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Dynamics of quantum correlation for three qubits in hierarchical environment

Song Yue Li Jun-Qi Liang Jiu-Qing

Song Yue, Li Jun-Qi, Liang Jiu-Qing. Dynamics of quantum correlation for three qubits in hierarchical environment. Acta Phys. Sin., 2021, 70(10): 100301. doi: 10.7498/aps.70.20202133
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Dynamics of quantum correlation for three qubits in hierarchical environment

Song Yue, Li Jun-Qi, Liang Jiu-Qing
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  • Much attention has been paid to the dynamics of quantum correlation in an open quantum system coupled to a single-layered environment for a long time. However, the system can be influenced by the multilayer environment or hierarchical environment in realistic scenarios, which is attracting increasing interest at present. In this context, we explore in this paper the dynamics of quantum correlation for a quantum system consisting of three independent qubits, each being immersed in a single mode lossy cavity which is further connected to another cavity. The influences of cavity-cavity coupling strength Ω and the decay rate of cavity Γ1 on the measures of quantum correlation, including negativity, Bell non-locality as well as entanglement witness, are investigated in detail in a strong coupling regime and a weak coupling regime. It is shown that the phenomena of sudden death and sudden birth can happen to both Bell non-locality and entanglement witness. When the decay rate Γ1 = 0 is given, with the increase of Ω these measures eventually reach their stationary values over time after a short period of damping oscillations, in which these stationary values will become larger for the larger Ω. At the same time, the values or the survival times of quantum correlation considered by us in the weak coupling regime are better than in the strong coupling case. In addition, the non-zero Γ1 has a great negative effect on quantum correlation. Hence, in order to suppress the loss of quantum correlation better, the effective manipulation of quantum weak measurement and measurement reversal operator is considered further. Some interesting results are obtained.
      Corresponding author: Li Jun-Qi, ljqsxu@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11105087)

    量子纠缠和Bell非定域性是两种不同类型的量子关联, 两者既有联系又有区别, 均作为量子力学的奇妙特性而备受关注[1,2]. 纠缠是一种至关重要的量子资源, 在量子隐形传态[3]、量子计量[4]等诸多方面扮演着关键角色. 目前, 已成功制备了各种纠缠态[5], 并提出了多种纠缠度量或判据, 如负性纠缠度、纠缠目击等等[6,7]. Bell非定域性可通过Bell不等式来判定: 如果一个量子态违背了Bell不等式, 则称该态具有Bell非定域性或真正的量子非定域性, 该特性不能通过任何经典局域隐变量理论来描述. 研究表明, 具有Bell非定域性的量子态均为纠缠态, 反之则不一定成立. 所以, Bell非定域性是比纠缠更强的一种非定域形式, 且可作为一种纠缠的判据. 目前, 已存在多种Bell型的不等式, 如CHSH不等式[8]、Svetlichny不等式[9]等, 并已有很多Bell型不等式的实验工作[10,11].

    环境无处不在, 它所导致的量子退相干现象会限制量子关联的实际应用. 所以, 研究开放系统中的量子纠缠和Bell非定域性的动力学问题具有重要意义[12,13]. 文献[14]发现纠缠可以在有限时间内突然消失, 该现象被称作纠缠猝死. 紧接着, 文献[15]报道了纠缠猝生现象. 纠缠猝死和猝生都已被实验所证实[16]. 纠缠的这些特殊行为点燃了人们的浓厚研究热情[17,18]. 虽然时过多年, 但至今它们依然引人注目[19,20]. 类似于纠缠, Bell非定域性也存在猝死和猝生现象[21,22], 且Bell非定域性猝死时间更早于纠缠猝死. 近期, 文献[21]从理论和实验角度分析了振幅阻尼信道下的两体纠缠、Bell非定域性和量子导引的动力学行为, 发现它们都可出现猝死现象. 此外, 一些工作还探讨了纠缠目击的动力学行为[23,24]. 纠缠目击是深受青睐的一种纠缠判据[7,24], 是实验上探测纠缠的主要途径之一.

    在实际情况中, 量子系统并非只与单一环境有作用, 它还会受到级联环境或多层环境的影响, 例如量子点中的电子自旋可同时感受到原子核和声子环境[25]. 于是, 与多层环境相关的研究成果不断涌现出来[26-29]. 例如, 基于由两个衰减腔组成的级联环境, 文献[27]和文献[28]分别讨论了两比特量子失协和熵不确定性关系的动力学行为. 近期, 两层环境中的量子加速问题也得到详细讨论[29]. 我们注意到, 绝大多数与级联环境相关的研究工作仅限定于单体或双体系统, 而三体以上的量子动力学问题鲜有讨论. 所以, 本文将基于分别与各自的级联环境相互耦合的三量子比特模型, 详细探讨系统各参数对纠缠和Bell非定域性的影响, 并进一步考察弱测量及其反转操作的调控作用. 弱测量及其反转操作是近些年广受关注的一项调控技术[30]. 该技术或方法可使量子态概率性地恢复到初态, 从而达到保护量子态的目的, 现已被成功用于抑制量子退相干[31]、保护量子关联[32]以及提高隐形传态保真度[33]等许多方面, 其有效性已在光学系统[30]和超导线路[34]中得到了实验证实.

    本文考虑三个全同并相互独立的量子比特${\rm{A, B, C}}$, 其中每个量子比特都被置于一个单模腔内, 该腔又与另外一个空腔相耦合. 系统总的哈密顿量为

    $ {H} = \mathop \sum \limits_{i = {\rm{A, B, C}}} {{H}_i}, $

    (1)

    (1)式中${H_i}$为第$i$个量子比特与其级联环境的哈密顿量, 具有如下形式[35]:

    $\begin{split} {{{H}}_i} = \;&\frac{{{\omega _0}}}{2}{{{\sigma }}_z} + {\omega _1}{{a}}_{i1}^\dagger {{{a}}_{i1}} + {\omega _2}{{a}}_{i2}^\dagger {{{a}}_{i2}} + {g_i}({{a}}_{i1}^\dagger {{{\sigma }}_ - } \\ &+ {{{a}}_{i1}}{{{\sigma }}_ + }) + {\varOmega _i}({{a}}_{i1}^\dagger {{{a}}_{i2}} + {{{a}}_{i1}}{{a}}_{i2}^\dagger ), \end{split}$

    (2)

    式中, 泡利算符$ {{\sigma }}_{z}=|1\rangle \langle 1 |-|0\rangle \langle 0 |$, ${\omega _0}$$ {{\sigma }}_{+}({{\sigma }}_{-})$分别为量子比特$i$的跃迁频率和升(降)算符, 而${{a}}_{i1}^\dagger ({{{a}}_{i1}})$${{a}}_{i2}^\dagger ({{{a}}_{i2}})$分别代表频率为${\omega _1}$的腔${R_{i1}}$以及频率为${\omega _2}$的腔${R_{i2}}$的产生(湮灭)算符. ${g_i}$表示量子比特$i$与腔${R_{i1}}$间的耦合强度, ${\varOmega _i}$是两单模腔${R_{i1}}$${R_{i2}}$间的耦合强度. 为了简便起见, 令${\omega _0} = {\omega _1} = $$ {\omega _2} = \omega$.

    在计入两个级联腔的耗散后, “第$i$个量子比特+级联腔”的主方程为[35]

    $\begin{split} \frac{{{\rm{d}}{{\rho }}(t)}}{{{\rm{d}}t}} =\;& - {\rm{i}}[{{{H}}_i},{{\rho }}(t)] - \sum\limits_{j = 1}^2 \frac{{{\varGamma _{ij}}}}{2}\left[{{a}}_{ij}^\dagger {{{a}}_{ij}}{{\rho }}(t) \right.\\ & \left.- 2{{{a}}_{ij}}{{\rho }}(t){{a}}_{ij}^\dagger + {{\rho }}(t){{a}}_{ij}^\dagger {{{a}}_{ij}}\right] , \end{split}$

    (3)

    这里, ${\varGamma _{i1}}({\varGamma _{i2}})$为腔${R_{i1}}({R_{i2}})$的光子衰减率. 在${g_i} > $$ {{{\varGamma _{i1}}} / 4}$${g_i} \leqslant {{{\varGamma _{i1}}} / 4}$条件下[35], 第$i$个量子比特与腔${R_{i1}}$组成的系统将分别处于强、弱耦合体系.

    初始时刻, 假设量子比特$i$处于激发态$\left| 1 \right\rangle $, 两个耦合腔${R_{i1}}$${R_{i2}}$都处于基态$\left| 0 \right\rangle $, 则总系统的初始态可写为$ {{\rho }}\left(0\right)=|100\rangle \langle 100 |$. 在单激子近似下, $t$时刻总系统的动力学方程可表示为

    $ {{\rho }}(t)=|\varPsi (t)\rangle \langle \varPsi (t)|+{P}_{0}(t)|000\rangle \langle 000|, $

    (4)

    其中$\left| {\varPsi (t)} \right\rangle = P(t)\left| {100} \right\rangle + {P_1}(t)\left| {010} \right\rangle + {P_2}(t)\left| {001} \right\rangle $, ${P_0}(t)$, $P(t)$${P_j}(t)(j = 1, 2)$分别代表真空态、量子比特$i$、腔${R_{i1}}$和腔${R_{i2}}$的概率振幅. 这些概率幅可根据薛定谔方程得到. 对方程(4)中的两个腔模求迹后, 可获得量子比特$i$在基矢空间$\left\{\left| 0 \right\rangle, \left| 1 \right\rangle \right\}$中的约化密度矩阵$ {{\rho }}_{i}(t)$[28]:

    $ {{\rho }}_{i}(t)=\left(\begin{array}{cc}{\left|P(t)\right|}^{2}{\rho }_{11}(0)& P(t){\rho }_{10}(0)\\ P{(t)}^{\ast }{\rho }_{01}(0)& 1-{\left|P(t)\right|}^{2}{\rho }_{11}(0)\end{array}\right),$

    (5)

    式中$P(t) = {\ell ^{ - 1}}\{ F(s)/G(s)\} (t)$, 其中${\ell ^{ - 1}}\{ \cdot \} (t)$为拉普拉斯逆变换, $F(s)=-4{{\varOmega }}_{i}^{2}-(2 s + {\varGamma _{i1}})(2 s + $$ {\varGamma _{i2}})$, $G(s) = 2 g_i^2(2 s + {\varGamma _{i2}}) + s[4\{ \varOmega _i^2 + {s^2}\} + 2 s{\varGamma _{i2}} + $$ {\varGamma _{i1}}(2 s + {\varGamma _{i2}})]$.

    结合方程(5)和文献[28]中的方法, 可直接构建出三量子比特${\rm{A, B, C}}$的动力学方程. 为此, 选取如下最大纠缠的GHZ态作为${\rm{A, B, C}}$的初始态:

    $ {\left| {{\rm{GHZ}}} \right\rangle _{{\rm{ABC}}}} = \sqrt {\frac{1}{2}} \left(\left| {111} \right\rangle + \left| {000} \right\rangle \right).$

    (6)

    经过简单计算, 可得三量子比特${\rm{A, B, C}}$在基矢$\{ \left| 1 \right\rangle = \left| {111} \right\rangle, \left| 2 \right\rangle = \left| {110} \right\rangle, \left| 3 \right\rangle = \left| {101} \right\rangle, \left| 4 \right\rangle = \left| {100} \right\rangle,$$\left| 5 \right\rangle = \left| {011} \right\rangle, \left| 6 \right\rangle = \left| {010} \right\rangle, \left| 7 \right\rangle = \left| {001} \right\rangle, \left| 8 \right\rangle = \left| {000} \right\rangle \}$下的约化密度矩阵$ {{\rho }}_{\rm{ABC}}(t)$的对角矩阵元:

    $ \begin{split} {\rho _{11}}(t) =\;& \frac{1}{2}{{\left| {P(t)} \right|}^6},{\rho _{22}}(t) = {\rho _{33}}(t) = {\rho _{55}}(t) \\ =\;& \frac{1}{2}{{\left| {P(t)} \right|}^4}\left( {1 - {{\left| {P(t)} \right|}^2}} \right),\\ {\rho _{44}}(t) =\;& {\rho _{66}}(t) = {\rho _{77}}(t) = \frac{1}{2}{{\left| {P(t)} \right|}^2}{{\left( {1 - {{\left| {P(t)} \right|}^2}} \right)}^2},\\ {\rho _{88}}(t) =\;& \frac{1}{2} + \frac{1}{2}{{\left( {1 - {{\left| {P(t)} \right|}^2}} \right)}^3}, \end{split} $

    和非零非对角矩阵元

    $ {\rho _{18}}(t) = \rho _{81}^*(t) = P{(t)^3}/2. $

    (7)

    为了得到(7)式, 设定了${g_i} = g$, ${\varOmega _i} = \varOmega $, 腔耗散${\varGamma _{i1}} = \varGamma $${\varGamma _{i2}} = {\varGamma _1}$.

    三体纠缠可采用负性纠缠度${N_3}$来量化, 其定义为[6]

    ${N_3} = {({N_{{\rm{A}} \text- {\rm{BC}}}}{N_{{\rm{B}} \text- {\rm{AC}}}}{N_{{\rm{C}} \text- {\rm{AB}}}})^{1/3}},$

    (8)

    这里, ${N_{I \text- JK}}(I = {\rm{A, B, C}};JK = {\rm{AB, AC, BC}})$是两体负性纠缠度, 其定义是${N}_{I\text-JK}=-2\displaystyle \sum\nolimits_{i}{{\sigma }}_{i} \left({{\rho }}_{\rm{ABC}}^{{\rm{T}}_{I}}\right)$. 其中, ${{\sigma }}_{i}\left({{\rho }}_{{\rm{ABC}}}^{{\rm{T}}_{I}}\right)$是对$ {{\rho }}_{\rm{ABC}}$的子系统$I$作偏转置后的密度矩阵$ {{\rho }}_{\rm{ABC}}^{{\rm{T}}_{I}}$的负本征值[6]. 结合(7)式和(8)式, 可得:

    $ \begin{split} {N_3} = \;&- ({\rho _{22}}(t) + {\rho _{77}}(t) \\ &- \sqrt {{{\left( {{\rho _{22}}(t) - {\rho _{77}}(t)} \right)}^2} + 4{{\left| {{\rho _{18}}(t)} \right|}^2}} ). \end{split}$

    (9)

    为了量化Bell非定域性, 选择MABK不等式作为探测工具. 对三量子比特来说, MABK算符为[9]

    $ \begin{split} {B} =\;& \frac{1}{2}({{{M}}_{\rm{A}}}{{{M}}_{\rm{B}}} M'_{\rm{C}} + {{{M}}_{\rm{A}}}{{ M'_{\rm{B}}}}{{{M}}_{\rm{C}}} \\ &+ {{{ M' _{\rm{A}}}}}{{{M}}_{\rm{B}}}{{{M}}_{\rm{C}}} - {{{ M' _{\rm{A}}}}}{{{ M'_{\rm{B}}}}}{{{ M' _{\rm{C}}}}}), \end{split} $

    (10)

    这里, ${{{M}}_K}$是第$K$个量子比特的测量算符, 且${{{M}}_K}$${{M}}'_K$表示对第$K$个量子比特作测量时的两种不同的测量方向. 方程(6)所给的量子态为初态时, 可选取[9]:

    $ \begin{split} &{{{M}}_{\rm{A}}} = {{{\sigma }}_y} \otimes {I_2} \otimes {I_2},{{{{ M' _{\rm{A}}}}}} = {{{\sigma }}_x} \otimes {I_2} \otimes {I_2},\\ &{{{M}}_{\rm{B}}} = {I_2} \otimes [\cos ({\theta _{\rm{B}}}){{{\sigma }}_y} - \sin ({\theta _{\rm{B}}}){{{\sigma }}_x}] \otimes {I_2},\\ &{{{{M'}}}_{\rm{B}}} = {I_2} \otimes [\sin ({\theta _{\rm{B}}}){{{\sigma }}_y} + \cos ({\theta _{\rm{B}}}){{{\sigma }}_x}] \otimes {I_2},\\ &{{{M}}_{\rm{C}}} = {I_2} \otimes {I_2} \otimes [\cos ({\theta _{\rm{C}}}){{{\sigma }}_y} - \sin ({\theta _{\rm{C}}}){{{\sigma }}_x}],\\ &{{{{M'}}}_{\rm{C}}} = {I_2} \otimes {I_2} \otimes [\sin ({\theta _{\rm{C}}}){{{\sigma }}_y} + \cos ({\theta _{\rm{C}}}){{{\sigma }}_x}], \end{split} $

    (11)

    式中, ${I_2}$$2 \times 2$维的单位矩阵, $ {{\sigma }}_{x, y}$是泡利矩阵. 结合(7)式、(10)式和(11)式, 可求得MABK算符${{B}}$在量子态$ {{\rho }}_{\rm{ABC}}(t)$中的期望值:

    $ \begin{split} \left|\langle {{B}}\rangle \right|=\;&|[2{\rho }_{81}(t)-2{\rho }_{18}(t)]{\rm{i}}\sin{\theta }_{\rm{BC}}\\ &-[2{\rho }_{81}(t)+2{\rho }_{18}(t)]\cos{\theta }_{\rm{BC}}|, \end{split}$

    (12)

    其中, 角度${\theta _{{\rm{BC}}}} = {\theta _{\rm{B}}} + {\theta _{\rm{C}}}$. 当$\left| {\left\langle {{B}} \right\rangle } \right| > 1$时, 量子系统将具有Bell非定域性.

    纠缠目击${\rm{(EWs)}}$是判定多体纠缠的一种有效工具. 但不同类型的纠缠, 需要使用不同的${\rm{EWs}}$算符[36]. 对于GHZ态来说, 常用的${\rm{EWs}}$算符是[7]:

    $ {{{w}}}_{\rm{GHZ}}=\frac{I}{2}-{{\rho }}_{\rm{GHZ}},$

    (13)

    根据(7)式和(13)式, 可得到${\rm{EWs}}$的期望值:

    $ \begin{split} {\rm{EWs}}=\;&{\rm{Tr}}[{{\rho }}_{\rm{GHZ}}(t){{{w}}}_{\rm{GHZ}}]=\frac{1}{2}(3{\rho }_{22}(t)\\ &+3{\rho }_{66}(t)-{\rho }_{18}(t)-{\rho }_{81}(t)). \end{split}$

    (14)

    $ - {\rm{EWs}} > 0$, 意味着存在实验上可直观探测的三体纠缠; 但$ - {\rm{EWs}} \leqslant 0$时, 无法判定量子态是否有纠缠.

    弱测量(WM)和测量反转(WMR)技术不仅可以有效保护纠缠[30], 还可提高量子费舍尔信息[37]. 对单量子比特来说, WM和WMR操作可分别表示为[30]:

    ${{{M}}_{{\rm{wk}}}} = \left| 0 \right\rangle \left\langle 0 \right| + \sqrt {1 - m} \left| 1 \right\rangle \left\langle 1 \right|,$

    ${{{M}}_{{\rm{rev}}}} = \sqrt {1 - k} \left| 0 \right\rangle \left\langle 0 \right| + \left| 1 \right\rangle \left\langle 1 \right|.$

    (15)

    本文所考虑的三个量子比特都被以同样的方式施加了WM和WMR操作. 具体来说, 在与环境相互作用之前, 对每一个量子比特都执行WM操作${{{M}}_{{\rm{wk}}}}$. 之后, 量子比特系统被置于环境中. 在$t$时刻, 再对每一个量子比特执行WMR操作${{{M}}_{{\rm{rev}}}}$. 为了简单起见, 设这三个量子比特的WM强度${m_i} \!=\! m~(0 \!\leqslant\! m \!<\! 1)$和WMR强度${k_i} \!=\! k(0 \leqslant k < 1)$. 这样, 经过WM及WMR操作后, 三量子比特的约化密度矩阵的非零元素为:

    $\begin{split} \;& {\rho _{11}}(t) =\frac{1}{N}\left[ {\frac{1}{2}{{(1 - m)}^3}{{\left| {P(t)} \right|}^6}} \right],\\ \;& {\rho _{22}}(t) ={\rho _{33}}(t) = {\rho _{55}}(t) \\ =& \frac{1}{N}\bigg[ \frac{1}{2}{{(1 - m)}^3} (1- k)(1 - {{\left| {P(t)} \right|}^2}){{\left| {P(t)} \right|}^4} \bigg],\\ \;&{\rho _{44}}(t) = {\rho _{66}}(t) = {\rho _{77}}(t) \\ =& \frac{1}{N}\bigg[ \frac{1}{2}{{(1 - m)}^3} (1- k)^2{{(1 - {{\left| {P(t)} \right|}^2})}^2}{{\left| {P(t)} \right|}^2} \bigg],\\ {\rho _{88}}(t) =\;\;& \frac{1}{N}\bigg[ \frac{1}{2}{{(1 - k)}^3} + \frac{1}{2}{{(1 - m)}^3}{{(1 - k)}^3} \\ & \times (1- {{\left| {P(t)} \right|}^2})^3 \bigg],\\ {\rho _{18}}(t) = \;&\frac{1}{N}\left[ {\frac{1}{2}{{(1 - m)}^{3/2}}{{(1 - k)}^{3/2}}{{\left| {P(t)} \right|}^3}} \right] = \rho _{81}^*(t), \end{split}$

    (16)

    这里$N \!=\! \dfrac{1}{2}{(1 \!-\! k)^3} + \dfrac{1}{2}{(1 \!-\! m)^3}\bigg[ 1 \!-\! k(1 - {{\left| {P(t)} \right|}^2}) \bigg]^3$. 将(16)式分别代入到(9)式、(12)式和(14)式中, 即可得到执行量子测量操作后的量子关联的解析表达式. 此外, 为了获得最优量子关联, 选择$k = 1 - (1 - m){\left| {P(t)} \right|^2}$[30].

    接下来来分析三个独立量子比特的量子关联动力学行为. 图1描述了给定理想空腔${R_{i2}}$(即$ {{\varGamma}}_{1}=0$)而改变两级联腔的耦合强度$\varOmega $时, 负性纠缠度${N_3}$、Bell函数$\left| {\left\langle {{B}} \right\rangle } \right| - 1$和纠缠目击$ - {\rm{EWs}}$在强耦合体系($g = 0.5\varGamma$, 见图1(a)图1(c))和弱耦合体系($g = 0.2\varGamma$, 见图1(d)图1(f))下随无量纲时间$\varGamma t$的变化曲线. 图1(a)(或图1(d))表明, 当$\varOmega = 0$时, $\left| {\left\langle {{B}} \right\rangle } \right| - 1$$ - {\rm{EWs}}$都会在短时间内出现猝死现象, 但${N_3}$仅单调衰减至零. 这里的猝死是指, 在探测纠缠方面Bell不等式和$ - {\rm{EWs}}$已失效. 相比$\left| {\left\langle {{B}} \right\rangle } \right| - 1$而言, $ - {\rm{EWs}}$能在更大的有限范围内探测纠缠. 对比图1(a)图1(d)可发现, 量子关联的量值或存活时间在弱耦合体系要优于强耦合体系. 随着$ {{\varOmega }}$的增加, 出现了一些有趣的行为(如图1所示): 1)${N_3}$在两种耦合体系下都会在一些阻尼震荡后达到一稳定值$N_{\rm{3}}^{\rm{s}}$, $N_3^{\rm{s}}$随着$\varOmega $的增大而提升, 且弱耦合体系的$N_{\rm{3}}^{\rm{s}}$明显高于强耦合体系. 尤其当$\varOmega = 2\varGamma $时, ${N_3}$能获得非常大的$N_{\rm{3}}^{\rm{s}}$. 另外, ${N_3}$在强耦合体系的震荡行为更明显. 2) 在强耦合体系(见图1(b)), $ - {\rm{EWs}}$在经历一段死亡时间后, 也可达到一稳定值, 而Bell函数要经过两段时间的猝死后, 方可最终达到一较小的稳定值. 这暗含着, 在表征纠缠方面$ - {\rm{EWs}}$要更优于MABK不等式. 特别在$\varGamma t > 4$后, $ - {\rm{EWs}}$一直能探测纠缠. 但在弱耦合体系(见图1(e)), $\left| {\left\langle {{B}} \right\rangle } \right| - 1$$ - {\rm{EWs}}$均呈现了俘获现象, 整个过程它们都能表征纠缠. 此时, 纠缠始终为真正的非定域性量子关联. 3)当$\varOmega = 2\varGamma $时, 在两种耦合体系下$\left| {\left\langle {{B}} \right\rangle } \right| - 1$$ - {\rm{EWs}}$都能获得很高的量值, 不会出现猝死现象.

    Figure 1.  Time evolution of Negativity ${N_3}$, Bell non-locality $\left| {\left\langle {{B}} \right\rangle } \right| - 1$ and entanglement witnesses $ - {\rm{EWs}}$ as the function of dimensionless time $\varGamma t$ for the different values of coupling strength $\varOmega $ in the strong coupling regime $g = 0.5\varGamma $((a)–(c)) and the weak coupling regime $g = 0.2\varGamma $ ((d)–(f)) with ${\varGamma _1} = 0$.

    下面来分析引入量子测量操作后的量子关联动力学行为. 图2图4分别是${N_3}$, $\left| {\left\langle {{B}} \right\rangle } \right| - 1$$ - {\rm{EWs}}$的变化图形. 图2图4中的所有(a)图和(c)图均为强耦合体系 ($g = 0.5\varGamma $), 所有(b)图和(d)图都是弱耦合体系($g = 0.2\varGamma $). 从图2可以看出, 耦合强度$\varOmega $越大越有利于纠缠的存在, 这与图1相符合. 一旦固定了$\varOmega $, 可通过增大$m$来提高纠缠. 同样, 图3也说明, $m$越大越有利于Bell非定域性的存在: 更大的$m$可更好地延长$\left| {\left\langle {{B}} \right\rangle } \right| - 1$的存活时间或使得它变成一稳定值(如图3(a)图3(c)). 这间接说明, 测量操作能让MABK不等式更好地表征纠缠. 还可以看出, 同样$m$调控下的$\left| {\left\langle {{B}} \right\rangle } \right| - 1$的稳定值在弱耦合体系总好于强耦合体系, 如图3(a)图3(b)所示. $ - {\rm{EWs}}$图4的变化行为也反映了测量操作的积极作用. 但图4(a)却出现了一个有趣现象, 即$\varOmega = 0$$m = 0.6$时, $ - {\rm{EWs}}$可分段周期性猝生和猝死, 猝生后可在一段时间内取值恒定. 事实上, 在强耦合体系, 当$\varOmega $较小而$m$较大时, 都存在此类现象. 值得一提的是, 根据图1得到$ - {\rm{EWs}}$探测纠缠的能力要好于MABK不等式. 然而, 对比图3(a)图4(a)却能发现, $m$值较大时, MABK不等式能比$ - {\rm{EWs}}$更好地探测纠缠. 不过, 随着$\varOmega $的增大, 在强耦合体系依然可以得到$ - {\rm{EWs}}$要优于MABK不等式的结论, 该结论此时与$m$的取值无关. 在弱耦合体系(如图3(b)图4(b)所示), $ - {\rm{EWs}}$在探测纠缠方面始终好于或等同于$\left| {\left\langle {{B}} \right\rangle } \right| - 1$, 测量操作仅起到提高量值或延长量子关联存活时间的作用.

    Figure 2.  Negativity $N_3$ versus dimensionless time $\varGamma t$ for the different values of coupling strength $\varOmega $ and the weak measurement strength $m$ in the strong coupling regime $g = 0.5\varGamma $((a) and (c)) and the weak coupling regime $g = 0.2\varGamma $((b) and (d)) with ${\varGamma _1} = 0$.
    Figure 3.  The change of Bell function $\left| {\left\langle {{B}} \right\rangle } \right| - 1$ as a function of $\varGamma t$ in the strong coupling regime $g = 0.5\varGamma $((a) and (c)) and the weak coupling regime $g = 0.2\varGamma $((b) and (d)). The values of other parameters are the same as those in Fig. 2.
    Figure 4.  Dynamics of entanglement witnesses $ - {\rm{EWs}}$ in the strong ((a), (b)) and the weak ((c), (d)) coupling regimes. The values of other parameters are the same as those in Fig. 2(a).

    图5为存在腔耗散${\varGamma _1} = 0.25\varGamma $、执行和未执行弱测量操作时量子关联随时间的演化行为, 参数$\varOmega = \varGamma $. 其中, 图5(a)图5(c)为强耦合体系($g = 0.5\varGamma $), 而图5(b)图5(d)为弱耦合体系($g = 0.2\varGamma $). 对比图1(b)图5(a)可以发现, 一旦级联空腔${R_{i2}}$具有了耗散, ${N_3}$, Bell非定域性和$ - {\rm{EWs}}$将不再出现关联俘获现象. 具体来说, ${N_3}$经过阻尼震荡后随时间逐渐衰减, Bell非定域性会在短时间内直接猝死, 而$ - {\rm{EWs}}$在猝死后还会出现短暂的复苏现象. 同理, 图1(e)图5(b)也说明, 非零耗散${\varGamma _1}$破坏了量子关联俘获, 致使它们都以阻尼震荡方式衰减. 图5(c)图5(d)则说明, 较大的$m$会使得量子关联得到很大提高, 且不再发生猝死行为.

    Figure 5.  Change curves of quantum correlation in the strong coupling regime $g = 0.5\varGamma $((a) and (c)) and the weak coupling regime $g = 0.2\varGamma $((b) and (d)), where (a) and (b) are the cases without measurement, while (c) and (d) are the cases with measurement. The parameters $\varOmega $ and ${\varGamma _1}$ are set to $\varGamma $ and $0.25\varGamma $, respectively.

    本文考虑初始被制备于最大纠缠GHZ态的三个全同量子比特系统, 详细讨论了强、弱耦合体系下三比特纠缠和Bell非定域性这两种量子关联的动力学行为及其调控问题. 其中, 每个量子比特都被置于各自的一个单模腔内, 该腔又与另外一个空腔相级联, 这两个级联腔共同构成了一个级联环境. 利用负性纠缠度和纠缠目击($ - {\rm{EWs}}$)来描述纠缠. 同时, 借助MABK不等式来量化Bell非定域性. 研究表明, 纠缠和Bell非定域性强烈依赖于腔-腔耦合强度$\varOmega $和空腔的耗散率${\varGamma _1}$. 具体来说, ${\varGamma _1} = 0$$\varOmega $接近零时, 除了负性纠缠度外, Bell非定域性和纠缠目击都能发生猝死现象. 通过增大$\varOmega $, 量子关联的量值能被大幅提高, 甚至可以出现量子俘获现象. 但非零${\varGamma _1}$在很大程度上削弱了量子关联, 甚至破坏了这种俘获现象. 给定$\varOmega $后, 量子弱测量和测量反转技术可以有效提高量子关联或延长量子关联的存活时间. 所以, 弱测量及其反转操作的引入, 可以更好地保护量子关联. 另外, 我们发现, 当未施加量子弱测量和测量反转时, $ - {\rm{EWs}}$在强、弱耦合体系下探测纠缠的能力都好于MABK不等式. 但在施加了量子测量操作后, 这一结论不总是成立. 结果显示, 在强耦合体系中, 当$\varOmega $较小且弱测量强度$m$较大时, MABK不等式探测纠缠的能力可以好于$ - {\rm{EWs}}$, 但这一现象不会在弱耦合体系发生. 当然, 随着$\varOmega $的增大, 即便在强耦合体系且$m$很大时, 又会得到$ - {\rm{EWs}}$探测纠缠的能力超过MABK不等式的结论. 此外还能知道, 量子关联在弱耦合体系的量值或存活时间始终要好于强耦合体系.

    [1]

    Paneru D, Cohen E, Fickler R, Boyd R W, Karimi E 2020 Rep. Prog. Phys. 83 064001Google Scholar

    [2]

    Su Z F, Tan H S, Li X Y 2020 Phys. Rev. A 101 042112Google Scholar

    [3]

    Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895Google Scholar

    [4]

    Kura N, Ueda M 2020 Phys. Rev. Lett. 124 010507Google Scholar

    [5]

    Hu X M, Xing W B, Liu B H, Huang Y F, Li C F, Guo G C 2020 Phys. Rev. Lett. 125 090503Google Scholar

    [6]

    Sabín C, García-Alcaine G 2007 Eur. Phys. J. D 48 435

    [7]

    Maity A G, Das D, Ghosal A, Roy A, Majumdar A S 2020 Phys. Rev. A 101 042340Google Scholar

    [8]

    Rosset D, Branciard C, Barnea T J, Pütz G, Brunner N, Gisin N 2016 Phys. Rev. Lett. 116 010403Google Scholar

    [9]

    Altintas F, Eryigit R 2010 Phys. Lett. A 374 4283Google Scholar

    [10]

    Anwer H, Nawareg M, Cabello A, Bourennane M 2019 Phys. Rev. A 100 022104Google Scholar

    [11]

    Shin D K, Henson B M, Hodgman S S, Wasak T, Chwedeńczuk J, Truscott A G 2019 Nature 10 4447

    [12]

    Kuo W T, Akhtar A A, Arovas D P, You Y Z 2020 Phys. Rev. B 101 224202Google Scholar

    [13]

    刑贵超, 夏云杰 2018 物理学报 67 070301Google Scholar

    Xing G C, Xia Y J 2018 Acta Phys. Sin. 67 070301Google Scholar

    [14]

    Yu T, Eberly J H 2004 Phys. Rev. Lett. 93 140404Google Scholar

    [15]

    López C E, Romero G, Lastra F, Solano E, Retamal J C 2008 Phys. Rev. Lett. 101 080503Google Scholar

    [16]

    Aguilar G H, Valdés-Hernández A, Davidovich L, Walborn S P, Souto Ribeiro P H 2014 Phys. Rev. Lett. 113 240501Google Scholar

    [17]

    Antonelli C, Shtaif M, Brodsky M 2011 Phys. Rev. Lett. 106 080404Google Scholar

    [18]

    Bellomo B, Lo Franco R, Compagno G 2007 Phys. Rev. Lett. 99 160502Google Scholar

    [19]

    Mohamed A B A, Eleuch H, Raymond Ooi C H 2019 Sci. Rep. 9 19632Google Scholar

    [20]

    Deordi G L, Vidiella-Barranco A 2020 Opt. Commun. 475 126233Google Scholar

    [21]

    Pramanik T, Cho Y W, Han S W, Lee S Y, Moon S, Kim Y S 2019 Phys. Rev. A 100 042311Google Scholar

    [22]

    Hu M L 2012 Ann. Phys. 327 2332Google Scholar

    [23]

    Weilenmann M, Dive B, Trillo D, Aguilar E A, Navascués M 2019 Phys. Rev. Lett. 124 200502

    [24]

    Zhou Y 2020 Phys. Rev. A 101 012301Google Scholar

    [25]

    Hanson R, Dobrovitski V V, Feiguin A E, Gywat O, Awschalom D D 2008 Science 320 352Google Scholar

    [26]

    Man Z X, Xia Y J, Rosario L F 2015 Phys. Rev. A 92 012315Google Scholar

    [27]

    Basit A, Ali H, Badshah F, Zhang H Y, Ge G Q 2017 Laser Phys. Lett. 14 125202Google Scholar

    [28]

    Bai X M, Xue N T, Liu N, Li J Q, Liang J Q 2019 Ann. Phys. 531 1900098Google Scholar

    [29]

    Xu K, Zhang G F, Zhou Y, Liu W M 2020 J. Opt. Soc. Am. B 37 933Google Scholar

    [30]

    Kim Y S, Lee J C, Kwon O, Kim Y H 2012 Nat. Phys. 8 117Google Scholar

    [31]

    Korotkov A N, Keane K 2010 Phys. Rev. A 81 040103(RGoogle Scholar

    [32]

    He Z, Zeng H S 2020 Quantum Inf. Process. 19 299Google Scholar

    [33]

    Qiu L, Tang G, Yang X Q, Wang A 2014 Ann. Phys. 350 137Google Scholar

    [34]

    Groen J P, Ristè D, Tornberg L, Cramer J, Degroot P C, Picot T, Johansson G, Dicarlo L 2013 Phys. Rev. Lett. 111 090506Google Scholar

    [35]

    Man Z X, Xia Y J, Rosario L F 2015 Sci. Rep. 5 13843Google Scholar

    [36]

    Gühne O, Tóth G 2009 Phys. Rep. 474 1Google Scholar

    [37]

    Xiao X, Yao Y, Zhong W J, Li Y L, Xie Y M 2016 Phys. Rev. A 93 012307Google Scholar

  • 图 1  耦合强度$\varOmega $取不同值时, 负性纠缠度${N_3}$、Bell非定域性$\left| {\left\langle {{B}} \right\rangle } \right| - 1$和纠缠目击$ - {\rm{EWs}}$在强耦合体系$g = 0.5\varGamma $ ((a)—(c))和弱耦合体系$g = 0.2\varGamma $((d)—(f))下随无量纲时间$\varGamma t$的变化曲线. 其中, ${\varGamma _1} = 0$

    Figure 1.  Time evolution of Negativity ${N_3}$, Bell non-locality $\left| {\left\langle {{B}} \right\rangle } \right| - 1$ and entanglement witnesses $ - {\rm{EWs}}$ as the function of dimensionless time $\varGamma t$ for the different values of coupling strength $\varOmega $ in the strong coupling regime $g = 0.5\varGamma $((a)–(c)) and the weak coupling regime $g = 0.2\varGamma $ ((d)–(f)) with ${\varGamma _1} = 0$.

    图 2  耦合强度$\varOmega $和弱测量强度m取不同值时, 负性纠缠度$N_3$在强耦合体系$g = 0.5\varGamma $ ((a)和(c))和弱耦合体系$g = 0.2\varGamma $((b)和(d))下随无量纲时间$\varGamma t$的变化曲线. 其中, ${\varGamma _1} = 0$

    Figure 2.  Negativity $N_3$ versus dimensionless time $\varGamma t$ for the different values of coupling strength $\varOmega $ and the weak measurement strength $m$ in the strong coupling regime $g = 0.5\varGamma $((a) and (c)) and the weak coupling regime $g = 0.2\varGamma $((b) and (d)) with ${\varGamma _1} = 0$.

    图 3  Bell函数$\left| {\left\langle {{B}} \right\rangle } \right| - 1$在强耦合体系$g = 0.5\varGamma $ ((a)和(c))和弱耦合体系$g = 0.2\varGamma $((b)和(d))下随无量纲时间$\varGamma t$的变化曲线. 其他参数取值与图2相同

    Figure 3.  The change of Bell function $\left| {\left\langle {{B}} \right\rangle } \right| - 1$ as a function of $\varGamma t$ in the strong coupling regime $g = 0.5\varGamma $((a) and (c)) and the weak coupling regime $g = 0.2\varGamma $((b) and (d)). The values of other parameters are the same as those in Fig. 2.

    图 4  纠缠目击$ - {\rm{EWs}}$在强 ((a), (c))、弱((b), (d))耦合体系下的动力学行为. 其他参数取值与图2相同

    Figure 4.  Dynamics of entanglement witnesses $ - {\rm{EWs}}$ in the strong ((a), (b)) and the weak ((c), (d)) coupling regimes. The values of other parameters are the same as those in Fig. 2(a).

    图 5  量子关联在强耦合体系$g = 0.5\varGamma $ ((a)和(c))和弱耦合体系$g = 0.2\varGamma $((b)和(d))下的变化曲线. 其中, (a)和(b)无弱测量操作, (c)和(d)有弱测量操作. 参数$\varOmega = \varGamma $${\varGamma _1} = 0.25\varGamma $

    Figure 5.  Change curves of quantum correlation in the strong coupling regime $g = 0.5\varGamma $((a) and (c)) and the weak coupling regime $g = 0.2\varGamma $((b) and (d)), where (a) and (b) are the cases without measurement, while (c) and (d) are the cases with measurement. The parameters $\varOmega $ and ${\varGamma _1}$ are set to $\varGamma $ and $0.25\varGamma $, respectively.

  • [1]

    Paneru D, Cohen E, Fickler R, Boyd R W, Karimi E 2020 Rep. Prog. Phys. 83 064001Google Scholar

    [2]

    Su Z F, Tan H S, Li X Y 2020 Phys. Rev. A 101 042112Google Scholar

    [3]

    Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895Google Scholar

    [4]

    Kura N, Ueda M 2020 Phys. Rev. Lett. 124 010507Google Scholar

    [5]

    Hu X M, Xing W B, Liu B H, Huang Y F, Li C F, Guo G C 2020 Phys. Rev. Lett. 125 090503Google Scholar

    [6]

    Sabín C, García-Alcaine G 2007 Eur. Phys. J. D 48 435

    [7]

    Maity A G, Das D, Ghosal A, Roy A, Majumdar A S 2020 Phys. Rev. A 101 042340Google Scholar

    [8]

    Rosset D, Branciard C, Barnea T J, Pütz G, Brunner N, Gisin N 2016 Phys. Rev. Lett. 116 010403Google Scholar

    [9]

    Altintas F, Eryigit R 2010 Phys. Lett. A 374 4283Google Scholar

    [10]

    Anwer H, Nawareg M, Cabello A, Bourennane M 2019 Phys. Rev. A 100 022104Google Scholar

    [11]

    Shin D K, Henson B M, Hodgman S S, Wasak T, Chwedeńczuk J, Truscott A G 2019 Nature 10 4447

    [12]

    Kuo W T, Akhtar A A, Arovas D P, You Y Z 2020 Phys. Rev. B 101 224202Google Scholar

    [13]

    刑贵超, 夏云杰 2018 物理学报 67 070301Google Scholar

    Xing G C, Xia Y J 2018 Acta Phys. Sin. 67 070301Google Scholar

    [14]

    Yu T, Eberly J H 2004 Phys. Rev. Lett. 93 140404Google Scholar

    [15]

    López C E, Romero G, Lastra F, Solano E, Retamal J C 2008 Phys. Rev. Lett. 101 080503Google Scholar

    [16]

    Aguilar G H, Valdés-Hernández A, Davidovich L, Walborn S P, Souto Ribeiro P H 2014 Phys. Rev. Lett. 113 240501Google Scholar

    [17]

    Antonelli C, Shtaif M, Brodsky M 2011 Phys. Rev. Lett. 106 080404Google Scholar

    [18]

    Bellomo B, Lo Franco R, Compagno G 2007 Phys. Rev. Lett. 99 160502Google Scholar

    [19]

    Mohamed A B A, Eleuch H, Raymond Ooi C H 2019 Sci. Rep. 9 19632Google Scholar

    [20]

    Deordi G L, Vidiella-Barranco A 2020 Opt. Commun. 475 126233Google Scholar

    [21]

    Pramanik T, Cho Y W, Han S W, Lee S Y, Moon S, Kim Y S 2019 Phys. Rev. A 100 042311Google Scholar

    [22]

    Hu M L 2012 Ann. Phys. 327 2332Google Scholar

    [23]

    Weilenmann M, Dive B, Trillo D, Aguilar E A, Navascués M 2019 Phys. Rev. Lett. 124 200502

    [24]

    Zhou Y 2020 Phys. Rev. A 101 012301Google Scholar

    [25]

    Hanson R, Dobrovitski V V, Feiguin A E, Gywat O, Awschalom D D 2008 Science 320 352Google Scholar

    [26]

    Man Z X, Xia Y J, Rosario L F 2015 Phys. Rev. A 92 012315Google Scholar

    [27]

    Basit A, Ali H, Badshah F, Zhang H Y, Ge G Q 2017 Laser Phys. Lett. 14 125202Google Scholar

    [28]

    Bai X M, Xue N T, Liu N, Li J Q, Liang J Q 2019 Ann. Phys. 531 1900098Google Scholar

    [29]

    Xu K, Zhang G F, Zhou Y, Liu W M 2020 J. Opt. Soc. Am. B 37 933Google Scholar

    [30]

    Kim Y S, Lee J C, Kwon O, Kim Y H 2012 Nat. Phys. 8 117Google Scholar

    [31]

    Korotkov A N, Keane K 2010 Phys. Rev. A 81 040103(RGoogle Scholar

    [32]

    He Z, Zeng H S 2020 Quantum Inf. Process. 19 299Google Scholar

    [33]

    Qiu L, Tang G, Yang X Q, Wang A 2014 Ann. Phys. 350 137Google Scholar

    [34]

    Groen J P, Ristè D, Tornberg L, Cramer J, Degroot P C, Picot T, Johansson G, Dicarlo L 2013 Phys. Rev. Lett. 111 090506Google Scholar

    [35]

    Man Z X, Xia Y J, Rosario L F 2015 Sci. Rep. 5 13843Google Scholar

    [36]

    Gühne O, Tóth G 2009 Phys. Rep. 474 1Google Scholar

    [37]

    Xiao X, Yao Y, Zhong W J, Li Y L, Xie Y M 2016 Phys. Rev. A 93 012307Google Scholar

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Metrics
  • Abstract views:  5439
  • PDF Downloads:  132
  • Cited By: 0
Publishing process
  • Received Date:  15 December 2020
  • Accepted Date:  09 January 2021
  • Available Online:  13 May 2021
  • Published Online:  20 May 2021

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