⇦Chinese Physics Letters, 2019, Vol. 36, No. 3, Article code 034202 High-Efficiency Quantum Routing in a Multi-Cross-Shaped Waveguide * Jin-Song Huang (黄劲松)**, Jing-Wen Wang (王敬文), Yao Wang (王尧), Yan-Ling Li (李艳玲) Affiliations School of Information Engineering, Jiangxi University of Science and Technology, Ganzhou 341000 Received 8 October 2018, online 23 February 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11247032 and 61765007, and the Scientific Research Foundation of the Jiangxi Provincial Education Department under Grant Nos GJJ170556 and GJJ180424.
**Corresponding author. Email: jshuangjs@126.com Citation Text: Huang J S, Wang J W, Wang Y and Li Y L 2019 Chin. Phys. Lett. 36 034202    Abstract Efficiently routing the quantum signals between different channels is essential in a quantum multichannel network. We investigate the quantum routing in a multi-cross-shaped waveguide coupled to driven three-level systems. Numerical results show that the high routing capacity transferring from the input channel to the other channels can be explicitly enhanced by effective reflection potentials. The proposed system may be utilized as a scalable quantum device to control single-photon routing. DOI:10.1088/0256-307X/36/3/034202 PACS:42.50.Ex, 42.79.Gn, 03.67.Lx, 78.67.-n © 2019 Chinese Physics Society Article Text In a quantum optical network, a quantum node is an essential component to coherently connect different quantum channels,[1-3] while quantum routers in the nodes are usually utilized to control the quantum signal transport in these channels. Recently, numerous investigations of quantum routing have been carried out in many systems including optomechanical systems,[4,5] cavity quantum electrodynamics (QED) systems,[6-9] etc. As an ideal channel model for photon transmission, coupled cavity arrays (CCAs) have also been widely applied in the study of photon transport and routing.[10-21] For example, Zhou et al.[20,21] investigated the quantum routing properties in an X-shaped waveguide constructed by two CCA channels. By coupling the two CCAs mediated by a driven quantum emitter, single photons can be transferred from the input channel to another channel. However, a low routing rate of no more than 0.5 from the input channel to another channel exists in the X-shaped waveguide, which may limit its more potential applications, since a high routing capability can efficiently distribute the transmission signals between channels. Inspired by these considerations, we propose a simple scheme to examine the quantum routing of single photons in an extended multi-cross-shaped waveguide, in which multichannels are constructed by cross-CCAs, and are coupled by the driven three-level atoms located in their corresponding cross-resonators. Using the discrete-coordinate scattering and the transfer matrices approach, photon scattering amplitudes in the double-cross and triple-cross waveguides are obtained analytically. Our results show that a high transfer rate of single photons from the incident channel into the other channels can be achieved in a wide frequency range by tuning the atom–resonator coupling. In sharp contrast to the single cross-shaped waveguide scheme[20,21] with a maximum transfer of 0.5, the photon probability routed into the other channels can significantly exceed 0.5, since effective reflection potentials generated by the cross-CCAs reflow the incident photons and more photons are redirected into the other waveguide channels. The possible extension of the S-cross-shaped ($S>3$) waveguide is also easy to perform by employing the transfer matrices, and thus the proposed system may provide a potential application in designing scalable optical devices with high routing capability.
cpl-36-3-034202-fig1.png
Fig. 1. A triple-cross-shaped waveguide made by an infinite CCA and three cross-infinite CCAs. Every atom embedded in the cross-resonators is used to couple the cross-channels. An incoming wave from the left side of CCA-a is reflected, transmitted, or transferred to CCA-b(c,d).
As a specific schematic diagram of the considered system, a triple-cross waveguide is shown in Fig. 1, which is constructed by an infinite CCA-a with three cross-CCAs (CCA-b, -c, and -d). The cavity modes of the four channels in the waveguide are described respectively by annihilation operators $a_{j_{\rm a}}$, $b_{j_{\rm b}}$, $c_{j_{\rm c}}$, and $d_{j_{d}}$, with the subscripts $j_{\rm a(b,c,d)}= -\infty,\ldots, +\infty$. The three ${\it \Lambda}$-type three-level atoms located at the $0(n,m)$th cross-cavities in CCA-a are characterized respectively by the ground states $|g_{1(2,3)}\rangle$, excited states $|e_{1(2,3)}\rangle$, and intermediate states $|s_{1(2,3)}\rangle$, and are simultaneously connected to CCA-b(c,d). The cavity modes $a_0(a_n,a_m)$ and $b_0(c_0,d_0)$ couple the atomic transitions $|g_{1(2,3)}\rangle\leftrightarrow|e_{1(2,3)}\rangle$ with the coupling strengths $g_{a_1}(g_{a_2},g_{a_3})$ and $g_{\rm b}(g_{\rm c},g_{d})$, respectively. Additionally, the atomic transitions $|e_{1(2,3)}\rangle\leftrightarrow|s_{1(2,3)}\rangle$ are resonantly driven by the classical field with the Rabi frequency ${\it \Omega}_{1(2,3)}$. When a photon is incident from the left of CCA-a, it propagates or is reflected along the waveguide, and it can also be routed into the other channels (channel-b, -c, -d). The total Hamiltonian of the composite system comprises three parts: $H= H_{\rm C}+H_{\rm A}+H_{\rm I}$, where $H_{\rm C}$ is the Hamiltonian of the four CCAs, $H_{\rm A}$ is the Hamiltonian of the three-level atoms, and $H_{\rm I}$ describes the interactions among the resonator modes, the classical fields, and the atoms. They can be expressed as $$\begin{align} H_{\rm C}=\,& \sum_{p={\rm a,b,c,d}}\sum_{j_{p}}[\omega_{p} {p}_{j_{p}}^† {p}_{j_{p}}-\xi_{p}({p}_{j_{p}}^† {p}_{{j_{p}}+1}+{\rm H.c.})],\\ H_{\rm A}=\,& \sum_{q=1,2,3}(\omega_{e_q} |e_q\rangle\langle e_q|+\omega_{s_q} |s_q\rangle\langle s_q|),\\ H_{\rm I}=\,&|e_1\rangle\langle g_1|(g_{a_1}a_0+g_bb_0)+|e_2\rangle\langle g_2|(g_{a_2}a_{n}+g_cc_0)\\ &+|e_3\rangle\langle g_3|(g_{a_3}a_{m}+g_{d}d_0)\\ &+\sum_{q=1,2,3} {\it \Omega}_q|e_q\rangle\langle s_q|+{\rm H.c.}~~ \tag {1} \end{align} $$ For simplicity, we assume that all CCAs have the same frequency $\omega_{p}$ (${p}={\rm a,b,c,d}$ throughout), and the same intercavity coupling constant $\xi_p$ between any two nearest-neighbor cavities. By performing the Fourier transformations $ p_{k}=\frac{1}{\sqrt{2\pi}} p _{j_{p}} e^{i k_{p} j_{p}}$ for CCA-a(b,c,d), respectively, the cosine dispersion relation of each CCA can be written as $$\begin{align} E_{k}^{[p]}= \omega_{p} -2\xi_{p} \cos{k_{p}l},~~ \tag {2} \end{align} $$ where the lattice constant $l$ is assumed to be unity. It indicates that each CCA occupies an energy band with the corresponding bandwidth $4\xi_p$. Single-photon routing properties in the system are represented by the transmission and reflection in CCA-a and the transfer into other CCAs. Since quantum routing in a single cross-shaped waveguide has been investigated,[21] we examine the double-cross and triple-cross cases in detail, respectively. We first consider the double-cross waveguide, whose Hamiltonian can be represented by removing the CCA-d and adopting $q=1,2$ in Eq. (1). As the total number of excitations is conserved in this hybrid system, the eigenstate in the single-excitation subspace is expressed as ($\hbar=1$ throughout) $$\begin{alignat}{1} |\psi\rangle =\,& \sum_{j_{\rm a}} \alpha({j_{\rm a}}) a_{j_{\rm a}}^† |\emptyset,g_1,g_2\rangle+\sum_{j_{\rm b}} \beta({j_{\rm b}})b_{j_{\rm b}}^†|\emptyset,g_1,g_2\rangle \\ &+\sum_{j_{\rm c}} \nu({j_{\rm c}})c_{j_{\rm c}}^†|\emptyset,g_1,g_2\rangle+u_{e1} |\emptyset,e_1,g_2\rangle\\ +&u_{s1}|\emptyset,s_1,g_2 \rangle+u_{e2} |\emptyset,g_1,e_2\rangle+u_{s2} |\emptyset,g_1,s_2\rangle,~~ \tag {3} \end{alignat} $$ where $|\emptyset\rangle$ describes the vacuum state of the double-cross waveguide; $\alpha({j_{\rm a}})$, $\beta({j_{\rm b}})$, and $\nu({j_{\rm c}})$ are the corresponding probability amplitudes in these channels; and $u_{e1}$, $u_{e2}$, $u_{s1}$, and $u_{s2}$ are the probability amplitudes in the atomic excited $|e_1\rangle$, $|e_2\rangle$, and intermediate states $|s_1\rangle$ and $|s_2\rangle$, respectively. From the eigen equation $H|\psi\rangle=E_k|\psi\rangle$, one can obtain a series of coupled stationary equations $$\begin{align} (E_{k}-\omega_{\rm a})\alpha({j_{\rm a}})=\,&-\xi_{\rm a}[\alpha({j_{\rm a}}-1)+\alpha({j_{\rm a}}+1)]\\ &+\delta_{j_{a0}}[V_{a0}\alpha(0)+G_1\beta(0)]\\ &+\delta_{j_{an}}[V_{an}\alpha(n)+G_2\eta(0)],\\ (E_{k}-\omega_{\rm b})\beta({j_{\rm b}})=\,&-\xi_{\rm b}[\beta({j_{\rm b}}-1)+\beta({j_{\rm b}}+1)]\\ &+\delta_{{j_{\rm b}}0}[V_{\rm b}\beta(0)+G_1^*\alpha(0)],\\ (E_{k}-\omega_{\rm c})\nu({j_{\rm c}})=\,&-\xi_{\rm c}[\nu({j_{\rm c}}-1)+\nu({j_{\rm c}}+1)]\\ &+\delta_{{j_{\rm c}}0}[V_{\rm c}\nu(0)+G_2^*\alpha(n)],~~ \tag {4} \end{align} $$ where we have taken $E_{k}=E_{k}^{\rm [a]}=E_{k}^{\rm [b]}=E_{k}^{\rm [c]}$ for simplicity. The couplings between the atoms and CCAs produce energy-dependent effective potentials $$\begin{align} V_{a0{\rm (b)}}(E_k) =\,&\frac{g_{a_1{\rm (b)}}^2{\it \Delta}_{s_1}}{{\it \Delta}_{e_1}{\it \Delta}_{s_1}-|{\it \Omega}_1|^2},\\ V_{an{\rm (c)}}(E_k) =\,&\frac{g_{a_2{\rm (c)}}^2{\it \Delta}_{s_2}}{{\it \Delta}_{e_2}{\it \Delta}_{s_2}-|{\it \Omega}_2|^2},\\ G_{1(2)}(E_k)=\,&\frac{g_{a_1(a_2)}g_{\rm b(c)}{\it \Delta}_{s_1(s_2)}} {{\it \Delta}_{e_1(e_2)}{\it \Delta}_{s_1(s_2)}-|{\it \Omega}_{1(2)}|^2},~~ \tag {5} \end{align} $$ where ${\it \Delta}_{e_1(e_2,s_1,s_2)}=E_k-\omega_{e_1(e_2,s_1,s_2)}$ is set. The wave functions of the incident photon with the energy $E_k$ are given by $$\begin{align} &\alpha(j_{\rm a}) =\left\{\begin{aligned} e^{ik_{\rm a}{j_{\rm a}}}+r_2 e^{-ik_{\rm a}{j_{\rm a}}}, & ~~~~{j_{\rm a}} < 0 \\ t_ne^{ik_{\rm a}{j_{\rm a}}}+r_n e^{-ik_{\rm a}{j_{\rm a}}}, & ~~~~0 < {j_{\rm a}} < n \\ t_2 e^{ik_{\rm a}{j_{\rm a}}}, &~~~~ {j_{\rm a}}>n \end{aligned} \right.~~ \tag {6} \end{align} $$ $$\begin{align} &\beta(j_{\rm b}) =\left\{\begin{aligned} t_{\rm b}^{\rm d}e^{-ik_{\rm b}{j_{\rm b}}}, & ~~~~{j_{\rm b}} < 0 \\ t_{\rm b}^u e^{ik_{\rm b}{j_{\rm b}}}, &~~~~ {j_{\rm b}}>0 \end{aligned} \right.~~ \tag {7} \end{align} $$ $$\begin{align} &\nu(j_{\rm c}) =\left\{\begin{aligned} t_{\rm c}^{\rm d}e^{-ik_{\rm c}{j_{\rm c}}}, & ~~~~{j_{\rm c}} < 0 \\ t_{\rm c}^u e^{ik_{\rm c}{j_{\rm c}}}, & ~~~~{j_{\rm c}}>0 \end{aligned} \right.~~ \tag {8} \end{align} $$ where $r_2$ and $t_2$ denote reflection and transmission amplitudes at the two sides of CCA-a, $t_{\rm b}^u(t_{\rm c}^u)$ and $t_{\rm b}^d(t_{\rm c}^d)$ are the up- and down-transfer amplitudes from CCA-a to CCA-b(c), and $r_n$ and $t_n$ represent the reflection and transmission amplitudes between the zeroth and $n$th resonators in CCA-a, respectively. Substituting Eqs. (6)-(8) into Eq. (4), the expressions of the scattering amplitudes can be obtained as follows: $$\begin{align} r_2=\,&-\frac{(Z_2+1)+ (Z_1-1)e^{2ik_{\rm a}n}} {(Z_1+1)(Z_2+1)- e^{2ik_{\rm a}n}},\\ t_2=\,&\frac{Z_1 Z_2}{(Z_1+1)(Z_2+1)- e^{2ik_{\rm a}n}},\\ t_{\rm b}=\,&\frac {Z_1 G^{*}_{1} e^{-ik_{\rm b}}(Z_2+1-e^{2ik_{\rm a}n})}{[(Z_1+1)(Z_2+1)- e^{2ik_{\rm a}n}](2i\xi_{\rm b}\sin k_{\rm b}-V_{\rm b})}, \end{align} $$ $$\begin{align} t_{\rm c}=\,&\frac{Z_1 Z_2 G^{*}_{2} e^{ik_{\rm a}n}} {[(Z_1+1)(Z_2+1)- e^{2ik_{\rm a}n}](2i\xi_{\rm c}\sin k_{\rm c}-V_{\rm c})},\\ Z_{1}=\,&\frac{-2i\xi_{\rm a}\sin k_{\rm a} (2i\xi_{\rm b}\sin k_{\rm b}-V_{\rm b})} {|G_{1}|^2+V_{a0}(2i\xi_{\rm b}\sin k_{\rm b}-V_{\rm b})},\\ Z_{2}=\,&\frac{-2i\xi_{\rm a}\sin k_{\rm a} (2i\xi_{\rm c}\sin k_{\rm c}-V_{\rm c})} {|G_{2}|^2+V_{\rm an}(2i\xi_{\rm c}\sin k_{\rm c}-V_{\rm c})}.~~ \tag {9} \end{align} $$ In the six-port system, the scattering properties are governed by the transmission $T_{\rm a}=|t_2|^2$, reflection $R_{\rm a}=|r_2|^2$, and transfers $T_{\rm b}=2|t_{\rm b}|^2$ and $T_{\rm c}=2|t_{\rm c}|^2$. For simplicity, we set $\omega_{\rm a} =\omega_{\rm b} =\omega_{\rm c}=\omega_0$, $\xi_{\rm a}=\xi_{\rm b} =\xi_{\rm c}=\xi$, $g_{a_1} =g_{a_2}=g_{\rm b}= g_{\rm c}=g$, $k_{\rm a} =k_{\rm b}= k_{\rm c}=k$, and $\omega_{e_1({s_1})}=\omega_{e_2({s_2})}$. As a review, we first consider the routing in the single-cross waveguide for $g_{a_2}=g_{\rm c}=0$, where the scattering amplitudes in Eq. (9) can be reduced to the forms of the X-shaped waveguide.[21] As seen in Fig. 2(a), the incident photon is transmitted to the right side of CCA-a at resonance ${\it \Delta}=E_k-\omega_{s_1}=0$, and there is no photon routed into channel-b, while for the photon with non-resonance energy, the transfer $T_{\rm b}$ to channel-b is performed at a rate of no more than 0.5, as examined in Ref. [21]. When adding another output channel-c, the input photons are distributed in three channels, and the conservation relation of the photon flow satisfies $|t_2|^2+ |r_2|^2+2|t_{\rm b}|^2+2|t_{\rm c}|^2= 1$. In sharp contrast to the single-cross case, the transfer $T_{\rm b}$ can exceed 0.5, while $T_{\rm c}$ into CCA-c is less than 0.5, as shown in Fig. 2(b). This scenario may result from the effect of an effective reflection potential in a supercavity created by two atoms in the zeroth and $n$th resonators.[13] Photons passing through the last atom in the supercavity are partly reflected by the effective potential $V_{\rm an}$ related to the coupling strength $g_{a_2}$, thus more photons are localized and redirected into other output channels. Consequently, the transfer rate to CCA-b increases and the total transfer can significantly exceed 0.5 up to 0.82, while the transfer into channel-c remains less than 0.5 due to the conservation relation. The considered system can be constructed by optical nanocavities and three-level $^{87}$Rb atoms, whose ${\it \Lambda}$-type structures are realized by the hyperfine states.[22] In the numerical calculations, the parameter choices are close to the experimental results in cavity QED systems,[22-25] and are taken to be $\xi=2\pi\times10$ MHz, $g=2\pi\times10$ MHz, $\omega_{\rm e}=\omega_0$, and $\omega_{s}-\omega_{0}=2\pi\times4$ MHz. For simplification, these parameters are denoted as $\omega_{\rm e}-\omega_0=0$, $\omega_{s}-\omega_{0} =0.4$, $g=1$, and $\xi=1$. All the parameters are scaled by the hopping coefficient $\xi$. When increasing the coupling $g_{a_2}$ and decreasing $g_{\rm b}$, as seen in Fig. 2(c), the total transfer still remains at around 0.82, and the transfer rate to CCA-c can exceed 0.5 to 0.72, while the transfer into channel-b changes to be less than 0.5. As a consequence, the routing rate distribution in different ports may be efficiently implemented by adjusting the couplings. The coupling strength can be tuned by changing the atom-cavity separation and the polarization of the atomic exciton.[26] Although it is difficult to vary the atomic cavity distance in reality, one can alter the coupling by applying an external field to orient the dipole moment direction of the atomic exciton.[26]
cpl-36-3-034202-fig2.png
Fig. 2. Transmission $T_{\rm a}$ (cyan dot-dashed curve), reflection $R_{\rm a}$ (pink dashed curve), transfers $T_{\rm b}$ (blue solid curve), $T_{\rm c}$ (red dashed curve), and the total transfer $Tr=T_{\rm b}+T_{\rm c}$ (green dot-dashed curve) as a function of the detuning ${\it \Delta}$. These parameters are respectively taken as ${\it \Omega}=0.6$, $g=1$, $\omega_s-\omega_0=0$, $\omega_e-\omega_0=0.4$, and $\xi=1$, and they are scaled by the hopping coefficient $\xi$. The resonator number is set as $n=4$. (a) Scattering for the single-cross waveguide. (b) Scattering for the double-cross waveguide. (c) Transfers in the different couplings $g_{\rm b}=0.2$ and $g_{\rm a_2}=1.9$. (d) Transfer $Tr$ with the dissipations: green dot-dashed curve for zero dissipation, black solid line for $\gamma_{\rm a}=\gamma_{\rm b}=\gamma_{\rm c}=0.008$ and $\gamma_{\rm e1(e2)}=\gamma_{\rm s1(s2)}=0.005$, and yellow dashed line for $\gamma_{\rm a}=\gamma_{\rm b}=\gamma_{\rm c}=0.1$ and $\gamma_{\rm e1(e2)}=\gamma_{\rm s1(s2)}=0.01$. Maximum transfer rate $T_{\rm b}^{\rm m}$ (blue solid line), $T_{\rm c}^{\rm m}$ (red dashed line), and the total maximum transfer $Tr^{\rm m}=T_{\rm b}^{\rm m}+T_{\rm c}^{\rm m}$ (green dot-dashed line) for different (e) couplings $g$ and (f) drives ${\it \Omega}$, respectively.
In real experiments the dissipations inevitably exist, since the system is also coupled to the outside environment. Similar to the above calculation process, the probability amplitudes with dissipations can be obtained after replacing the corresponding frequencies $\omega_{{\rm a}({\rm b},{\rm c})}$, $\omega_{e_1(e_2)}$, $\omega_{s_1(s_2)}$ with $\omega'_{{\rm a}({\rm b},{\rm c})}$, $\omega'_{e_1(e_2)}$, $\omega'_{s_1(s_2)}$, respectively. Here $\omega'_{a(b,c,{e_1},{e_2},s_1,s_2)}=\omega_{a(b,c,{e_1},{e_2},s_1,s_2)} -i\gamma_{a(b,c,{e_1},{e_2},s_1,s_2)}$ and $\gamma_{a(b,c,{e_1},{e_2},s_1,s_2)}$ are the decay rates of the waveguide and atoms. Compared with that without dissipations, all scattering probabilities are reduced due to the leakages to the external environment. For a low-Q cavity with $\gamma_{\rm a}=\gamma_{\rm b}=\gamma_{\rm c}=0.1$, a drastic decrease of the transfer (yellow curve) is presented in Fig. 2(d), while a high transfer rate of 0.79 is also available for a high-Q cavity (e.g. $Q=4.4\times10^7$)[27] with considerably low decays $\gamma_{\rm a}=\gamma_{\rm b}=\gamma_{\rm c}=0.008$. To gain a deeper insight into the dependence of the routing capability on coupling and drive, Figs. 2(e) and 2(f) show the maximum transfer rates $T_{\rm b}^{\rm m}$, $T_{\rm c}^{\rm m}$ and the total maximum transfer $Tr^{\rm m}=T_{\rm b}^{\rm m}+T_{\rm c}^{\rm m}$. As shown in Fig. 2(e), when $g=0$, CCA-b, and CCA-c are decoupled from CCA-a, and the input photon cannot be redirected into these output channels completely. Increasing the coupling $g$ favors the transfer into channel-b and -c, since these effective reflection potentials mentioned above improve the routing capacity. In sharp contrast to the single-cross waveguide, the total transfer rate $Tr^{\rm m}>0.5$ occurs with the increase of the coupling $g$, and a high transfer $Tr^{\rm m}=0.82$ also emerges for ${\it \Omega}=0.6$. However, maximum transfers are affected slightly for the drive ${\it \Omega}$, as shown in Fig. 2(f); this may result from the common coupling transition channels used by the cross-CCAs, since no additional energy supplement from the drive is necessary. Further, the triple-cross waveguide shown in Fig. 1 is investigated. As the basic calculation process of photon scattering in the triple-cross case is similar but complicated, we choose the concise transfer-matrix approach to obtain the scattering amplitudes. We first review the transmission amplitude $t_1$ and the reflection amplitude $r_1$ for the single-cross case, $$\begin{align} t_1=\frac{Z_1}{Z_1+1},~~~r_1=-\frac{1}{Z_1+1},~~ \tag {10} \end{align} $$ where $Z_1$ has been defined above, and the single-cross transfer matrix $M_1$ can be expressed as $$\begin{align} M_1=\,&\frac{1}{t_1} \begin{bmatrix} t_1^2-r_1^2&r_1\\-r_1&1 \end{bmatrix}\\ =\,&\frac{1}{Z_1}\begin{bmatrix} Z_1-1&-1\\1&Z_1+1 \end{bmatrix}.~~ \tag {11} \end{align} $$ Then, the transfer matrix $F_{\rm S}$ for the S-cross-shaped waveguide is given by $$\begin{align} F_{\rm S}=\prod_{s=1}^{S}M_{s}\tau_{d_{s}},~~ \tag {12} \end{align} $$ where $$\begin{align} \tau_{d_{s}}=\begin{bmatrix} e^{i\chi_{s}}&0\\0&e^{-i\chi_{s}} \end{bmatrix},~~ \tag {13} \end{align} $$ with $\chi_{s}=k[d_{s}-d_{s-1}]$, $(s\geq2)$, and $\chi_1=0$. Here $M_{s}$ is the transfer matrix of the single-cross waveguide at position $d_{s}$. The reflection and transmission amplitudes in the S-cross case can thus be expressed as $$\begin{align} \begin{bmatrix} t_{\rm S}\\0 \end{bmatrix} = \begin{bmatrix} e^{-i kd_{\rm S}}&0\\0&1 \end{bmatrix}F_{\rm S}\begin{bmatrix} 1\\r_{\rm S} \end{bmatrix}.~~ \tag {14} \end{align} $$ It is easy to verify that the calculation results for $S=2$ coincide with that in Eq. (9). For the triple-cross waveguide, the scattering amplitudes can be obtained by solving Eq. (14) with $S=3$, $$\begin{alignat}{1} t_3=\,& -{Z_1Z_2Z_3e^{-i\vartheta_{3}}} [(Z_2-1)e^{i\vartheta_{3}}\\ &+(Z_3+1)e^{-i\vartheta_{2}}+(Z_1+1)e^{i\vartheta_{2}}\\ &-(Z_1+1)(Z_2+1)(Z_3+1) e^{-i\vartheta_{3}} ]^{-1},\\ r_3=\,&[(Z_2+1)(Z_3+1)e^{-i\vartheta_{3}} +(Z_2-1)(Z_1-1)e^{i\vartheta_{3}} \\ &-e^{i\vartheta_{2}}+(Z_1-1)(Z_3+1)e^{-i\vartheta_{2}}] [(Z_2-1)e^{i\vartheta_{3}}\\ &+(Z_3+1)e^{-i\vartheta_{2}}+(Z_1+1)e^{i\vartheta_{2}}\\ &-(Z_1+1)(Z_2+1)(Z_3+1) e^{-i\vartheta_{3}} ]^{-1},~~ \tag {15} \end{alignat} $$ where $\vartheta_{3}=\chi_3+\chi_2$, $\vartheta_{2}=\chi_3-\chi_2$, and $Z_3=\frac{-2i\xi_{\rm a}\sin k_{\rm a}(2i\xi_{\rm c}\sin k_{d}-V_{d})} { G^2_3+V_{\rm am}(2i\xi_{\rm c}\sin k_{d}-V_{d})}$, with $V_{\rm am} =\frac{g_{a_3}^2{\it \Delta}_{s_3}}{{\it \Delta}_{e_3}{\it \Delta}_{s_3}-|{\it \Omega}_3|^2}$, $V_{d} =\frac{g_{d}^2{\it \Delta}_{s_3}}{{\it \Delta}_{e_3}{\it \Delta}_{s_3}-|{\it \Omega}_3|^2}$, $G_3 =\frac{g_{a_3}g_{d} {\it \Delta}_{s_3}}{{\it \Delta}_{e_3}{\it \Delta}_{s_3}-|{\it \Omega}_3|^2}$, and ${\it \Delta}_{e_3(s_3)}=E_k-\omega_{e_3(s_3)}$.
cpl-36-3-034202-fig3.png
Fig. 3. Transmission $T_{\rm a}$ (cyan dash-dotted line), reflection $R_{\rm a}$ (pink dashed line), and transfers $Tr$ (blue solid line) in the triple-cross waveguide for (a) $g=0.3$ and (b) $g=0.8$. (c) Transfers $Tr$ as functions of detuning ${\it \Delta}$ and coupling $g$. Here $d_1=n=4$, and $d_2=m=8$, ${\it \Omega}=0.8$. The other parameters are the same as those in Fig. 2.
Similar to the double-cross waveguide, Figs. 3(a) and (b) show that the high transfer rate $Tr>0.5(Tr=1-T_{\rm a}-R_{\rm a}=1-|t_3|^2-|r_3|^2)$ from the input channel-a into the other channels is also exhibited here, and more high transfer peaks emerge by increasing the coupling strength, since effective potentials generated in the cross-resonators reflect the traveling photons and route more photons into the other output channels. For simplicity, we have set $g_{a_3}=g_{d}=g$, $k_{d}=k$, ${\it \Omega}_3={\it \Omega}$, $E_k^{[d]}=E_k$, and ${\it \Delta}_{e_3(s_3)}={\it \Delta}_{e_1(s_1)}$. In addition, Fig. 3(c) displays how ${\it \Delta}$ and $g$ affect $Tr$ under the conditions of $d_2=nl=4$ and $d_3=ml=8$ in the ($g,{\it \Delta}$) plane. As seen, for $g=0$, the total transfer $Tr$ into the output channels is still less than 0.5. When increasing $g$, more strip regions of $Tr>0.5$ generally emerge in a wide frequency range, and the orange window of high transfer $Tr>0.8$ is also magnified in the scattering spectra. Note that the experimental couplings range from 0 to about 2,[25] and the numerical results show that the maximum transfer can reach 0.83, and it reduces with the decays. In summary, the quantum routing of single photons in a multi-cross-shaped waveguide has been examined, and these results reveal that a high routing capability between channels can be implemented in a wide frequency range by increasing the atom–resonator coupling, because the generated effective potentials reflow the incident photons and redirect more photons into the other channels. As the multi-cross-shaped waveguide case can be generalized using the transfer matrices, the proposed system may be used as a scalable quantum device to perform high-efficiency routing. References The quantum internetPerformance analysis of quantum access network using code division multiple access modelQuantum photonic network on chipOptomechanical systems as single-photon routersSingle-photon multi-ports router based on the coupled cavity optomechanical systemEfficient Routing of Single Photons by One Atom and a Microtoroidal CavityDemonstration of a Single-Photon Router in the Microwave RegimeAll-Optical Switching and Router via the Direct Quantum Control of Coupling between Cavity ModesAll-optical routing of single photons by a one-atom switch controlled by a single photonMott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled CavitiesA proposal for the implementation of quantum gates with photonic-crystal waveguidesCoherent control of photon transmission: Slowing light in a coupled resonator waveguide doped with Λ atomsQuantum supercavity with atomic mirrorsControlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanismIntrinsic cavity QED and emergent quasinormal modes for a single photonQuantum switch for single-photon transport in a coupled superconducting transmission-line-resonator arrayLehmann-Symanzik-Zimmermann reduction approach to multiphoton scattering in coupled-resonator arraysScattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavityDesigning Fano-Like Quantum Routing via Atomic Dipole-Dipole InteractionsQuantum Routing of Single Photons with a Cyclic Three-Level SystemSingle-photon router: Coherent control of multichannel scattering for single photons with quantum interferencesVacuum-Stimulated Raman Scattering Based on Adiabatic Passage in a High-Finesse Optical CavityReversible State Transfer between Light and a Single Trapped AtomCritically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities in Rb 87 Engineering a controllable coupling between two quantized cavity modes via an ensemble of four-level atoms in the diamond configurationA single-photon transistor using nanoscale surface plasmonsVacuum Rabi splitting in semiconductors
[1] Kimble H J 2008 Nature 453 1023
[2] Hu L X, Yang C and He G Q 2017 Chin. Phys. B 26 060304
[3] Zhang Q Y, Xu P and Zhu S N 2018 Chin. Phys. B 27 054207
[4] Agarwal G S and Huang S 2012 Phys. Rev. A 85 021801
[5] Li X, Zhang W Z, Xiong B and Zhou L 2016 Sci. Rep. 6 39343
[6] Aoki T, Parkins A S, Alton D J, Regal C A, Dayan B, Ostby E, Vahala K J and Kimble H J 2009 Phys. Rev. Lett. 102 083601
[7] Hoi I C, Wilson C M, Johansson G, Palomaki T, Peropadre B and Delsing P 2011 Phys. Rev. Lett. 107 073601
[8] Xia K and Twamley J 2013 Phys. Rev. X 3 031013
[9] Shomroni I, Rosenblum S, Lovsky Y, Brechler O, Guendelman G and Dayan B 2014 Science 345 903
[10] Rossini D and Fazio R 2007 Phys. Rev. Lett. 99 186401
[11] Angelakis D G, Santos M F, Yannopapas V and Ekert A 2007 Phys. Lett. A 362 377
[12] Zhou L, Lu J and Sun C P 2007 Phys. Rev. A 76 012313
[13] Zhou L, Dong H, Sun C P and Nori F 2008 Phys. Rev. A 78 063827
[14] Gong Z R, Ian H, Zhou L and Sun C P 2008 Phys. Rev. A 78 053806
[15] Dong H, Gong Z R, Ian H, Zhou L and Sun C P 2009 Phys. Rev. A 79 063847
[16] Liao J Q, Huang J F, Liu Y X, Kuang L M and Sun C P 2009 Phys. Rev. A 80 014301
[17] Shi T and Sun C P 2009 Phys. Rev. B 79 205111
[18] Li M X, Yang J, Lin G W, Niu Y P and Gong S Q 2018 Chin. Phys. B 27 054206
[19] Huang J S, Zhang J H, Wang Y and Xu Z H 2018 Chin. Phys. Lett. 35 034201
[20] Zhou L, Yang L P, Li Yong and Sun C P 2013 Phys. Rev. Lett. 111 103604
[21] Lu J, Zhou L, Kuang L M and Nori F 2014 Phys. Rev. A 89 013805
[22] Hennrich M, Legero T, Kuhn A and Rempe G 2000 Phys. Rev. Lett. 85 4872
[23] Boozer A D, Boca A, Miller R, Northup T E and Kimble H J 2007 Phys. Rev. Lett. 98 193601
[24] Safronova M S and Safronova U I 2011 Phys. Rev. A 83 052508
[25] Chimczak G, Bartkiewicz K, Ficek Z and Tanaś R 2016 arXiv:1608.05282 [quant-ph]
[26] Chang D E, Søensen A S, Demler E A and Lukin M D 2007 Nat. Phys. 3 807
[27] Khitrova G, Gibbs H M, Kira M, Koch S W and Scherer A 2006 Nat. Phys. 2 81