Fig. 1. Schematic of our hybrid system. (a) Three identical optical cavities with frequency ωc are arranged in a row, and the central cavity couples to the left cavity and the right cavity with the identical coupling strength J, through exchanging photons via optical fibers [68]. Each cavity dispersively couples to the corresponding MR with a coupling strength g0. The additional second-order nonlinear pump is applied to each MR, which can be realized by modulating the spring constant in time. The central mechanical resonator is additionally coupled to the other two bilateral MRs with the same coupling rate Jm. A single NV center is placed inside the central cavity and interacts with this cavity mode with the coupling strength g. (b) Energy-level diagram illustrating the blue and red sideband transitions for the tripartite interaction quantum system.

Fig. 2. (a) Spin–phonon coupling enhancement Λ/λ and (b) cooperativity enhancement C versus the squeezing parameter r and the photon number ncav of the cavity mode a^0, with g0=0.001g, J=10g, λ/2π=0.1  MHz, g/2π=1  GHz, the effective mechanical dissipation ΓmS/2π∼1  MHz, and the NV spin decay rate γ/2π∼15  MHz.

Fig. 3. Dynamical population of the phonon number b^0†b^0 and the spin operator σ^z according to (a), (b) the J-C model and (c), (d) the anti J-C model, with different ncav and r. The parameters are g0=0.001g and J=10g, the effective mechanical dissipation is ΓmS∼0.001g, and the NV spin decay rate is γ∼0.02g. This system is initially prepared in state |ϕ(0)⟩=|1⟩m|0⟩s.

Fig. 4. (a), (b) Dynamical population of the spin operator σ^z, number operators of (c), (d) the optical mode a^0†a^0 and (e), (f) the phonon mode b^0†b^0. (a), (c), and (e) correspond to the blue sideband condition, and (b), (d), and (f) correspond to the red sideband condition, with different squeezing parameter r. The parameters are g0∼γ, J=2.8×103γ, and g∼70γ, the effective mechanical dissipation and the cavity decay rate are assumed to be ΓmS∼0.001γ and κ∼0.1γ, and the NV spin decay rate is γ/2π∼15  MHz. This tripartite system is initially prepared, respectively, in states |ψ(0)⟩=|1⟩o|0⟩m|0⟩s (blue sideband) and |ψ(0)⟩=|1⟩o|1⟩m|0⟩s (red sideband).

Fig. 5. Dynamical fidelity of the target entangled GHZ state for four NV spins, in which, the initial state is |ψsystem(0)⟩=|0⟩m|0000⟩s, and the target GHZ state is |ψτNV⟩=[e−iπ/4|0000⟩s+eiπ/4|1111⟩s]/2. The parameters are the squeezing parameter r≃4.0, g0∼0.001g, J∼10g, g/2π∼1.0  GHz, n¯cav∼104, the effective mechanical dissipation ΓmS∼0.001γ, and the NV spin decay rate γ/2π∼15  MHz.

Fig. 6. Dynamical population of this mechanical supermode b^0 with the assumption of its initial average phonon number ⟨b^0†b^0⟩≃50, in which the parameters are the squeezing parameter r≃2.0, n¯cav∼100, g0∼0.001g, g∼66γ, J∼10g, the effective mechanical dissipation ΓmS∼0.001γ, and the NV spin decay rate γ/2π∼15  MHz.

Fig. 7. Dynamical evolution of the mechanical population of the left local mode ⟨b^LS†b^LS⟩ (green dashed line), the right local mode ⟨b^RS†b^RS⟩ (black solid line), and the supermode ⟨b^0†b^0⟩ (blue solid line) in the time interval [0,2/γ] ([500/γ, 2000/γ] in the inset), assuming both b^RS and b^LS are initially in the single phonon state. Other parameters are the same as in Fig. 6.