Fig. 1. The energies of degenerate quantum states and the superposition state of odd parity of left(right)-displaced number states varies as the spin-orbit coupling strength . It is seen that for , the superposition state has the lowest energy which is the best approximation for the ground state in our interest. And for the cases of , the energies of the two quantum states have pitchforks.The relevant parameters is Ω=1.4 and the results are in agreement with those in Ref.[19]. 简并量子态 能量 与左右平移奇宇称叠加态 能量 随SO耦合强度 的变化　可见 叠加态 能量最低, 更接近基态; 而对于激发态 , 二者能量随参数变化出现交叉; 相关参数取值为 , 与文献[19]精确解的结果基本一致

Fig. 2. The coarse dynamics evulution of momentum distribution of single particle (left for 3D; right for 2D) with and . The initial state is set as . Momentum is defined by . 原子动量分布概率的粗粒动力学演化 (3D, 左侧; 2D, 右侧)　相关参数取值为 , , 初态为 , 动量

Fig. 3. The coarse dynamics evolution of position distribution of single particle (left for 3D; right for 2D) with the same parameters and the initial state in Fig. 2 and . 原子空间位置分布概率的粗粒动力学演化(3D, 左侧; 2D, 右侧)　 相关参数取值及初态同图2, 位置

Fig. 4. Time evolution of with the initial state being and the parameters and . The time is scaled by the tunneling period . 原子极化 随时间演化初态为 , 参数取值为 和 , 时间以因子 标度