Controllable scattering of a single photon inside a one-dimensional coupled resonator waveguide with second-order nonlinearity

Y. H. Zhou; X. Y. Zhang; Dan Dan Zou; Qi-Cheng Wu; Biao Ling Ye; Y. L. Fang; H. Z. Shen; H. Z. Shen; Chui-Ping Yang; Chui-Ping Yang

doi:10.1364/OE.380250

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

(1) $$H=H_a+H_b+H_i,$$

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)

(2)$$\begin{aligned} H_{a} & =\omega_a\sum_ja_j^{\dagger}a_j-\xi\sum_j(a_j^{\dagger}a_{j+1}+a_{j+1}^{\dagger}a_{j}),\\ H_{b} & =\omega_bb_0^{\dagger}b_0,\\ H_{i} & =g(a_0^{\dagger}b_0^2+b_0^{\dagger2}a_0), \end{aligned}$$

where $a_j$ ($a_j^{\dagger}$) is the annihilation (creation) operator of the $j$th cavity in the CRW with the same resonance frequency $\omega _a$, and $b_0$ ($b_0^{\dagger}$) is the annihilation (creation) of the additional cavity with frequency $\omega _b$. Here $\xi$ is the hopping energy between the two nearest-neighbor resonators of the CRW. And $g$ denotes the coefficient of second-order nonlinearity, which mediates the conversion of the photon in cavity $a_0$ into two photons in cavity $b_0$ and the two photons in cavity $b_0$ into a single photon cavity $a_0$. Besides the schematic of two resonators coupled by the second-order nonlinearity shown in Fig. 1, the model can also be described by a doubly resonant resonator $a_0$ filled with $\chi ^{(2)}$ nonlinear material [55] without the additional resonator $b_0$. Hereafter, we will describe the above system by the second-order nonlinear system for convenience. From the perspective of single-photon switch, we select $g$ as the parameter to control the transmission property of an incident single photon. In this paper, we use a cyclic three-level artificial atom of a superconducting flux quantum circuit interacting with a two-mode superconducting transmission-line resonator to realize the second-order nonlinear system [59], where the second-order nonlinear coupling strength $g$ can be adjusted by adjusting the position of the qutrit in the transmission-line resonator, and the value of $g/(2\pi )$ can reach 5MHz. The CRW can also be realized by using coupled superconducting transmission line resonators [31]. Experimentally, large-scale ultrahigh-Q coupled cavity arrays based on the transmission line resonators have been realized [60].

Fig. 1. Schematic illustration of the coherent transport of a single photon in a coupled-resonator waveguide, where one resonator of the waveguide coupling to an additional resonator via $\chi ^{(2)}$ nonlinearity.

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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

(3)$$|E_k\rangle=\sum_ju_k(j)a_j^{\dagger}|0\rangle_a|0\rangle_b+u_b|0\rangle_a|2\rangle_b,$$

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

(4)$$\begin{aligned} (E_k-\omega_a)u_k(j)-\sqrt{2}gu_b\delta_{j,0} & =-\xi[u_k(j-1)+u_k(j+1)],\\ (E_k-2\omega_b)u_b & = \sqrt{2}gu_k(0). \end{aligned}$$

By eliminating $u_b$, we obtain the discrete scattering equation

(5)$$(E_k-\omega_a+V_g)u_k(j)={-}\xi[u_k(j-1)+u_k(j+1)],$$

where

(6)$$V_g={-}\frac{2g^2\delta_{j,0}}{E_k-2\omega_b}.$$

The effective potential $V_g$, resulting from the interaction between the additional resonator and the CRW located at site $j = 0$, modifies the single-photon transport property in the resonator waveguide. When the coefficient $g$ of second-order nonlinearity equals to zero, the effective potential $V_g$ vanishes.

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

(7)$$u_k(j)= \left\{ \begin{array}{lr} e^{ikj}+re^{{-}ikj}, & j\;<\;0, \\ te^{ikj}, & j\;>\;0, \end{array} \right.$$

where $t$ and $r$ are the transmission and reflection amplitudes, respectively. And the $e^{ikj}$ denotes the wave traveling to the right, and $e^{-ikj}$ denotes the wave traveling to the left side. In the CRW, $E_k$ is characterized by

(8)$$E_k=\omega_a-2\xi \cos k,$$

which is crucial for the derivation of the transmission and reflection coefficients, we have provided the derivation in the appendix. By considering the continuity condition $u_k(0^+)=u_k(0^-)$ and Eqs. (5)–(8), we can obtain the transmission and reflection amplitudes given in the following expressions

(9)$$\begin{aligned} t & = \frac{2i\xi (E_k - 2 \omega_b) \sin k}{-2 g^2 + 2 i \xi(E_k - 2 \omega_b) \sin k},\\ r & = \frac{2 g^2}{-2 g^2 + 2 i \xi(E_k - 2 \omega_b) \sin k}. \end{aligned}$$

We define a transmission rate $T=|t|^2$ and reflection rate $R=|r|^2$. The relation $T+R=1$ can be easily verified, which makes the reflection rate $R$ can be deduced if we obtain the transmission rate $T$. So we only analyze the transmission rate $T$ in this paper.

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

(10)$$\begin{aligned} k & = n\pi,\\ k & = \arccos(\frac{\omega_a-2\omega_b}{2\xi}). \end{aligned}$$

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).

Fig. 2. Transmission rate $T$ as a function of $k$ and $g$ for different $\omega _a$ and $\omega _b$. (a) $\omega _a=3$ and $\omega _b=1.5$. (b) $\omega _a=3$ and $\omega _b=1$. (c) $\omega _a=6$ and $\omega _b=2$. All the parameters are rescaled by the hopping energy $\xi$ in this paper.

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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

(11)$$t=\frac{2i\xi \Delta_b\sqrt{1-(\frac{\Delta_a}{2\xi})^2}}{-2g^2+2i\xi\Delta_b\sqrt{1-(\frac{\Delta_a}{2\xi})^2}},$$

where $\Delta _a=E_k-\omega _a$ and $\Delta _b=E_k-2\omega _b$. In Fig. 3(a), we plot $T$ as a function of $\Delta _a$ and $\Delta _b$ with $g=1$. If the condition $\Delta _b=0$ or $\Delta _a=\pm 2\xi$ is satisfied, the transmission rate reaches the minimum value $T=0$. In this case, the photon is completely reflected. While $g^4\ll \xi ^2\Delta _b^2[1-(\Delta _a/2\xi )^2]$, the photon can pass through the CRW.

Fig. 3. (a) $T$ as a function of $\Delta _a$ and $\Delta _b$ with $g=1$. (b) $T$ as a function of $g$ and $\Delta$. (c) $T$ as a function of $\Delta$ with $g=0.4$. In both (b) and (c), $\Delta =\Delta _a=\Delta _b$, $\Delta _a=E_k-\omega _a$ and $\Delta _a=E_k-2\omega _b$.

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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

(12)$$H=H_a+\omega_b\sigma^+\sigma+J(a_0^{\dagger} \sigma+\sigma^+ a_0),$$

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

(13)$$t=\frac{2i\xi(E_k-\omega_b)\sin k}{-J^2+2i\xi(E_k-\omega_b)\sin k}.$$

The corresponding effective potential $V_J$ is

(14)$$V_J={-}\frac{J^2\delta_{j,0}}{E_k-\omega_b}.$$

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).

Fig. 4. The comparison of the transmission rate of the second-order nonlinearity system with the two-level system, where the solid line denotes the second-order nonlinearity system and the dotted line denotes the two-level system. (a) $g=0.3$, $J=0.3$. (b) $g=1$, $J=1$. In both (a) and (b), the parameters are $\omega _a=6$ and $\omega _b=3$ for second-order nonlinear system, and $\omega _a=6$ and $\omega _b=6$ for the two-level system.

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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

(15) $$H=H_a+H_b'+H_i',$$

where $H_a$ is given in Eq. (2), $H_b'$ and $H_i'$ are given by

(16)$$\begin{aligned} H_b'& =\omega_bb_0^{\dagger}b_0+\omega_cc_0^{\dagger}c_0,\\ H_i'& =g'(a_0^{\dagger}b_0c_0+c_0^{\dagger}b_0^{\dagger}a_0), \end{aligned}$$

where $g'$ denotes the coefficient of three-wave-mixing interactions.

For the one-excitation subspace, the eigenstates of the system has the form of

(17)$$|E_k\rangle=\sum_ju_k(j)a_j^{\dagger}|0\rangle_a|0\rangle_b|0\rangle_c+u_b|0\rangle_a|1\rangle_b|1\rangle_c,$$

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

(18)$$\begin{aligned} (E_k-\omega_a)u_k(j)-g'u_b\delta_{j,0} & = -\xi[u_k(j-1)+u_k(j+1)],\\ (E_k-\omega_b-\omega_c)u_b & = g'u_k(0)\delta_{j,0}. \end{aligned}$$

By using the previous method, we can obtain the transmission amplitude $t$ as

(19)$$t=\frac{2i\xi \sin k (E_k - \omega_b-\omega_c)}{-g^2 + 2 i \xi(E_k - \omega_b-\omega_c) \sin k}.$$

The transmission amplitude and reflection amplitude satisfy the relation $|r|^2+|t|^2=1$.

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

(20)$$\begin{aligned} k & = n\pi,\\ k & = \arccos(\frac{\omega_a-\omega_b-\omega_c}{2\xi}), \end{aligned}$$

under the relation $\omega _a=\omega _b+\omega _c$, the above conditions reduce to $k=n\pi /2$, which agree well with the blue line in Fig. 5. While $\omega _a\neq \omega _b+\omega _c$, the second condition shown in Eq. (20) will change the positions of the valleys and the maximum of the transmission rate will increase. We now compare the scattering properties of the three-wave mixing system with the previous second-order nonlinear system. The valleys of the three-wave mixing system shown in Eq. (20) will reduce to that of the second-order nonlinear system shown in Eq. (10) when $\omega _b=\omega _c$ is satisfied. However, the transmission amplitude of three-wave mixing system cannot reduce to second-order nonlinear system due to the different the eigenstates for one-excitation subspace, i.e., the Eq. (17) cannot describe the eigenstates in Eq. (3) when $\omega _b=\omega _c$.

Fig. 5. $T$ as a function of $k$ with $g=1$ for different $\omega _a$, $\omega _b$ and $\omega _c$.

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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:

(A1)$$\begin{aligned} a_k&=\frac{1}{\sqrt{N}}\sum_ja_je^{{-}ikj},\\ a_j&=\frac{1}{\sqrt{N}}\sum_ka_ke^{ikj}. \end{aligned}$$

By substituting Eq. (A1) into the Hamiltonian $H_a$, we can obtain

(A2)$$\begin{aligned} &\omega_a\sum_ja_j^{\dagger}a_j-\xi\sum_j(a_{j+1}^{\dagger}a_{j}+a_j^{\dagger}a_{j+1}),\\ =&\sum_j{\bigg\{}\omega_a\frac{1}{\sqrt{N}}\sum_ka_k^{\dagger} e^{{-}ikj}\frac{1}{\sqrt{N}}\sum_{k'}a_{k'}e^{ik'j} -\xi{\bigg[}\frac{1}{\sqrt{N}}\sum_ka_k^{\dagger} e^{{-}ik(j+1)}\frac{1}{\sqrt{N}}\sum_{k'}a_{k'}e^{ik'j}\\ &+\frac{1}{\sqrt{N}}\sum_ka_k^{\dagger} e^{{-}ikj}\frac{1}{\sqrt{N}}\sum_{k'}a_{k'}e^{ik'(j+1)}{\bigg]} {\bigg\}}\\ =&\sum_j\omega_a\frac{1}{\sqrt{N}}\sum_{kk'}a_k^{\dagger} a_{k'} e^{{-}i(k-k')j} -\xi\sum_j{\bigg[}\frac{1}{N}\sum_{kk'}a_k^{\dagger} a_{k'} e^{{-}i(k-k')j}e^{{-}ik}\\ &+\frac{1}{N}\sum_{kk'}a_k^{\dagger} a_{k'} e^{{-}i(k-k')j}e^{ik'}{\bigg]}. \end{aligned}$$

Considering the relation

(A3)$$\frac{1}{N}\sum_je^{{-}i(k-k')j}=\delta_{k,k'},$$

Eq. (A2) reduces to

(A4)$$\begin{aligned} &\omega_a\sum_{kk'}a_k^{\dagger} a_{k'}\delta_{k,k'} -\xi(\sum_{kk'}a_k^{\dagger} a_{k'}\delta_{k,k'}e^{{-}ik}+\sum_{kk'}a_k^{\dagger} a_{k'}\delta_{k,k'}e^{ik'})\\ =&\omega_a\sum_ka_k^{\dagger} a_{k}-\xi(\sum_{k}a_k^{\dagger} a_{k}e^{{-}ik}+\sum_{k}a_k^{\dagger} a_{k}e^{ik})\\ =&(\omega_a-2\xi \cos k)\sum_ka_k^{\dagger} a_{k}. \end{aligned}$$

Based on Eq. (A4), one can easily obtain the corresponding energy eigenvalues, as shown in Eq. (8).

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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Controllable scattering of a single photon inside a one-dimensional coupled resonator waveguide with second-order nonlinearity

Abstract

1. Introduction

2. Physical model

3. Results

3.1 Analytic solutions of transmission and reflection amplitudes

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

4. Comparison with the two-level system

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

5.1 The analytic solutions of transmission amplitude

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

6. Conclusion

Appendix

Funding

Disclosures

References

Cited By

Figures (5)

Equations (24)

Optics Express