1. Introduction

Different quantum systems may be combined in a hybrid device for exploring new physical phenomenon or developing new quantum technology [1–9]. As an important component of hybrid systems, microwave cavities or resonators are often used in state-of-the-art science and technology [10]. Over the past years, microwave cavities or photons have been widely used for non-networked small-scale quantum computing in local area. To achieve a networked large-scale quantum computation, it is necessary to implement quantum state transmission between spatially separated microwave cavities.

Microwave photons can be effectively generated and manipulated. While, optical photons can be coherently transmitted by optical fibers. The combination of both enables long-distance quantum communication and quantum information processing [11–15]. As optical photons do not directly interact with microwave photons, a transducer is often required to exchange quantum states between microwave photons and optical photons [16–18]. Thus, quantum transducers are particularly attractive and play an important role in quantum information science and technology [15,19–22].

In recent years, several theoretical proposals for a transducer have been put forward by using various physical systems, such as optomechanical systems [23], NV ensembles [24], atoms [15,25], electro-optical systems [26], magnons [27,28], superconducting circuits [29], and so on. With the help of a transducer, the coherent conversion between microwave and optical photons has been widely studied and has also been demonstrated experimentally [29–35].

NV ensembles are one of the most promising candidates for a microwave-to-optical transducer because they can strongly couple to both microwave and optical cavities [24]. The strong coupling of NV ensembles with superconducting and optical cavities has been realized [36–39]. On the other hand, NV ensembles have been considered as good quantum memory elements in quantum information processing. Long lifetime of NV ensembles has been experimentally reported [40]. Moreover, various quantum operations such as information transfer, entanglement preparation, and quantum gates have been realized with NV ensembles [41–49].

As shown below, the goal of this work is to propose a method for implementing long-distance transmission of arbitrary quantum states between two spatially separated microwave cavities. In this proposal, we use a NV ensemble as a quantum transducer which enables the local transfer of quantum states between a microwave cavity and an optical cavity. In addition, we use an optical fiber to connect two optical cavities, which enables the long-distance transfer of quantum states between the two optical cavities. This method has the following features and advantages.

(1) Remote transmission can be realized by using this method.

(2) The local microwave-optical conversion of photons can be implemented using a single-step operation, and the whole scheme only needs three steps to complete the long-distance transmission of arbitrary quantum states between two spatially separated microwave cavities.

(3) Because the NV ensembles are unexcited during the entire process, decoherence from the NV ensembles is greatly suppressed.

(4) Arbitrary quantum states of microwave photons can be high-fidelity transferred between the spatially separated microwave cavities.

(5) The two optical cavities and the receiver microwave cavity have no requirements for the initial states, and they do not need to be prepared in specific initial states.

(6) The method is robust against the decay caused by environment, the parameter fluctuation, and the additive white Gaussian noise (AWGN).

The structure of this paper is as follows. In Sec. 2, we introduce our hybrid system and derive the effective Hamiltonian. In Sec.3, we explicitly show how to transfer an arbitrary quantum state between two spatially separated microwave cavities. In Sec. 4, we discuss the possible experimental implementation of our proposal and numerically calculate the operational fidelity for transferring an arbitrary quantum state of a microwave photonic qubit in a realistic case. In Sec. 5, we discuss the interferences caused by the real environment. We separately investigate the robustness against stochastic parameter fluctuation and the influences caused by the additive white Gaussian noise (AWGN), which is the most common noise in real experiments. A brief conclusion is given in Sec. 6.

2. Hybrid system and effective Hamiltonian

We consider a hybrid system in Fig. 1 which contains of two parts 1 and 2. Each part contains a microwave cavity, an NV ensemble, and an optical cavity. Each microwave cavity here is a superconducting coplanar waveguide (CPW) resonator. In each part, an NV ensemble is coupled to a microwave cavity via a magnetic coupling and simultaneously interacts with an optical cavity through an optical transition. The two optical cavities (or the two parts 1 and 2) are linked by an optical fiber.

 

Fig. 1. Schematic of a hybrid system. For each part, an optical cavity with an embedded NV ensemble is placed above a superconducting coplanar waveguide (CPW) cavity. The two optical cavities are linked by an optical fiber.

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As shown in Fig. 2(a), the NV defect center in diamond consists of a substitutional nitrogen atom (N) associated with a vacancy (V) in an adjacent lattice site of the diamond crystalline matrix. As shown in Fig. 2(b), the energy-level structure of an NV center is formed by a ground state $^{3}A$, an excited state $^{3}E$, and a metastable state $^{1}A$. Both $^{3}A$ and $^{3}E$ are spin triplet states while the metastable $^{1}A$ is a spin singlet state [50]. Owing to the $C_{3v}$ symmetry of the NV center, the ground-state spin sublevels $m_{s}=\pm 1$ are degenerate and the zero-field splitting from $m_{s}=0$ is $D_{gs}=2.87\,\mathrm {GHz}$ [51]. For simplicity, we label the ground state $|^{3}A\rangle \bigotimes |m_{s}=0\rangle$, the first excited state $| ^{3}A\rangle \bigotimes |m_{s}=+1\rangle$, and the second excited state $|^{3}E\rangle \bigotimes |m_{s}=0\rangle$ of a spin in each ensemble as $|0\rangle$, $|1\rangle$, and $|2\rangle$, respectively.

 

Fig. 2. (a) Atomic structure of an NV center. A single substitutional nitrogen atom (N) is accompanied by a vacancy (V) at a nearest neighbor lattice position. (b) Simplified energy-level diagram of an NV center. The energy-level structure consists of a ground state $^{3}A$, an excited state $^{3}E$, and a metastable state $^{1}A$. Both $^{3}A$ and $^{3}E$ are spin triplet states while the metastable $^{1}A$ is a spin singlet state. Here $D_{gs}$ and $D_{es}$ correspond, respectively, to the zero-field splitting between $m_{s}=0$ and $m_{s}=\pm 1$ in the triplet ground state $^{3}A$ and in the triplet excited state $^{3}E$. (c) The interactions in each part of the hybrid system. For each part, a microwave cavity couples the $|0\rangle _{j}\leftrightarrow |1\rangle _{j}$ transition, an optical cavity couples the $|0\rangle _{j}\leftrightarrow |2\rangle _{j}$, while a classical field drives the $|1\rangle _{j}\leftrightarrow |2\rangle _{j}$ transition of spin $j$ in each NV ensemble, with a Rabi frequency $\Omega$.

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As usual, in the low excitation limit, the collective spin excitations can be mapped to a boson mode labelled as c and then could be used as a medium for the conversion. In Fig. 1, each part can be described by a Hamiltonian composed of two Jaynes-Cummings (JC) interactions. One is the interaction between the microwave cavity mode $a_{1(2)}$ and the collective spin mode $c_{1(2)}$, while the other is the interaction between the optical cavity mode $b_{1(2)}$ and the collective spin mode $c_{1(2)}$. Here and below, the subscript 1 represents part 1 while the subscript 2 represents part 2 in the hybrid system (Fig. 1).

For each part, we consider the NV ensemble contains $N$ spins. We define the three states $|0\rangle$, $|1\rangle$, and $|2\rangle$ of spin $j$ in the NV ensemble as $|0\rangle _{j}$, $|1\rangle _{j}$, and $|2\rangle _{j}$, respectively (Fig. 2(c)). Here, $j=1,2,\ldots N$. In each part, a microwave cavity couples the $|0\rangle _{j}\leftrightarrow |1\rangle _{j}$ transition, an optical cavity couples the $|0\rangle _{j}\leftrightarrow |2\rangle _{j}$, while a classical field drives the $|1\rangle _{j}\leftrightarrow |2\rangle _{j}$ transition of spin $j$ in each NV ensemble, with a Rabi frequency $\Omega$. The detunings for these transitions are $\Delta _{j,1}=\omega _{j,10}-\omega _{m},\,\Delta _{j,2}=\omega _{j,20}-\omega _{o},$ and $\Delta _{j,3}=\omega _{j,12}-\omega _{p}$, Here $\omega _{j,10}$, $\omega _{j,20}$, and $\omega _{j,21}$ are the $|0\rangle _{j}\leftrightarrow |1\rangle _{j}$, $|0\rangle _{j}\leftrightarrow |2\rangle _{j}$, and $|1\rangle _{j}\leftrightarrow |2\rangle _{j}$ transition frequencies of the $j$th spin for each ensemble, respectively. While $\omega _{m},\,\omega _{o}$ and $\omega _{p}$ are the frequencies of the microwave cavity, the optical cavity, and the laser pulse, respectively.

Due to the fluctuating magnetic environment, there might be fluctuations in the detuning of the level $|1\rangle _{j}$. However, these environmental induced fluctuations are much smaller than the frequency detunings, and can be safely ignored [52]. The $j$th spin located at the position $r_{j}$ is coupled to the two cavities (a microwave cavity and an optical cavity) with the coupling strengths $G_{m,j}\propto B_{1}(r_{j})$ and $G_{o,j}\propto E_{2}(r_{j})$, where $B_{1}(r_{j})$, and $E_{2}(r_{j})$ are the zero-point magnetic and electric fields of the microwave and the optical cavity modes, respectively. As the collective enhanced couplings are employed, here we introduce

Since the two parts 1 and 2 in the hybrid system are the same, we first derive the effective Hamiltonian of part 1 for the sake of simplicity. In the interaction picture, after making the rotating-wave approximation, the Hamiltonian of the two cavities and the NV ensemble in part 1 is (assuming $\hbar =1$)

Under the large detuning, in the low excitation limit, and for a large $N$, we could obtain the following effective Hamiltonian describing part 1 [58]:

Similar to Part 1, the effective Hamiltonian of Part 2 can be expressed as

Here $J_{\mathrm {eff}}$ is the coupling strength between the two optical cavities, induced by the fiber. Taking into account a finite length $L$ of the fiber implies a quantization of the modes of the fiber with frequency spacing given by $2\pi c/L$ [59]. Thus, the number of modes, which would significantly interact with the cavities’ modes, is on the order of $n=(L\Gamma _{b})/(2\pi c)$, where $\Gamma _{b}$ is the decay rate of the cavities’ fields into a continuum of fiber modes [59]. Here we will focus on the case: short fiber limit [60], for which essentially only one (resonant) mode of the fiber will interact with the model of each optical cavity ($n\leq 1$). This condition applies in most realistic experimental situations: for instance, the fiber length meets $L\leq 1\mathrm {m}$ and the decay rate of the optical cavity satisfies $\Gamma _{b}\simeq 1\,\mathrm {GHz}$ [60]. It is also noted that under specific conditions [59,61], the fiber mode can be adiabatically eliminated to create an effective cavity-cavity coupling through exchanging cavity photons. Thus, the coupling strength $J_{\mathrm {eff}}$ between the two optical cavities, induced by the modes of a fiber, can be estimated as $J_{\mathrm {eff}}\simeq \sqrt {8\pi \Gamma _{b}c/L}$, which can be increased by decreasing the reflectivity of the cavity mirrors connected to the fiber [59–62].

In the next section, we will show in details how to use the effective Hamiltonian $H_{\mathrm {eff}}$ to transfer quantum states between two spatially separated microwave cavities.

3. Quantum state transfer between spatially separated microwave cavities

Now we show how to transfer quantum states between two spatially separated microwave cavities. Generally, the state transfer includes three steps: step 1, turn on the coupling $G_{\mathrm {eff,1}}$ (while $G_{\mathrm {eff,2}}= 0$) in a time interval $0<t<T_{1}$, which uses the spin mode $c_{1}$ to swap the quantum states between the microwave mode $a_{1}$ and the optical mode $b_{1}$ in part 1; step 2, turn off the coupling $G_{\mathrm {eff,1}}$ and $G_{\mathrm {eff,2}}$ in a time interval $T_{1}<t<T_{2}$, so that quantum states are transferred from the optical mode $b_{1}$ in part 1 to the distant optical mode $b_{2}$ in part 2; step 3, turn on the coupling $G_{\mathrm {eff,2}}$ (while $G_{\mathrm {eff,1}}= 0$) in a time interval $T_{2}<t<T_{3}$, which uses the spin mode $c_{2}$ to swap the quantum states between the optical mode $b_{2}$ and the microwave mode $a_{2}$ in part 2. Notice that in the above steps, the coupling between the fiber and the optical cavities can be manipulated through the application of an optical switch [62,63]. Due to potential mismatch between the modes of the fiber and the optical cavities, a coupling loss is typically introduced. By adjusting the coupling loss appropriately, a switching control can be achieved. Here the coupling loss can be adjusted by incorporating a lens to match the mode of the optical cavities, similar to the Q-adjusting cavity scheme which control the leakage of the cavity by adjusting the coupling between the cavity and the environment [59,61]. The effective Hamiltonian of the system reads

3.1 Quantum state transfer from a microwave cavity to an optical cavity

In part 1, the collective spin mode $c_{1}$ of the NV ensemble interacts with the microwave mode $a_{1}$ and the optical mode $b_{1}$, respectively. In the following, we show that the spin mode can be used to swap the quantum states between the microwave mode $a_{1}$ and the optical mode $b_{1}$.

In the time interval $0<t<T_{1}$, according to Eq. (8), the dynamics of the operators can be derived by solving the Heisenberg equations as

From Eq. (8), one can see that for the time interval $0<t<T_{1}$, the Hamiltonian given in Eq. (8) is the Hamiltonian $H_{\mathrm {eff,1}}$ given above. After the evolution time $t=\pi /2|G_{\mathrm {eff,1}}|$, the state $|\psi \rangle _{a_{1}}|\phi \rangle _{b_{1}}|\eta \rangle _{b_{2}}|\xi \rangle _{a_{2}}$ of the system evolves into

3.2 Quantum state transfer between spatially separated optical cavities via the optical fiber

In the time interval $T_{1}<t<T_{2}$, we turn off both couplings $G_{\mathrm {eff,1}}$ and $G_{\mathrm {eff,2}}$, the quantum states could be exchanged between the optical modes $b_{1}$ and $b_{2}$ with assistance of the optical fiber. This state exchange can be described as follows:

For the detailed derivation, please see Appendix B. From Eq. (12), one can see that an arbitrary quantum state $|\psi \rangle$ of the optical cavity $b_{1}$ is transferred onto the optical cavity $b_{2}$.

3.3 Quantum states transfer from an optical cavity to a microwave cavity

In the time interval $T_{2}<t<T_{3}$, we turn on the coupling $G_{\mathrm {eff,2}}$ while turn off the coupling $G_{\mathrm {eff,1}}$. The quantum states could be exchanged between the optical mode $b_{2}$ and the microwave mode $a_{2}$ as follows:

In Appendix B, we provide a detailed derivation of Eq. (13). From Eq. (13), one can see that, through the third step, we successfully transfer the quantum state $|\psi \rangle$ from the optical cavity $b_{2}$ to the microwave cavity $a_{2}$. Hence, this three-step procedure completes a perfect state transmission between two spatially separated microwave cavities, which is useful in networked long-distance quantum communication and quantum information processing.

4. Analysis and numerical simulation

To better understand the inner principle of the scheme, we numerically simulate the transfer process. For simplicity, we set $G_{m,1}=G_{m,2}=G_{m}$, $G_{o,1}=G_{o,2}=G_{o}$, and $\Omega _{1}=\Omega _{2}=\Omega$. Since any quantum system would suffer from decoherence, we consider the dissipations from the microwave cavities, the NV ensemble collective spin modes, and the optical cavities. In the low-temperature environment, namely, the thermal photon occupation number is nearly zero, the dynamics of the system can be described by the following master equation

Accordingly, the ideal target state is given by

The state transmission efficiency can be evaluated by the average fidelity of the operation, which is defined as

We perform numerical simulations by solving the master Eq. (14). The parameters we chosen in the numerical simulations are $\sqrt {N}G_{m}=2\pi \times 10\,\mathrm {MHz}$ and $\sqrt {N}G_{o}=2\pi \times 350\,\mathrm {MHz}$ with the number of NV centers $N\sim 10^{12}$ in each NV ensemble [38,67]. We assume the inhomogeneous broadening of the transition frequencies with $\Delta _{1}=2\pi \times 10\,\mathrm {MHz}$, $\Delta _{2}=2\pi \times 1\,\mathrm {GHz}$, and $\Delta _{3}=2\pi \times 500\,\mathrm {MHz}$, which can be achieved experimentally [43,47,48,68]. In addition, we choose $\Omega =2\pi \times 100\,\mathrm {MHz}$, $\kappa _{a_{1(2)}}=2\pi \times 3\,\mathrm {KHz}$, $\kappa _{b_{1(2)}}=2\pi \times 100\,\mathrm {KHz}$, and $\gamma _{c_{1(2)}}=2\pi \times 10\,\mathrm {KHz}$ [69]. At the temperature of $T\sim 20\,\mathrm {mK}$, the equilibrium thermal photon occupation number is $n<0.01$, which can be neglected [16].

We first consider the transmission in the ideal case that the decoherence process is not taken into account. As the black-solid line shown in Fig. 3, the spin mode can be used to successfully transfer the quantum state from the microwave cavity $a_{1}$ to the distant microwave cavity $a_{2}$ when selecting a suitable evolution time. In a realistic situation, the decoherence should be taken into account. Thus, we consider the dissipation process with the decay rates $\kappa _{a_{1(2)}}=2\pi \times 3\,\mathrm {KHz}$ for the microwave cavities, $\gamma _{c_{1(2)}}=2\pi \times 10\,\mathrm {KHz}$ for the collective spin modes, and $\kappa _{b_{1(2)}}=2\pi \times 100\,\mathrm {KHz}$ for the optical cavities. As the red-dash line shown in Fig. 3, a high fidelity (0.988) for the state transmission can still be achieved. Therefore, our method works well for the long-distance transmission of an arbitrary quantum state of a microwave photonic qubit between two spatially separated microwave cavities in a realistic case.

 

Fig. 3. Average fidelity of the operation. The parameters used in the numerical simulation are $\sqrt {N}G_{m}=2\pi \times 10\,\mathrm {MHz}$, $\sqrt {N}G_{o}=2\pi \times 350 \,\mathrm {MHz}$, $N\sim 10^{12}$, $\Delta _{1}=2\pi \times 10\,\mathrm {MHz}$, $\Delta _{2}=2\pi \times 1\,\mathrm {GHz}$, $\Delta _{3}=2\pi \times 500\,\mathrm {MHz}$, $\Omega =2\pi \times 100\,\mathrm {MHz}$, $\kappa _{a_{1(2)}}=2\pi \times 3\,\mathrm {KHz}$, $\kappa _{b_{1(2)}}=2\pi \times 100\,\mathrm {KHz}$, and $\gamma _{c_{1(2)}}=2\pi \times 10\,\mathrm {KHz}$.

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5. Interference of the real environment

5.1 Robustness to the stochastic parameter fluctuation

In order to find out whether the scheme is effective in practice, one should take the robustness against stochastic parameter fluctuation into consideration. In this section, we set the amplitude noise as $\beta H_{k}$ and the time derivative of the Brownian motion $W_{t}$ as $\alpha (t)$. Therefore, a stochastic Schrödinger equation in a close system can be written as

Thus, the dynamics of the whole system (including the dissipation and noise terms) is

In the following, to investigate the robustness against stochastic parameter fluctuation, we consider the amplitude noises caused by the JC interaction. Hence, the $\mathcal {N}\rho$ in this case can be rewritten as

According to Eq. (23), we numerically simulate the robustness of the scheme versus the stochastic parameter fluctuation. For the error Hamiltonian $H_{k}$, we select the deviation as $\beta \in [-1{\% },1{\% }]$, which is quite normal in real experiments [73,74]. As shown in Fig. 4, the influence caused by the stochastic parameter fluctuation is negligibly small and the fidelity only reduces $2{\% }$ at most. In this sense, the scheme is robust against stochastic noise errors caused by the JC interaction.

 

Fig. 4. Average fidelity of the operation versus time with different parameter fluctuations. The parameters used in the numerical simulation are $\sqrt {N}G_{m}=2\pi \times 10\,\mathrm {MHz}$, $\sqrt {N}G_{o}=2\pi \times 350 \,\mathrm {MHz}$, $N\sim 10^{12}$, $\Delta _{1}=2\pi \times 10\,\mathrm {MHz}$, $\Delta _{2}=2\pi \times 1\,\mathrm {GHz}$, $\Delta _{3}=2\pi \times 500\,\mathrm {MHz}$, $\Omega =2\pi \times 100\,\mathrm {MHz}$, $\kappa _{a_{1(2)}}=2\pi \times 3\,\mathrm {KHz}$, $\kappa _{b_{1(2)}}=2\pi \times 100\,\mathrm {KHz}$, and $\gamma _{c_{1(2)}}=2\pi \times 10\,\mathrm {KHz}$.

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5.2 Robustness to the additive white Gaussian noise

In real experiments, there may exist several kinds of noises, which disturb wave shapes of Rabi frequencies of pulses. It is worthwhile to study the influences caused by the noise. Here, we investigate the influences caused by the additive white Gaussian noise (AWGN), which is the most common noise in real experiments [75]. The Rabi frequencies under the influence of the AWGN of pulse amplitudes can be described as [76]

We consider two different conditions based on Eq. (24). For each group of SNR, 50-time numerical simulations are performed, and the results for corresponding $R_{SN1}$ and $=R_{SN2}$ are plotted in Fig. 5. From Fig. 5, one can see that the scheme has great robustness against the AWGN. Even with a quite large $R_{SN1}$ and $R_{SN2}$, e.g., $R_{SN1}=R_{SN2}=5$, the fidelity of the target state can still reach $0.99$. Thus, the method is robust against the additive white Gaussian noise.

 

Fig. 5. Average fidelity of the operation versus simulation counts with SNR. (a) $R_{SN1}=R_{SN2}=5$, (b) $R_{SN1}=R_{SN2}=10$, (c) $R_{SN1}=R_{SN2}=20$, (d) $R_{SN1}=R_{SN2}=50$. The parameters used in the numerical simulation are $\sqrt {N}G_{m}=2\pi \times 10\,\mathrm {MHz}$, $\sqrt {N}G_{o}=2\pi \times 350 \,\mathrm {MHz}$, $N\sim 10^{12}$, $\Delta _{1}=2\pi \times 10\,\mathrm {MHz}$, $\Delta _{2}=2\pi \times 1\,\mathrm {GHz}$, $\Delta _{3}=2\pi \times 500\,\mathrm {MHz}$, $\Omega =2\pi \times 100\,\mathrm {MHz}$, $\kappa _{a_{1(2)}}=2\pi \times 3\,\mathrm {KHz}$, $\kappa _{b_{1(2)}}=2\pi \times 100\,\mathrm {KHz}$, and $\gamma _{c_{1(2)}}=2\pi \times 10\,\mathrm {KHz}$.

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6. Conclusion

We have proposed a method for long-distance transmission of arbitrary quantum states between two spatially separated microwave cavities. The method has these features and advantages. First, the local coherent microwave-optical conversion of photons can be implemented using a single-step operation. Second, only three steps are needed to complete the transmission of an arbitrary quantum states between two spatially separated microwave cavities. Third, the NV ensembles are unexcited during the entire process, thus the decoherence from the NV ensembles is greatly suppressed. Fourth, the two optical cavities and the receiver microwave cavity have no requirements for the initial states, and they do not need to be prepared in specific initial states. Last, the method is robust against the decay caused by environment, the parameter fluctuation, and the additive white Gaussian noise. Our numerical simulations demonstrate that arbitrary quantum states of a microwave photonic qubit can be high-fidelity transfered between two spatially separated microwave cavities. This method could be a good candidate to realize long-distance transmission of arbitrary quantum states between two spatially separated microwave cavities, which has important applications in long-distance networked quantum communication and quantum information processing.