图 1  (a) 磁-腔QED系统示意图. YIG小球位于腔中短边中心处, 该处微波腔的TE₁₀₂模式的磁场沿y轴方向, 静磁场沿x轴方向; (b) 利用Comsol模拟微波腔TE₁₀₂的网格划分示意图; (c) 微波腔TE₁₀₂模式的磁场方向和磁场强度

Fig. 1.  (a) Schematic diagram of a magnetic-cavity QED system. A YIG sphere is located at the center of one short edge in the cavity, where magnetic field of microwave cavity mode TE₁₀₂ is along y-axis direction, and static magnetic field is along x-axis direction; (b) schematic diagram of grid of microwave cavity mode TE₁₀₂ by Comsol simulation; (c) magnetic field direction and magnetic field intensity of the microwave cavity mode TE₁₀₂.

图 2  (a) 平均光子数、(b) FMR模的平均磁子数和 (c) MS模的平均磁子数随失谐量${\varDelta _{{m_1}}}$和${\varDelta _{\rm{c}}}$的变化关系, 其中$\gamma /2{\rm{\pi = 1}}{\rm{.0\;MHz}}$. 取${\varepsilon _{\rm{p}}} = \gamma $, $\varOmega = 1.5\gamma $, 其他参数已在正文中给出

Fig. 2.  (a) Average photon number, (b) FMR mode average magnon number, and (c) MS mode average magnon number versus detunings ${\varDelta _{{m_1}}}$and${\varDelta _{\rm{c}}}$, where, $\gamma /2\rm{\pi} = 1{\rm{.0\;MHz}}$. We take${\varepsilon _{\rm{p}}} = \gamma $, $\varOmega = 1.5\gamma $. Other parameters are given in the text.

图 3  两体纠缠的负值度　(a) ${E_{a{m_1}}}$, (b) ${E_{a{m_2}}}$和(c) ${E_{{m_1}{m_2}}}$随失谐量${\varDelta _{{m_1}}}$和${\varDelta _{\rm{c}}}$的变化关系. 其中${\varepsilon _{\rm{p}}} = \gamma $, $\varOmega = 1.5\gamma $. (d)负值度${E_{a{m_1}}}$ (实线)、${E_{a{m_2}}}$ (虚线)和${E_{{m_1}{m_2}}}$ (点划线)随非线性增益$\varOmega $的变化. 其中${\varepsilon _{\rm{p}}} = \gamma $, ${\varDelta _{\rm{c}}} = - 2.5\gamma $, ${\varDelta _{{m_1}}} = 23\gamma $. 其他参数与图2一致

Fig. 3.  Density plot of logarithmic negativity related to bipartite entanglement (a) ${E_{a{m_1}}}$, (b) ${E_{a{m_2}}}$, and (c) ${E_{{m_1}{m_2}}}$ versus detunings ${\varDelta _{{m_1}}}$ and ${\varDelta _{\rm{c}}}$, where ${\varepsilon _{\rm{p}}} = \gamma $, $\varOmega = 1.5\gamma $. (d) Logarithmic negativity ${E_{a{m_1}}}$ (solid), ${E_{a{m_2}}}$ (dashed), and${E_{{m_1}{m_2}}}$ (dot-dashed) versus the nonlinear gain coefficient $\varOmega $, we take ${\varepsilon _{\rm{p}}} = \gamma $, ${\varDelta _{\rm{c}}} = - 2.5\gamma $ and${\varDelta _{{m_1}}} = 23\gamma $. The other parameters are the same as in Fig.2.

图 4  (a) 不同非线性增益情况下, 三体纠缠的最小剩余共生纠缠$R_\tau ^{{\rm{min}}}$随失谐量${\varDelta _{{m_1}}}$的变化关系; (b) 最小剩余共生纠缠$R_\tau ^{{\rm{min}}}$随温度T的变化, 其中${\varepsilon _{\rm{p}}} = \gamma $, ${\varDelta _{\rm{c}}} = - 2.5\gamma $. 虚线、实线、点线和点划线分别对应非线性相互作用强度$\varOmega = 1.65\gamma $, $\varOmega = 1.5\gamma $, $\varOmega = 1.0\gamma $和$\varOmega = 0.5\gamma $的情况. 在图4(b)中, 取${\varDelta _{{m_1}}} = 23\gamma $, 其他参数与图2一致

Fig. 4.  (a) Tripartite entanglement in terms of the minimum residual contangle $R_\tau ^{{\rm{min}}}$ versus detuning ${\varDelta _{{m_1}}}$; (b) robust against temperature of the minimum residual contangle $R_\tau ^{{\rm{min}}}$, Where ${\varepsilon _{\rm{p}}} = \gamma $, ${\varDelta _{\rm{c}}} = - 2.5\gamma $. The dashed line, solid line, dotted line, and dash-dot line indicate nonlinear interaction strength $\varOmega = 1.65\gamma $, $\varOmega = 1.5\gamma $, $\varOmega = 1.0\gamma $, and $\varOmega = 0.5\gamma $, respectively. At the same time, we take ${\varDelta _{{m_1}}} = 23\gamma $ for Fig. 4 (b). The other parameters are the same as in Fig. 2.

图 5  三体纠缠的最小剩余共生纠缠$R_\tau ^{{\rm{min}}}$随失谐量${\varDelta _{{m_1}}}$和耗散系数${\kappa _2}$的变化关系; 设定${\varepsilon _{\rm{p}}} = \gamma $, $\varOmega = 1.5\gamma $, ${\varDelta _{\rm{c}}} = - 2.5\gamma $, 从左到右, ${\kappa _2}$所取的数值分别为$1.7\gamma $, $2.1\gamma $, $2.5\gamma $, $2.9\gamma $以及$3.3\gamma $. 其他参数与图2一致

Fig. 5.  Tripartite entanglement in terms of the minimum residual contangle $R_\tau ^{{\rm{min}}}$ versus detuning ${\varDelta _{{m_1}}}$ and dissipation rates ${\kappa _2}$, setting ${\varepsilon _{\rm{p}}} = \gamma $, $\varOmega = 1.5\gamma $ and ${\varDelta _{\rm{c}}} = - 2.5\gamma $, the lines denote $1.7\gamma $, $2.1\gamma $, $2.5\gamma $, $2.9\gamma $, and $3.3\gamma $for ${\kappa _2}$ from left to right. The other parameters are the same as in Fig. 2.