Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 61875145) and the Foundation for Key Disciplines of the Thirteenth Five-Year Plan of Jiangsu Province, China (Grant No. 20168765)
Received Date:26 July 2020
Accepted Date:24 August 2020
Available Online:22 October 2020
Published Online:05 January 2021
Abstract:The time evolution of multipartite quantum coherence is studied in a three-body spin system with an asymmetric interaction. The l1 norm measurement is used to estimate the degree of quantum coherence in the spin system. The decoherence of all components of quantum coherence in the three-body spin system is analyzed by the exact diagnolization and numerical method based on quantum master equation. The environmental noise induced by the spontaneous decay can be simulated by the quantum amplitude damping model. It is found that the time evolution of quantum coherence component is closely related to the quantum property of the whole initial state. When the initial state is a separable pure one, the asymmetric interactions can conduce to the enhancement of the degree of multipartite quantum coherence in a short time interval. Under the influence of quantum noise, the degree of quantum coherence decreases gradually. We pay much attention to the spatial distribution of the degree of quantum coherence in a many-body system. The additivity relationship of bipartite component and tripartite coherence can exist if the initial state is chosen to be a Werner-like state. This kind of the coherence additivity between all bipartite components and global coherence can be extended to an arbitrary N-body Werner-like state. But this additivity relationship depends on the l1 norm coherence measurement. Owing to the asymmetric interaction and noise, the degree of tripartite quantum coherence is more than the sum of all degrees of bipartite quantum coherence. The difference between the degree of tripartite coherence and the sum of all degrees of bipartite coherence is increased in a short time interval. The environmental noise can also suppress the difference in the coherence degree. The degree of the nearest neighboring bipartite coherence decreases more quickly than those of other bipartite coherences. The asymmetric interaction gives rise to the improvement in the degree of bipartite coherence and tripartite coherence. The coherence of the next-nearest neighboring two systems can be robust against the environmental noise. These results are helpful in preparing the multipartite quantum resources. We can utilize the system of coupled micro-cavities to realize the quantum spin system with controllable asymmetric interaction. In this way, the global coherence and bipartite coherence can be manipulated effectively by the quantum electromagnetic technology. Keywords:multipartite quantum coherence/ asymmetric spin systems/ quantum decoherence/ quantum master equation
如果系统初态选择类GHZ态, 那么在环境噪声影响下, 任意t时刻的量子相干度为$C_{123}^{{\rm{GHZ}}} = 2{(1 - p)^{ \textstyle\frac{3}{2}}}\sqrt {q(1 - q)}$, 其他两体量子相干仍然为零. 当自旋相互作用为零时, 初态处于类Werner态的系统, 其量子相干空间分布仍然满足(6)式所描述的加和性. 而且, 量子相干度随着时间呈现指数衰减规律. 噪声参数p越大, 量子相干数值衰减得越快. 然而, 当自旋之间存在非对称性相互作用时, 利用量子主方程的数值计算, 发现三体量子相干度与任意两组分量子相干之和存在差异. 当初态为 $\left| {001} \right\rangle $时, 图1描述了三体量子相干度${C_{123}}$、近邻两组分量子相干${C_{12}}$和${C_{23}}$、以及次近邻两体量子相干${C_{13}}$的含时演化. 如图1所示, 当时间较短时, 状态演化处于初始阶段, 一些能级组分态$\left| {{\psi _{6, 7}}} \right\rangle $的混合, 使得类Werner态组分在量子态中占有一定比重, 所以量子相干度的数值从零逐渐增大. 其中, 对于相互作用较小的两个自旋, 其量子相干度${C_{13}}$和${C_{23}}$在演化初期增长较快. 但是, 随着演化进一步发展, 环境噪声又会导致量子相干度的衰减, 量子退相干现象明显. 图 1 当初态为$ \left| {001} \right\rangle $时, 量子相干组分演化, 参量选取为$ D = 0.2, $$ \gamma = 0.2, $$ \varGamma = 0.5, $$ n = 0.2$, 黑色实线为$ {C_{123}}$, 红色虚线和绿色点划线分别为$ {C_{12}}, {C_{13}}$, 蓝色点线为$ {C_{23}}$ Figure1. The dynamics of all fractions of quantum coherence for the initial state $ \left| {001} \right\rangle $. The parameters are chosen to be$ D = 0.2, $$ \gamma = 0.2, $$ \varGamma = 0.5, $$ n = 0.2$. The black solid line denotes $ {C_{123}}$, the red dashed and green dot-dashed line are $ {C_{12}}, {C_{13}}$ respectively, and the blue dotted line represents $ {C_{23}}$.
为了进一步研究多体量子相干与两体量子相干的关系, 定义一种量子相干组分差值${C_{\rm{r}}} = {C_{123}} - ({C_{12}} + {C_{23}} + {C_{13}})$. 如图2所示, 分析了非均匀自旋相互作用参量D和$\gamma $对量子相干性质的影响. 图2(a)的曲线变化表明, 随着自旋与轨道耦合作用D的增加, 量子相干组分差值${C_{\rm{r}}}$也会增大, 但是环境噪声又会大大抑制量子相干, 导致量子退相干. 同样, 图2(b)的曲线变化表明, 各项异性参量$\gamma $的增加也会引起量子相干组分差值${C_{\rm{r}}}$的增加. 这个结论说明, 非均匀相互作用更有利于三体量子相干度的增加. 图 2 当初态为$ \left| {001} \right\rangle $时, 量子相干组分差值的演化, 参量选取为$ \varGamma = 0.5, $$ n = 0.2$ (a) 当$ \gamma = 0.2$时, 黑色实线对应参数$ D = 0.3, $ 红色虚线对应$ D = 0.5;$ (b)当$ D = 0.2$时, 黑色实线对应参数$ \gamma = 0.3, $ 红色虚线对应$ \gamma = 0.7$ Figure2. The dynamics of the difference of quantum coherence for the initial state $ \left| {001} \right\rangle $. The parameters are chosen to be $ \varGamma = 0.5, $$ n = 0.2$: (a) When $ \gamma = 0.2$, the black solid line denotes $ D = 0.3$ the red dashed line is $ D = 0.5;$ (b) When $ D = 0.2$, the black solid line denotes$ \gamma = 0.3, $ the red dashed line is $ \gamma = 0.7$.
当初态选择W态$\left(a = b = c = {1}/{{\sqrt 3 }}\right)$时, 量子相干的各个组分随时间演化行为如图3所示. 所有量子相干组分都呈现出单调衰减现象. 这与(12)式的解析结果相对应. 图 3 当初态为W态时, 量子相干组分演化, 参量$ D = 0.2, $$ \gamma = 0.2, $$ \varGamma = 0.5, $$ n = 0.2$, 黑色实线为$ {C_{123}}$, 红色虚线和绿色点划线分别为$ {C_{12}}, {C_{13}}$, 蓝色点线为$ {C_{23}}$ Figure3. The dynamics of all fractions of quantum coherence for the initial W state. The parameters are chosen to be $ D = 0.2, $$ \gamma = 0.2, $$ \varGamma = 0.5, $$ n = 0.2$. The black solid line denotes $ {C_{123}}$, the red dashed and green dot-dashed line are $ {C_{12}}, {C_{13}}$ respectively, and the blue dotted line represents $ {C_{23}}$.
但是, 由于自旋相互作用的非对称性, 所以量子相干组分之间存在差异, 相应差值${C_{\rm{r}}}$也会不为零. 图4(a)和图4(b)分别表现出, 自旋与轨道耦合作用D和各项异性参量$\gamma $的增加都会引起三体量子相干度的增大, 使得相应量子相干组分差值在演化初期呈现增长现象. 但是, 由于环境噪声的影响, 无论是三体量子相干度, 还是任意两体量子相干度, 都在较长时间里发生衰减, 直至完全消失. 图 4 当初态为W态时, 量子相干组分差值的演化, 参量选取为$ \varGamma = 0.5, $$ n = 0.2$ (a) 当$ \gamma = 0.2$时, 黑色实线对应参数$ D = 0.3, $ 红色虚线对应$ D = 0.5;$ (b)当$ D = 0.2$时, 黑色实线对应参数$ \gamma = 0.3, $ 红色虚线对应$ \gamma = 0.7$ Figure4. The dynamics of the difference of quantum coherence for the initial W state. The parameters are chosen to be $ \varGamma = 0.5, $$ n = 0.2$; (a) When $ \gamma = 0.2$, the black solid line denotes $ D = 0.3$ the red dashed line is $ D = 0.5;$ (b) When $ D = 0.2$, the black solid line denotes$ \gamma = 0.3, $ the red dashed line is $ \gamma = 0.7$.