1. Introduction

Cavity quantum electrodynamics (cavity QED), which studies interactions between single photons and single atoms engineered through the use of optical cavities or resonators, provides a way of realizing quantum-state preparation and manipulation that are of immediate relevance to quantum information processing (QIP). Cavity QED systems comprised of single atoms in optical resonators could also be key physical systems for constructing a quantum network, in which atoms are used for storage of quantum information at local nodes of the network, photons provide a way of distributing quantum information between distant nodes, and atom-photon interactions enable transfer between matter and light, and also manipulation, of the quantum information [1, 2]. In this context, cavity QED systems with optical Fabry-Pérot cavities have contributed an impressive range of experimental demonstrations, such as state transfer between light and atoms, nondestructive detection of optical photons, and controlled quantum dynamics [3–10]. On the other hand, in order to realize photonic architectures for scalable QIP and quantum networks based on propagating photons and effects in cavity QED, various systems with monolithic whispering-gallery-mode (WGM) resonators are now under active investigation [11–18]. The WGM resonators such as microspheres, microdisks, and microtoroids coupled via evanescent fields to tapered optical fibers generally exhibit the properties of ultralow intrinsic cavity-mode losses and high-efficiency transfer of photons into and out of the cavity modes. Meanwhile, the ultrasmall mode volumes of the WGMs supported by these resonators could lead to very large single-photon electric fields and, consequently, very large photon-atom coupling strengths. Nowadays, the strong-coupling regime of cavity QED, defined as the regime where the photon-atom coupling rate is bigger than both the rates for cavity decay and dipole decay of the atom, has been demonstrated for a single alkali atom in the evanescent field of a microtoroidal [19] or microbottle [20] resonator, and for a single quantum dot (QD) embedded in a microdisk resonator [21,22]. Also, some operations in the bad-cavity regime have been demonstrated with microtoroidal resonators [23, 24], which provide key techniques for the realization of entanglement and quantum gates with atoms [25,26] or QDs [27–29].

Quantum router is a quantum-mechanical counterpart of the classical router used to steer the quantum signal from its source to intended destination. In contrast to a classical router, the quantum router exploits various quantum dynamics and phenomena such as quantum superposition or entanglement, which is a key element needed for the construction of future complex quantum networks. In some cases, classical information is used to control the path of quantum signals [30]. For a fully functional quantum router, however, both the control and signal must be quantum. Thus it can direct a signal qubit to its desired signal mode controlled by the state of a control qubit while keeping the signal qubit intact. Another important element in quantum networks is quantum decoupler, a device implementing an inverse operation to quantum routing. Construction of photonic quantum routers and decouplers has been the subject of both theoretical and experimental investigation in recent years. In 2013, Lemr et al. [31, 32] presented linear-optical protocols of quantum routing together with several criteria the router has to fulfill in order to be fully functional. Two similar processes named “quantum jointing” and “quantum splitting” have been demonstrated by Vitelli et al. [33] with linear optics. Recently, photon-atom interactions in cavity QED systems have also been considered for the construction of single-photon routers [24,34–36]. In addition to the simple routing for the purposes of quantum networking, quantum router and decoupler are also very important quantum devices in QIP, such as in quantum message authentication and Bell-state manipulation [37], quantum entanglement generation [38,39], hyperentangled-state manipulation [40,41], quantum secure direct communication [42], and quantum network manipulation [43]. In particular, the quantum router and decoupler can work as quantum wave divider and combiner in duality quantum computer [44–46], which shows very important applications in designing quantum algorithms [47–50].

The nitrogen-vacancy (N-V) defect center in diamond has been identified as a promising solid-state candidate for QIP [51–53]. It features a lot of desirable properties such as optical controllability and good electron-spin coherence even at room temperature, and the experimental feasibility of this system has been well established in recent years. Experiments have demonstrated individual electronic spin initialization, manipulation, and measurement [54–56]. The possibility of an optical Λ-type scheme of orthogonally polarized photons at 637 nm in a NV center system has also been demonstrated, which leads to a coherent interface between an electron spin and an optical photon [57, 58]. In particular, much interest to date has been focused on QIP using solid-state counterparts of the cavity QED systems in which the NV centers are coupled with WGM resonators. For example, entanglement of separate NV centers could be set up via coupling to WGM resonators [59–63]. The similar physics has also been shown useful for constructing quantum logic gates [64–70] and realizing entangled-state manipulations [71–76]. In recent experiments, the coupling between the NV center and the WGM has been demonstrated by using different resonators, such as microspheres [77–79], GaP microdisks [80], and SiN photonic crystals [81]. Moreover, the considerable enhancement of the resonant zero phonon line (ZPL) relevant to the emitted photons from NV centers has also been demonstrated by coupling the NV centers to WGM microring resonators [82].

A single microresonator that is coupled to two waveguides can behave as a resonant drop filter, which has a large number of applications in QIP. Driving such a drop-filter structure with an incident field is mathematically equivalent to driving a double-sided cavity [83]. Thus some QIP proposals with double-sided cavity configuration could be implemented with the drop-filter structure. Another important property of the structure is that it is very suited for implementing photonic multiple-DOF QIP based on cavity QED. In 2014, Wang et al. [84] presented two schemes of constructing universal hybrid three-qubit quantum gates, resorting to the system of a single NV center interacting with a WGM microtoroidal resonator that couples with two tapered optical fibers functioning as a drop-filter structure. Subsequently, Ren et al. [85] showed that the dipole induced transparency (DIT) of a NV center embedded in a photonic crystal cavity coupled to two waveguides can be used to construct photonic hyperparallel hybrid quantum gates. Drop filtering has been demonstrated in some experiments, e.g., using microdisks [86], photonic crystals [87, 88], and microtoroids [89–91]. Recently, highly efficient optical switching between two optical fibers controlled by a single atom was demonstrated using a WGM microbottle resonator that is coupled to the atom and interfaced by two tapered fiber couplers, assisted by the vacuum Rabi splitting in the strong-coupling regime of cavity QED [92].

In this work, we present a scheme for single-photon quantum routing controlled by the other photon using a hybrid cavity QED system and the single-photon input-output process regarding the system. The system consists of a single NV center coupled with a microtoroidal WGM resonator that couples to two tapered optical fibers functioning as a drop-filter structure. Using this scheme, the signal-photon qubit can be directed to its desired signal mode controlled by the control-photon qubit, with the original signal qubit being unchanged. Different from the cases in which classical information is used to control the path of quantum signals, both the control and signal photons are quantum in our implementation. Therefore, our scheme leads to a fully functional quantum router, which could play a key role in QIP and quantum networks. Compared with the probabilistic quantum routing protocols based on linear optics, our scheme is deterministic and also scalable to multiple photons. We also present a scheme for single-photon quantum decoupling from an initial state with polarization and spatial-mode encoding, which can implement an inverse operation to the quantum routing. We discuss the feasibility of our schemes by considering current or near-future techniques, and show that both the schemes can operate effectively in the bad-cavity regime, defined as the regime where the resonator extrinsic decay rate is bigger than the coupling strength of the NV center to the WGM. So, our schemes relax the constraint on using the strong-coupling regime of cavity QED, allowing implementation in a more practical parameter regime for NV centers. The future technical advances in increasing the coupling strength and reducing the intrinsic photon loss would further improve the implementation efficiency and the fidelity. We believe that our schemes would be useful for implementing large-scale QIP and constructing NV-resonator-based quantum networks.

This paper is organized as follows: In Sec.2, we describe the theoretical model of the hybrid system used in our schemes and present the single-photon input-output relation regarding the system under ideal conditions. In Sec.3, we present a single-photon quantum routing scheme controlled by the other photon using the system and the corresponding input-output relation. The following Sec.4 describes a scheme for single-photon quantum decoupling from an initial state with polarization and spatial-mode encoding, which implements an inverse process to the quantum routing. Finally, in Sec.5, we discuss the feasibility of our schemes by considering current or near-future techniques and summary the paper.

2. Theoretical model and a linearized analysis

A schematic of the system used in our schemes is shown in Fig. 1, in which a negatively charged NV center in diamond is fixed on the exterior surface of a microtoroidal WGM resonator and coupled by the evanescent field to the resonator. The NV center has six electrons from the nitrogen and three carbons surrounding the vacancy. Due to spin-spin interactions, the ground state of the NV center is a spin triplet with 2.87 GHz zero-field splitting between levels |0〉 (m_s = 0) and |±1〉 (m_s = ±1). As defined by group theory [57,58], the six excited states can be expressed as |A₁〉 = |E₋〉 |+1〉−|E₊〉 |−1〉, |A₂〉 = |E₋〉 |+1〉+|E₊〉 |−1〉, |E_x〉 = |X〉 |0〉, |E_y〉 = |Y〉 |0〉, |E₁〉 = |E₋〉 |−1〉−|E₊〉 |+1〉, and |E₂〉 = |E₋〉 |−1〉 +|E₊〉 |+1〉, with |E₋〉, |E₊〉, |X〉, and |Y〉 being orbital states. In the absence of any perturbation such as by crystal strain or an external magnetic field, the six excited states are eigenstates of the full Hamiltonian including spin-orbit and spin-spin interactions. The spin-orbit interaction splits the states |A₁〉 and |A₂〉 from the others by at least 5.5 GHz. The spin-spin interaction increases the energy gap between |A₂〉 and |A₁〉 to 3.3 GHz. In the low-strain limit, the |A₂〉 state exhibits the robust symmetric properties, and decays to the ground-state sublevels |−1〉 and |+1〉 with equal probability [57,58]. We can therefore encode the qubits in the sublevels |±1〉 and employ the |A₂〉 state as an ancillary to construct an optical Λ -type scheme. The transition |−1〉 ↔ |A₂〉 could only couple with a σ₊ -polarized photon, while the transition |+1〉 ↔ |A₂〉 could only couple with a σ₋-polarized photon.

 

Fig. 1 (a) Schematic of the system used in our schemes. A single NV center is fixed on the exterior surface of a WGM microtoroidal resonator and coupled by the evanescent field to the resonator. The resonator is evanescently coupled to two tapered fiber fibers (waveguides) and behaves as a drop-filter structure. The two waveguides are marked with 1 and 2 and the four ports are marked with a₁, b₁, a₂, and b₂, respectively. κ_e represents the extrinsic decay rate from the resonator into each waveguide mode. κ_i denotes the intrinsic decay rate of the resonator into loss mode. γ is the NV center dipolar decay rate. (b) Energy level configuration of the NV center.

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As sketched in Fig. 1, the WGM resonator is evanescently coupled to two tapered optical fibers (waveguides) and behaves as a drop-filter structure. The two waveguides are marked with 1 and 2 and the four ports are marked with a₁, b₁, a₂, and b₂, respectively. We consider a single-photon pulse with frequency ω_p incident along the port a₁ and the NV center initially prepared in the ground state, the Langevin equations of motion for the lowering operators of the resonator and the NV center (i.e., â and ${\widehat{\sigma }}^{-}$) can be described as

Under the resonator condition (ω₀ = ω_c = ω_p), Eq. (2) can be rewritten as

For the uncoupled cases (i.e., g = 0), as the incident photon is in resonance with the bare resonator, we have r₀(ω_p) → −1 and t₀(ω_p) → 0 with κ_e ≫ κ_i. When g²/(κ_eγ) ≫ 1 and κ_e ≫ κ_i, we have r(ω_p) → 0 and t(ω_p) → 1. This situation is illustrated in Fig. 2, where |r(ω_p)| and |t(ω_p)| are plotted as a function of g²/(κ_eγ) with ω_p = ω_c = ω₀ and κ_i = 0.01κ_e. Thus the single-photon input-output process regarding the system, under ideal conditions, can be described as follows: If the NV center is initially prepared in state |−1〉 the only possible transition is |−1〉 ↔ |A₂〉 driven by a σ₊-polarized photon. In this case, a single photon with σ₊ polarization being sent in port a₁ would be transmitted along the waveguide to port b₁, while a photon with σ₋ polarization in port a₁ would be reflected to port b₂ and obtain a π phase shift due to the polarization mismatch. In contrast, if the NV center is initially prepared in |+1〉, the corresponding transition is |+1〉 ↔ |A₂〉 driven by a σ₋-polarized photon. In this case, a photon with σ₊ polarization being sent in port a₁ would be reflected to port b₂ and obtain a π phase shift, while a photon with σ₋ polarization in port a₁ would be transmitted to port b₁. Similarity, a photon being sent in port a₂ would be reflected to port b₁ in the resonant but uncoupled cases, or be transmitted to port b₂ in the resonant and coupled cases.

 

Fig. 2 Absolute values of the reflection and transmission coefficients (|r(ω_p)| and |t(ω_p)|) as a function of g²/(κ_eγ) with ω_p = ω_c = ω₀ and κ_i = 0.01κ_e. |r(ω_p)| and |t(ω_p)| correspond to the red dashed and blue solid lines, respectively.

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This process is similar to the DIT shown in [83,85], which can operate in the bad-cavity limit, for which κe > g (but g ≫ γ so that g²/(κ_eγ) ≫ 1). This case is illustrated in Fig. 3, where |r(ω_p)| and |t(ω_p)| are plotted as a function of normalized frequency detuning (ω_c − ω_p)/κ_e for several different values of g/κ_e. Good operation (i.e., |r(ω_p)| → 0 and |t(ω_p)| → 1) on resonance (ω_p = ω₀) of the NV-resonator-waveguide system can be achieved even g < κ_e, indicating that our schemes relax the constraint on using the strong-coupling regime of cavity QED, allowing implementation in a more practical parameter regime for NV centers. On the other hand, photon polarization is commonly defined with respect to the z axis (see Fig. 1). In our schemes, photon qubits are encoded in states |R〉 and |L〉 where R and L represent the right and left circular polarizations with respect to the z axis, respectively. a₁ and a₂ are chosen as two input ports of the system with opposite directions of propagation of a photon incident. Upon reflection, both the polarization and propagation direction of the photon would be flipped [84, 85, 94, 95]. For example, if the NV center is initially prepared in |−1〉 the only possible transition is |−1〉 ↔ |A₂〉 driven by a photon in state |Ra₁〉 or |La₂〉. As a result, an incident photon in the state |La〉, experiencing the input-output process, would be transformed to the state −|Rb₂〉 Under ideal conditions, the single-photon input-output relation regarding the system can be summarized as

 

Fig. 3 Absolute values of the reflection and transmission coefficients (|r(ω_p)| and |t(ω_p)|) as a function of normalized frequency detuning (ω_c − ω_p)/κ_e for (a) g = 0, (b) g = 0.1κ_e, (c) g = 0.5 κ_e, and (d) g = κ_e with ω₀ = ω_c, γ = 1.5 × 10⁻³ κ_e, and κ_i = 0.01κ_e. |r(ω_p)| and |t(ω_p)| correspond to the red dashed and blue solid lines, respectively.

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3. Single-photon quantum routing controlled by the other photon

The function of a quantum router is to direct a signal qubit to its desired signal mode controlled by the state of a control qubit while keeping the signal qubit unaffected. The hybrid system and the single-photon input-output relation introduced above can be used to make a single-photon quantum router and the quantum circuit is shown in Fig. 4. Here the signal qubit is encoded in polarization DOF of a single photon as

The initial state of the two photon, labelled s and c, can then be written as the product |ϕ〉^s ⊗ |φ〉. The over quantum circuit, shown in Fig. 4, has two input modes c_in and s_in. The NV center is initially prepared in state $\frac{1}{\sqrt{2}}\left(|-1〉+|+1〉\right)$.

 

Fig. 4 Schematic of the single-photon quantum router for the implementation of signal-photon quantum routing controlled by a control photon. CPBS represents polarizing beam splitter in the circular basis, which transmits the right circularly polarized photon |R〉 and reflects the left circularly polarized photon |L〉. H denotes half-wave plate which is set to 22.5° to induce the Hadamard transformations on the polarization of photons as $|R〉\to \frac{1}{\sqrt{2}}\left(|R〉+|L〉\right)$ and $|L〉\to \frac{1}{\sqrt{2}}\left(|R〉-|L〉\right)$. P is phase shifter that contribute a π phase shift to the photon passing through it. X is half-wave plate which is used to perform a polarization bit-flip operation ${\sigma }_x^p=|R〉〈L|+|L〉〈R|$ on the photon passing through it. DL is delay line used to erase the time difference between the different components.

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The signal-photon quantum routing controlled by the control photon can be implemented with the following procedure: First, the control photon in the state |φ〉^c is taken to enter the setup along input mode c_in. After the control photon passes through the circuit (see the red dashed lines in Fig. 4), the state of the composite system composed of the control photon and the NV center is transformed from |Φ₀〉 to |Φ₁〉. Here

The evolution process can be described as follows:

After a Hadamrad operation $[|-1〉\to \frac{1}{\sqrt{2}}\left(|-1〉+|+1〉\right),|+1〉\to \frac{1}{\sqrt{2}}\left(|-1〉-|+1〉\right)]$ is performed on the ground-state spin of the NV center, the control photon and the NV center become entangled in state

Then, the signal photon in the state |φ〉^s enters the setup along input mode s_in. After the signal photon passes through the circuit (see the green solid lines in Fig. 4), the state of the composite system composed of the NV center and the two photons is transformed from |Φ₃〉 to |Φ₄〉. Here

If the outcome of the measurement is |−〉, we obtain

This state can be transformed back into Eq. (11) by adding a π phase shift on the spacial mode 2 of the outgoing signal photon. That is, after the conditional unitary operations are performed on the signal photon, the two-photon state is deterministically changed from |ϕ〉^s ⊗ |φ〉 to Eq. (11), which is generally a superposition state of two modes in ports 1 and 2. The signal photon can therefore be directed to port 1, port 2, or both controlled by the control-photon qubit, with the original signal-photon qubit being unchanged.

4. Single-photon quantum decoupling from an initial state with polarization and spatial-mode encoding

The system and the single-photon input-output relation can also be useful for implementing single-photon quantum decoupling, by which two qubits encoded in two DOFs of a single photon is transferred into the internal quantum state of two photons, which can be seen as an inverse process to quantum routing. The quantum circuit is shown in Fig. 5. We assume to have an incident signal photon encoding two qubits in its polarization and spatial-mode DOFs as

 

Fig. 5 Schematic of the single-photon quantum decoupler for the implementation of signal-photon quantum decoupling. All the devices are the same as that in Fig. 4.

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We label this photon as s, with subscripts in₁ and in₂ denoting two spatial modes. Each of these modes may be considered as a single polarization-encoded qubit. We also need an ancillary photon, labeled as a, initialed in state |R〉^a, so that the initial two-photon state is |ψ〉^s ⊗ |R〉^a. The NV center is initially prepared in state $\frac{1}{\sqrt{2}}\left(|-1〉+|+1〉\right)$ First, the signal photon in the state |ψ〉^s enters the setup along input modes in₁ and in₂. After the photon passes through the circuit (see the red dashed lines in Fig. 5), the state of the composite system composed of the signal photon and the NV center is transformed from |Ψ₀〉 to |Ψ₁〉. Here

Meanwhile, the polarization-encoded qubit is transferred to the ground-state spin of the NV center. Then, the ancillary photon in the state |R〉^a enters the setup along input mode a_in. After the ancillary photon passes through the circuit (see the green solid lines in Fig. 5), the state of the composite system composed of the NV center and the two photon is transformed from |Ψ₂〉 to |Ψ〉. Here

When the signal photon comes out of the output mode 1 (i.e., no photon comes out of the output mode 2), the purpose of quantum decoupling can be achieved by measuring the NV center in the orthogonal basis {|+〉, |−〉}. If the outcome of the measurement is |+〉, the ancillary photon is in the state

If the outcome of the measurement is |−〉, the ancillary-photon state is

This state can be transformed back into Eq. (17) by a simple phase-flip operation ${\sigma }_z^p=|R〉〈R|-|L〉〈L|$. That is, after the conditional operations are performed on the ancillary photon, the two-photon state is changed to |φ〉^s ⊗ |ϕ〉^a, which has the same two qubits as the incident signal, but encoded in two photons instead of one. Two separate photonic qubits are therefore decoupled from an initial photonic state |ψ〉^s with polarization and spatial-mode encoding.

The proposed scheme for quantum decoupling has a 50% rate of success. It might be possible to bring the success rate to 100% by detecting the actual signal-photon output mode by, for instance, a quantum non-demolition approach, and then applying appropriate unitray operations to the ancillary photon, same as in [33]. In this way, we can in principle turn the scheme into a deterministic one.

5. Discussion and summary

Using the analysis in Sec. 2, we can evaluate the efficiencies and fidelities of our schemes. Efficiency of our quantum routing scheme is defined as the probability of both the signal and control photons being detected after the routing process in a practical experimental environment. Fidelity of the scheme is defined as $F={\left|〈{\mathrm{\Phi }}_f|\mathrm{\Phi }〉\right|}^2$, where |Φ〉 is the ideal final state of the two photons after the routing process and |Φ_f〉 is the final two-photon state with considering the experimental imperfections. In our implementation, the single-photon input-output relation can be stringently expressed as

Eq. (19) could get back to Eq. (4) under ideal conditions. The parameters g, κ_e, κ_i, and γ would affect the reflection and transmission coefficients and, consequently, the efficiency and fidelity of the scheme. If we assume perfect NV center operations and $\beta =\delta =1/\sqrt{2}$, the efficiency η₁ and the fidelity F₁ of our quantum routing scheme can be straightforwardly calculated as

We numerically calculated η₁ and F₁ as a function of g²/(κ_eγ) and κ_i/κ_e under the resonant condition and the results are shown in Figs. 6(a) and 6(b). For our quantum decoupling scheme, efficiency is defined as the probability of both the signal (outgoing from the output port 1) and ancillary photons being detected after the decoupling process in a real experiment. Fidelity is defined as $F={\left|〈{\mathrm{\Psi }}_c|\mathrm{\Psi }〉\right|}^2$, where |Ψ〉 = |φ〉 ^s ⊗ |ϕ〉^a is the ideal final state of the two photon and |Ψ_f〉 is the final two-photon state with considering the experimental imperfections. Omitting the NV center related infidelities, the efficiency η₂ and the fidelity F₂ of our quantum decoupling scheme is given by

 

Fig. 6 Efficiencies and fidelities of our quantum routing (η₁, F₁) and decoupling (η₂, F₂) schemes as a function of g²/(κ_eγ) and κ_i/κ_e with $\beta =\delta =1/\sqrt{2}$ and ω_p = ω_c = ω₀.

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A variety of cavity QED system have been pursued for NV centers over past years, and the coupling strength of the NV center to an optical resonator is generally on the order of hundreds of megahertz [78–81]. However, the strong-coupling regime is still hard to achieve due to some effects, such as spectral diffusion and a large phonon side band. Recent experimental efforts have focused on composite microcavity systems where NV centers in high-purity bulk diamond couple evanescently to optical resonators, such as silica microspheres [78, 79], diamond-GaP microdisks [80], and hybrid photonic crystals [81,96], which can retain the excellent properties of NV centers. Considering the existing or near-future techniques, if we set the coupling strength g = 2π × 500 MHz [60] and the dipolar decay rate γ = 2π × 15 MHz [57], the efficiencies and fidelities of our schemes are η₁ = 98.19%, F₁ = 97.08% and η2 = 48.56%, F₂ = 98.53% with κ_e = 2π × 0.5 GHz and κ_i = 0. They are η₁ = 96.49%, F₁ = 94.30% and η₂ = 47.21%, F₂ = 97.11% with κ_e = 2π × 1 GHz and κ_i = 0, and they are η₁ = 95.88%, F₁ = 93.35% and η₂ = 46.73%, F₂ = 96.62% with κ_e = 2π × 1 GHz and κ_i = 0.01κ_e. According to a recent experiment [80], the coupling of a NV center and a WGM resonator could reach the relevant cavity-QED parameters g = 2π × 300 MHz (a prediction value based on ideal placement of the NV centre at the surface of a diamond attached to the WGM resonator) and γ = 2π × 13 MHz. In this case, the efficiencies and fidelities are η₁ = 91.66%, F₁ = 86.30% and η₂ = 43.45%, F₂ = 92.90% with κ_e = 2π × 1 GHz and κ_i = 0.01κ_e. These results imply that our schemes can operate effectively in the bad-cavity regime.

High-fidelity initialization, coherent manipulation, and single-shot readout of the electronic spin associated with a single NV center in diamond have been demonstrated. In particular, the electronic spin can be high-fidelity (∼ 93.2%) readout and addressed at low temperature (T=8.6 K) based on resonant optical excitation techniques [56]. The state |A₂〉 connects |±1〉, and connects |0〉. After spin manipulation by a microwave pulse and resonant excitation transition |0〉 ↔ |E_x,y〉, the presence or absence of fluorescence decay reveals the spin state [56,57]. In the schemes, two photons are sent to the setups in sequence. The time interval between two photons should be much shorter than the electron-spin coherence time to achieve high fidelity, but it should be long enough to make the weak excitation approximation valid. Recent experiments have also shown long spin coherence time (T^e ~ ms) [55, 97], which can be longer than the cavity photon lifetime and the time interval of the input photons (approximately nanoseconds). To achieve the bad-cavity regime as opposed to the bad-emitter regime where pure dephasing dominates the NV center ZPL [93], the experiments need to be done at cryogenic temperatures. Recently, Gould et al. [98] demonstrated GaP-on-diamond disk resonators which resonantly couple ZPL photons from single NV centers to single-mode waveguides. In these devices, the probability of a single NV center emitting a ZPL photon into the waveguide mode after optical excitation can reach 9%, due to a combination of resonant enhancement of the ZPL emission and efficient coupling between the resonator and waveguide. These results indicate that our proposal implemented with the NV center might be feasible, and the experimental and technical imperfections will be largely improved with the further technical advances.

Before ending, we intend to compare this work with some previous studies. In the hybrid system, a microtoroidal WGM resonator is evanescently coupled to two tapered optical fibers and behaves as a drop-filter structure. Compared with the drop-filter cavity-waveguide systems in photonic crystals [83, 85], an important aspect of this particular system, related to the input-output coupling efficiency of photons, is the ability to tune the extrinsic decay rate κ_e of the resonator to the waveguides by adjusting the distances between the fibers and the microtoroid. The WGM resonator that interfaces with two tapered fibers has been demonstrated in [89–92]. As for the NV center, another advantage of the NV center coupling to the WGM resonator is the possibility of fixing the NV center, which is better for implementation compared to moving atoms coupling to cavities. The system is associated with both the polarization and spatial-mode DOFs of photons, which lends itself well to photonic multi-DOF QIP. By encoding qubits simultaneously in polarization and spatial-mode DOFs of single photons, the information-carrying capacity of photons can be increased largely [41]. Therefore, investigating the potential applications of this system in the field of multiple-DOF QIP is of great interest. On the other hand, different from the cases in which classical information is used to control the path of quantum signals [30], both the control and signal photons are quantum in our quantum routing scheme. Therefore, our scheme leads to an all-optical fully quantum router, which could play a key role in QIP and quantum networks. Compared with the probabilistic quantum routing and decoupling protocols based on linear optics [31–33], our schemes are, in principle, deterministic and also scalable to multiple photons. Moreover, our schemes can operate effectively in the bad-cavity regime, which relaxes the constraint on using the strong-coupling regime of cavity QED as in [92].

In summary, we have shown the possibility of implementing single-photon quantum routing and decoupling by using a hybrid cavity QED system and the input-output process of single photons. Compared with previous studies, our schemes have some advantages and are feasible with current or near-future techniques. The future technical advances in increasing the coupling strength and reducing the intrinsic photon loss will further improve the implementation efficiency and the fidelity. Therefore, we believe that our schemes would be useful for implementing large-scale QIP and constructing future complex NV-resonator-based quantum networks.