## Abstract

We note that most of the studies of the single photon scattering inside a one-dimensional coupled resonator waveguide are based on the waveguide coupling with the atom systems. In this paper, we will study the single photon scattering enabled by another system, i.e., the second-order nonlinearity, which can act as a single photon switch to control the single photon transmission and reflection inside the one-dimensional coupled resonator waveguide. The transmission rate is calculated to analyze the single-photon scattering properties. In addition, a more complicated second-order nonlinear form, i.e., three-wave mixing, is discussed to control single photon transmission inside the one-dimensional coupled resonator waveguide.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In order to realize quantum information and quantum computation, several systems have been proposed to build quantum networks. Among all the systems, quantum optical systems have attracted considerable attention, because they preserve quantum states for a long lifetime and rarely interact with static fields from environment, so photons are excellent candidate for quantum information and quantum computation [1–7].

As an important ingredient of building optical quantum networks, quantum optical switch has become a research focus due to the applications in quantum information and communication technology. Among all kinds of quantum switches, the switch in a low-dimensional waveguide has attracted much attention, because the waveguide can act a quantum channel to transfer the quantum bit from one node to another one. The basic idea is that the atoms embedded in the waveguide can control the transmission and reflection of the photon by the scattering process. The waveguide system used to realize the switch can be divided into two categories based on different physical mechanisms [8]: one is one-dimensional continuum waveguide, the feature of this system is that the dispersion relation is linear, which has been investigated widely [9–18]. The other system is coupled-resonator waveguide (CRW) [19,20], which can be realized by photonic crystals [21,22] or superconductor transmission line resonators [23–25]. The nonlinear dispersion of the CRW can be used to realize single-photon quasibound states [26–28] and photon-atom bound state [29,30]. The switch we concerned in this paper is based on the photon transport properties in the CRW, which has been widely explored [26,27,31–38] by embedding with a two-level or three-level atom. Recently, the theory studies were expanded to the photon scattering from a system of multilevel quantum emitters [39,40].

The second-order nonlinearity [41] is an important phenomenon in nonlinear optics. The conversion of a single photon with high frequency into two photons with low frequency is called parametric down-conversion. The opposite process of two photons converting into a single photon corresponds to parametric up-conversion. Second-order nonlinearity can be realized by a high-$\chi ^{(2)}$ nonlinear material such as III-V semiconductors (e.g., GaAs, GaP, GaN, AlN, etc.) [42–44]. Recently, fast-emerging LiNbO3 waveguides can also be used as the high high-$\chi ^{(2)}$ nonlinear material platform [45–47]. The applications include low-power spectroscopic sources, up-conversion of mid-IR to visible for imaging applications [48], and fundamental researches, such as strong coupling of single photons [49], a single-photon blockade [50–53] and induced transparency [54,55]. The experimental researches include the parametric down-conversion photon-pair source on a nanophotonic chip [56], on-chip strong coupling and efficient frequency conversion between telecom and visible optical modes [57], and second-harmonic generation in aluminum nitride microrings [58].

In this paper, we introduce the second-order nonlinearity into the CRW to realize the quantum optical switch. We analyze the coherent transport of a single photon, which propagates in the CRW and is scattered by an additional resonator, where the additional resonator is coupled to the a CRW via second-order nonlinearity. The reflection and transmission coefficients for a single photon propagating in such a system are deduced by using the discrete coordinates approach. It is shown that both perfect transmission points and reflection positions can be controlled via adjusting the system parameters. The contribution of the present scheme can be summarized as: (i) A new nonlinear system is proposed to study the controllable scattering of a single photon inside a one-dimensional CRW. (ii) In comparison with the CRW coupling to a two-level system under the different resonance conditions, the second-order nonlinear system has a better transmission property than the two-level system because the former can better control the reflection of a single photon when the interaction is weak. In addition, the three-wave mixing form of second-order nonlinearity is also discussed to study the single photon transport inside the CRW.

The remainder of this paper is organized as follows. In Sec. 2, we introduce the physical model. In Sec. 3, we illustrate the theory of the control of the single-photon transport inside a one-dimensional CRW with second-order nonlinearity, where the analytic solutions of transmission and reflection amplitudes are obtained, and the controllable scattering properties are studied. In Sec. 4, we compare the second-order nonlinear system with the two-level system. In Sec. 5, we show the control of single-photon transport inside a one-dimensional CRW with the three-wave mixing. Discussion and conclusion are given in Sec. 6.

## 2. Physical model

We consider a hybrid system containing a CRW coupled to an additional resonator via second-order nonlinearity, which is illustrated in Fig. 1. The total Hamiltonian of such a system can be written as

where $H_a$ describes the free photon transport in the CRW, $H_b$ expresses the additional resonator, and $H_i$ denotes the interaction between the coupled resonator waveguide and the additional resonator via second-order nonlinearity. $H_a$, $H_b$, and $H_i$ are given as (we set $\hbar =1$ hereafter)## 3. Results

#### 3.1 Analytic solutions of transmission and reflection amplitudes

We now restrict our discussion to the single excitation subspace for the CRW. When the single-photon transports from the left of the CRW to the resonator $a_0$, the photon will come to the additional resonator $b_0$ with a certain probability and be converted into two photons. While the two photons in resonator $b_0$ come back to resonator $a_0$, they will be converted to one by the second-order nonlinearity. In this case, we can guarantee that there is always one photon in the CRW. For the one-excitation subspace for the CRW, the eigenstates of the system has the form

where $|0\rangle _a|0\rangle _b$ indicates the vacuum states of the CRW and the additional resonator, and $|0\rangle _a|2\rangle _b$ indicates the vacuum state of the CRW and two-photon state of the additional resonator, $u_k(j)$ denotes the probability amplitude of a single photon in the $j$th resonator of the CRW, and $u_b$ denotes the probability amplitude with two photons in the additional resonator.Substituting the stationary eigenstate of Eq. (3) and the Hamiltonian in Eq. (1) into the eigenequation $H|E_k\rangle =E_k|E_k\rangle$, we get a set of equations for the coefficients

We assume that the single photon is injected from the left side of the CRW, then a usual solution for the scattering equation is

#### 3.2 Controllable scattering of a single photon inside a CRW with second-order nonlinearity

In this section, we will study the single-photon scattering properties in the one-dimensional CRW with second-order nonlinearity. For convenience, we rescale all the parameters with respect to the hopping energy $\xi$ in this paper.

In Fig. 2, we plot the transmission rate $T$ as a function of $k$ and $g$ with different $\omega _a$ and $\omega _b$. The values of the transmission rate $T$ and reflection rate $R$ ($R=1-T$) can range from zero to unity under the condition $\omega _a=2\omega _b$, which is shown in Fig. 2(a). The shape of $T=1$ looks like a series of circular regions. The joint positions of the circular regions can be calculated as

If one of the joint position conditions is satisfied, the nodes appear. When $\omega _a=2\omega _b$, the second condition reduce to $k=(n+1/2)\pi$, thus the joint positions of Eq. (10) becomes $k=n\pi /2$, which is shown in Fig. 2(a). The case of $\omega _a\neq 2\omega _b$ and $\mid \omega _a-2\omega _b\mid /2\xi \leq 1$ is shown in Fig. 2(b). Compared with Fig. 2(a), the changes of $\omega _a$ and $\omega _b$ will move the positions decided by the second condition of Eq. (10). If $\mid \omega _a-2\omega _b\mid /2\xi\;>\;1$, the position $k=\arccos (\frac {\omega _a-2\omega _b}{2\xi })$ doesn’t work, which is shown in Fig. 2(c).In order to further investigate the scattering properties of the second-order nonlinearity coupled system, we rewrite the $t$ of Eq. (9) in another form

Figure 3(b) exhibits the photon transport in the CRW with $\Delta$ and $g$ for $\Delta =\Delta _a=\Delta _b$. The results indicate that the perfect photon transmission takes place in the region of a small $g$. Finally, we show $T$ as a function of $\Delta$ with $g=0.4$, the perfect reflection appears at $\Delta =0$ and $\Delta =\pm 2\xi$, which can be predicted as before.

## 4. Comparison with the two-level system

Next, we try to compare the single-photon scattering of the present system with the system of a one-dimensional CRW coupling to a two-level system, the Hamiltonian of the latter system can be written as

where $\sigma =|g\rangle \langle e|$ is the lowering operators for the two-level system and $H_a$ is shown in Eq. (2). In [31], the coherent transport of a single photon in a one-dimensional CRW is analyzed with a two-level system located inside one of the resonators of the waveguide, the Hamiltonian is the same with the above Hamiltonian. Hereafter, we will describe the above system by the two-level system.The transmission amplitude is calculated as

The corresponding effective potential $V_J$ is The fundamental difference of the transmission of the two systems is the effective potential. For a single incident photon with a specific momentum, the potential determines the properties of a scattering process.We now compare the scattering properties of the two systems under the different resonance conditions shown in Fig. 4, where the transmission rate $T$ is plotted as a function of $k$ for the same $g$ and $J$ with $\omega _a=2\omega _b$ for the second-order nonlinear system and $\omega _a=\omega _b$ for two-level system. Figure 4(a) indicates that when $g$ and $J$ have a small value, the maximum value of the transmission rate $T$ can reach one and the minimum value on the point $k=n\pi /2$ can reach zero for the two systems, while the second-order nonlinear system has a wider reflection region, which means that the second-order nonlinear system has a better transmission property for realizing the single photon switches, because the second-order nonlinear system can better control the reflection of a single photon when $g$ and $J$ have a small value. When $g$ and $J$ increase, the maximum value of the transmission rate for the two systems decrease simultaneously and the two-level system has a better scattering property for the high transmissivity, which is depicted in Fig. 4(b).

## 5. Control of single-photon transport with three-wave mixing

#### 5.1 The analytic solutions of transmission amplitude

Second-order nonlinearity has a more complicated form, i.e., three-wave mixing. In this section, we will study the single-photon transport inside a one-dimensional CRW combining with a three-wave-mixing system. The schematic is also shown in Fig. 1, where the additional resonator is a doubly resonant cavity with the frequencies $\omega _b$ and $\omega _c$ for mode $b$ and mode $c$, respectively. The total Hamiltonian of such a system can be written as

where $H_a$ is given in Eq. (2), $H_b'$ and $H_i'$ are given byFor the one-excitation subspace, the eigenstates of the system has the form of

Substituting the eigenstate of Eq. (17) and the Hamiltonian of Eq. (15) into the eigenequation $H|E_k\rangle =E_k|E_k\rangle$, we can get the coefficients equations

#### 5.2 Controllable scattering of a single photon inside a CRW with three-wave mixing

The single-photon scattering properties are studied in the CRW with three-wave mixing in this section, we find that this system has the similar scattering properties with the system shown in Eq. (1). In Fig. 5, we plot $T$ as a function of $k$ with different frequencies. The results show that, when the relation $\omega _a=\omega _b+\omega _c$ is satisfied, the transmission line has the symmetrical structure, which is similar to that shown in Fig. 4(b) with the second-order nonlinear system. The positions of the valleys can be calculated as

## 6. Conclusion

We have investigated the coherent controlling of a single photon transport in a CRW coupled to an additional resonator via the second-order nonlinearity. The reflection and transmission coefficients for a single photon propagating in such a system are deduced by using the discrete coordinates approach. It is shown that the single photon can be perfectly reflected or transmitted by tuning the the system parameters, and the transmission points can also be controlled. We compared the results with the two-level system and find the advantages. Moreover, the three-wave mixing form of the second-order nonlinearity is also discussed to study the single photon transmit inside the CRW. The combination of the second-order nonlinearity and CRW will provide a new way for realizing quantum devices and promote its development.

## Appendix

We now derive the dispersion relation in Eq. (8) in this appendix:

## Funding

Key R&D Program of Guangdong province (2018B0303326001); Jiangxi Education De-partment Fund (GJJ180873);National Natural Science Foundation of China (11705025, 11774076, 11804228, 11965017);the Jiangxi Natural Science Foundation (20192ACBL20051); Fundamental Research Funds for the Central Universities (2412019FZ044); Science Founda-tion of the Education Department of Jilin Province during the 13th Five Year Plan Period (JJKH20190262KJ); Fundamental Research Funds for the Central Universities (3132019181); Fundamental Research Funds for the Central Universities (2412017QD005); NKRDP of China (2016YFA0301802).

## Disclosures

The authors declare no conflicts of interest.

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