Appendix A: Two-Photon Scattering Eigenstate
In this Appendix, we provide a step to derive the scattering states for two incident photons in the right-going mode. Decomposing the right-going mode to the even and odd modes by \(a_{R}^{\dag } (x)=[a_{e}^{\dag } (x)+a_{o}^{\dag } (x)]/\sqrt {2}\), the incoming state [(8)] can be written as
$$\begin{array}{@{}rcl@{}} |\psi_{i}\rangle &=&\sum\limits_{n}[\frac{1}{2}\int dx_{1}dx_{2}\phi_{k}(x_{1},x_{2})\frac{1}{\sqrt{2}}a_{e}^{\dag} (x_{1})a_{e}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b} \\ &&+\frac{1}{2}\int dx_{1}dx_{2}\phi_{k}(x_{1},x_{2})\frac{1}{\sqrt{2}} a_{o}^{\dag} (x_{1})a_{o}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b} \\ &&+\frac{1}{\sqrt{2}}\int dx_{1}dx_{2}\phi_{k}(x_{1},x_{2})\frac{1}{\sqrt{2} }a_{e}^{\dag} (x_{1})a_{o}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b}] \end{array} $$
(A.1)
where |ϕ〉 is no photon in the waveguide. The general two-photon scattering state of the system in the spaces with even and odd modes can be written as (10) with
$$\begin{array}{@{}rcl@{}} \psi_{ee} &=& \sum\limits_{n}\int\int dx_{1} dx_{2}\phi_{ee}(x_{1},x_{2},n)\frac{1}{ \sqrt{2}}a^{\dag}_{e}(x_{1})a^{\dag}_{e}(x_{2})|0\rangle_{c}|\phi\rangle|n \rangle_{b}\\ &&+\sum\limits_{n}\int dx\phi_{ae}(x,n)a^{\dag}_{e}(x)a^{\dag}|0\rangle_{c}|\phi\rangle|\widetilde{n} (1)\rangle_{b} \\ &&+\sum\limits_{n}\phi_{aa}\frac{1}{\sqrt{2}} a^{\dag}a^{\dag}|0\rangle_{c}|\phi \rangle|\widetilde{n}(2)\rangle_{b} \end{array} $$
(A.2)
$$\begin{array}{@{}rcl@{}} \phi_{oe} &=& \sum\limits_{n}\int\int dx_{1} dx_{2}\phi_{oe}(x_{1},x_{2},n)a^{\dag}_{o}(x_{1})a^{\dag}_{e}(x_{2})|0 \rangle_{c}|\phi\rangle|n\rangle_{b}\\ && +\sum\limits_{n}\int dx\phi_{oa}(x,n)a^{\dag}_{o}(x)a^{\dag}|0\rangle_{c}|\phi\rangle|\widetilde{n} (1)\rangle_{b} \end{array} $$
(A.3)
$$\begin{array}{@{}rcl@{}} \phi_{eo} &=& \sum\limits_{n}\int\int dx_{1} dx_{2}\phi_{eo}(x_{1},x_{2},n)a^{\dag}_{o}(x_{1})a^{\dag}_{e}(x_{2})|0 \rangle_{c}|\phi\rangle|n\rangle_{b}\\ && +\sum\limits_{n}\int dx\phi_{ao}(x,n)a^{\dag}_{o}(x)a^{\dag}|0\rangle_{c}|\phi\rangle|\widetilde{n} (1)\rangle_{b} \end{array} $$
(A.4)
$$\begin{array}{@{}rcl@{}} \phi_{oo}&=&{\sum}_{n}\int \int dx_{1}dx_{2}\phi_{oo}(x_{1},x_{2},n)\frac{1}{ \sqrt{2}}a_{o}^{\dag} (x_{1})a_{o}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b} \end{array} $$
(A.5)
where ϕ
i
j
(i,j = e,o,a) is the amplitude of the two photons with one photon in mode i and the other in mode j, subscript e(o) stands for the even (odd) mode and subscript a stands for the cavity mode. In order to satisfy the statistical property of photons, the amplitudes satisfy the relations ϕ
e
e
(x
1, x
2) = ϕ
e
e
(x
2, x
1), ϕ
o
o
(x
1, x
2) = ϕ
o
o
(x
2, x
1),ϕ
e
o
(x
1, x
2) = ϕ
o
e
(x
2, x
1), and ϕ
o
a
(x) = ϕ
a
o
(x). Under the two-photon transport of the frequency E = v
g
(k
1 + k
2) + n
0
ω
m
, by solving Schr\(\ddot {o}\)dinger equation H|Ψ〉 = E|Ψ〉, we obtain the amplitudes for the two-photon scattering state
$$\begin{array}{@{}rcl@{}} \phi_{ee}(x_{1},x_{2},n) \!&\,=\,&\!\frac{1}{\sqrt{2}}[\phi_{e,k_{1}}(x_{1})\phi_{e,k_{2}}(x_{2})+\phi_{e,k_{1}}(x_{2})\phi_{e,k_{2}}(x_{1}) \\ &&+\{\theta (x_{2}-x_{1})\theta (x_{1})Be^{iE\frac{x_{c}}{v_{g}}} e^{\{i[E-2(\omega_{eff}-n\omega_{m}-\delta )]-(\kappa -{\Gamma} )\}\frac{x}{ 2v_{g}}}\,+\,(x_{2}\longleftrightarrow x_{1})\}\\ && \end{array} $$
(A.6)
$$\begin{array}{@{}rcl@{}} \phi_{oe}(x_{1},x_{2},n)=\frac{1}{\sqrt{2}}[\phi_{o,k_{1}}(x_{1})\phi _{e,k_{2}}(x_{2})+\phi_{e,k_{1}}(x_{2})\phi_{o,k_{2}}(x_{1})] \end{array} $$
(A.7)
$$\begin{array}{@{}rcl@{}} \phi_{oo}(x_{1},x_{2})=\frac{1}{\sqrt{2}}[\phi_{o,k_{1}}(x_{1})\phi _{o,k_{2}}(x_{2})+\phi_{o,k_{1}}(x_{2})\phi_{o,k_{2}}(x_{1})] \end{array} $$
(A.8)
$$\begin{array}{@{}rcl@{}} \phi_{oa}(x_{i},m)=\rho_{k_{1}}e^{ik_{1}x_{i}}+\rho_{k_{1}}e^{ik_{2}x_{i}} \end{array} $$
(A.9)
$$\begin{array}{@{}rcl@{}} \phi_{ae}(x_{i},m) &=&\theta (-x_{i})(\mu_{k_{1}}e^{ik_{1}x_{i}}+\mu_{k_{2}}e^{ik_{2}x_{i}})+ \\ &&\theta (x_{i})\{\eta_{k_{1}}e^{i[k_{1}+(n_{0}-m)\frac{\omega_{m}}{v_{g}} ]}x_{i}+\eta_{k_{2}}e^{i[k_{2}+(n_{0}-m)\frac{\omega_{m}}{v_{g}} ]})x_{i}+\xi (m)e^{\lambda (m)x_{i}}\}\\ \end{array} $$
(A.10)
$$\begin{array}{@{}rcl@{}} \phi_{aa}(m^{\prime} ) &=&-\sum\limits_{n}U_{n_{1}m_{2}^{\prime} }^{*}\frac{\sqrt{2} \overline{V}\phi_{ae}(0^{-},n)}{2U+2\omega_{eff}-i(\kappa+{\Gamma})+m^{\prime} \omega_{m}-E-4\delta} \end{array} $$
(A.11)
with
$$\begin{array}{@{}rcl@{}} \phi_{e,k_{i}}(x_{j},n) &=& \frac{1}{\sqrt{2\pi}}[\delta_{nn_{0}} \theta(-x_{j})+t_{k_{i}}\theta(x_{j})]e^{i[k_{i}+(n_{0}-n)\frac{\omega_{m}}{ v_{g}}]x_{j}} \end{array} $$
(A.12)
$$\begin{array}{@{}rcl@{}} \phi_{o,k_{i}}(x_{j},n) &=& \frac{\delta_{nn_{0}}}{\sqrt{2\pi}} e^{i[k_{i}+(n_{0}-n)\frac{\omega_{m}}{v_{g}}]x_{j}} \end{array} $$
(A.13)
where
$$\begin{array}{@{}rcl@{}} \mu_{k_{1}}(m) &=& \frac{\overline{V}U_{n_{0}m_{1}}^{*}}{2\pi}[\frac{1}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{2} +(n_{0}-m)\omega_{m}+\delta} ] \end{array} $$
(A.14)
$$\begin{array}{@{}rcl@{}} \mu_{k_{2}}(m) &=& \frac{\overline{V}U_{n_{0}m_{1}}^{*}}{2\pi}[\frac{1}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{1} +(n_{0}-m)\omega_{m}+\delta }] \end{array} $$
(A.15)
$$\begin{array}{@{}rcl@{}} t_{k_{1}}(n) &=& \sum\limits_{m}U_{nm_{1}}U_{n_{0}m_{1}}^{*}[\frac{-\omega_{eff}+i \frac{\kappa-{\Gamma}}{2}+v_{g}k_{2} +(n_{0}-m)\omega_{m}+\delta}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{2} +(n_{0}-m)\omega_{m}+\delta} ] \end{array} $$
(A.16)
$$\begin{array}{@{}rcl@{}} t_{k_{2}}(n) &=& \sum\limits_{m}U_{nm_{1}}U_{n_{0}m_{1}}^{*}[\frac{-\omega_{eff}+i \frac{\kappa-{\Gamma}}{2}+v_{g}k_{1} +(n_{0}-m)\omega_{m}+\delta}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{1} +(n_{0}-m)\omega_{m}+\delta} ] \end{array} $$
(A.17)
$$\begin{array}{@{}rcl@{}} \eta_{k_{1}}(m^{\prime} ) &=& \sum\limits_{n}U_{nm_{1}^{\prime} }^{*}\frac{ \overline{V}}{2\pi}\frac{t_{k_{1}}} {-\omega_{eff}+i\frac{\kappa-{\Gamma}}{2} +v_{g}k_{2}+(n-m^{\prime} )\omega_{m}+\delta} \end{array} $$
(A.18)
$$\begin{array}{@{}rcl@{}} \eta_{k_{2}}(m^{\prime} ) &=& \sum\limits_{n}U_{nm_{1}^{\prime} }^{*}\frac{ \overline{V}}{2\pi}\frac{t_{k_{2}}} {-\omega_{eff}+i\frac{\kappa-{\Gamma}}{2} +v_{g}k_{1}+(n-m^{\prime} )\omega_{m}+\delta} \end{array} $$
(A.19)
$$\begin{array}{@{}rcl@{}} \xi(n) &=& \sum\limits_{m}U_{nm_{1}}\frac{\sqrt{2}\overline{V}}{iv_{g}} \phi_{aa}(m)+\mu_{k_{1}}(n)+\mu_{k_{2}}(n)-\eta_{k_{1}}(n)-\eta_{k_{2}}(n) \end{array} $$
(A.20)
$$\begin{array}{@{}rcl@{}} B(n) &=& \sum\limits_{m^{\prime} }U_{nm_{1}^{\prime} }\frac{\overline{V}}{\sqrt{2} iv_{g}}\xi \end{array} $$
(A.21)
$$\begin{array}{@{}rcl@{}} \lambda(m^{\prime} ) &=& \frac{1}{iv_{g}}(\omega_{eff}-i\frac{\kappa+{\Gamma}}{ 2}+m^{\prime} \omega_{m}-E-\delta) \end{array} $$
(A.22)
and \(\rho _{k_{i}}=\mu _{k_{i}}/\sqrt {2}\)
$$\begin{array}{@{}rcl@{}} U_{nm_{1}} &=& \sum\limits_{l}e^{\frac{1}{2}(\frac{g_{0}}{\omega_{m}})^{2}}(-\frac{ g_{0}}{\omega_{m}})^{l+m-n}(-\frac{g_{0}}{\omega_{m}})^{l} \frac{(l+m)!}{ (l+m-n)!l!\sqrt{n!m!}} \end{array} $$
(A.23)