1. Introduction

Photons are excellent candidates for transferring and manipulating quantum information since as qubit carriers photons can transport fast and sustain quantum coherence for vast distances in comparison with other possible information carriers such as electrons. Processing quantum information in a quantum network requires photons to be operated at the level of a single photon. Remarkable progress has been made in manipulating quantum data encoded in photon states experimentally [1–10]. A waveguide confine light in a transverse direction and maintain uniform in the longitudinal direction, which forms an effective one-dimensional (1D) continuum for propagating photons. The strong photon-emitter interaction becomes possible when 1D waveguide is coupled with quantum emitters, which opens an avenue for manipulating few-photon states, e.g. the total reflection caused by the quantum interference between the guided photon and that emitted from the quantum emitters [11], the multi-photon bound state in a 1D waveguide of a linearized modal dispersion [12,13], Fano resonance [14] and the outside finite band of 1D coupled-resonator waveguide [15–18].Many quantum devices have been proposed, such as quantum super cavities [19–21], a Mach-Zehnder interferometer [22], diodes [23,24], frequency converters [25–27].

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 1D waveguide [28–37]. The possibility of obtaining chiral light-matter interactions [38–44] promises new prospects for manipulation and control of photon states. Quantum routers based on the chiral coupling have been proposed [45–48]. In an ordinary waveguide, a quantum emitter interacts with photons in either of the two propagation directions, and the probability of successful routing is probabilistically and decreases with increasing the number of routing ports, which restricts the scalability. Due to the chiral coupling between the quantum emitter and the optical field, the probability that the photon will be released from the quantum emitter into the left and right propagation modes of the one-dimensional waveguide is not equal, the photons can therefore be reliably routed to an arbitrarily selected output port. In this paper, we proposed one- and bi-photon routers by studying quantum propagation of photons in waveguides chiral coupled to a three-level quantum system (3LS) in $\Lambda$ configuration. It is shown that a scattering spectrum is characteristic of electromagnetically induced transparency (EIT), and the chiral coupling insures the definitively routing of photon to a waveguide instead of continuing in the same waveguide. The photons routed to another waveguide are anti-bunched. The classical field applied to the 3LS turns off and on the tunneling paths. Moreover, it also controls the number of two-photon bound states and the interference pattern of bi-photon bound states.

The paper is organized as follows. In Sec. 2, we propose the model $-$ a driven $\Lambda$-type 3LS chirally coupling to two waveguides. In Sec. 3, we discuss the scattering process of one photon and show how external classical field applied to the 3LS routes the injected photon in a waveguide to other waveguid. In Sec. 4, we derive an analytic solution of the output bi-photon wave packet and discuss the routing probability for a bi-photon incident state. In Sec. 5, we calculate the second correlation function of the output bi-photon wave packet, and the interference of the two bi-photon bound states is presented. Finally, a summary has been made.

2. Model for a three-level emitter coupling to two 1D waveguides

As shown in Fig. 1, a 3LS at coordinate $x=0$ is coupled to two identical 1D waveguides, labeled as $a$ and $b$ respectively. The waveguides form a four-port arrangement in which each input-output port is labeled with the numbers 1-4. We ignore thermal fluctuations and losses. The 3LS is a $\Lambda$-type atom with a ground state $\left \vert g\right \rangle$, a metastable state $\left \vert 1\right \rangle$ and an excited state $\left \vert 2\right \rangle$. The transition $\left \vert g\right \rangle \leftrightarrow \left \vert 2\right \rangle$ is dipole coupled to two 1D waveguides, which absorbs and emits a photon with rates $\gamma _{pl}$ and $\gamma _{pr}$ into the left- and right-going continuum in waveguide $p\in \left \{ a,b\right \}$. The emission of the 3LS into guided modes of waveguide $p$ is depicted by $\gamma _{p}=\gamma _{pr}+\gamma _{pl}$. The Hamiltonian of the total system consists of three parts:

The transition between $\left \vert 2\right \rangle$ and $\left \vert 1\right \rangle$ of the 3LS is driven by a classical field. In the dipole approximation, the 3LS Hamiltonian reads

The real-space field operators $\hat {p}_{R}(x)$ and $\hat {p}_{L}(x)$ ($\hat {p }_{R}^{\dagger}(x)$ and $\hat {p}_{L}^{\dagger}\left ( x\right )$) annihilate (create) a left- and right-going photon at position $x$ in the waveguide $p$, and $[\hat {p}_{R}(x)),\hat {p} _{R}^{\prime \dagger }(x^{\prime })]=[\hat {p}_{L}(x)),\hat {p}_{L}^{\prime \dagger }(x^{\prime })]=\delta _{pp^{\prime }}\delta (x-x^{\prime })$. If $\omega _{2}$ is far away from the cutoff frequency of the waveguide modes, the waveguide dispersion can be linearized as $\omega _{k}=v|k|$ with the group velocity $v>0$. The dipole Hamiltonian under the rotating-wave approximation

The Hamiltonian for the free propagation of the guided photon is transformed into

We have set $v=1$ in the following discussion. By defining creation operators

Here, the 3LS couples to the $C$ mode of the field with a coupling strength characterized by the total decay of the 3LS $\gamma =\gamma _{a}+\gamma _{b}$. The number of excitations in this system is conserved.

 

Fig. 1. A three-level system interacts with two independent waveguides labeled by $a$ and $b$. The blue line represents the chiral coupling between the three-level atom and the left and right going photons of the bottom waveguide $a$ ($b$), and the coupling rates are $\gamma _{ar}$ ($\gamma _{br}$) and $\gamma _{al}$ ($\gamma _{bl}$).

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3. Routing single 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 3LS, the eigenstate of $H_{c}$ takes the form

By removing $u_{1}$, $u_{2}$, the scattering equation for the amplitude of the single photon reads

The photon coming from one side of the 3LS in one waveguide can be scattered into four different ports, resulting in a reflected, transmitted and transferred waves. Consider an monochromatic photon incident from the left side of the 3LS in waveguide $a$ with state

We first decompose the operator $\hat {a}_{_{R}}^{\dagger }(x)$ into a linear combination of operators $\hat {C}$, $\hat {D}$ and $\hat {O}_{a}$. Since waves in $D$ and $O$ spaces propagate freely, the outgoing wave in $C$ space is obtained by multiplying the incident wave by $t_{k}$. By transforming back to the right- and left-going operators, the scattering amplitudes read

Here $t_{ak}\left ( t_{bk}\right )$ and $r_{ak}\left ( r_{bk}\right )$ are the transmission amplitude and reflection amplitude of photons in waveguide $a\left ( b\right )$. The corresponding four scattering coefficients are defined as

It can be observed that the scattering amplitudes in Eq. (19) can be obtained from the scattering amplitudes in Eq. (17) by interchanging $\gamma _{ar}\leftrightarrow \gamma _{al}$, which can be understood from the symmetry of the system. Since the reflection probabilities $|r_{ak}|^2=|r_{ak}^{\prime }|^2$ in any case, the reciprocal scattering [49–52] in the incident waveguide only appears at $\gamma _{ar}=\gamma _{al}$ or $\gamma _{b}=0$. The 3LS is fully transparent to the incident photons with carried frequency satisfying $\Delta _{1}=0$.

The right- and left-moving modes are two degenerate modes of waveguides. Photons moving towards the 3LS will be absorbed by the 3LS and then be emitted into waveguides. If the guided photon is emitted into the modes of waveguide $b$, the photon is routed to waveguide $b$ from waveguide $a$, the routing probability is denoted by $T_{bk}+F_{bk}$. When the applied field is turned off, as shown in Fig. 2(b) (the black dash-dot line), the peak of the routing probability is achieved when incident waves resonate with the transition $\left \vert g\right \rangle \leftrightarrow \left \vert 2\right \rangle$, and the spectra are the Lorentzian-shaped line, our system reduces to a two-level quantum emitter coupled to two identical 1D waveguides. As $\Omega$ increases, the applied field makes the spectra split into a doublet with a separation of $2\Omega$ since the spectra have two maximum at $k=\omega _{\pm }$ and a dip at $\Delta _{1}=0$. It allows us to observe the EIT based on the Autler-Townes splitting. The width of this transparency window increases when the frequency of the applied field is off-resonant with the transition of the 3LS. The spectra are symmetric around $\Delta _{1}=0$ only for $\omega _{1}=\omega _{2}$ and asymmetric otherwise. When the incoming photon is far off-resonant from the dressed state of the 3LS, the photon does not interact with the atom, and can not be routed.

 

Fig. 2. (a) The routing probability $T_{bk}+F_{bk}$ as a function of $k /\gamma$ and $\gamma _{b}/\gamma$, with $\omega _{1}=\omega _{2}=100\gamma, \Omega =2\gamma, \gamma _{ar}/\gamma =1-\gamma _{b}/\gamma$. (b) The routing probability as a function of $k /\gamma$. We take $\omega _{1}=\omega _{2}=100\gamma,\Omega =0, \gamma _{ar}=\gamma /4,\gamma _{b}=\gamma /2$ for black dash-dot line, $\omega _{1}=\omega _{2}=100\gamma,\Omega =\gamma,\gamma _{ar}=\gamma /4, \gamma _{b}=\gamma /2$ for red long-dashed line, $\omega _{1}=\omega _{2}=100 \gamma,\Omega =2\gamma,\gamma _{ar}=\gamma _{b}=\gamma /2$ for blue solid line, and $\omega _{1}=98\gamma,\omega _{2}=100\gamma,\Omega =2\gamma,\gamma _{ar}= \gamma /2,\gamma _{b}=\gamma /6$ for green short-dashed line.

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The routing process is an absorption of the photon from the incident waveguide and sequent emission of the photon to the non-incident waveguide. It requires 3LS interacting with modes of incident photons as well as modes of non-incident waveguide, otherwise, routing is impossible. For photons incident from the left side of the 3LS, it requires $\gamma _{ar}\neq 0$ and $\gamma _{b}\neq 0$. The asymmetry in each waveguide is usually quantified by $D_{p}=(\gamma _{pr}-\gamma _{pl})/\gamma _{p}$. The maximum value of routing probability is obtained as $50\%$ in the symmetrical-coupling case ($D_{j}=0$). With the increase of $\gamma _{ar}$, the routing probability increases, so the chirality in the incident waveguide is desirable for a fixed $\gamma _{b}$. Increasing the coupling between the 3LS and the waveguide $b$ is a further way to improve the routing probability, so it is desirable to reduce the imbalance of emission in different waveguide for a given decay $\gamma$. Completely routing is only possible via the dressed state if $\gamma _{ar}=\gamma _{b}=\gamma /2$, in this case, the couplings in the incident waveguide are maximally chiral, the coupling of the 3LS to the other waveguide could be symmetrical or asymmetrical. In order to completely route the photons into a selected port in waveguide $b$, as for port 4 (3), the maximum asymmetric coupling $D_{b}=1(-1)$ is required. We note that if our system is regarded as a four-port device, the photons could be routed to any selected port by adjusting the coupling strengths and the frequency of the classical driving field.

4. Routing via two-photon plane waves

Since any photons incident far away from the 3LS in waveguides can be decomposed to the $O_{a}$, $O_{b}$, $D$ and $C$ modes, $O_{a}$, $O_{b}$ and $D$ modes are decoupled to the 3LS. As a result, we just need to find the photon scattering state in $C$ space. The general two-excitation eigenstate of Hamiltonian $H_{c}$ can be expressed as

At the region that all photons are away from the 3LS, the solution of $\phi _{cc}\left ( x_{1},x_{2}\right )$ consists of two parts: plane wave with wave vectors $k_{1}$, $k_{2}$ and bi-photon bound states. The wavefunction of the bi-photon bound states reads

And expressions of $\bar {C}_{i}$ are as follow

Here, two bi-photon bound states are found as long as $\Omega \neq 0$ and $\Omega \neq \gamma /4$, we attribute this difference to the field coupling to the transitions from the ground state to the two dressed states.

Consider that two photons are incident from the left side of the 3LS described by

In the centroid and relative coordinate system, $\phi _{k_{1}k_{2}}\left ( x_{1},x_{2}\right )$ can be rewritten as

After the photon is absorbed by the 3LS, each can be re-emitted into the modes in any waveguide. We use $\rho _{mn}$ to denote the output wave functions with $m,n\in \lbrack 1,4]$. Since the input photons coming from the same port are indistinguishable, we have ten output wave functions: four of them describe two photons traveling in the same direction in the same waveguide

Two of them describe two photons traveling along different directions of the same waveguide

Two of them describe two photons propagating in the same direction but of different waveguides

The rest describes two photons propagating in different directions of different waveguides

Obviously, the probabilities for two photons being scattered in the same direction are not dependent on the center of mass coordinate $x_{c}$; and the probabilities for two photons being scattered in different directions are not dependent on the relative coordinate $x_{r}$. It can be observed that wave functions for photons incident from the right side of waveguide $a$ can be obtained from wave functions in Eq. (31)–(34), by interchanging $\gamma _{pr}\leftrightarrow \gamma _{pl}$($p\in \left \{ a,b\right \}$) and interchanging $\rho$’s subscript $1\leftrightarrow 2$, $3\leftrightarrow 4$.

At the two-photon resonance, i.e. $k_{1}=k_{2}=\omega _{1}$, the coefficients $\bar {C}_{i}$ vanish and only $t_{ak}=1$, both photons are all routed to port 2 due to the transparent window. In order to discuss the transfer rate of photon from one waveguide $p$ into a waveguide $\bar {p}$ instead of continuing in the same waveguide, we introduce the definition of routing probability

 

Fig. 3. The probability $P_{b}$ of the input photons from port 1 routed to port 3 as a function of energy $k_{1}$ and $k_{2}$ at the maximally chiral couplings $\gamma _{ar}=\gamma _{br}=\gamma /2$. Other parameters are setting as follow: $\omega _{1}=\omega _{2}=100\gamma$, (a) $\Omega =0$ (b) $\Omega =2\gamma$

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It is well-known that the routed photons are anti-bunched because an excited state cannot emit two photons at the same time. In our system, the two dressed states act as two excited states, it is impossible for two photons to be routed simultaneously whose frequencies are equal and in resonance with one of dressed states, however, is it possible for one photon to be routed via the upper dressed state and the other photon to be routed via the lower dressed state at the same time?

5. interference of two-photon bound states

We now consider the second-order correlation function

Then correlation functions in waveguides read

To answer the question raised in the last section, we discuss $\tau =0$, then Eq. (38) become

Both $\rho _{22}$ and $\rho _{44}$ are characterized by the interference between the plane wave and the wave of the bound state. Under the condition that the frequency of incoming photons is resonant with anyone of the dressed states, we obtain $g_{b}^{\left ( 2\right ) }\left ( 0\right ) =0$ since $\rho _{44}\left ( x,x\right ) =0$ is deduced from Eq. (31d), which indicates that two photons are antibunched. So it is impossible either for one photon to be routed via the upper dressed state and the other photon to be routed via the lower dressed state at the same time, or for two photons to be routed via one of the dressed state simultaneously.

However, things become different in waveguide $a$. When the frequency of incoming photons is resonant with anyone of the dressed states, the contribution of the plane waves vanish, i.e., only bi-photon bound states exist in each amplitude, so one may expect an oscillation with $x_{_{c}}$ or $x_{r}$ from Eqs. (23) and (24) as long as $\Omega$ exceeds some value, for example, an oscillation with $x_{_{c}}$ or $x_{r}$ as long as $\Omega >\gamma /4$ when $\omega _{1}=\omega _{2}$. In Fig. 4(a-b), we plot the second-order correlation function $g_{a}^{\left ( 2\right ) }\left ( 0\right )$ as a function of the coordinator $x$ when incident photons resonate with the transitions between the ground state and the dressed states for different $\Omega$. Contrary to what one expects, $g_{a}^{\left ( 2\right ) }\left ( 0\right )$ decreases as $x$ increases when $k_{1}=k_{2}=\omega _{\pm }$ (see the red dashed and black dotted lines), which is independent of $\Omega$. When $k_{1}=\omega _{+}$($\omega _{-}$), $k_{2}=\omega _{-}$($\omega _{+}$) and $\Omega >\gamma /4$, $g_{a}^{\left ( 2\right ) }\left ( 0\right )$ still decreases with the increase of $x$, but the decay rate of $g_{a}^{\left ( 2\right ) }\left ( 0\right )$ oscillates, which is different from the previous two cases. In Fig. 4(c-d) we plot the second-order correlation function $g_{a}^{\left ( 2\right ) }\left ( \tau \right )$ as a function of the coordinator $\tau$ when incident photons resonate with dressed states. The red dashed and black dotted lines also coincide which correspond to $g_{a}^{\left ( 2\right ) }\left ( \tau \right )$ with $k_{1}=k_{2}=\omega _{\pm }$. And the oscillation with $\tau$ occurs under the condition $k_{1}=\omega _{+}$($\omega _{-}$), $k_{2}=\omega _{-}$($\omega _{+}$) and $\Omega >\gamma /4$, however the maxima are reduced in magnitude as $\tau$ increases. This phenomenon is a consequence of a coherent superposition of two probability amplitudes [53], which correspond to two bi-photon paths

 

Fig. 4. the second-order correlation function (a-b) $g_{a}^{\left ( 2\right ) }\left ( 0\right )$ as a function of $x$ and (c-d) $g_{a}^{\left ( 2\right ) }\left (\tau \right )$ as a function of $\tau$ at the maximally chiral couplings $\gamma _{ar}= \gamma _{br} = \gamma /2$ when $k_{1}=k_{2} = \omega _{-}$ (red dashed line), $k_{1}=k_{2}= \omega _{+}$ (black dotted line), and $k_{1}= \omega _{-},k_{2}= \omega _{+}$ (blue solid line). Other parameters are setting as follow: $\omega _{1}= \omega _{2}=100 \gamma$.

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Output states for bi-photon scattering are the common superposition state of plane wave and bound state, interference pattern can be clarified as interference between plane waves, between plane wave and bound wave, and between bound waves. We found there are maximums and minimums in the second order correlation function, but the maximums decline gradually to zero. We note that the maximums of the second order correlation function of the plane waves remain unchanged, and the maximums of the second order correlation function of the plane wave and bound state cannot decline gradually to zero.

6. Conclusion

We have studied the system consisting of two 1D waveguides coupling to a $\Lambda$-type quantum emitter. Plane-wave and bound-state solutions are derived from the problem of quantum propagation of photons in waveguides. Plane waves give all contributions to direct photons to a waveguide instead of continuing in the same waveguide. The 3LS behaves as a quantum multichannel router. The perfect few-photon routing requires chiral waveguide-emitter coupling. When a classical field is absent, one can turn on the multichannel routing by adjusting the transition frequency of the 2LS between $\left \vert g\right \rangle$ and $\left \vert 2\right \rangle$ to match the desired to propagate states. To turn off the multichannel routing, one can tune the atomic transition energy far away from the energy of the incident photons, but such multichannel routing based on large detuning is not perfect. By applying a classical field, on the one hand, one can perfectly turn off the quantum routing via the Autler-Townes splitting, on the other hand, more tunneling paths are opened for routing. However, two photons can not be routed via dressed states simultaneously. Moreover, applied classical field produces two bi-photon bound states, the interference of bi-photon bound states can be found with field strength $\Omega >\gamma /4$ when incident photons resonate with dressed states separately. We note that there are radiative processes in real system, the radiative processes not only influence the free propagation of photons, but also result in the inelastic scattering of the single photon. So observations of our manuscript definitely become weak, moreover, they might be disappeared.