## Abstract

In this study, we design a T-shaped quantum router that comprises two-level systems (TLSs), an infinite coupled resonator waveguide (CRW), and a semi-infinite CRW. The loss (absorption) and gain (amplification) of the energy levels of the TLSs can be considered as energy exchange between the system and its environment. Considering loss in the ground state and gain in the excited state of the TLSs and loss of cavities, the system is non-energy-conserving and non-Hermitian. Loss in the system consists of loss of cavities and TLSs. The total transmission probabilities (TPs) of photons in the system are equal to 1 or lower when the system has loss only. Loss causes a bounce-back phenomenon in the TPs. The TPs have a divergent point when the TLSs have gain, and we obtain this divergent condition. The reflection probability has a minimal point only when photons are incident from the semi-infinite CRW and the system has loss. The TPs of the non-Hermitian router are increased by gain, decreased by loss, and conserved under certain conditions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum routers are crucial elements of quantum networks [1–16]. The photons in quantum routers carry quantum information along quantum channels and transfer the information from one channel to others, distributing it in the network [1]. Quantum channels are required to transport photons, which act as flying q ubits. A quantum network comprises quantum routers, channels, and nodes. Researchers have recently made theoretical and experimental progress in the development of quantum routers. Theoretically, single-photon routers can be designed using nanomechanical systems [2] and employ a quantum state turnplate [4]; chiral photon-atom interactions [5] are applied in the microwave regime [6], and there are multiple output ports [7]. The entanglement routers in a wireless quantum network can be designed on the basis of arbitrary two-qubit systems [8]. Experimentally, scholars have achieved quantum routing of single-photon pulses in a Mach-Zehnder interferometer [9]. And a Hermitian quantum router comprising cavities or circuits can be extended to become a non-Hermitian quantum router.

Closed systems are commonly solved, but there is growing experimental demand for the solution of open systems. The interactions between the system and its environment can be described using non-Hermitian terms [17–29]. In non-Hermitian open systems, interesting phenomena have been discovered such as non-Hermitian Bloch oscillations [37, 38], unidirectional reflectionlessness [40–44], coherent perfect absorbers [47–49], optical solitons [39, 45, 46], non-Hermitian topological insulators [53, 54], phonon lasers [50–52], and “exceptional ring” effects [55, 58]. Unidirectional absorption, lasing, and wave propagation are proposed [56, 57]. The non-Hermitian interactions can be realized in optomechanical systems [30–34]. The scattering properties of PT-symmetric andanti-PT symmetric structures are discussed [35, 36]. Experimentally, phase transition leads to a loss of induced optical transparency in coupled waveguides [60]. Atoms can control the transport processes of photons. Adding atoms to a non-Hermitian system and determining the subsequent gain and loss in the atoms’ energy levels are worthy of investigation. In a system of N-type four-level atomic ensembles, the effects of energy levels on the phase of photons and transport properties have been carefully studied [68–70], and photonic diodes and transistors may be formed with atoms which are trapped in a one-dimensional optical lattice. In three-level systems, the probe-field refractive index [64] can have a non-Hermitian profile [71], create coherent perfect absorbers and lasers [63], and discuss non-adiabatic and adiabatic transitions at the level crossing [67].

A single-photon router can be formed from two coupled resonator waveguides (CRWs) coupled with a two-or three-level atom located at the intersection cavities of the two CRWs [10–14]. Accordingly, infinite and semi-infinite CRWs can have a T-shaped structure. When two-level systems (TLSs) are embedded at the intersection of two CRWs, the TLSs in the ground state can absorb photons and enter the excited state through resonance absorption. The TLSs in the excited state can then emit photons through spontaneous radiation and consequently return to the ground state because the TLSs are coupled with different cavities of the two CRWs. Photons incident from one CRW may transfer to the other CRW through absorption and emission processes. Finally, the T-shaped quantum router is formed [10]. Loss in the ground state and gain in the excited state of TLSs and loss of cavities can be considered energy exchange between the system and environment. Total transmission probabilities (TPs) are equal to 1 or less when the system has loss only and are conserved when the pump frequency of the TLSs and frequency of the photons are nearly resonant. The existence of loss can cause the TPs to bounce back. When the excited state has gain only, the total TPs can be equal to 1, be larger than 1, or be divergent. When the system has both loss and gain, the TPs are divergent for photons that are incident from both CRWs but minimal when photons are incident from the semi-infinite CRW. The divergent condition of the TPs can be determined by the gain, and the minimal condition can be discovered by the loss. The TPs of the non-Hermitian router are increased (decreased) for a specific gain (loss) and incident photon energy.

This paper is organized as follows. In the second part, we construct the T-shaped non-Hermitian quantum router. In the third and fourth parts, the transport properties of photons incident from semi-infinite and infinite CRWs, respectively, are analyzed. In the final part, we discuss the effect of loss and gain on the non-Hermitian router.

## 2. Model setup

The model in this study comprises an infinite CRW-a and a semi-infinite CRW-b coupled with the TLSs to form a T-shaped non-Hermitian quantum router, as illustrated in Fig. 1(a). The CRWs consist of a chain of Fabry-Pérot cavities. The cavities are made of two mirrors. The two mirrors are not perfect reflectors. The photons in neighboring cavities thus interact. Photons of only a few frequencies can be supported because the system has only a few cavities. When the system has numerous cavities, photons of a range of frequencies can move within the CRW, resulting in a photonic energy band. The TLSs in our system are not natural atoms but artificial atoms. The physical implementation of the TLSs can be coupled with superconducting transmission line resonators with one resonator coupled to a charge qubit based on a direct current superconducting quantum interference device [14]. Considering the interactions between the adjacent cavities of two CRWs, the propagation processes of photons are described by the tight-bonding model. The two CRWs intersect at one cavity of the CRW-a and the end cavity of the CRW-b, where the TLSs are located. The non-Hermitian action is implemented by considering the gain in the excited state and loss in the ground state of the TLSs and the loss of cavities.

The system comprises two CRWs $({\widehat{H}}_{C}^{[a]},{\widehat{H}}_{C}^{[b]})$, TLSs $({\widehat{H}}_{T})$, and the interactions between the two cavities at the intersection of two CRWs and TLSs (${\widehat{H}}_{TC}$, which are written under the rotating wave approximation). The total energy of the system is $\widehat{H}={\widehat{H}}_{C}^{[a]}+{\widehat{H}}_{C}^{[b]}+{\widehat{H}}_{T}+{\widehat{H}}_{TC}(\hslash =1)$.

We use the CRW-d (*d* = *a, b*), referring to the CRW-a and CRW-b. The eigen-frequency of the cavity in the CRW-d is *ω _{d}*. The hopping energies between the adjacent cavities of the CRW-d are

*ξ*. ${d}_{{j}_{d}}^{\u2020}$ is the creation operator of photons in the CRW-d. The subscript

_{d}*j*is the

_{d}*j*th cavity of the CRW-d.

_{d}*j*(

_{a}*j*) changes from the negative infinity to the positive infinity (from one to the positive infinity). The crosspoint is at the 0th cavity of the CRW-a (

_{b}*j*= 0) and the first cavity of the CRW-b (

_{a}*j*= 1). The coupling strength between TLSs and the 0th cavity of the CRW-a (the first cavity of the CRW-b) is

_{b}*g*(

_{a}*g*). The frequency of the excited state |

_{b}*e*〉 of TLSs is

*ω*, whereas the ground state |

_{e}*g*〉 is

*ω*. Because we are mainly concerned with the energy differences,

_{g}*ω*is assumed to be zero.

_{g}The gain in the excited state (the loss in the ground state) of TLSs is *γ _{e}*(

*γ*). Both

_{g}*γ*and

_{e}*γ*are higher than or equal to zero, the positive (negative) sign before

_{g}*γ*(

_{e}*γ*) indicating the gain (loss). The TLSs do not exist naturally. The ground state of the TLSs may be not the natural ground state, and exists loss unavoidably. Interactions between the system and the environment lead to energy dissipation, causing randomization of the phase and decay. The Lindblad equation can be employed to describe the evolution of open quantum systems [59]. Cavities’ loss should be accounted for, and this type of loss affects the performance of the system. Loss of cavities other than the two cavities at the crosspoint can decrease the overall TPs. This can be considered a classical effect and this is trivial. We are mainly interested in the transfer property of photons. We assume that the two cavities’ loss at the crosspoint are larger than all other cavities. The loss of cavities at the crosspoint of the CRW-a and -b are referred to by

_{g}*γ*and

_{a}*γ*. To induce gain in the excited state, we use the method introduced in [63–65], in which two quantum pump lights are input at the intersection of the two CRWs to form an auxiliary system. The TLS can be an abstraction of the three-level atoms, as shown in Fig. 1(b). We additionally select a unique metastable state |

_{b}*s*〉 (the frequency

*ω*). The transitions between the |

_{s}*s*〉 and the excited state |

*e*〉 do not couple with the two cavities. The transitions between the |

*s*〉 and the ground state |

*g*〉 are forbidden. Two quantum pump lights couple to the transitions |

*e*〉↔|

*a*〉 and |

*e*〉↔|

*g*〉, with Rabi frequency Ω

_{1}and Ω

_{2}. The one-photon detuning Δ =

*ω*−

_{e}*ω*− Ω

_{a}_{1}is much larger than the two-photon detuning

*δ*= Ω

_{2}− Ω

_{1}−(

*ω*−

_{a}*ω*). We inject atoms in the |

_{g}*a*〉 state to ensure that the atoms in the cavity remain in the state |

*a*〉. The atoms can then release signal light of frequency

*ω*=

_{eg}*ω*−

_{e}*ω*from state |

_{g}*a*〉 to state |

*e*〉 and then to state |

*g*〉, exhibiting two-photon gain. Three level atoms can be regarded as TLSs under some conditions [66]. We treat this type of the atom as an effective TLS in the system. Theoretically and principally, the gain in the excited state can be obtained through the |

*s*〉 and pump lights. The gain of other type of the three-level atom [67] and the four level N-type atomic system [68–71] are also discussed.

Single photons can be present in the any cavity of two CRWs or activate TLSs from the ground state to the excited state. Photons may be located at the crosspoint but do not activate TLSs. Thus, the eigenstate in the single-excitation subspace has the form below,

*j*th (

_{a}*j*th) cavity of the CRW-a (CRW-b).

_{b}*U*is the probability amplitude that photons activate the TLSs from the ground state to the excited state. |

_{e}*g,*0〉 (|

*e,*0〉) indicates that TLSs are in the ground (excited) state when no photons are present in the system.

Using the Schrödinger equation $\widehat{H}|E\u3009=E|E\u3009$, we obtain the scattering equation,

The local deltalike energy-dependent potential is ${V}_{d}(E)={g}_{d}^{2}/(E-{\omega}_{e}-i{\gamma}_{e})-i({\gamma}_{d}+{\gamma}_{g})$*, d* = *a, b*, relating to *g _{a}* or

*g*only. The effective dispersive coupling potential is

_{b}*G*(

_{d}*E*) =

*g*/(

_{a}g_{b}*E*−

*ω*−

_{e}*iγ*)−

_{e}*i*(

*γ*+

_{d}*γ*), dependent on both

_{g}*g*and

_{a}*g*. The loss of the system and gain of TLSs are placed into the potential and do not appear in the scattering equation explicitly. Because the loss of two cavities at the intersection are almost the same, we assume

_{b}*γ*=

_{a}*γ*≡

_{b}*γ*. The loss of the system is

_{ab}*γ*≡

_{s}*γ*+

_{ab}*γ*. Because ${a}_{0}^{\u2020}{a}_{0}{a}_{0}^{\u2020}|g0\u3009$ and $|g\u3009\u3008g|{a}_{0}^{\u2020}|g0\u3009$ have the same result. Then the loss of two cavities and the ground state of TLSs are then partially undistinguishable. If we change

_{g}*γ*and

_{ab}*γ*simultaneously while keeping

_{g}*γ*unchanged, TPs are unchanged. When TLSs have neither gain nor loss, the complex delta-like potentials become actual delta potentials [10].

_{s}#### 2.1. Photons are incident from the semi-infinite CRW-b

When photons are incident from the CRW-b, they are reflected but not transmitted in the CRW-b because of the hard boundary. Photons propagate across the TLSs, being absorbed and emitted by them. They may transfer into the CRW-a, propagating leftward and rightward. The wave functions are

We obtain ${t}_{l}^{a}={t}_{r}^{a}\equiv {t}_{b}^{a}$ by using the continuous condition of the wave function of the CRW-a. The dispersion relation of two CRWs is *E* = *ω _{d}* +

*ω*− 2

_{g}*ξ*(

_{d}cos*k*),

_{d}*d*=

*a, b*[10–12]. The group velocity is

*v*= 2

_{gd}*ξ*(

_{d}sin*k*), and

_{d}*k*ranges from 0 to

_{d}*π*. Hereafter, we assume that the energy bands of two CRWs overlap (

*ω*=

_{a}*ω*≡

_{b}*ω*and

*ξ*=

_{a}*ξ*≡

_{b}*ξ*). Because only in this case can the maximum transfer probability be obtained [10–12]. Applying the dispersion relation, we obtain

*k*=

_{a}*k*≡

_{b}*k*and

*v*=

_{ga}*v*≡

_{gb}*v*.

_{g}Solving wave functions and scattering functions, the amplitudes of transfer and reflection probabilities are as follows,

The transfer and reflection probabilities are ${T}_{b}^{a}=2|{t}_{b}^{a}{|}^{2}$ and *R _{b}* = |

*r*|

_{b}^{2}.

When the denominator of the TPs is zero and the numerator is nonzero, the TPs are divergent: 1) ${\gamma}_{e}={\gamma}_{1}\equiv -i{g}_{b}^{2}/(\xi {e}^{-ik}-i{\gamma}_{g})+{g}_{a}^{2}/({v}_{g}-{\gamma}_{g})$, and 2) the incident energy of photons resonates with the pump energy of TLSs (*E* = *ω _{e}*). The TPs have a divergent point whenever TLSs have gain. When TLSs have gain only, condition 1) changes into ${\gamma}_{1}=2{g}_{b}^{2}\mathrm{sin}{(k)}^{2}/{v}_{g}+{g}_{a}^{2}/{v}_{g}-i{g}_{b}^{2}\mathrm{sin}(2k)/{v}_{g}$. From the [10], we know that the decay rate of the CRW-b (CRW-a) is expressed as $2{g}_{b}^{2}{\mathrm{sin}}^{2}(k)/{v}_{g}({g}_{a}^{2}/{v}_{g})$ [72]. The real part of divergent condition 1) without loss and the decay rate of the [10] are the same. This may indicate that transport probabilities of photons are increased when the gain equals to the decay rate.

As the two couple strengthes between the TLSs and two cavities are almost the same, we assume *g _{a}* =

*g*≡

_{b}*g*.

*When the system has neither loss nor gain*, the total TPs are always conserved [10–12]. All photons are transferred into the CRW-a and not reflected in the CRW-b when $E={\omega}_{e}^{\prime}\equiv {\omega}_{e}-{g}_{b}^{2}\mathrm{cos}(k)/\xi $. In this case, photons are fully absorbed and emitted by TLSs, and all are transferred into the CRW-a (half leftward and half rightward).

*When the system has loss only*, total $\text{TPs}({R}_{b}+{T}_{b}^{a})$ are equal to 1 or lower. When $E={\omega}_{e}^{\u2033}\equiv {\omega}_{e}-{g}_{b}^{2}\xi \mathrm{cos}(k)/[{\xi}^{2}+{\gamma}_{g}^{2}-2\xi {\gamma}_{g}\mathrm{sin}(k)]$, that the energy of photons and the frequency of TLSs are nearly equal, the TPs are conserved. ${\omega}_{e}^{\u2033}$ is reduced to ${\omega}_{e}^{\prime}$ when

*γ*equals to zero.

_{g}*R*increases and ${T}_{b}^{a}$ decreases (photons reflect more and transfer less) as the extent of loss increases. The incident photons are absorbed by the TLSs, pumping the TLSs from the ground state to the excited state. This process is enhanced when $E={\omega}_{e}^{\u2033}$, and the number of the TLSs in the ground state decreases. The effects of loss in the ground state on the TPs are then negligible. All photons are reflected when the group velocity of photons becomes zero (the incident energy reach the boundaries of the energy band). Because the photons are motionless. Except for the near-resonate condition and the case that happens when the incident energy reaches the boundary of the energy band, the TPs are conserved. In the other cases, the TPs are not conserved, and few photons are transferred to CRW-a although the TLSs couple to two CRWs, as illustrated in Fig. 2(a).

_{b}*R*first decreases and then increases, exhibiting a bounce back phenomenon. During the bounce-back process, energy first transfers from the system to the environment, and then from the environment to the system.

_{b}*R*is minimal under the conditions that 1) the loss nearly equals to the coupling strength between the neighboring cavities (

_{b}*γ*≈

_{s}*ξ*) and 2) the frequencies of the photons and cavities are nearly resonant (

_{b}*E*≈

*ω*), as displayed in Fig. 2(b). Except for the near-resonate condition and the case that happens when the incident energy reaches the boundary of the energy band, photons are incident on the hard boundary and may escape to the environment through the cavities.

*When the system has the gain only*, all photons are reflected except for those at the divergent point. The TLSs decay from the excited state to the ground state when

*E*=

*ω*and the system has gain, and numerous photons are radiated. Then the TPs are divergent with the condition

_{e}*γ*=

_{e}*γ*

_{1}. In this process, the energy transfers from the environment to the system.

*When the system has gain and loss*, we consider the simple case that the gain and loss are equal, assuming

*γ*=

_{e}*γ*≡

_{s}*γ*. ${T}_{b}^{a}$ is zero except for the divergent point [Fig. 3(a)].

*R*has a divergent and a minimal point, determined by

_{b}*ω*and

_{e}*ω*respectively, as shown in Fig. 3(b). When the gain satisfies the divergent condition 1), nearly all photons are reflected, and total TPs are equal to 1 or lower except for the divergent point, as illustrated in Fig. 3(c). In Fig. 3(d) when

*E*=

*ω*, the reflected probability bounces back after the divergent point when loss is high. This bounce-back phenomenon in the TPs is typically present in lossy non-Hermitian systems [60–62].

_{e}#### 2.2. Photons are incident from the infinite CRW-a

When photons are incident from the CRW-a, they are reflected and transmitted, different from the CRW-b case. Photons can be absorbed and emitted by the TLSs when they pass through the intersection cavities. Photons may only move upwards into the CRW-b.

The wave functions are

Using the continuous condition of wave functions of the CRW-a at *j _{a}* = 0, we obtain 1 +

*r*=

_{a}*t*. The amplitudes of transfer, reflection, and transmit probabilities can be determined as

_{a}The transfer, reflection, and transmit probabilities are ${T}_{a}^{b}=|{t}_{a}^{b}{|}^{2}$, *R _{a}* = |

*r*|

_{a}^{2}and

*T*= |

_{a}*t*|

_{a}^{2}. The probability that photons are in the CRW-a is

*P*≡

_{a}*T*+

_{a}*R*. We can see that ${t}_{a}^{b}$ equals ${t}_{b}^{a}$. This is because the T-shaped system can be considered as three semi-infinite CRWs. The probability that photons are transferred into one semi-infinite chain of the CRW is the same. Photons incident from the CRW-b are transferred leftward and rightward, while only upward from the CRW-a case. Then the transfer probability that photons are incident from the CRW-b is thus twice for the CRW-a case.

_{a}We consider the situation in which TLSs couple to two CRWs. *When the system has neither loss nor gain*, total TPs are equal to 1. In case of $E={\omega}_{e}^{\prime}$, half of the photons are reflected, and half of them are transferred, but none are transmitted. *When the system has loss only*, total TPs are equal to 1 or lower. When $E={\omega}_{e}^{\u2033}$, photons are almost all reflected [Fig. 4(b)], with the remaining photons transferring into the CRW-b, and none are transmitted [Fig. 4(a)], maintaining reserved. In the near-resonant case, the TLSs in the ground state absorb photons and jump to the excited state because of their interactions with the CRW-b, after which they return to the ground state because of their couplings with the CRW-a. This physical process is efficient when there is no loss but may not occur if the TLSs have loss. The number of transferred photons drops to zero as the degree of loss increases. For photons of different incident energies, except that the incident energy equals to the pump frequency nearly and the boundary of the energy band, *T _{a}* decreases and

*R*increases as the extent of loss increases. However,

_{a}*P*first decreases and then increases. This bounce-back phenomenon is demonstrated by

_{a}*P*but concealed in

_{a}*T*or

_{a}*R*. All photons are reflected as they reach the boundary of the energy band. When photons are incident from the CRW-a, they may be transmitted and reflected in the CRW-a. Therefore

_{a}*T*and

_{a}*R*have no obvious minimal point.

_{a}*When the system has gain only*, all photons are transmitted except for the divergent point. This is different compared with when photons are incident from the CRW-b, because photons meet the hard boundary and cannot be transmitted but only reflected. When the incident energy of photons reaches the boundary of the energy band, all photons are transmit, which is different from the loss-only case. Because in loss-only case, the TLSs may lose efficacy and photons pass straight through TLSs.

*When the system has loss and gain*, the TPs can be divergent under certain conditions but have no obvious minimal point. However, the TPs can have minimal point if photons are incident from the CRW-b. This is because that

*T*decreases and

_{a}*R*increases as the extent of the loss and gain increases except at the divergent point. When photons are incident from the CRW-b, in which there is a hard boundary,

_{a}*R*amounts to

_{b}*P*, indicating the probability that photons are present in each CRW. Then

_{a}*R*has a minimal point, but

_{b}*T*and

_{a}*R*do not. When the loss and gain satisfy

_{a}*γ*=

_{e}*γ*

_{1}, almost all photons are transmitted except for the divergent point and the boundary case, and the total TPs are below 1 with the effect of loss, as illustrated in Fig. 5(c).

*ω*is the frequency of the photons. When

_{p}*E*=

*ω*,

_{e}*R*experience a bounce back. That

_{a}*T*decreases before reaching the divergent point is a unique feature (discussed later), as shown in Fig. 5(d).

_{a}The loss of two cavities at the intersection affects the transport properties of the non-Hermitian router. Because the loss of the system consists of loss of the cavities and TLSs, we consider the case in which the loss of the system is larger than its gain. Because *γ _{g}* =

*γ*=

_{ab}*γ*is included in the case

_{e}*γ*= 2

_{s}*γ*, we consider the case that

_{e}*γ*= 2

_{s}*γ*. When

_{e}*E*=

*ω*, the TPs have a divergent point when the system has gain, and the increasing loss has little effect on the TPs. However, when

_{e}*E*=

*ω*, the increasing loss does affect the TPs. As illustrated in Fig. 6(a), when

*E*=

*ω*, ${T}_{b}^{a}$,

*T*and

_{a}*R*decrease, whereas

_{a}*R*increase, because

_{b}*γ*increases and we maintain

_{e}*γ*=

_{s}*γ*. As the loss of system increases, keeping

_{e}*γ*= 2

_{s}*γ*, the minimal point of

_{e}*R*is left-shifted, as displayed in Fig. 6(b). The total TPs first decrease and then increase, showing a bounce-back phenomenon.

_{b}We illustrate the effects of solely loss and gain and also their combination and how the non-Hermitian quantum router works. When the system has loss, the TPs are conserved and exhibit a bounce-back phenomenon and *R _{b}* has a minimal point. The main contribution of gain is to increase the TPs under the divergent condition, and to cover the probabilitiy conservation case if the system has loss. The combination of loss and gain causes shifts of the divergent and minimal points. The TLSs mediate between the loss and gain and the transport properties of photons. Next, we explain how the non-Hermitian router works. The router switches on when TLSs couple with the two CRWs and $E={\omega}_{e}^{\u2033}$. We increase the transfer probability of the router by adjusting the incident energy and the extent of gain to match the divergent condition. To decrease the TPs of the router, we make the parameters match those in the minimal condition when photons are incident from the CRW-b, and identify an area in which

*T*+

_{a}*R*is relatively small for the CRW-a case. The router switches off providing the TLSs decouple with one of the two CRWs. The on-off switching of the router is thus controlled by the coupling strength between the TLSs and the two CRWs and is modified by the loss of cavities and the ground state of TLSs and gain in the excited state of TLSs.

_{a}## 3. Conclusion

In this paper, we analyze transport properties of a T-shaped non-Hermitian quantum router comprising an infinite CRW-a, a semi-infinite CRW-b, and TLSs. Photons are conserved and totally reflected when $E={\omega}_{e}^{\u2033}$ and the system has loss only. *R _{b}* and

*P*exhibit a bounce-back phenomenon.

_{a}*R*is at a minimum when

_{b}*E*≈

*ω*and

*γ*≈

_{s}*ξ*. The TPs are divergent when

_{b}*E*=

*ω*and

_{e}*γ*=

_{e}*γ*

_{1}. The main contributions of gain are to increase the TPs under the divergent condition and to cover the probability conservation case if the TLSs have loss. The combination of loss and gain results in a shift of the divergent and minimal points. The on-off switching of the router is controlled by the coupling strength between the TLSs and two CRWs and is modified by loss and gain. The TPs of the router is increased (decreased) for proper values of the incident energy of photons and the gain (loss).

## Funding

Nankai University Baiqing Plan Foundation.

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