching" is obtained by adjusting the waveguide modes and the effective refractive indices at different wavelengths.   14 An advantage of such an approach is that polarized entangled photon pairs with different wavelengths can be generated in a wide range of frequencies by choosing the parameters of the structure appropriately. Fig. 4(a)     With the rapid development of fabrication techniques, our theoretical design is easily realized in experiments based on the method described in Refs. [69,70]. Fig. 6 shows an experimental implementation for generating entangled photon pairs. A high pass filter is used to filter out the pump light, after which the signal light and idler light are separated by a tilted dichromatic mirror. A fiber coupler and a single-mode fiber are used to collect the light for the detectors and a time-to-amplitude converter used for coincidence measurement. Such a scheme is used extensively in experimental measurements for entangled photon pairs [73]. In Ref. [73], it has been demonstrated the generation of entanglement for the signal and idler polarized photons at different wavelengths in periodically poled materials. The phenomena disclosed in the present work are identical with those described in Ref. [73].

Conclusion
We proposed a new method to efficiently produce polarization-entangled photon pairs in LN waveguides based on the TM bound state in the TE continuum modes. Differing from periodically-poled LN waveguides, a complicated polarization process is not needed in the design. Compared with the corresponding case without BICs, the generation rate of the entangled photon pair can be improved by several orders of magnitude in properly designed waveguides only a few millimeters long. At the same time, the generation rate increases linearly with the waveguide length. The phenomena can appear in a very wide spectrum range from the visible to THz regions. We believe that the calculated results are of great significance for the development of active integrated quantum chips for various wavelength ranges.

Funding
National Key R & D Program of China (2017YFA0303800); National Natural Science Foundation of China (91850205).

Disclosures
The authors declare no conflicts of interest.

Appendix
In the appendix, we give the expressions for The free field propagation matrix is:  (28)