Optical Frequency Down-Conversion With Bandwidth Compression Based on Counter-Propagating Phase Matching

Optical quantum network plays an important role in large scale quantum communication. However, different components for photon generation, transmission, storage and manipulation in network usually cannot interact directly due to the wavelength and bandwidth differences, and thus interfaces are needed to overcome such problems. We propose an optical interface for frequency down-conversion and bandwidth compression based on the counter-propagating quasi-phase-matching difference frequency generation process in the periodically-poled lithium niobate on insulator waveguide. We prove that a separable spectral transfer function can be obtained only by choosing proper pump bandwidth, thus relaxing the limitation of material, dispersion, and working wavelength as a result of the counter-propagation phase-matching configuration. With numerical simulations, we show that our design results in a nearly separable transfer function with the Schmidt number very close to 1. With proper pump bandwidth, an photon at central wavelength of 550 nm with a bandwidth ranging from 50 GHz to 5 THz can be converted to a photon at central wavelength of 1,545 nm with a much narrower bandwidth of 33 GHz.


INTRODUCTION
Photons play an important role in quantum information science, such as long distance quantum communication [1,2], linear optical quantum computation [3,4] and interface to quantum memories [5,6]. However, in these applications, different devices and systems usually require different photon central frequencies and bandwidths. In order to combine all these systems in one quantum network, photon frequency interface capable of converting frequency and bandwidth is indispensable.
Electro-optical modulation is an efficient way to shift photon frequency [7][8][9], which is commonly used in pulse manipulation, however, its conversion range is limited to several GHz. Sum frequency generation (SFG) [10][11][12] and difference-frequency generation (DFG) [13] in nonlinear optical process are beneficial to frequency conversion between different frequency bands and have been utilized as an interface between the visible and communication wavelength bands [14][15][16]. Moreover, a bandwidth compression factor of 40 was achieved by utilizing SFG process with chirped input photon and anti-chirped strong pump laser [17]. However, interfaces generated by this way usually suffer from low conversion efficiency due to the need of ultra wide phase-matching bandwidth. How to convert frequency and compress bandwidth effectively at the same time is a big challenge. Recently, Allgaier et al. made an approach towards both goals with dispersion-engineered SFG [18], where photon at communication wavelength was converted to the visible range with a bandwidth compression factor of 7.47 and an internal conversion efficiency of 61.5%. Such method relies on modulating the dispersion and group velocity relationship among the input, pump, and output photons, and thus has limited choices on the working wavelengths and materials.
On the other hand, counter-propagating quasi-phase-matching (QPM) spontaneous parametric down-conversion (SPDC) process has been extensively studied due to its unique spectral properties [19,20], such as narrow bandwidth [21,22] and frequency uncorrelated [23][24][25][26] photon pairs. In contrast to the traditional co-propagating process, in the counter-propagating process the phase-matching function is greatly affected by the counter propagation of the signal and idler photons, and hence such method can be applied in a large range of nonlinear materials and working wavelengths. In this paper, we propose to use the counter-propagating QPM DFG to realize an optical interface for frequency down-conversion and bandwidth compression. We design an experimental feasible waveguide structure based on the thin-film lithium niobate on insulator (LNOI) platform, which has been considered as a revolutionary platform for integrated photonics [27] and aroused a great deal of interest in recent years, as it allows a strong optical confinement and thus brings strong nonlinear effects [28][29][30].

METHODS
The configuration of a counter-propagating QPM DFG process is shown in Figure 1. With a pulse laser as the pump in a y-direction waveguide, a co-propagating high-frequency input photon is converted into a low-frequency output photon in the counterpropagation direction. In the undepleted pump approximation, the effective Hamiltonian describing the DFG process can be written as [31].
where θ is the coupling parameter having absorbed all the constants, and the frequencies are constrained by the energyconservation relation of ω i − ω p ω o , with the subscripts i, p, and o representing the input, pump, and output photons, respectively. The normalized DFG transfer function can be expressed as is the spectral amplitude of pump light and Φ(ω i , ω o ) denotes the phase matching function.
With broadband pump laser, the transfer function can be written as Schmidt decomposition form [32].
where {ϕ j (ω i )} and {ψ j (ω o )} are two sets of orthogonal spectral amplitude functions and κ j are the real Schmidt coefficients satisfying j κ 2 j 1. Thus the effective Hamiltonian can be rewritten as with broadband mode operators A j dωϕ j (ω)â i (ω) and C j dωψ j (ω)â o (ω). Compared with the effective Hamiltonian of an optical beam splitter (BS) H BS θâĉ † + H.c. [33], the DFG process can be considered as a set of independent BSs which convert A j to C j with effective coupling parameter θκ j , namely, A j → cos (θκ j )A j + i sin (θκ j )C j with conversion efficiency sin 2 (θκ j ) [31]. Hence, for the multimode input photon j α j A † j |0〉, with α 2 j 1, the total conversion efficiency is given by j |α j | 2 sin 2 (θκ j ). We can see that, given a fixed pump light power, the maximum total conversion efficiency is achieved when the Schmidt number K 1, i.e., the transfer function is separable, according to the Cauchy-Schwarz inequality. Hence, it is important to design a separable transfer function for efficient frequency conversion. In the following, we propose a method to obtain a separable transfer function by using the counter-propagating QPM DFG process.
The phase matching function can be expressed as with L denoting the poling length. For the counter-propagating DFG process, the phase mismatch Δk is given by where k G 2πm/Λ is the mth order reciprocal wave vector with Λ denoting the poling period. We define frequency offsets Δω j ≡ Ω j − ω j , with j i, p, o, where Ω j are central frequencies satisfying perfect phase-matching condition Δk 0. Thus, according to the energy-conservation relation, we have Ω i − Ω p Ω o , and Δω i − Δω p Δω o . Then, by expanding Δk to the first order in Δω j and Δω o around central frequencies, we obtain where u j , j i, o, p are the group velocities at central frequencies.
For comparison, the phase mismatch of the co-propagating DFG process given by Δk co We can see that in the traditional co-propagating process the coefficients of frequency offsets only depend on the difference of the reciprocal of group velocities, and thus in such process the phase-matching function is usually engineered by selecting working frequencies and structures to control the dispersion and group velocities [34]. While, in the counter-propagating process, the coefficients also depend on the sum of the group velocities, enabling intrinsic features for phase-matching engineering [20][21][22][23][24][25][26].
To further characterize the transfer function, we define two characteristic bandwidth scales Thus, the phase matching function given by Eq. 4 can be rewritten as where we defined a function ofΦ(x) ≡ sinc(x)e −ix . By rewriting the pump amplitude function α(ω i − ω o ) asα(Δω i − Δω o ), we can write the transfer function against frequency offsets as In the following, we prove that when the pump light bandwidth δω p satisfies δω 2 ≪ δω p ≪ δω 1 , the transfer function can approach a separable function of Δω i , Δω o . In analogy to the analysis in Ref. [20], we first recast the argument of functionΦ with where Δω p /δω 1 has been neglected because it is on the order δω p / δω 1 ≪ 1. Then we recast the argument of functionα as where Δω i /δω 1 + Δω o /δω 2 is the argument of the sinc function inΦ as given in Eq. 9, and thus it is limited to values on the order of ∼ 10, namely, inside the bandwidth of sinc function, due to the product relationship ofα andΦ shown in Eq. 10. Consequently, provided that δω 2 /δω p is small enough, we can have δω 2 (Δω i /δω 1 + Δω o /δω 2 ) much smaller than δω p , the bandwidth ofα, and therefore this term is negligible in the argument ofα. Hence we have the following approximatioñ In addition, considering δω 2 /δω 1 ≪ 1, we can further make approximation on Eq. 14 as Consequently, the transfer function given by Eq. 10 approaches the factorized form Such separable function also means that the correlation between the input and output photons is eliminated. It is clear that the frequency of the input photon can vary within the bandwidth of the pump light, and thus the bandwidth of the input photon can be as large as that of the pump light, namely, δω i δω p . While, the bandwidth of the output photon is only determined by the phase-matching function irrespective of the pump bandwidth, which can be obtained from the full width at half maximum of the spectral density function |Φ Δω o /δω 3 | 2 |sinc Δω o /δω 3 | 2 , given by Hence, we can get the bandwidth compression factor as

RESULTS
The schematic of the LNOI waveguide is shown in Figure 2 consisting of three layers of silicon (Si), silica (SiO 2 ), and lithium Frontiers in Physics | www.frontiersin.org September 2021 | Volume 9 | Article 752137 niobate (LN), respectively. The LN layer is made from X-cut LNOI film and the waveguide propagates in y-direction with a sidewall angle of φ 60°, and a length of L 10 mm. The waveguide height h and its bottom ridge width w restrict the transverse distribution of the guide mode and can be adjusted in structure design and fabrication process. By varying the height and width, light dispersion can be tuned in the LNOI waveguide. In order to characteristic the property of the bandwidth compressor, numerically simulations of group index and effective refractive index are obtained by utilizing the Mode Solution software with the material dispersion of LN given by Ref. [35].
Here we aim to design a counter-propagating DFG process that converts the broadband input photons centered at 550 nm in TE 00 mode to narrowband output photons centered at 1,545 nm in TM 00 mode with a pulsed laser light centered at 854 nm in TM 00 mode as the pump. Such frequency conversion process with the chosen wavelengths may connect the quantum communication channels with single-photon emitters around 550 nm, such as the charge-neutral nitrogen-vacancy center in diamond [36] and the CdSe quantum dots [37,38]. The structure parameters are h 0.6 μm and w 0.8 μm. Single mode condition can be achieved at 1,545 nm with the field distribution of TE 00 and TE 00 modes shown in Figure 3. With simulated effective index of the waveguide, we can calculate the poling period to be Λ 0.402 μm for satisfying the first-order QPM condition of Δk 0 according to Eq. 5. The simulated results of group index n g c/u of TE 00 and TM 00 modes with wavelength ranging from 500 to 1,600 nm are shown in Figure 4. Explicitly, the simulated group indexes of the pump light at 854 nm in TM 00 mode, the input light at 550 nm in TE 00 mode, and the output light at 1,545 nm in TM 00 mode are n g,p 2.533, n g,i 2.532, and n g,o 2.401, respectively. Therefore, we can obtain δω 1 60 THz and δω 2 12.2 GHz.
In order to show the spectrum relation between the input and output photons, we simulate the transfer function given by Eq. 10. By assuming a Gaussian spectrum pump, the simulation  results when pump bandwidths δω p 50 GHz and δω p 5 THz are shown in Figure 5, 6, respectively. The corresponding Schmidt numbers K are estimated to be 1.037, and 1.041, respectively. Hence, we can see that the transfer function is very close to a separable one. The bandwidth of the output photon can be estimated from Eq. 17, namely, δω o 33 GHz. Then according to Eq. 18 we can express the bandwidth compression factor as and consequently, in our simulation range of δω p 50 GHz ∼ 5 THz, we can obtain a compression factor ranging from 1.5 to 150. Then we give a simulation of the conversion efficiency. The DFG process with a separable transfer function can be treated as a BS model and the conversion efficiency is given by sin 2 θ [31].
Here the coupling parameter can be expressed as where d 2d 31 /(mπ) is the nonlinear coefficient, and P p is the pump peak power, with n j (j p, i, o) representing the effective refractive index of pump, input and output lights at central frequencies, respectively. The parameter N is the normalization factor of transfer function given by The effective interaction area B can be written as where g j (x, z) (j p, i, o) is the normalized spatial distribution of the cross-sectional area of pump, input and output fields, respectively. Through numerical simulation using the Mode solution software, we estimated B to be 0.472 μm 2 . With these calculations, we can estimate a pump peak power of 2.04 W in the case of unity conversion efficiency. If setting the pump pulse width to be 200 fs with a repetition rate of 80 MHz, we can calculate the average pump power to be 0.032 mW, which is much lower than the previous experiment results [39,40].
It should be noted that the ideal unity conversion efficiency in a single process could be achieved only in the limit of short interaction length or long pump pulse [41]. In broadband mode case, time-ordering corrections may affect the conversion efficiency [42,43], which are resulted from the noncommutativity of the interaction Hamiltonian at different times. A maximum conversion efficiency of 87.7% has been obtained in a SFG process [44]. Moreover, Reddy et al. [45] proposed a scheme to overcome the time-ordering correction limitation by cascading two frequency conversion processes with 50% conversion efficiency.

DISCUSSION
We would like to discuss the experimental feasibility of our design. The LNOI waveguide structure is experimentally feasible with current LNOI fabrication techniques [27][28][29][30]. The poling period on the order of 0.402 μm is still challenging at present. However, we can use a higher-order reciprocal wave vector to obtain a bigger poling period at the cost of lower efficiency. For example, if using the third-order reciprocal wave vector, we would get a poling period of 1.206 μm with the nonlinear coefficient reduced to d/3. Such poling period is possible with current fabrication techniques [46].
In conclusion, we proposed a scheme to realize optical frequency down-conversion and bandwidth compression via the counter-propagating QPM DFG process, which can provide a quantum network interface for devices working at different central frequencies and bandwidths. We proved that, due to the counter-propagation configuration, a separable spectrum transfer function can be obtained only by choosing the pump bandwidth in a range between two characteristic bandwidth scales, rather than satisfying constrained dispersion and group velocity relations, and thus this method is not strictly limited by the material, dispersion, and working wavelength. Moreover, under this condition, the input photon can have a bandwidth the same with that of the pump light, while the bandwidth of the out photon is only determined by the phase-matching function irrespective of the pump bandwidth. Such feature enables a large bandwidth compression factor as well as facilitates the application in the interface between photons with different spectral shapes. We designed a periodically-poled LNOI waveguide to realize the scheme. The simulation result shows a nearly separable transfer function with the Schmidt number estimated very close to 1. By changing the pump bandwidth, a bandwidth compression factor ranging from 1.5 to 150 can be obtained. We also calculate a pump Frontiers in Physics | www.frontiersin.org September 2021 | Volume 9 | Article 752137 5 average power of 0.032 mW to achieve unit conversion efficiency. In addition, the counter-propagating output feature is also of great benefit to compressing co-propagating noises. Finally, our approach opens up a way for efficient optical interface connecting photons with different frequency and spectrum bandwidth benefiting from the counter-propagating nonlinear process. We hope our approach can stimulate more such investigations.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.