In recent years, significant progress has been made toward quantum information processing based on nitrogen-vacancy centers (NV centers) in diamond. Extended ground state electron spin coherence times of up to 350 μs have been observed. Full control over electron spin has been achieved using optically detected magnetic resonance, and electron-nuclear qubit transfer, crucial for long quantum memory times, has been demonstrated. However, these demonstrations have so far only involved the manipulation of isolated NV centers. For large-scale quantum information processing or quantum repeater systems, it will be essential to connect NV centers using flying qubits such as photons. To achieve this goal, silicon-based optical waveguides are necessary to facilitate the transfer of quantum information between the electron spin of NV centers and photons. Quantum information transfer between separate quantum nodes in a coupled system of NV centers and silicon-based optical waveguides represents the core technology for realizing quantum networks and quantum communication. In this paper, we propose a theoretical framework for achieving quantum information transfer between two separated quantum nodes within such a coupled system.

In this system, a silicon-based optical waveguide and the coupled NV center spin ensembles can be regarded as quantum nodes. Two separate quantum nodes are connected by an empty silicon-based optical waveguide. The NV center spin ensemble in each quantum node interacts with a silicon-based optical waveguide resonator controlled by an external microwave pulse. This quantum node functions to send, store, and receive optical quantum information. The empty silicon-based optical waveguide in the middle serves as a transmission channel connecting two separate quantum nodes, which allows photons carrying quantum information to propagate between them. In the process of quantum information transmission, microwave photons act as carriers of quantum information, which transfers it from one silicon-based waveguide resonator to another adjacent silicon-based waveguide resonator, thereby achieving the function of transmitting quantum information. The specific implementation plan involves first performing a canonical transformation on the Hamiltonian of the system, which is equivalent to a Jaynes?Cummings (JC) coupling model between two NV centers and the same silicon-based optical waveguide resonator. Quantum information is then encoded using NV center spin-photon hybrid bits. Ultimately, quantum information transfer between two separate quantum nodes is achieved by precisely controlling the resonant frequency of the silicon-based optical waveguide resonator and the evolution time of the system. For NV center spin-photon hybrid bit encoding, under coherent evolution conditions of the system, high-fidelity transmission of quantum states between quantum nodes can be realized through theoretical calculations and numerical simulations.

In our system, through careful selection of system parameters and precise control of the evolution time, we transmit the quantum state from the first quantum node to the second. This process restores the first quantum node to its ground state, which effectively transfers quantum information between these separate nodes. Our operational timeframe for realizing quantum state transfer between two different nodes is about t=π/g≈0.05μs in the case of g/2π=1GHz. Given the cavity quality factor Q=10⁸, the decay rate of the silicon-based optical waveguides is κ=c/λQ=2π×5MHz, and the characteristic spontaneous decay rate γ from the state e to g could be estimated as γ=2π×13MHz, which implies an effective dephasing time 1/γ×2%≈0.6μs. Consequently, nearly 10 quantum state transfer operations are feasible under present experimental conditions. Recent experimental advancements with isotopically pure diamond samples have demonstrated extended dephasing time of 2 ms. This also implies that the influence of the intrinsic damping and dephasing in NV centers is potentially negligible in the present NV center and silicon-based optical waveguide system.

Under the condition of resonance interaction, considering the decay rate of the silicon-based optical waveguides κ and the spontaneous decay rate of the NV center κ, in the case of α=β=1/2, in the strongly coupled system gmax?κ,γ, assuming that κ=γ=0.01g, where g is the coupling strength between the NV center spin ensemble and the silicon-based optical waveguide resonator, the time-dependent curve of the fidelity F of quantum state transfer between quantum nodes is simulated and shown in Fig. 5. In the diagram, the dotted line denotes the initial state, while the solid line represents the final state. It can be observed from the diagram that the fidelity of quantum state transfer can be as high as 0.9699. Despite unequal coupling strengths between the two quantum nodes, the fidelity of quantum state transfer decreases slightly to 0.9479, as illustrated in Fig. 6. This scheme can also be extended to three, five, or even more quantum nodes in a hybrid system coupled with NV center spin ensembles and silicon-based optical waveguide resonators. These multiple quantum nodes can form a distributed quantum network. Through external system manipulation, these quantum nodes can entangle with each other or achieve quantum teleportation. In summary, we provide a highly feasible theoretical solution for achieving quantum information transmission between separated quantum nodes, which holds potential application value in the field of quantum information research.