Tunable quantum routing via asymmetric intercavity couplings

Abstract

Routing efficiently the photon signals between different channels is essential in an optical quantum network. Recent chiral waveguide–emitter coupling techniques are widely applied to improve the routing capabilities in waveguide systems. As a possible alternative, we put forward a proposal to control the quantum routing of photons by adjusting asymmetric intercavity couplings on both sides of a cross-cavity in an X-shaped coupled-resonator waveguide. With the robust and tunable intercavity couplings, high transfer rate between quantum channels and expected probability distributions of two ports in the same output channel can be easily implemented. The asymmetric intercavity coupling may be utilized as a new handle to efficiently control the single-photon routing.

Appendix: Scattering amplitudes in the dissipative case

In the appendix, the scattering amplitudes in the dissipative case are given. The basic calculation process is similar to the above one in the text. When the dissipations are considered, the corresponding frequencies $\omega _a$, $\omega _b$, $\omega _e$, and $\omega _f$ are replaced by $\omega '_a$, $\omega '_b$, $\omega '_e$, and$\omega '_f$, respectively. Here, $\omega '_{a(b,e,f)}$ $=\omega _{a(b,e,f)}-i\gamma _{a(b,e,f)}$, and $\gamma _{a(b,e,f)}$ is the decay rate of CCA-a (CCA-b, atomic excited state $|e\rangle$, and $|f\rangle$). The corresponding effective potentials and dispersive coupling strength are described as

$$\begin{aligned} V'_a(E_k)= & {} \frac{g^2_a\varDelta '_f}{ \varDelta '_e\varDelta '_f-|\varOmega |^2},\nonumber \\ V'_b(E_k)= & {} \frac{g^2_b\varDelta '_e}{\varDelta '_e\varDelta '_f-|\varOmega |^2},\nonumber \\ G'(E_k)= & {} \frac{\varOmega g_ag_b }{\varDelta '_e\varDelta '_f-|\varOmega |^2}, \end{aligned}$$

(A1)

with $\varDelta '_e=E_k-\omega '_e$ and $\varDelta '_f=E_k-\omega '_f$.

After some algebra, the exact form of the scattering amplitudes is obtained as

$$\begin{aligned} r_a= & {} \frac{\xi _a\alpha (0)-\xi _a+i\gamma _ae^{-ik_a} }{ \xi _a-i\gamma _ae^{ik_a} } ,\nonumber \\ t_a= & {} \frac{(1+\lambda _a)\xi _a\alpha (0)}{ \xi _a-i\gamma _ae^{ik_a} },\nonumber \\ t_b^u= & {} \frac{(1+\lambda _b)\xi _b\beta (0)}{ \xi _b-i\gamma _be^{ik_b} },\nonumber \\ t_b^d= & {} \frac{\xi _b\beta (0)}{ \xi _b-i\gamma _be^{ik_b} }, \end{aligned}$$

(A2)

with

$$\begin{aligned} \alpha (0)= & {} \frac{ Q_b\xi _a (e^{ik_a}\frac{\xi _a-i\gamma _ae^{-ik_a}}{\xi _a-i\gamma _ae^{ik_a}}-e^{-ik_a}) }{(Q_a-V'_a)(Q_b-V'_b)-|G'|^2},\nonumber \\ \beta (0)= & {} \frac{ G'^*\xi _a(e^{ik_a}\frac{\xi _a-i\gamma _ae^{-ik_a}}{\xi _a-i\gamma _ae^{ik_a}}-e^{-ik_a}) }{(Q_a-V'_a)(Q_b-V'_b)-|G'|^2},\nonumber \\ Q_b= & {} i\gamma _b-2\xi _b\cos {k_b}+\frac{\xi _b^2 e^{ik_b}(2+2\lambda _b+\lambda ^2_b) }{\xi _b-i\gamma _b e^{ik_b}},\nonumber \\ Q_a= & {} i\gamma _a-2\xi _a\cos {k_a}+\frac{\xi _a^2 e^{ik_a}(2+2\lambda _a+\lambda ^2_a) }{\xi _a-i\gamma _a e^{ik_a}},\nonumber \\ \end{aligned}$$

(A3)

When the dissipations are not included, the scattering coefficients in Eq. (A2) can be returned to the forms in Eq. (12) in the text. While the dissipations are added, the conservation relation of the photon flow doesn’t maintain and changes into $|t_a|^2 + |r_a|^2 + |t_b^u|^2 +|t_b^d|^2< 1$.