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Few-photon routing via chiral light-matter couplings

  • Ya Yang(杨亚) , ,
  • Jing Lu(卢竞) ,
  • Lan Zhou(周兰)
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  • Synergetic Innovation Center for Quantum Effects and Applications, Key Laboratory for Matter Microstructure and Function of Hunan Province, Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, School of Physics and Electronics, Hunan Normal University, Changsha 410081, China

Author to whom any correspondence should be addressed.

Received date: 2021-11-15

  Revised date: 2021-12-27

  Accepted date: 2021-12-28

  Online published: 2022-03-10

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© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

A quantum router is one of the essential elements in the quantum network. Conventional routers only direct a single photon from one quantum channel into another. Here, we propose a few-photon router. The active element of the router is a single qubit chirally coupled to two independent waveguides simultaneously, where each waveguide mode provides a quantum channel. By introducing the operators of the scatter-free space and the controllable space, the output state of the one-photon and two-photon scattering are derived analytically. It is found that the qubit can direct one and two photons from one port of the incident waveguide to an arbitrarily selected port of the other waveguide with unity, respectively. However, two photons cannot be simultaneously routed to the same port due to the anti-bunch effect.

Cite this article

Ya Yang(杨亚) , Jing Lu(卢竞) , Lan Zhou(周兰) . Few-photon routing via chiral light-matter couplings[J]. Communications in Theoretical Physics, 2022 , 74(2) : 025101 . DOI: 10.1088/1572-9494/ac46a6

1. Introduction

Quantum channels, which serve mainly to distribute quantum information, are building blocks of scalable quantum information processing and quantum communication. In a quantum channel, the information is usually carried by single photon, which is an ideal carrier of a flying qubit for long-distance communication due to the fact that photons have vanishingly small cross sections for direct coupling. However, digital signal processing requires photon–photon interactions on the level of a single quantum. Since the nonlinearities in nonlinear crystals are tiny at the single photon level, it would be fulfilled by the mediation of material systems. Although a single atom is able to have an appreciable influence on the light field, a strong matter-light interaction is desirable for not being masked by the dissipative coupling to the environment. The attainment of strong coupling was pioneered by putting an atom in tiny high-Q cavities [14], where the electric field enclosed in three dimensions, and antibunching has been observed [57]. As the technology proceeds to being smaller on chip structures, strong coupling is also achieved between a trapped atom and the single-photon flying through the waveguide. Then the coherent of single photons controlled by interaction with an atom inside a waveguide have been a subject of considerable interest in recent years [821], since a waveguide confining photons to a one-dimensional (1D) geometry increases the atom-photon interaction and the field remains uniform in the longitudinal direction of propagation. With waveguides becoming basic elements for complex quantum networks, quantum devices operating on the level of a single quantum are necessary for signal processing in waveguides. Quantum router is one of the essential elements in the quantum network, which directs single photons from a coherent input to a separate output. Single-photon router is proposed both theoretically and experimentally by exploiting the strong coupling between an atom and the confined field [2234]. The chiral atom-photon couplings has even been used to direct a single photon from the input port to an arbitrarily selected output port deterministically. However, few attentions have been paid on the routing of two photons from the input port to anarbitrarily selected output port. Here, we proposed a one- and two-photon router. The active element of the router is a single qubit chirally coupled to two independent waveguides simultaneously. The study of the scattering of few photons by the qubit shows that we can route one and two photons from an input port of the incident waveguide to an arbitrarily selected port of the other waveguide with a ratio of unity, and the routed photons are anti-bunched.
This paper is organized as follows: in section 2, we present the theoretical model for a qubit coupled to two identical 1D waveguides. The scatter-free subspace and the controllable subspace has been introduced. In section 3, we study the scattering process of one photon in the real-space and give the expressions for scattering coefficients, the properties of the nonreciprocity are discussed. In section 4, we study the scattering process in the double excitation subspace in the real space. After obtain the output wave-function for the photon-pair incident from one port of a waveguide, the anti-bunch and bunch effect are discussed for photon-pair appearing to the same port, and the conditions of incident photons routed to desired ports are studied. Finally, we conclude with a brief summary of the results.

2. Model for a qubit coupling to two 1D waveguides

The system we consider is depicted in figure 1: a pointlike scatterer e.g. a two-level quantum emitter or qubit at coordinate x = 0, is coupled to two identical 1D waveguides, which we label a and b respectively, We neglect thermal fluctuations and losses. The qubit absorbs and emits photons with rates γpl and γpr into the left- and right-going modes along waveguide p = a, b. The emission of the qubit into guided modes of waveguide p is depicted by γp = γpr + γpl. The Hamiltonian of the total system consists of three parts
$\begin{eqnarray}H={H}_{A}+{H}_{p}+{H}_{\mathrm{Ap}}.\end{eqnarray}$
The qubit Hamiltonian is given by
$\begin{eqnarray}{H}_{A}={\rm{\Omega }}{\sigma }_{\mathrm{ee}},\end{eqnarray}$
where Ω is the bare energy separation between the ground state ∣g⟩ and the excited state ∣e⟩ of the qubit and σee = ∣e⟩⟨e∣. If Ω is far away from the cutoff frequency of the waveguide modes, the waveguide dispersion can be linearized as ωk = vk∣ with the group velocity v > 0. The real-space field operators ${\hat{R}}_{p}(x)$ and ${\hat{L}}_{p}(x)$ (${\hat{R}}_{p}^{\dagger }(x)$ and ${\hat{L}}_{p}^{\dagger }(x)$) annihilate (create) a left- and right-going photon at position x in the waveguide p, and satisfy the bosonic commutation rules $[{\hat{R}}_{p}(x),{\hat{R}}_{{p}^{{\prime} }}^{{}^{{\prime} }\dagger }({x}^{{\prime} })]=[{\hat{L}}_{p}(x),{\hat{L}}_{{p}^{{\prime} }}^{{}^{{\prime} }\dagger }({x}^{{\prime} })]={\delta }_{{{pp}}^{{\prime} }}\delta (x-{x}^{{\prime} })$, $[{\hat{R}}_{p}(x),{\hat{L}}_{{p}^{{\prime} }}^{{}^{{\prime} }\dagger }({x}^{{\prime} })]=0$, while the rest of the commutators vanish. The Hamiltonian of free photons propagating in waveguides is given by
$\begin{eqnarray}{H}_{p}=-{\rm{i}}v\sum _{p=a,b}\int {\rm{d}}x\left[{\hat{R}}_{p}^{\dagger }\left(x\right){\partial }_{x}{\hat{R}}_{p}\left(x\right)-{\hat{L}}_{p}^{\dagger }\left(x\right){\partial }_{x}{\hat{L}}_{p}\left(x\right)\right].\end{eqnarray}$
The variable x takes values in the continuum −∞ < x < +∞. The dipole Hamiltonian under the rotating-wave approximation
$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{Ap}} & = & \displaystyle \sum _{p=a,b}\sqrt{v{\gamma }_{\mathrm{pr}}}\int {\rm{d}}x\delta \left(x\right)\left[{\hat{R}}_{p}^{\dagger }\left(x\right){\sigma }_{-}+{\rm{h}}.{\rm{c}}.\right]\\ & & +\displaystyle \sum _{p=a,b}\sqrt{v{\gamma }_{\mathrm{pl}}}\int {\rm{d}}x\delta \left(x\right)\left[{\hat{L}}_{p}^{\dagger }\left(x\right){\sigma }_{-}+{\rm{h}}.{\rm{c}}.\right],\end{array}\end{eqnarray}$
is adopted to describe the coupling between the qubit and the field as long as the coupling strengths $\sqrt{v{\gamma }_{\mathrm{pr}}}$ and $\sqrt{v{\gamma }_{\mathrm{pl}}}$ are much smaller than the excitation energy Ω. σ = ∣g⟩⟨e∣ is the lowering operator of the qubit. The direct coupling between waveguides is neglected for simplify. This simplification can be realized as long as the distance between the two waveguides is much larger than the characteristic length of a two-level quantum emitter.
Figure 1. (a) Schematic of a few-photon router with four ports: a two-level system interacts with two independent waveguides labeled by a and b. The blue (green) line represents the chiral coupling between the two-level atom and the left and right going photons of the bottom waveguide a (b), and the coupling rates are γar (γbr) and γal (γbl), respectively. (b) An equivalent few-photon router by introducing the scatter-free channels and the controllable channel: two-level system only interacts with the lower C mode with strength determined γ.
We further introduce the following transformation
$\begin{eqnarray}{\hat{E}}_{p}^{\dagger }\left(x\right)=\sqrt{\displaystyle \frac{{\gamma }_{\mathrm{pr}}}{{\gamma }_{p}}}{\hat{R}}_{p}^{\dagger }\left(x\right)+\sqrt{\displaystyle \frac{{\gamma }_{\mathrm{pl}}}{{\gamma }_{p}}}{\hat{L}}_{p}^{\dagger }\left(-x\right),\end{eqnarray}$
$\begin{eqnarray}{\hat{O}}_{p}^{\dagger }\left(x\right)=\sqrt{\displaystyle \frac{{\gamma }_{\mathrm{pl}}}{{\gamma }_{p}}}{\hat{R}}_{p}^{\dagger }\left(x\right)-\sqrt{\displaystyle \frac{{\gamma }_{\mathrm{pr}}}{{\gamma }_{p}}}{\hat{L}}_{p}^{\dagger }\left(-x\right),\end{eqnarray}$
where γp = γpl + γpr, the free Hamiltonian for the waveguides is transformed into
$\begin{eqnarray}{H}_{p}=-{\rm{i}}v\sum _{p=a,b}\int {\rm{d}}{x}\left[{\hat{E}}_{p}^{\dagger }\left(x\right){\partial }_{x}{\hat{E}}_{p}\left(x\right)+{\hat{O}}_{p}^{\dagger }\left(x\right){\partial }_{x}{\hat{O}}_{p}\left(x\right)\right],\end{eqnarray}$
and the interaction Hamiltonian between the qubit and photons reads
$\begin{eqnarray}{H}_{\mathrm{Ap}}=\displaystyle \sum _{p=a,b}\sqrt{v{\gamma }_{p}}\int {\rm{d}}x\delta \left(x\right)\left[{\hat{E}}_{p}^{\dagger }\left(x\right){\sigma }_{-}+{\rm{h}}.{\rm{c}}.\right].\end{eqnarray}$
By defining the creation ${\hat{C}}^{\dagger }$ and ${\hat{D}}^{\dagger }$ as a linear combination of the real-space field operators of two waveguides
$\begin{eqnarray}{\hat{C}}^{\dagger }\left(x\right)=\sqrt{\displaystyle \frac{{\gamma }_{a}}{\gamma }}{\hat{E}}_{a}^{\dagger }\left(x\right)+\sqrt{\displaystyle \frac{{\gamma }_{b}}{\gamma }}{\hat{E}}_{b}^{\dagger }\left(x\right),\end{eqnarray}$
$\begin{eqnarray}{\hat{D}}^{\dagger }\left(x\right)=\sqrt{\displaystyle \frac{{\gamma }_{b}}{\gamma }}{\hat{E}}_{a}^{\dagger }\left(x\right)-\sqrt{\displaystyle \frac{{\gamma }_{a}}{\gamma }}{\hat{E}}_{b}^{\dagger }\left(x\right),\end{eqnarray}$
the Hamiltonian of the total system is the sum of the Hamiltonian Hc of the controllable space and Hdo of the scatter-free space, i.e. H = Hc + Hdo where
$\begin{eqnarray}\begin{array}{rcl}{H}_{c} & = & -{\rm{i}}v\displaystyle \int {\rm{d}}x{\hat{C}}^{\dagger }\left(x\right)\displaystyle \frac{\partial }{\partial x}\hat{C}\left(x\right)+{\rm{\Omega }}{\sigma }_{\mathrm{ee}}\\ & & +\sqrt{v\gamma }\displaystyle \int {\rm{d}}x\delta \left(x\right)\left[{\hat{C}}^{\dagger }\left(x\right){\sigma }_{-}+{\rm{h}}.{\rm{c}}.\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{do}} & = & -{\rm{i}}v\displaystyle \int {\rm{d}}x{\hat{D}}^{\dagger }\left(x\right)\displaystyle \frac{\partial }{\partial x}\hat{D}\left(x\right)\\ & & -{\rm{i}}v\displaystyle \sum _{p=a,b}\displaystyle \int {\rm{d}}x{\hat{O}}_{{p}}^{\dagger }\left(x\right)\displaystyle \frac{\partial }{\partial x}{\hat{O}}_{{p}}\left(x\right).\end{array}\end{eqnarray}$
Here, the qubit couples to the C mode of the field with an effective coupling constant characterized by the total decay of the qubit γ = γa + γb. It can be found that the number of excitations in this system is conserved.

3. Transport property of one photon

As the photons in the O and D spaces propagate freely, the scattering process mainly occurs in the C space. Since a photon lying at x = 0 can be absorbed by the qubit, the eigenstate of Hc takes the form
$\begin{eqnarray}\left|{\rm{\Psi }}\right\rangle =\int {\rm{d}}x{\varphi }_{c}\left(x\right){\hat{C}}^{\dagger }\left(x\right)\left|\varnothing \right\rangle +{u}_{e}{\sigma }_{+}\left|\varnothing \right\rangle ,\end{eqnarray}$
where $\left|\varnothing \right\rangle =\left|g0\right\rangle $ is the combined qubit-field vacuum state with no photon in the field and the qubit in the ground state, ${\hat{C}}^{\dagger }\left(x\right)\left|\varnothing \right\rangle $ is the state that the qubit is in the ground state and a photon is generated from the vacuum of the field at position x, ${\sigma }_{+}\left|\varnothing \right\rangle $ is the state with the qubit in the excited state and no photon in the field. ${\varphi }_{c}\left(x\right)$ and ue are their corresponding amplitudes. The time-independent Schrödinger equation leads to the coupled equations for the amplitudes
$\begin{eqnarray}\varepsilon {\varphi }_{c}\left(x\right)=-{\rm{i}}v{\partial }_{x}{\varphi }_{c}\left(x\right)+\sqrt{v\gamma }{u}_{e}\delta \left(x\right),\end{eqnarray}$
$\begin{eqnarray}\varepsilon {u}_{e}={\rm{\Omega }}{u}_{e}+\sqrt{v\gamma }{\varphi }_{c}\left(0\right).\end{eqnarray}$
A photon with wave vector k = ϵ/v > 0 is sent towards the qubit from the left side will be only absorbed by the qubit at x = 0, which indicates that ${\varphi }_{c}\left(x\right)={{\rm{e}}}^{{\rm{i}}{kx}}$ at the region with x ≠ 0. Since there is no energy loss, substituting the scattering ansatz ${\varphi }_{c}\left(x\right)={{\rm{e}}}^{{\rm{i}}{kx}}[{\rm{\Theta }}(-x)+{\bar{t}}_{{\rm{k}}}{\rm{\Theta }}(x)]$ (Θ(x) is the Heaviside step function) into the equation (11a) yields
$\begin{eqnarray}{\bar{t}}_{k}=\displaystyle \frac{{kv}-{\rm{\Omega }}-{\rm{i}}\gamma /2}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2}.\end{eqnarray}$
The photon coming from one side of the qubit in one waveguide can be scattered into four different ports, resulting in a reflected, transmitted and transferred waves. We aim to determine the scattering coefficients for an incident monochromatic photon. Photons incident from the left side of the qubit in waveguide p are described by the state
$\begin{eqnarray*}\left|{\varphi }_{p}^{\mathrm{in}}\right\rangle =\displaystyle \frac{1}{\sqrt{2\pi }}\int {\rm{d}}x{{\rm{e}}}^{{\rm{i}}{kx}}{\hat{R}}_{p}^{\dagger }\left(x\right)\left|\varnothing \right\rangle .\end{eqnarray*}$
We first decompose the operator ${\hat{R}}_{p}\left(x\right)$ into a linear combination of operators $\hat{C}$, $\hat{D}$ and ${\hat{O}}_{p}$. Since waves in D and O spaces propagate freely, the outgoing wave in C space is obtained by multiplying the incident wave by ${\bar{t}}_{k}$. By transforming back to the right- and left-going operators, the scattering amplitudes read
$\begin{eqnarray}{t}_{\mathrm{pk}}^{p}=\displaystyle \frac{{kv}-{\rm{\Omega }}+{\rm{i}}\left({\gamma }_{\mathrm{pl}}-{\gamma }_{\mathrm{pr}}+{\gamma }_{\bar{p}}\right)/2}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
$\begin{eqnarray}{t}_{\mathrm{pk}}^{\bar{p}}=\displaystyle \frac{-{\rm{i}}\sqrt{{\gamma }_{\mathrm{pr}}{\gamma }_{\bar{{\rm{p}}}{\rm{r}}}}}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
$\begin{eqnarray}{r}_{\mathrm{pk}}^{p}=\displaystyle \frac{-{\rm{i}}\sqrt{{\gamma }_{\mathrm{pr}}{\gamma }_{\mathrm{pl}}}}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
$\begin{eqnarray}{r}_{\mathrm{pk}}^{\bar{p}}=\displaystyle \frac{-{\rm{i}}\sqrt{{\gamma }_{\mathrm{pr}}{\gamma }_{\bar{{\rm{p}}}{\rm{l}}}}}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
where $p\ne \bar{p}$ and $p,\bar{p}\in \left\{a,b\right\}$. These results are also obtained in [30]. Here, ${t}_{\mathrm{pk}}^{p}$ and ${r}_{\mathrm{pk}}^{p}$ are the transmitted and reflected amplitudes of the photon in the waveguide p. ${t}_{\mathrm{pk}}^{\bar{p}}$ and ${r}_{\mathrm{pk}}^{\bar{p}}$ are the reflected and transmitted amplitudes of the photon in the waveguide $\bar{p}$ coming from waveguide p. It can be verified that the probability is conserved as ${T}_{\mathrm{pk}}^{p}+{T}_{\mathrm{pk}}^{\bar{p}}\,+{F}_{\mathrm{pk}}^{\bar{p}}+{F}_{\mathrm{pk}}^{p}=1$, where we have defined
$\begin{eqnarray}{T}_{\mathrm{pk}}^{p}={\left|{t}_{\mathrm{pk}}^{p}\right|}^{2},{T}_{\mathrm{pk}}^{\bar{p}}={\left|{t}_{\mathrm{pk}\ }^{\bar{p}}\right|}^{2},\end{eqnarray}$
$\begin{eqnarray}{F}_{\mathrm{pk}}^{p}={\left|{r}_{\mathrm{pk}}^{p}\right|}^{2},{F}_{\mathrm{pk}}^{\bar{p}}={\left|{r}_{\mathrm{pk}\ }^{\bar{p}}\right|}^{2}.\end{eqnarray}$
For photons incident from the right side of the qubit in waveguide p
$\begin{eqnarray*}\left|{\varphi }_{p}^{\mathrm{in}}\right\rangle =\displaystyle \frac{1}{\sqrt{2\pi }}\int {\rm{d}}x{{\rm{e}}}^{-{\rm{i}}{kx}}{\hat{L}}_{p}^{\dagger }\left(x\right)\left|\varnothing \right\rangle ,\end{eqnarray*}$
one can also obtain the four scattering amplitudes in a similar way
$\begin{eqnarray}{\tilde{t}}_{\mathrm{pk}}^{p}=\displaystyle \frac{{kv}-{\rm{\Omega }}+{\rm{i}}\left({\gamma }_{\mathrm{pr}}-{\gamma }_{\mathrm{pl}}+{\gamma }_{\bar{p}}\right)/2}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
$\begin{eqnarray}{\tilde{t}}_{\mathrm{pk}}^{\bar{p}}=\displaystyle \frac{-{\rm{i}}\sqrt{{\gamma }_{\mathrm{pl}}{\gamma }_{\bar{{\rm{p}}}{\rm{l}}}}}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
$\begin{eqnarray}{\tilde{r}}_{\mathrm{pk}}^{p}=\displaystyle \frac{-{\rm{i}}\sqrt{{\gamma }_{\mathrm{pl}}{\gamma }_{\mathrm{pr}}}}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2},\end{eqnarray}$
$\begin{eqnarray}{\tilde{r}}_{\mathrm{pk}}^{\bar{p}}=\displaystyle \frac{-{\rm{i}}\sqrt{{\gamma }_{\mathrm{pl}}{\gamma }_{\bar{{\rm{p}}}{\rm{r}}}}}{{kv}-{\rm{\Omega }}+{\rm{i}}\gamma /2}.\end{eqnarray}$
Compared the scattering amplitudes in equations (14) to those in equations (13) where the incident photons are from the left side, the scattering amplitudes where the incident photons are from the right side can be obtained by interchanging γprγpl, ${\gamma }_{\bar{{\rm{p}}}{\rm{r}}}\leftrightarrow {\gamma }_{\ \bar{{\rm{p}}}{\rm{l}}}$ in equations (13). It can be found from equations (13) and (14) that the reflectances in the incident waveguide are the same due to ${r}_{\mathrm{pk}}^{p}={\tilde{r}}_{\mathrm{pk}}^{p}$. In the symmetry case with γpr = γpl, ${\gamma }_{\bar{\mathrm{pr}}}={\gamma }_{\bar{\mathrm{pl}}}$, the scattering of single photon is reciprocal. The reciprocal scattering in the incident waveguide can be found with the following two cases: (1) the couplings are symmetry γpr = γpl in waveguide p regardless whether ${\gamma }_{\bar{p}}=0$ or not; (2) ${\gamma }_{\bar{p}}=0$ regardless whether γpr = γpl or not, i.e. the qubit is decoupled with waveguide $\bar{p}$. In figure 2(a), we have plotted the transmittance ${T}_{\mathrm{pk}}^{p}$ (${\tilde{T}}_{\mathrm{pk}}^{p}$) as a function of $2\left(\varepsilon -{\rm{\Omega }}\right)/\gamma $. The transmission zero occurs at γpr = γpl and ${\gamma }_{\bar{p}}=0$, see the blue short dashed line. It is the result of destructive interference between the photon transmitted directly to the right and the photon re-emitted after being absorbed by the two-level atom. However, when qubit-waveguide coupling is chiral, the situation even changes in the incident waveguide. The total transmission happens when the qubit only interacting with the right-going field in the waveguide p. As ${\gamma }_{\bar{p}}\ne 0$, transmission zeros disppear, see the black solid line and the red dotted–dashed line. In figure 2(b), we have plotted the transfer flow as a function of $2\left(\varepsilon -{\rm{\Omega }}\right)/\gamma $. The spontanous emission rate ${\gamma }_{\bar{p}}\ne 0$ guarantees the appearance of transfer. It can be observed from figure 2(b) that the maximum transfer rate is 0.5 which occurs at resonance ϵ = Ω and the spontanous emission rates ${\gamma }_{\bar{p}}={\gamma }_{p}$, which indicates that to improve the routing probability requires ${\gamma }_{\bar{p}}=\gamma /2$ and adjusting γpr,γpl so that γplγpr. When γplγpr and ${\gamma }_{\bar{p}}\ne 0$, $\left|{t}_{\ \mathrm{pk}}^{p}\right|\ne \left|{\tilde{t}}_{\mathrm{pk}}^{p}\right|$, which indicates a nonreciprocal scattering behavior of single photon in the incident waveguid. In figure 3, we have plotted the routing probability as a function of $2\left(\varepsilon -{\rm{\Omega }}\right)/\gamma $ by fixing ${\gamma }_{\bar{p}}=\gamma /2$ for γplγpr. It can be observed that the routing probability achieves its maximum value 1 at γpl = 0 for the photons incident from the left side, or at γpr = 0 for the photons incident from the right side. When the qubit does not absorb the left-going (right-going) incident photon in the incident waveguide featured by γpl(γpr) = 0, there is no reflection in the waveguide p since ${r}_{\mathrm{pk}}^{p}$(${\tilde{r}}_{\mathrm{pk}}^{p}$) = 0, the probability for finding the photon in the waveguide $\bar{p}$ can be maximized when the energy of the photon is in resonance with the qubit, however, a destructive interference is needed to cancel the transmission in the incident waveguide, which is obtained by letting ${\gamma }_{\bar{p}}={\gamma }_{\mathrm{pl}}\left({\gamma }_{\mathrm{pr}}\right)$, then the photon is routed by unity. Furthermore, if the qubit does not interact with the left-going (right-going) photon in waveguide $\bar{p}$, i.e. ${\gamma }_{\bar{{\rm{p}}}{\rm{l}}}=0$, the amplitude ${r}_{\mathrm{pk}}^{\bar{p}}$(${\tilde{t}}_{\mathrm{pk}}^{\bar{p}}$) cancels out, which means that single photon is deterministically directed from the waveguide p to the right-side of the waveguide $\bar{p}$. Thus, a single-photon router can be constructed.
Figure 2. (a) Transmission flow ${T}_{\mathrm{pk}}^{p}\left({\tilde{T}}_{\mathrm{pk}}^{p}\right)$ as a function of energy $2\left(\varepsilon -{\rm{\Omega }}\right)/\gamma $. (b) Transfer flow ${T}_{\mathrm{pk}}^{\bar{p}}+{F}_{\mathrm{pk}}^{\bar{p}}$ (${\tilde{T}}_{\mathrm{pk}}^{\bar{p}}+{\tilde{F}}_{\mathrm{pk}}^{\bar{p}}$) as a function of $2\left(\varepsilon -{\rm{\Omega }}\right)/\gamma $.
Figure 3. (a) ${T}_{\mathrm{pk}}^{\bar{p}}+{F}_{\mathrm{pk}\ }^{\bar{p}}$ and (b) ${\tilde{T}}_{\mathrm{pk}}^{\bar{p}}+{\tilde{F}}_{\mathrm{pk}}^{\bar{p}}$ plotted as functions of $2\left(\varepsilon -{\rm{\Omega }}\right)/\gamma $ by fixing ${\gamma }_{\bar{p}}=\gamma /2$ for γ prγpl.

4. Transport property of two photons

In this section, we consider the scattering process in the double-excitation subspace. Since any photons incident far away from the qubit in waveguides can be decomposed to the Oa, Ob, D and C modes, photons in the Oa, Ob, and D modes propagate freely, so we need to find the two-photon scattering state in the C space. An eigenstate for Hamiltonian Hc is a superposition of two-excitation states:
$\begin{eqnarray}\begin{array}{rcl}{\left|{\rm{\Phi }}\right\rangle }_{\mathrm{CC}} & = & \displaystyle \int {\rm{d}}{x}_{1}{\rm{d}}{x}_{2}{\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right)\displaystyle \frac{{C}^{\dagger }\left({x}_{1}\right){C}^{\dagger }\left({x}_{2}\right)}{\sqrt{2}}\left|\varnothing \right\rangle \\ & & +\displaystyle \int {\rm{d}}{xe}\left(x\right){C}^{\dagger }\left(x\right){\sigma }_{+}\left|\varnothing \right\rangle ,\end{array}\end{eqnarray}$
where $e\left(x\right)$ is the probability amplitude with one photon in waveguides while the atom in the excited state, and ${\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right)$ is the wave function of the two-photon state in the waveguides and meets the permutation symmetry ${\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right)\,={\phi }_{\mathrm{cc}}\left({x}_{2},{x}_{1}\right)$. The real-space equations of the amplitudes read
$\begin{eqnarray}\begin{array}{rcl}E{\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right) & = & -{\rm{i}}v\left({\partial }_{{x}_{1}}+{\partial }_{{x}_{2}}\right){\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right)\\ & & +\sqrt{\displaystyle \frac{v\gamma }{2}}\left[\delta \left({x}_{1}\right)e\left({x}_{2}\right)+\delta \left({x}_{2}\right)e\left({x}_{1}\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\left(E-{\rm{\Omega }}\right)e\left(x\right) & = & {\rm{i}}v{\partial }_{x}e\left(x\right)\\ & & +\sqrt{\displaystyle \frac{v\gamma }{2}}\left[{\phi }_{\mathrm{cc}}\left(0,x\right)+{\phi }_{\mathrm{cc}}\left(x,0\right)\right].\end{array}\end{eqnarray}$
The boson statistics allowed us to confine the solution on x1 < x2 and the solution for x1 > x2 can be obtained by interchanging x1$\Longleftrightarrow $x2. At the region that all photons are away from the qubit, there are two kinds of solutions [8, 10]: plane waves with wave vectors k and ${k}^{{\prime} }$ and energy $E=v\left(k+{k}^{{\prime} }\right)$
$\begin{eqnarray}{\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right)={B}_{\alpha }{{\rm{e}}}^{\ {\rm{i}}{{kx}}_{1}}{{\rm{e}}}^{{\rm{i}}{k}^{{\prime} }{x}_{2}}+{A}_{\alpha }{{\rm{e}}}^{{\rm{i}}{k}^{{\prime} }{x}_{1}}{{\rm{e}}}^{{\rm{i}}{{kx}}_{2}},\end{eqnarray}$
and two-photon bound state
$\begin{eqnarray}{\phi }_{\mathrm{cc}}\left({x}_{1},{x}_{2}\right)={C}_{\alpha }{{\rm{e}}}^{\ {\rm{i}}E\left({x}_{1}+{x}_{2}\right)/\left(2v\right)}{{\rm{e}}}^{-\beta \left|{x}_{2}-{x}_{1}\right|/v},\end{eqnarray}$
where regions 0 < x1 < x2, x1 < 0 < x2, and x1 < x2 < 0 are denoted by α = 1, 2, 3 respectively. The interaction between the qubit and field gives the boundary conditions [8]
$\begin{eqnarray}0={\phi }_{\mathrm{cc}}\left({0}^{+},{x}_{2}\right)-{\phi }_{\mathrm{cc}}\left({0}^{-},{x}_{2}\right)+{\rm{i}}\sqrt{\displaystyle \frac{\gamma }{2v}}e\left({x}_{2}\right),\end{eqnarray}$
$\begin{eqnarray}0=\left({\rm{\Omega }}-E-{\rm{i}}v{\partial }_{{x}_{2}}\right)e\left({x}_{2}\right)+\sqrt{2v\gamma }{\phi }_{\mathrm{cc}}\left(0,{x}_{2}\right),\end{eqnarray}$
for waves at the position x2 > x1 = 0, and
$\begin{eqnarray}0={\phi }_{\mathrm{cc}}\left({x}_{1},{0}^{+}\right)-{\phi }_{\mathrm{cc}}\left({x}_{1},{0}^{-}\right)+{\rm{i}}\sqrt{\displaystyle \frac{\gamma }{2v}}e\left({x}_{1}\right),\end{eqnarray}$
$\begin{eqnarray}0=\left({\rm{\Omega }}-E-{\rm{i}}v{\partial }_{{x}_{1}}\right)e\left({x}_{1}\right)+\sqrt{2v\gamma }{\phi }_{\mathrm{cc}}\left({x}_{1},0\right),\end{eqnarray}$
for waves at the position x2 = 0 > x1 as well as the matching condition
$\begin{eqnarray}e\left({0}^{+}\right)=e\left({0}^{-}\right).\end{eqnarray}$
The conditions help us to find
$\begin{eqnarray}{B}_{1}={\bar{t}}_{k}{B}_{2},{B}_{2}={\bar{t}}_{{k}^{{\prime} }}{B}_{3},\end{eqnarray}$
$\begin{eqnarray}{A}_{1}={\bar{t}}_{p}{A}_{2},{A}_{2}={\bar{t}}_{k}{A}_{3},\end{eqnarray}$
$\begin{eqnarray}{B}_{3}=\displaystyle \frac{{k}^{{\prime} }v-{kv}+{\rm{i}}\gamma }{{k}^{{\prime} }v-{kv}-{\rm{i}}\gamma }{A}_{3},\end{eqnarray}$
for the plane waves in equation (17) and β = γ/2,
$\begin{eqnarray}{C}_{1}=\displaystyle \frac{E-2{\rm{\Omega }}-2{\rm{i}}\gamma }{E-2{\rm{\Omega }}}{C}_{2},\end{eqnarray}$
$\begin{eqnarray}{C}_{2}=\displaystyle \frac{E-2{\rm{\Omega }}}{E-2{\rm{\Omega }}+2{\rm{i}}\gamma }{C}_{3},\end{eqnarray}$
$\begin{eqnarray}{t}_{E}\equiv \displaystyle \frac{{C}_{1}}{{C}_{3}}=\displaystyle \frac{E-2{\rm{\Omega }}-2{\rm{i}}\gamma }{E-2{\rm{\Omega }}+2{\rm{i}}\gamma },\end{eqnarray}$
for bound state in equation (18).
The solution of the eigenstate in the real space allows us to consider the two-photon scattering by the qubit. A two-photon state incoming from the left in waveguide a is given by
$\begin{eqnarray}\begin{array}{rcl}\left|{\varphi }_{\mathrm{Ra}}^{\mathrm{in}}\right\rangle & = & \displaystyle \int {\rm{d}}{x}_{1}{\rm{d}}{x}_{2}\left[{X}_{k,{k}^{{\prime} }}\left({x}_{1},{x}_{2}\right)+{Y}_{k,{k}^{{\prime} }}\left({x}_{1},{x}_{2}\right)\right]\\ & & \times \displaystyle \frac{1}{2}{R}_{a}^{\dagger }\left({x}_{1}\right){R}_{a}^{\dagger }\left({x}_{2}\right)\left|\varnothing \right\rangle .\end{array}\end{eqnarray}$
The amplitude with the center-of-mass coordinate ${x}_{c}\,=\left({x}_{1}+{x}_{2}\right)/2$ and the relative coordinate x = x1x2 reads
$\begin{eqnarray}{X}_{k,{k}^{{\prime} }}\left({x}_{c},x\right)=\displaystyle \frac{1}{2\pi }{{\rm{e}}}^{{\rm{i}}\displaystyle \frac{E}{v}{x}_{c}}{{\rm{e}}}^{{\rm{i}}\displaystyle \frac{{\rm{\Delta }}}{v}x},\end{eqnarray}$
$\begin{eqnarray}{Y}_{k,{k}^{{\prime} }}\left({x}_{c},x\right)=\displaystyle \frac{1}{2\pi }{{\rm{e}}}^{{\rm{i}}\displaystyle \frac{E}{v}{x}_{c}}{{\rm{e}}}^{-{\rm{i}}\displaystyle \frac{{\rm{\Delta }}}{v}x},\end{eqnarray}$
where the total energy of the two photons $E=\left(k+{k}^{{\prime} }\right)v$ and the energy difference between two photons ${\rm{\Delta }}=\left(k-{k}^{{\prime} }\right)v/2$. For later convenience, we introduce
$\begin{eqnarray}b\left({y}_{1},{y}_{2}\right)=\displaystyle \frac{\sqrt{2}}{2\pi }\displaystyle \frac{4{\gamma }^{2}{{\rm{e}}}^{{\rm{i}}\tfrac{E}{v}{y}_{1}}{{\rm{e}}}^{{\rm{i}}\tfrac{\left(E-2{\rm{\Omega }}\right)}{2v}\left|{y}_{2}\right|}{{\rm{e}}}^{-\tfrac{\gamma }{2v}\left|{y}_{2}\right|}}{4{{\rm{\Delta }}}^{2}-{\left(E-2{\rm{\Omega }}+{\rm{i}}\gamma \right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}{S}_{k,{k}^{{\prime} }}\left({y}_{1},{y}_{2}\right)=\displaystyle \frac{\sqrt{2}}{2\pi }{{\rm{e}}}^{{\rm{i}}\displaystyle \frac{E}{v}{y}_{1}}\cos \left(\displaystyle \frac{{\rm{\Delta }}}{v}{y}_{2}\right).\end{eqnarray}$
Each of photons impinges on the qubit, being eventually scattered off. Each one might be reflected to the left side of the qubit (denoted as port 1) and transmitted to the right side of the qubit (denoted as port 2) in the same waveguide, or be directed to the left side of the qubit (denoted as port 3) and transmitted to the right side of the qubit (denoted as port 4) instead of continuing in the same waveguide. Then the output state contains ten terms: Both photons are reflected to port 1 or 3 with amplitudes ${\rho }_{11}\left({x}_{1},{x}_{2}\right)$ or ${\rho }_{33}\left({x}_{1},{x}_{2}\right);$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{11}\left({x}_{c},x\right) & = & {r}_{\mathrm{ak}}^{a}{r}_{{\mathrm{ak}}^{{\prime} }}^{a}{S}_{k,{k}^{{\prime} }}\left(-{x}_{c},x\right)\\ & & -\displaystyle \frac{{\gamma }_{\mathrm{ar}}{\gamma }_{\mathrm{al}}}{{\gamma }^{2}}b\left(-{x}_{c},x\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{33}\left({x}_{c},x\right) & = & {r}_{\mathrm{ak}}^{b}{r}_{{\mathrm{ak}}^{{\prime} }}^{b}{S}_{k,{k}^{{\prime} }}\left(-{x}_{c},x\right)\\ & & -\displaystyle \frac{{\gamma }_{\mathrm{ar}}{\gamma }_{\mathrm{bl}}}{{\gamma }^{2}}b\left(-{x}_{c},x\right).\end{array}\end{eqnarray}$
Both are transmitted to port 2 or 4 with amplitudes ${\rho }_{22}\left({x}_{1},{x}_{2}\right)$ and ${\rho }_{44}\left({x}_{1},{x}_{2}\right);$
$\begin{eqnarray}{\rho }_{22}\left({x}_{c},x\right)={t}_{\mathrm{ak}}^{a}{t}_{{\mathrm{ak}}^{{\prime} }}^{a}{S}_{k,{k}^{{\prime} }}\left({x}_{c},x\right)-\displaystyle \frac{{\gamma }_{\mathrm{ar}}^{2}}{{\gamma }^{2}}b\left({x}_{c},x\right),\end{eqnarray}$
$\begin{eqnarray}{\rho }_{44}\left({x}_{c},x\right)={t}_{\mathrm{ak}}^{b}{t}_{{\mathrm{ak}}^{{\prime} }}^{b}{S}_{k,{k}^{{\prime} }}\left({x}_{c},x\right)-\displaystyle \frac{{\gamma }_{\mathrm{ar}}{\gamma }_{\mathrm{br}}}{{\gamma }^{2}}b\left({x}_{c},x\right).\end{eqnarray}$
One is reflected to port 1 and the other is directed to ports 3, 4 with amplitudes ${\rho }_{13}\left({x}_{1},{x}_{2}\right)$ and ${\rho }_{14}\left({x}_{1},{x}_{2}\right);$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{13}\left({x}_{c},x\right) & = & \sqrt{2}{r}_{\mathrm{ak}}^{a}{r}_{{\mathrm{ak}}^{{\prime} }}^{b}{S}_{k,{k}^{{\prime} }}\left(-{x}_{c},x\right)\\ & & -\displaystyle \frac{{\gamma }_{\mathrm{ar}}\sqrt{2{\gamma }_{\mathrm{ar}}{\gamma }_{\mathrm{bl}\ }}}{{\gamma }^{2}}b\left(-{x}_{c},x\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{14}\left({x}_{c},x\right) & = & \sqrt{2}{r}_{\mathrm{ak}}^{a}{t}_{{\mathrm{ak}}^{{\prime} }}^{b}{S}_{k,{k}^{{\prime} }}\left(-\displaystyle \frac{x}{2},2{x}_{c}\right)\\ & & -\displaystyle \frac{{\gamma }_{\mathrm{ar}}\sqrt{2{\gamma }_{\mathrm{al}}{\gamma }_{\mathrm{br}\ }}}{{\gamma }^{2}}b\left(-\displaystyle \frac{x}{2},2{x}_{c}\right).\end{array}\end{eqnarray}$
One is transmitted to port 2 and the other is directed to port 3 or 4 with amplitudes ${\rho }_{23}\left({x}_{1},{x}_{2}\right)$ or ${\rho }_{24}\left({x}_{1},{x}_{2}\right);$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{23}\left({x}_{c},x\right) & = & {r}_{{\mathrm{ak}}^{{\prime} }}^{b}{t}_{\mathrm{ak}}^{a}{X}_{k,{k}^{{\prime} }}\left(\displaystyle \frac{x}{2},2{x}_{c}\right)+{r}_{\mathrm{ak}}^{b}{t}_{\mathrm{ak}{\ }^{{\prime} }}^{a}{Y}_{k,{k}^{{\prime} }}\left(\displaystyle \frac{x}{2},2{x}_{c}\right)\\ & & -\displaystyle \frac{\sqrt{2{\gamma }_{\mathrm{ar}}^{3}{\gamma }_{\mathrm{bl}}}}{{\gamma }^{2}}b\left(\displaystyle \frac{x}{2},2{x}_{c}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{24}\left({x}_{c},x\right) & = & {t}_{{\mathrm{ak}}^{{\prime} }}^{b}{t}_{\mathrm{ak}}^{a}{X}_{k,{k}^{{\prime} }}\left({x}_{c},x\right)+{t}_{\mathrm{ak}}^{b}{t}_{{\mathrm{ak}}^{{\prime} }}^{a}{Y}_{k,{k}^{{\prime} }}\left({x}_{c},x\right)\\ & & -\displaystyle \frac{\sqrt{2{\gamma }_{\mathrm{ar}}^{3}{\gamma }_{\mathrm{br}}}}{{\gamma }^{2}}b\left({x}_{c},x\right).\end{array}\end{eqnarray}$
One is reflected to port 1 and the other is transmitted to port 2 with amplitude
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{12}\left({x}_{c},x\right) & = & {r}_{{\mathrm{ak}}^{{\prime} }}^{a}{t}_{\mathrm{ak}}^{a}{X}_{k,{k}^{{\prime} }}\left(\displaystyle \frac{x}{2},2{x}_{c}\right)+{r}_{\mathrm{ak}}^{a}{t}_{\mathrm{ak}{\ }^{{\prime} }}^{a}{Y}_{k,{k}^{{\prime} }}\left(\displaystyle \frac{x}{2},2{x}_{c}\right)\\ & & -\displaystyle \frac{\sqrt{2{\gamma }_{\mathrm{ar}}^{3}{\gamma }_{\mathrm{al}}}}{{\gamma }^{2}}b\left(\displaystyle \frac{x}{2},2{x}_{c}\right).\end{array}\end{eqnarray}$
One is directed to port 3 and the other is directed to port 4 with amplitude
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{34}\left({x}_{c},x\right) & = & \sqrt{2}{r}_{{\mathrm{ak}}^{{\prime} }}^{b}{t}_{\mathrm{ak}}^{b}{S}_{k,{k}^{{\prime} }}\left(\displaystyle \frac{x}{2},2{x}_{c}\right)\\ & & -\displaystyle \frac{{\gamma }_{\mathrm{ar}}\sqrt{2{\gamma }_{\mathrm{br}}{\gamma }_{\mathrm{bl}\ }}}{{\gamma }^{2}}b\left(\displaystyle \frac{x}{2},2{x}_{c}\right).\end{array}\end{eqnarray}$
The sum of the amplitude ρnm multiplying the corresponding state constructs the output state $| {\varphi }_{\mathrm{Ra}}^{\mathrm{out}}\rangle $ of the two photons.
It can be found from equations (26), (27), (28a) and (29b) that the probabilities for two photons appearing to the same port (denoted by ${R}_{\mathrm{jj}}=| {\rho }_{\mathrm{jj}}\left({x}_{c},x\right){| }^{2}$, ${T}_{\mathrm{jj}}=| {\rho }_{\mathrm{jj}}\left({x}_{c},x\right){| }^{2}$) and the same side of the qubit are independent of the center-of-mass coordinate. However, the probabilities for two photons traveling to different sides of the qubit are independent of relative coordinate, which can be seen from equation (28b), (29a), (30) and (31). The spatial features of the output photons are characterized by the interference between the plane wave and the wave of the bound state. When the energies of the two incident photons are equal and on resonance with the atom (i.e. ${vk}={{vk}}^{{\prime} }={\rm{\Omega }}$), only anti-bunching can be found for output photons which change the initial propagating direction to the same port, the direction change of photons are caused by the absorption and emission of photons by the qubit, but qubit can only absorb one photon and emit it later once. And the bunched output photons can only be observed when they keep the initial propagating direction. In figures 4(a)–(d), we have plotted the probability as a function of the scaled relative coordinate γx. When ${vk}={{vk}}^{{\prime} }={\rm{\Omega }}$, T22 (green short dashed line in figures 4(a)–(c)) decays exponentially as ∣x∣ increases, and thus they are in a bound state. When the total energy of the photon-pair is kept on resonance with the qubit while the energy difference between the photons increases (E − 2Ω = 0, Δ = γ/2), the peak at x = 0 reduces zero, then the photon-pair transmitted to port 2 change from bunching to antibunching. However, the photon-pair traveling away from their incident direction are always antibunching. In figure 4(a), R33 and T44 is always zero in the absence of the other waveguide. In figure 4(b), R11 and R33 vanish due to no coupling between the qubit and the left-going modes of both waveguides. However, when the qubit is allowed to emit a photon to the left-going modes of the waveguide b, T44 decreases. To understand how the change from bunching to antibunching depend on the parameters, we have plotted T22 for γar = γ/2 in figure 4(e) and for a fixed total energy E = 2Ω in figure 4(f). The solid lines in figures 4(e)–(f) present the condition for T22(0) = 0. It can be found that at the condition of γar = γ/2, the photon-pair exhibit a bunching effect as long as ∣Δ∣ < γ/2 no matter what the value of total energy E is, and at the condition of E = 2Ω, bunching effect appears to the photon-pair as long as γar < γ/4.
Figure 4. (a)–(d) The probability as a function of the scaled relative coordinate γx. When ${vk}={{vk}}^{{\prime} }={\rm{\Omega }}$ for (a)–(c) and E − 2Ω = 0, Δ = γ/2 for (d), where the damping rates are setting as follow: (a) γar = γal = γ/2, γbr = γbl = 0; (b) γar = γbr = γ/2, γal = γbl = 0; (c) γar = γ/2, γal = 0, γbr = γ/6, γbl = γ/3, (d) γar = γ/2, γal = 0, γbr = γ/6, γbl = γ/3. The condition of the vanishing probability for two photons transmitted to port 2 at x = 0 when (e) γar = γ/2 and (f) E − 2Ω = 0.
To discuss the routing of the photon-pair from the incident waveguide into the other waveguide, we introduce the definition of the detection probabilities as
$\begin{eqnarray}{P}_{\mathrm{nm}}=\displaystyle \frac{\int {\rm{d}}{x}_{1}{\rm{d}}{x}_{2}{\left|{\rho }_{\mathrm{nm}}\left({x}_{1},{x}_{2}\right)\right|}^{2}}{\langle {\varphi }_{\mathrm{Ra}}^{\ \mathrm{out}}| {\varphi }_{\mathrm{Ra}}^{\mathrm{out}}\rangle },\end{eqnarray}$
which denote the photons detected at ports n and m. The number of the probabilities is ten, and the ten probabilities add up to unity. The total probability for the photon-pair to be routed is depicted by P33 + P44 + P34Pb. As the integral over coordinates ranges from negative infinite to positive infinite, the contribution of the bound state is negligible, thus the single-photon scattering coefficients are the key to maximize the routing probability. When the energy of each incident photon is equal to the bare energy separation of the qubit (${vk}={{vk}}^{{\prime} }={\rm{\Omega }}$), the detection probability has a very simple expression
$\begin{eqnarray}{P}_{11}\propto {\left({F}_{\mathrm{ak}}^{a}\right)}^{2},{P}_{22}\propto {\left({T}_{\mathrm{ak}}^{a}\right)}^{2},\end{eqnarray}$
$\begin{eqnarray}{P}_{33}\propto {\left({F}_{\mathrm{ak}}^{b}\right)}^{2},{P}_{44}\propto {\left({T}_{\mathrm{ak}}^{b}\right)}^{2},\end{eqnarray}$
$\begin{eqnarray}{P}_{12}\propto 2{T}_{\mathrm{ak}}^{a}{F}_{\mathrm{ak}}^{a},{P}_{34}\propto 2{T}_{\mathrm{ak}}^{b}{F}_{\mathrm{ak}}^{b},\end{eqnarray}$
$\begin{eqnarray}{P}_{13}\propto 2{F}_{\mathrm{ak}}^{a}{F}_{\mathrm{ak}}^{b},{P}_{24}\propto 2{T}_{\mathrm{ak}}^{a}{T}_{\mathrm{ak}}^{b},\end{eqnarray}$
$\begin{eqnarray}{P}_{14}\propto 2{T}_{\mathrm{ak}}^{a}{F}_{\mathrm{ak}}^{b},{P}_{23}\propto 2{F}_{\mathrm{ak}}^{a}{T}_{\mathrm{ak}}^{b}.\end{eqnarray}$
From the discussion in section 2, the routing probability Pb = 1 can be achieved if a single photon is fully routed which is achieved at γb = γar = γ/2. Furthermore, the photon-pair is directed to port 3(4) with unity when γbr(γbl) = 0 since only left-going (right-going) modes interact with the qubit.

5. Discussion and conclusion

We have studied the coherent scattering process of few photons in two waveguide modes chirally coupled to a qubit. The output states of the photons are derived analytically in real-space for an incident one- and two-photon state after scattering off the qubit by introducing the scatter-free channel and the controllable channel. (1) In the one-photon case, the scattering coefficients are obtained. The properties of nonreciprocity are discussed. It is found that the maximum probability for the photon to be routed is 50% in the nonreciprocal case, however, the photon can be directed from the incident waveguide to an arbitrarily selected port of the other waveguide with unity in the reciprocal case. (2) In the two-photon case, the wavefunctions to each port are obtained. By notice that the probabilities for two photons appearing to the same port and the same side of the qubit are determined only by the relative coordinate of the two photons, and the probabilities for two photons traveling to different sides of the qubit are determined only by the center-of-mass coordinate, we found that photons which changing the initial propagating direction to the same port only display anti-bunching effect, the photon-pair transmitted in the incident waveguide can change from bunching to antibunching. Although two photons cannot be simultaneously routed to the same port since the qubit can only absorb one photon and emit it later once, the qubit can direct two photons from one port of the incident waveguide to an arbitrarily selected port of the other waveguide with unity, which means a single photon source can be achieved in the other waveguide.

This work was supported by NSFC Grants No. 11 975 095, No. 12 075 082, No.11935006, and the science and technology innovation Program of Hunan Province (Grant No. 2020RC4047).

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Outlines

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