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自旋轨道耦合量子点系统中的量子相干

王志梅 王虹 薛乃涛 成高艳

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自旋轨道耦合量子点系统中的量子相干

王志梅, 王虹, 薛乃涛, 成高艳

Quantum coherence in spin-orbit coupled quantum dots system

Wang Zhi-Mei, Wang Hong, Xue Nai-Tao, Cheng Gao-Yan
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  • 文章访问数:  670
  • PDF下载量:  46
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-11-16
  • 修回日期:  2022-01-13
  • 上网日期:  2022-01-26
  • 刊出日期:  2022-04-05

自旋轨道耦合量子点系统中的量子相干

  • 1. 太原师范学院物理系, 晋中 030619
  • 2. 太原师范学院, 计算物理与应用物理研究所, 晋中 030619
  • 3. 华中科技大学, 武汉国家光电实验中心, 武汉 430074
  • 通信作者: 王志梅, 120705547@qq.com
    基金项目: 国家自然科学基金(批准号: 11747057)和山西省高等学校科技创新项目(批准号: 2020L0526)资助的课题.

摘要: 研究了自旋轨道耦合量子点中的量子相干效应. 运用输运电子的全计数统计方法计算系统的平均电流、散粒噪声和偏斜, 发现体系存在自旋轨道耦合作用时, 散粒噪声值随自旋轨道耦合常数的增加而减小. 更重要的是, 电流、噪声和偏斜随磁通周期性波动, 并且波动周期不受自旋轨道耦合强度大小、自旋极化率以及动力学耦合不对称的影响.

English Abstract

    • 量子点是一种通过适当的偏置金属栅极, 将电子限制在二维电子气体小区域内的半导体器件. 量子点输运[1,2]为观察自旋相关和强相关系统的基本物理现象奠定了基础, 如近藤效应[3-5]、库仑效应和自旋阻塞效应[6-8]等. 介观纳米结构的量子输运揭示了许多与量子干涉、离散能级和多体关联相关的特性. 根据所研究的具体系统, 已经发展了一些理论方法, 如Landauer-Buttiker理论和非平衡格林函数方法[9], 这两种方法在探究声子非弹性散射和处理多电子库仑相互作用下的介观系统等方面的问题时并不具有广泛性. 在某些特定情况下, 解决这些问题的一种比较简单的办法是运用率方程方法. 然而, 这种方法要求偏置电压大、温度为零, 这极大地限制了其适用性[10-13]. 因为量子输运在本质上其实是一个随机的过程, 原则上, 通过探究相应的分布函数可以充分理解其随机过程, 而一阶和二阶累积矩完全足以描述分布函数为高斯分布的一些物理量[14]. 然而, 电流或电导的分布一般来说不是高斯分布. 这就需要所有的电流累积量(即全计数统计)包括在内[15-17], 以便完全展现出所有阶电荷输运之间的相关性. 特别是, 由于单个电子隧穿技术高度敏感片上检测技术的发展, 所有转移粒子数量的统计累积现在已经可以通过实验进行提取[18]. 正常态电子的全计数统计已经在理论[19,20]和实验[21-33]中得到了解决. 介观系统中电流波动的研究使我们能够获得电子相关信息, 而高阶矩能够更全面地描述输运特性. 全计数统计可以给定系统所有传输特性的完整信息[34]. 目前, 全计数统计方法已在很多体系中进行了探究, 如正常超导体混合结构[35,36]、超导弱链接[37]、隧道结[38]、混沌腔[39]、纠缠电子[40]和自旋相关系统[41]、库仑阻塞系统等. 另外, 全计数统计的实验测量方案已被提出[42,43].

      在两个隧穿的事件之间, 量子点态会经历量子相干演化. 与隧穿速率相比, 快速的相干演化可以很容易地支配系统的整体动力学[44]. 介观系统是量子相干和退相干极好的探测平台, 这是基础物理学和量子器件实现的最重要和最具挑战性的问题之一.Aharonov-Bohm干涉仪[45,46]是探测相干的标准介观工具, 它的振幅是一种很好的相干性量度. 介观量子相干之所以重要, 是因为它为处理量子自由度的技术开拓了广阔的前景[47].

      介观物理学的发展为传输和处理信息的装置中使用电子自旋提供了理论支持. 1990年, Datta和Das描述了如何运用电场进行调制电流, 并展示了场相关的自旋轨道耦合在这一机制中所起的重要作用. Rashba自旋轨道耦合作用激发了很多的预测、发现和创新概念. 通过在空间中移动电子来操控自旋方向, 运用自旋方向来控制电子轨迹, 发现了一些新的拓扑材料类别.Rashba自旋轨道耦合作用是由限制势的反演不对称性造成的. 二维电子气中的Rashba耦合强度可以通过改变栅极场实现高达50%的改变[48-52]. 这一发现再次激发了材料学家和物理学家进一步深入探究反相不对称结构材料的想法. 自旋电子学已经是固态物理中的一个重要研究领域, 而实验研究方面的进展为介观系统中自旋偏置诱导输运的探究开辟了新的可能性. 例如, 自旋偏压可以通过控制铁磁和非磁电极偏压接触处的自旋积累来实现[53-55].

      自旋极化电流的产生和控制是半导体自旋电子学探究的一个关键课题[56], 因此, 大量的理论和实验方面的研究都投入在介观系统中, 其中最主要的技术之一是自旋注入, 它主要依靠光学技术和磁性材料或磁场的使用. 然而, 光学自旋注入技术很难与电子器件集成, 通过铁磁体与非磁性半导体结自旋注入的效率通常很小[57,58], 针对这些, 最近的一些工作将与自旋相关的输运放置在一些环形或双通道结构. Aharonov-Bohm环[59]、Stern-Gerlach环[60]、Aharonov-Casher环[61]、Aharonov-Bohm干涉仪[62]及双通道半导体器件[63], 这些环形导体或双通道器件通常用于研究介观系统中的量子相干效应. Rashba自旋轨道作用可以避免使用任何磁性材料或场. 由于二维电子系统中电场的反演不对称性所产生的Rashba效应, 自旋向上的电子与自旋向下的电子在通过上臂和下臂时会获得不同的相位, 从而产生有趣的与自旋相关的相干现象.

      目前, 关于自旋轨道耦合诱导的量子相干相关方面的研究尚少, 本文将运用量子主方程方法重点研究自旋轨道耦合量子点系统中的量子相干效应.

    • 考虑自旋轨道耦合的量子点系统, 其哈密顿量可写为

      $ H = {H_{\text{D}}} + {H_{{\text{leads}}}} + {H_{\text{T}}}\text{, } $

      其中${H_{\text{D}}} = \displaystyle\sum\nolimits_{\sigma j} {{\varepsilon _j}d_{j\sigma }^\dagger {d_{j\sigma }}} + \displaystyle\sum\nolimits_{\sigma {\sigma '}} U {n_{1\sigma }}{n_{2{\sigma '}}} + {H_{{\text{SO}}}}$为量子点哈密顿量; 电极哈密顿量为${H_{{\text{leads}}}} = $$ \displaystyle\sum\nolimits_{\alpha , k, \sigma } {{\varepsilon _{\alpha k}}} a_{\alpha k\sigma }^\dagger {a_{\alpha k\sigma }}({\alpha = {\text{L, R}}})$; 量子点和电极间的隧穿耦合哈密顿量为${H_{\text{T}}} = \displaystyle\sum\nolimits_{\alpha kj} ( {t_{\alpha jk}}{p_{\alpha j}}a_{\alpha k\sigma }^\dagger {d_{j\sigma }} $$ + \text{H.c}. )$, ${p_{\alpha j}}({p_{\alpha j}}^* = {p_{j\alpha }})$是第$j$个量子点到$\alpha $电极电子跃迁时磁通诱导的相位, 当系统具有反平行磁通时(${\varPhi _1} = \zeta {\varPhi _0}$${\varPhi _2} = - {\varPhi _1}$, ${\varPhi _0} = ch/e$是磁通量子单位, $\zeta $是无量纲磁通数), 这里假定两磁通方向与电极-量子点所在的平面垂直, 相位可表示为${p_{{\text{L1}}}} = {{\text{e}}^{ - {\text{i\pi }}\zeta }}$, ${p_{{\text{L2}}}} = {{\text{e}}^{{\text{i\pi }}\zeta }}$, ${p_{{\text{R1}}}} = {{\text{e}}^{ - {\text{i\pi }}\zeta }}$${p_{{\text{R2}}}} = {{\text{e}}^{{\text{i\pi }}\zeta }}$, ${H_{{SO} }}$为自旋轨道耦合哈密顿量, 对应的二次量子化的形式为

      $ {H_{{\text{SO}}}} = {\alpha _{{\text{SO}}}}\left[ ( {1 - {\text{i}}} )d_{2 \uparrow }^\dagger {d_{1 \downarrow }} - ( {1 + {\text{i}}} )d_{2 \downarrow }^\dagger {d_{1 \uparrow }} + \text{H.c.} \right], $

      其中${\alpha _{{\text{SO}}}}$为自旋轨道耦合常数, $d_{j\sigma }^\dagger $(${d_{j\sigma }}$)是电子在第$ j $个量子点的产生(湮灭)算符, ${n_{j\sigma }} = d_{j\sigma }^\dagger {d_{j\sigma }}$($j = 1, 2$)是粒子数算符, 自旋算符$\sigma = \uparrow , \downarrow $, U是电子间库仑作用, ${\varepsilon _{\alpha k}}$为波失$k$相应能量, $a_{\alpha k\sigma }^\dagger $(${a_{\alpha k\sigma }}$)分别为左右电极电子产生(湮灭)算符, ${t_{\alpha jk}}$为电极和能级间的耦合常数. 跃迁率${\varGamma _{\alpha j\sigma }}{{ = 2\pi }}{{\text{g}}_{\alpha \sigma }}{\left| {{t_{\alpha j}}} \right|^2}$(${{\text{g}}_{\alpha \sigma }}$是态密度), $ {\varGamma _{{\text{L}} \uparrow \left( \downarrow \right)}} = {\varGamma _{\text{L}}}\left( {1 \pm p} \right) $, ${\varGamma _{{\text{R}} \uparrow \left( \downarrow \right)}} = {\varGamma _{\text{R}}}\left( {1 \pm p} \right)$(发射与收集电极平行); ${\varGamma _{{\text{L}} \uparrow \left( \downarrow \right)}} = {\varGamma _{\text{L}}}\left( {1 \pm p} \right)$, $\varGamma _{{\text{R}} \uparrow \left( \downarrow \right)} = $$ {\varGamma _R}\left( {1 \mp p} \right)$(发射与收集电极反平行). $p = \dfrac{{\left( {{g_ \uparrow } - {g_ \downarrow }} \right)}}{{\left( {{g_ \uparrow } + {g_ \downarrow }} \right)}}$表示极化率. 本文讨论自旋轨道耦合量子点系统, 这里假定同一量子点上的双占据态因为无穷大的库仑力而不允许存在, 这样, 共有9个有效的占据态可作为基矢, 分别为$ \left| {0, 0} \right\rangle $, $ \left| {0, \sigma } \right\rangle $, $ \left| {\sigma , 0} \right\rangle $, $ \left| {\sigma , {\sigma '}} \right\rangle $, 这里自旋$\sigma \left(\sigma '\right)=\uparrow , \downarrow$, 体系对应的能量本征值和本征态为

      $ \begin{split} &{E}_{0,0}=0,|{\psi }_{00}\rangle =|0,0\rangle ; \;\;{E}_{\sigma ,\sigma '}={\varepsilon }_{1}+{\varepsilon }_{2}+U,|{\psi }_{\sigma \sigma '}\rangle =|\sigma ,\sigma'\rangle ; \\ & {E}_{6}={\varepsilon }_{1}+{\varepsilon }_{2}-\sqrt{{\left({\varepsilon }_{1}-{\varepsilon }_{2}\right)}^{2}+8{\alpha }_{\text{so}}^{2}}/2,\;\;|{\psi }_{6}\rangle ={a}_{1}|\uparrow ,0\rangle +{c}_{1}|0,\downarrow \rangle ; \\ & {E}_{7}={\varepsilon }_{1}+{\varepsilon }_{2}-\sqrt{{\left({\varepsilon }_{1}-{\varepsilon }_{2}\right)}^{2}+8{\alpha }_{\text{so}}^{2}}/2, \;\;|{\psi }_{7}\rangle ={b}_{2}|\downarrow ,0\rangle +{c}_{2}|0,\uparrow \rangle ; \\ & {E}_{8}={\varepsilon }_{1}+{\varepsilon }_{2}+\sqrt{{\left({\varepsilon }_{1}-{\varepsilon }_{2}\right)}^{2}+8{\alpha }_{\text{so}}^{2}}/2,\;\; |{\psi }_{8}\rangle ={a}_{3}|\uparrow ,0\rangle +{c}_{3}|0,\downarrow \rangle ; \\ & {E}_{9}={\varepsilon }_{1}+{\varepsilon }_{2}+\sqrt{{\left({\varepsilon }_{1}-{\varepsilon }_{2}\right)}^{2}+8{\alpha }_{\text{so}}^{2}}/2,\;\; |{\psi }_{9}\rangle ={b}_{4}|\downarrow ,0\rangle +{c}_{4}|0,\uparrow \rangle .\end{split} $

      归一化系数

      $ \begin{split} &{a_1} = \sqrt 2 \left( { - 1 + {\text{i}}} \right)\left( {\varDelta - \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)\Bigg/\left[ {2\sqrt {{{\left( {\varDelta - \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)}^2} + 8{\alpha _{\rm so}^2}} } \;\right],\\ &{a_3} = \sqrt 2 \left( { - 1 + {\text{i}}} \right)\left( {\varDelta + \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)\Bigg/\left[ {2\sqrt {{{\left( {\varDelta + \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)}^2} + 8{\alpha _{\rm so}^2}} } \;\right],\\ &{b_2} = \sqrt 2 \left( {1 + {\text{i}}} \right)\left( {\varDelta - \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)\Bigg/\left[ {2\sqrt {{{\left( {\varDelta - \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)}^2} + 8{\alpha _{\rm so}^2}} } \;\right],\\ &{b_4} = \sqrt 2 \left( {1 + {\text{i}}} \right)\left( {\varDelta + \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)\Bigg/\left[ {2\sqrt {{{\left( {\varDelta + \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)}^2} + 8{\alpha _{\rm so}^2}} }\; \right],\\ &{c_1} = {c_2} = 2\sqrt 2 {\alpha _{\rm so}}/\sqrt {{{\left( {\varDelta - \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)}^2} + 8{\alpha _{\rm so}^2}} ,\\ & {c_3} = {c_4} = 2\sqrt 2 {\alpha _{\rm so}}/\sqrt {{{\left( {\varDelta + \sqrt {{\varDelta ^2} + 8{\alpha _{\rm so}^2}} } \right)}^2} + 8{\alpha _{\rm so}^2}} , \end{split}$

      其中$\varDelta = {\varepsilon _1} - {\varepsilon _2}$为能级差. 以上述的9个本征态为基矢, 可以获得量子点系统主方程的矩阵元(见附录).

    • 将电极与量子点之间的弱耦合哈密顿量${H_{\text{T}}}$作为微扰, 系统的量子主方程为

      $ \begin{split}\;& \frac{{\partial {{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right)}}{{\partial t}} = - {\text{i}}L{{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right) - \frac{1}{2}\bigg\{ \mathop \sum \limits_{\alpha j{\text{σ }}} \left[ {d_{j\sigma }^\dagger A_{\alpha j\sigma }^{\left( - \right)}{{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right) + {{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right)A_{\alpha j\sigma }^{\left( + \right)}d_{j\sigma }^\dagger + {\text{H}}{\text{.c}}{\text{. }}} \right] \\ & \qquad- \mathop \sum \limits_{\sigma j} \left[ {A_{{\text{L}}j\sigma }^{\left( - \right)}{{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right)d_{j\sigma }^\dagger + A_{{\text{R}}j\sigma }^{\left( - \right)}{{\boldsymbol{\rho}} ^{\left( {n - 1} \right)}}\left( t \right)d_{j\sigma }^\dagger + {\text{d}}_{j\sigma }^\dagger {{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right)A_{{\text{L}}j\sigma }^{\left( + \right)} + d_{j\sigma }^\dagger {{\boldsymbol{\rho}} ^{\left( {n + 1} \right)}}\left( t \right)A_{{\text{R}}j\sigma }^{\left( + \right)} + {\text{H}}{\text{.c}}{\text{. }}} \right]\bigg\} ,\end{split} $

      其中${{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right)$为隧穿到收集电极上$ n $个电子的约化密度矩阵, $L $为Liouvillian超算符并且定义为$L\left( t \right){\hat O }= $$ [ {{H_D}( t ), {\hat O}} ]$, 谱函数$A_{\alpha j\sigma }^{( + )} = \displaystyle \sum\nolimits_i {2{\text{π }}{g_\alpha }} t_{\alpha j}^ * p_{\alpha j}^ * {t_{\alpha i}}{p_{\alpha i}}n_\alpha ^{( + )}{d_{i\sigma }}$, $A_{\alpha j\sigma }^{( - )}= \displaystyle \sum\nolimits_i {2{\text{π }}{g_\alpha }} {t_{\alpha j}}{p_{\alpha j}}t_{\alpha i}^ * p_{\alpha i}^ * n_\alpha ^{( - )}{d_{i\sigma }}.$ $n_\alpha ^{( + )} = {f_\alpha }$, $n_\alpha ^{( - )} = $$ 1 - {f_\alpha }$, 费米分布$ {f_\alpha } = 1/\left[ {1 + {{\text{e}}^{\left( {\varepsilon - {\mu _\alpha }} \right)/\left( {KT} \right)}}} \right] $.

      为了更加有效地计算量子主方程, 这里引入累积生成函数$F(\chi) = - \ln \displaystyle\sum\nolimits_n {P(n, t)} {{\text{e}}^{{\text{i}}n\chi }}$, 其中$\chi$表示计数场, $P(n,t)$为转移粒子数概率分布. 累积生成函数$F(\chi)$起到了一个非常关键的作用, 它与约化密度矩阵的关系是${{\text{e}}^{ - F\left( \chi \right)}} = {\text{Tr}}\left[ {\displaystyle\sum\nolimits_n {{{\boldsymbol{\rho}} ^{\left( n \right)}}\left( t \right){{\text{e}}^{{\text{i}}n\chi }}} } \right]$.

      粒子数分辨的量子主方程(4)可表示为

      $ {\dot {\boldsymbol{\rho}} ^{\left( n \right)}} = A{{\boldsymbol{\rho }}^{\left( n \right)}} + B{{\boldsymbol{\rho}} ^{\left( {n + 1} \right)}} + C{{\boldsymbol{\rho}} ^{\left( {n - 1} \right)}}. $

      $S(\chi , t) = \displaystyle\sum\nolimits_n {{{\boldsymbol{\rho}} ^{(n)}} (t){{\text{e}}^{{\text{i}}n\chi }}}$, $\dot S = {L_\chi }S$, ${L_\chi } = A + {{\text{e}}^{ - {\text{i}}\chi }}B $$ + {{\text{e}}^{{\text{i}}\chi }}C$, 进行傅里叶变换可以得到${L_\chi }$的准确形式.

      在低频限制下, $F\left( \chi \right) = - \lambda \left( \chi \right)t$, $\lambda \left( \chi \right)$${L_\chi }$的本征值, 而且当$\chi \to 0$, $\lambda \left( \chi \right) \to 0$, 这样, 经过一系列的推导可获得

      $ \lambda \left( \chi \right) = \frac{1}{t}\sum\limits_{k = 1}^\infty {{C_k}} \frac{{{{\left( {{\text{i}}\chi } \right)}^k}}}{{k!}} . $

      将方程(6)代入久期方程$\left| {{L_\chi } - \lambda \left( \chi \right)I} \right| = 0$, 然后将${\left( {{\text{i}}\chi } \right)^k}$展开获得第k阶累积矩

      $ {C_k} = - {\left( { - {\text{i}}\frac{\partial }{{\partial \chi }}} \right)^k}F\left( \chi \right)\left| {_{\chi \to 0}} \right. $

      平均电流$\left\langle I \right\rangle = {{e{C_1}}}/{t}$与第一阶累积矩有关, 散粒噪声$ {{2{e^2}{C_2}}}/{t}$和第二阶累积矩有关, 偏斜与第三阶累积矩${C_3}$有关, 描述分布的不对称性[64].

      通常情况下, 散粒噪声可以用${F_{\text{a}}} = {C_2}/{C_1}$来描述, ${F_{\text{a}}} > 1$对应的是超泊松散粒噪声, 而$F_{a}<1$对应的则是次泊松散粒噪声. 偏斜度用${S_{\text{k}}} = {C_3}/{C_1}$表示.

      本文以系统的9个本征态为基矢, 运用量子主方程的方法求解电流、散粒噪声和偏斜, 考虑量子点体系的非对角元项, 可获得一个$33 \times 33$的矩阵, 具体的矩阵元表达式见附录.

    • 本文运用全计数统计的方法探索自旋轨道耦合量子点系统中的量子相干效应. 因为量子点密度矩阵元的表达式比较复杂, 无法直接给出累积矩的表达式, 本文将给出数值化的结果分析. 全文提到的所有量的单位都是meV.

      平均电流$\left\langle I \right\rangle $, 散粒噪声${F_{\text{a}}}$以及偏斜${S_{\text{k}}}$运用全计数统计的方法可以获得. 磁通数$\zeta $反映在非对角元上, 两体相互作用与非对角元密切相关[65,66]. 从系统哈密顿量可知自旋轨道耦合项连接了两体作用, 进而可以诱导量子相干.

      图1可以看到, 平均电流$\left\langle I \right\rangle $(图1(a)), 散粒噪声${F_{\text{a}}}$(图1(b))和偏斜${S_{\text{k}}}$(图1(c))随磁通$\zeta $周期性波动, 振荡周期为0.5.黑色实线、红色虚线和蓝色点线分别为不同自旋轨道作用(${\alpha _{{\text{SO}}}} = 0.3$, 0.6, 0.9)下电流, 噪声和偏斜的波动图. 从图1(a)图1(b)可以看到, ${\alpha _{{\text{SO}}}} = 0.3$时, 电流波动图的峰值为$\left\langle I \right\rangle = $$ 0.74$, 噪声波动图的峰值为${F_{\text{a}}} = 3.17$; ${\alpha _{{\text{SO}}}} = 0.6$时, 电流波动图的峰值为$\left\langle I \right\rangle = 0.81$, 噪声峰值为${F_{\text{a}}} = $$ 2.56$; ${\alpha _{{\text{SO}}}} = 0.9$时, 电流波动图的峰值为$\left\langle I \right\rangle = 0.87$, 噪声峰值为${F_{\text{a}}} = 2.08$. 很明显, 随着自旋轨道耦合常数的增加, 电流在增大, 而散粒噪声的值在减小. 这是由自旋轨道耦合作用诱导的自旋反演引起的. 自旋轨道耦合为电子的隧穿提供了一个新路径. 隧穿进发射电极的电子隧穿出量子点到达收集电极将改变它的自旋极化, 而且根据隧穿率公式${\varGamma _{{\text{L}} \uparrow \left( \downarrow \right)}} = {\varGamma _{\text{L}}}\left( {1 \pm p} \right)$${\varGamma _{{\text{R}} \uparrow \left( \downarrow \right)}} = {\varGamma _{\text{R}}}\left( {1 \pm p} \right)$可以看出, 自旋向上的电子隧穿比自旋向下的电子隧穿快, 由于这种隧穿的不平衡, 散粒噪声值将减小[3]. 散粒噪声和偏斜的波谷值随自旋轨道耦合强度的增加有下降的趋势, 这是由动力学自旋阻塞引起的. 但是, 可以发现自旋轨道耦合常数的大小并不影响振荡周期. 另外, 从图1(c)可以看到偏斜${S_{\text{k}}}$的值是从负值变到正值.

      图  1  自旋轨道耦合常数${\alpha _{{\text{SO}}}}$不同时, (a)平均电流$\left\langle I \right\rangle $, (b)散粒噪声${F_{\text{a}}}$和(c)偏斜${S_{\text{k}}}$随磁通$\zeta $振荡图. ${\varepsilon }_{1}=1, $$ {\varepsilon }_{2}=3, {\varGamma }_{\text{L}}={\varGamma }_{\text{R}}=0.01, p=0.1$

      Figure 1.  (a) Average current$\left\langle I \right\rangle $, (b) shot noise${F_{\text{a}}}$and (c) skewness${S_{\text{k}}}$ fluctuation diagram in different spin-orbit coupling strength${\alpha _{{\text{SO}}}}$. ${\varepsilon }_{1}=1, {\varepsilon }_{2}=3, {\varGamma }_{\text{L}}={\varGamma }_{\text{R}}=0.01, $$ p=\mathrm{0.1}$

      图2描述的是非对称动力耦合中的量子相干. 黑色实线、红色虚线和蓝色点线分别为$ {\varGamma _R} = {\varGamma _{\text{L}}} $, ${\varGamma _{\text{R}}} = 0.6{\varGamma _{\text{L}}}$${\varGamma _{\text{L}}} = 0.6{\varGamma _{\text{R}}}$下电流、噪声和偏斜随磁通$\zeta $的波动图. 可以看到, 量子点与电极的耦合不对称不影响振荡周期, 振荡周期依然为0.5. 但是, 波动图中一个周期内多了一个小波谷, 当${\varGamma _{\text{R}}} = $$ 0.6{\varGamma _{\text{L}}}$, 新增的噪声波谷值${F_{\text{a}}} = 3.32$, 当${\varGamma _{\text{L}}} = $$ 0.6{\varGamma _{\text{R}}}$, 新增的噪声波谷值${F_{\text{a}}} = 3.43$, 这是因为${\varGamma _{\text{R}}}/{\varGamma _{\text{L}}}$的增加可能导致${\varGamma _{1 L}} > {\varGamma _{2{\text{L}}}}$, ${\varGamma _{{\text{1 R}}}} \gg {\varGamma _{{\text{2 R}}}}$, 有效的快慢通道被发展产生了聚束效应. 另外, 从图2(c)可以看到偏斜${S_{\text{k}}}$的值在$\dfrac{{2 n + 1}}{2}T\left( {n = 0, 1, 2 \cdots } \right)$处会出现一个小波峰.

      图  2  非对称动力耦合下, (a)平均电流$\left\langle I \right\rangle $, (b)散粒噪声${F_{\text{a}}}$和(c)偏斜${S_{\text{k}}}$随磁通$\zeta $振荡图. ${\varepsilon }_{1}=1,\; {\varepsilon }_{2}=3, $$ \; {\alpha }_{\text{SO}}=0.3,\; p=0.1$

      Figure 2.  (a) Average current$\left\langle I \right\rangle $, (b) shot noise${F_{\text{a}}}$ and (c) skewness ${S_{\text{k}}}$ fluctuation diagram for asymmetric dot-electrode coupling. ${\varepsilon }_{1}=1,\; {\varepsilon }_{2}=3,\; {\alpha }_{\text{SO}}=0.3,\; p=\mathrm{0.1}$

      图3给出了不同自旋极化率下的量子相干, 黑色实线、红色虚线和蓝色点线分别为 $p = 0.1$, $p = 0.5$$p = 0.9$情况下电流、噪声和偏斜随磁通$\zeta $的波动图. 振荡周期不随自旋极化率的变化而改变, 周期仍然为0.5. 但是, 很明显, 随着自旋极化率$p$的增大, 散粒噪声${F_a}$的值在明显增大, 这从图3(b)可以看出, 当极化率$p = 0.1$, 噪声峰值${F_{\text{a}}} = $$ 3.17$; 当极化率$p = 0.5$, 噪声峰值${F_{\text{a}}} = 3.69$; 当极化率$p = 0.9$, 噪声峰值${F_{\text{a}}} = 4.19$. 极化率的增大使得自旋向上电子的隧穿率增加, 而自旋向下电子的隧穿率减小, 这可以根据

      图  3  自旋极化率$p$不同时, (a)平均电流$\left\langle I \right\rangle $, (b)散粒噪声${F_{\text{a}}}$和(c)偏斜${S_{\text{k}}}$随磁通$\zeta $振荡图. ${\varepsilon }_{1}=1, \;{\varepsilon }_{2}=3, $$ \;{\alpha }_{\text{SO}}=0.3,\; {\varGamma }_{\text{L}}={\varGamma }_{\text{R}}=0.01$

      Figure 3.  (a) Average current$\left\langle I \right\rangle $, (b) shot noise${F_{\text{a}}}$ and (c) skewness${S_{\text{k}}}$ with magnetic flux oscillation with different spin polarization$p$.${\varepsilon }_{1}=1,\; {\varepsilon }_{2}=3,\; {\alpha }_{\text{SO}}=0.3,\; {\varGamma }_{\text{L}}={\varGamma }_{\text{R}}= $$ \mathrm{0.01}$

      $ \dfrac{{\varGamma _{\text{L}}^ \uparrow }}{{\varGamma _{\text{L}}^ \downarrow }} = \dfrac{{1 + p}}{{1 - p}}, \qquad \dfrac{{\varGamma _{\text{R}}^ \uparrow }}{{\varGamma _{\text{R}}^ \downarrow }} = \dfrac{{1 + p}}{{1 - p}} $

      来解释($p > 0$), 并且在文献[3]中可以直观地看到隧穿率与自旋极化率的这个关系, 这诱导了自旋向上电子与自旋向下电子隧穿过程的竞争, 导致自旋聚束效应和明显的超泊松噪声.

    • 本文采用量子主方程方法研究了量子点体系中自旋轨道耦合作用引起的量子相干效应. 研究发现, 当存在自旋轨道作用时, 电流、散粒噪声和偏斜随磁通周期性波动. 自旋轨道作用诱导量子相干产生, 但是自旋轨道耦合的大小不影响振荡周期. 此外, 动力学耦合不对称和自旋极化率的变化均不影响振荡周期. 动力学耦合不对称会使波动图多一个波谷, 这与快慢输运通道的竞争有关. 而自旋极化率的增加会使波动图的峰值增大, 超泊松行为明显, 这是因为自旋向上的电子与自旋向下的电子在隧穿过程中竞争而引起的自旋聚束效应. 通过测量累积矩可以探索系统中的自旋轨道耦合强度, 这将对与自旋有关的器件设计有很重要的科学意义. 由于本文主要研究零频累积矩, 接下来的工作将主要通过全计数统计方法计算有限频累积矩, 这将对整个系统的输运特性有更全面和深入的认识及了解.

    • 系统主方程矩阵元如下$. $ 其中对角元项为

      $ \begin{split} \dot \rho _{11}^{\left( n \right)} =\;& \left\langle {{\psi _{00}}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{00}}} \right\rangle = - \frac{1}{2}({a_1}{g_1}\varGamma _{{\text{L}}11}^ \uparrow + {m_1}{a_3}\varGamma _{{\text{L}}11}^ \uparrow + {h_3}{c_2}\varGamma _{{\text{L}}22}^ \uparrow + {n_3}{c_4}\varGamma _{{\text{L}}22}^ \uparrow + {h_2}{b_2}\varGamma _{{\text{L}}11}^ \downarrow + {n_2}{b_4}\varGamma _{{\text{L}}11}^ \downarrow \\ &+{g_4}{c_1}\varGamma _{{\text{L}}22}^ \downarrow + {m_4}{c_3}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{00}^{\left( n \right)} + \frac{1}{2}[({g_1}h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {g_4}h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{76}^{\left( {n - 1} \right)} \\ &+ ({m_1}g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + {m_4}g_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{68}^{\left( {n - 1} \right)} + ({g_1}n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {g_4}n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{96}^{\left( {n - 1} \right)} \\ &+({m_1}m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + {m_4}m_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{88}^{\left( {n - 1} \right)} + ({m_1}h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {m_4}h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{78}^{\left( {n - 1} \right)} \\ &+({h_3}h_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {h_2}h_2^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{77}^{\left( {n - 1} \right)} + ({m_1}n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {m_4}n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{98}^{\left( {n - 1} \right)}\\ &+ ({h_3}n_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {h_2}n_2^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{97}^{\left( {n - 1} \right)} + ({h_3}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + {h_2}g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{67}^{\left( {n - 1} \right)} \\ &+ ({n_3}h_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {n_2}h_2^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{79}^{\left( {n - 1} \right)} + ({h_3}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + {h_2}m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{87}^{\left( {n - 1} \right)} \\ &+({g_1}g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + {g_4}g_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{66}^{\left( {n - 1} \right)} + ({n_3}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + {n_2}g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{69}^{\left( {n - 1} \right)}\\ &+({g_1}m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + {g_4}m_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{86}^{\left( {n - 1} \right)} + ({n_3}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + {n_2}m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{89}^{\left( {n - 1} \right)} \\ &+ ({n_3}n_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {n_2}n_2^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{99}^{\left( {n - 1} \right)}] + {\text{H}}{\text{.c}}{\text{. }} \\ \dot \rho _{22}^{\left( n \right)} =\;& \left\langle {{\psi _{ \uparrow \uparrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \uparrow \uparrow }}} \right\rangle = - \frac{1}{2}(h_3^ * c_2^ * \varGamma _{{\text{R}}11}^ \uparrow + n_3^ * c_4^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \uparrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{22}^{\left( n \right)} \\ &+\frac{1}{2}[(h_3^ * b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + n_3^ * b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{42}^{\left( n \right)} + (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{32}^{\left( n \right)} \\ &+h_3^ * {h_3}\varGamma _{{\text{L}}11}^ \uparrow \rho _{77}^{\left( n \right)} + h_3^ * {n_3}\varGamma _{{\text{L}}11}^ \uparrow \rho _{79}^{\left( n \right)} - h_3^ * {g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{76}^{\left( n \right)} - h_3^ * {m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{78}^{\left( n \right)} \\ &- g_1^ * {h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{67}^{\left( n \right)} - g_1^ * {n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{69}^{\left( n \right)} + g_1^ * {g_1}\varGamma _{{\text{L}}22}^ \uparrow \rho _{66}^{\left( n \right)} + g_1^ * {m_1}\varGamma _{{\text{L}}22}^ \uparrow \rho _{68}^{\left( n \right)} \\ &+ n_3^ * {h_3}\varGamma _{{\text{L}}11}^ \uparrow \rho _{97}^{\left( n \right)} + n_3^ * {n_3}\varGamma _{{\text{L}}11}^ \uparrow \rho _{99}^{\left( n \right)} - n_3^ * {g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{96}^{\left( n \right)} - n_3^ * {m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{98}^{\left( n \right)} \\ &- m_1^ * {h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{87}^{\left( n \right)} - m_1^ * {n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{89}^{\left( n \right)} + m_1^ * {g_1}\varGamma _{{\text{L}}22}^ \uparrow \rho _{86}^{\left( n \right)} + m_1^ * {m_1}\varGamma _{{\text{L}}22}^ \uparrow \rho _{88}^{\left( n \right)}] + {\text{H}}{\text{.c}}{\text{. }} \\ \dot \rho _{33}^{\left( n \right)} = \;&\left\langle {{\psi _{ \uparrow \downarrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \uparrow \downarrow }}} \right\rangle = - \frac{1}{2}(g_4^ * c_1^ * \varGamma _{{\text{R}}11}^ \uparrow + m_4^ * c_3^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \downarrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{33}^{\left( n \right)} \\ &+\frac{1}{2}[(g_4^ * a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + m_4^ * a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{23}^{\left( n \right)} + (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{53}^{\left( n \right)} \\ &+ (g_4^ * {g_4}\varGamma _{{\text{L}}11}^ \uparrow + g_1^ * {g_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{66}^{\left( n \right)} + (g_4^ * {m_4}\varGamma _{{\text{L}}11}^ \uparrow + g_1^ * {m_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{68}^{\left( n \right)} \\ &+ (m_4^ * {g_4}\varGamma _{{\text{L}}11}^ \uparrow + m_1^ * {g_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{86}^{\left( n \right)} + (m_4^ * {m_4}\varGamma _{{\text{L}}11}^ \uparrow + m_1^ * {m_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{88}^{\left( n \right)}] + {\text{H}}{\text{.c}}{\text{. }} \end{split}$

      $ \begin{split} \dot \rho _{44}^{\left( n \right)} =\;& \left\langle {{\psi _{ \downarrow \uparrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \uparrow }}} \right\rangle = - \frac{1}{2}(h_3^ * c_2^ * \varGamma _{{\text{R}}11}^ \downarrow + n_3^ * c_4^ * \varGamma _{{\text{R}}11}^ \downarrow + h_2^ * b_2^ * \varGamma _{{\text{R}}22}^ \uparrow + n_2^ * b_4^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{44}^{\left( n \right)}\\ &+ \frac{1}{2}[(h_3^ * b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + n_3^ * b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{54}^{\left( n \right)} + (h_2^ * c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + n_2^ * c_4^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{24}^{\left( n \right)} \\ &+(h_3^ * {h_3}\varGamma _{{\text{L}}11}^ \downarrow + h_2^ * {h_2}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{77}^{\left( n \right)} + (h_3^ * {n_3}\varGamma _{{\text{L}}11}^ \downarrow + h_2^ * {n_2}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{79}^{\left( n \right)} \\ &+(n_3^ * {h_3}\varGamma _{{\text{L}}11}^ \downarrow + n_2^ * {h_2}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{97}^{\left( n \right)} + (n_3^ * {n_3}\varGamma _{{\text{L}}11}^ \downarrow + n_2^ * {n_2}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{99}^{\left( n \right)}] + {\text{H}}{\text{.c}}{\text{. }} \\ \dot \rho _{55}^{\left( n \right)} =\;& \left\langle {{\psi _{ \downarrow \downarrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \downarrow }}} \right\rangle = - \frac{1}{2}(g_4^ * c_1^ * \varGamma _{{\text{R}}11}^ \downarrow + m_3^ * c_3^ * \varGamma _{{\text{R}}11}^ \downarrow + h_2^ * b_2^ * \varGamma _{{\text{R}}22}^ \downarrow + n_2^ * b_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{55}^{\left( n \right)}\\ &+\frac{1}{2}[(g_4^ * a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + m_4^ * a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{35}^{\left( n \right)} + (h_2^ * c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + n_2^ * c_4^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{45}^{\left( n \right)}\\ &+ g_4^ * {g_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{66}^{\left( n \right)} + g_4^ * {m_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{68}^{\left( n \right)} - g_4^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{67}^{\left( n \right)} - g_4^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{69}^{\left( n \right)} \end{split} \qquad \qquad$

      $ \begin{split} &+m_4^ * {g_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{86}^{\left( n \right)} + m_4^ * {m_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{88}^{\left( n \right)} - m_4^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{87}^{\left( n \right)} - m_4^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{89}^{\left( n \right)} \\ &+ - h_2^ * {g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{76}^{\left( n \right)} - h_2^ * {m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{78}^{\left( n \right)} + h_2^ * {h_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{77}^{\left( n \right)} + h_2^ * {n_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{79}^{\left( n \right)}\\ &- n_2^ * {g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{96}^{\left( n \right)} - n_2^ * {m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{98}^{\left( n \right)} + n_2^ * {h_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{97}^{\left( n \right)} + n_2^ * {n_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{99}^{\left( n \right)}] + {\text{H}}{\text{.c}}{\text{. }}\\ \dot \rho _{66}^{\left( n \right)} =\;& \left\langle {{\psi _6}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _6}} \right\rangle = - \frac{1}{2}[(a_1^ * g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{R}}22}^ \downarrow + {a_1}{g_1}\varGamma _{{\text{L}}22}^ \uparrow + {a_1}{g_1}\varGamma _{{\text{L}}22}^ \downarrow + {c_1}{g_4}\varGamma _{{\text{L}}11}^ \uparrow + {c_1}{g_4}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{66}^{\left( n \right)} \\ &+ (a_1^ * m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * m_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{86}^{\left( n \right)} + (a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{76}^{\left( n \right)} + ({a_1}{m_1}\varGamma _{{\text{L}}22}^ \uparrow\\ &+ {a_1}{m_1}\varGamma _{{\text{L}}22}^ \downarrow + {c_1}{m_4}\varGamma _{{\text{L}}11}^ \uparrow + {c_1}{m_4}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{68}^{\left( n \right)} + (a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{96}^{\left( n \right)} \\ &- ({a_1}{h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {c_1}{h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{67}^{\left( n \right)}] + \frac{1}{2}[({a_1}{n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {c_1}{n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{69}^{\left( n \right)} \\ &+ (a_1^ * {a_1}\varGamma _{{\text{L}}11}^ \uparrow + c_1^ * {c_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{00}^{\left( n \right)} - {a_1}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} + {a_1}a_1^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} \\ &-{a_1}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{53}^{\left( {n - 1} \right)} + ({a_1}a_1^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_1}c_1^ * \varGamma _{{\text{R}}11}^ \uparrow )\rho _{33}^{\left( {n - 1} \right)} \\ &- {c_1}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{23}^{\left( {n - 1} \right)} + {c_1}c_1^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} - {c_1}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{35}^{\left( {n - 1} \right)}] + {\text{H}}{\text{.c}}{\text{. }} \end{split}$

      $ \begin{split}& {g_1} = \frac{{{c_3}}}{{{a_1}{c_3} - {a_3}{c_1}}} ,\;\; {g_4} = \frac{{ - {a_3}}}{{{a_1}{c_3} - {a_3}{c_1}}} ,\;\; {h_2} = \frac{{{c_4}}}{{{b_2}{c_4} - {b_4}{c_2}}} , \;\; {h_3} = \frac{{ - {b_4}}}{{{b_2}{c_4} - {b_4}{c_2}}} ,\;\; \\ &{m_1} = \frac{{ - {c_1}}}{{{a_1}{c_3} - {a_3}{c_1}}} , \;\;{m_4} = \frac{{{a_1}}}{{{a_1}{c_3} - {a_3}{c_1}}} , {n_2} = \frac{{ - {c_2}}}{{{b_2}{c_4} - {b_4}{c_2}}} , {n_3} = \frac{{{b_2}}}{{{b_2}{c_4} - {b_4}{c_2}}}. \end{split} \qquad \qquad \qquad \qquad \qquad $

      $ \begin{split} \dot \rho _{77}^{\left( n \right)} = \;&\left\langle {{\psi _7}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _7}} \right\rangle = - \frac{1}{2}[(b_2^ * h_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_2^ * h_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {b_2}{h_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_2}{h_2}\varGamma _{{\text{L}}22}^ \downarrow + {c_2}{h_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_2}{h_3}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{77}^{\left( n \right)} \\ &+(b_2^ * n_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_2^ * n_3^ * \varGamma _{{\text{R}}22}^ \uparrow )_{97}^{\left( n \right)} + (b_2^ * g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_2^ * g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{67}^{\left( n \right)} + ({b_2}{n_2}\varGamma _{{\text{L}}22}^ \uparrow \\ &+{b_2}{n_2}\varGamma _{{\text{L}}22}^ \downarrow + {c_2}{n_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_2}{n_3}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{79}^{\left( n \right)} - ({b_2}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow + {c_2}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow )\rho _{76}^{\left( n \right)} \\ &-({b_2}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow + {c_2}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow )\rho _{78}^{\left( n \right)} + (b_2^ * m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_2^ * m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{87}^{\left( n \right)}] \\ &+ \frac{1}{2}[(b_2^ * {b_2}\varGamma _{{\text{L}}11}^ \downarrow + c_2^ * {c_2}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{00}^{\left( n \right)} + ({b_2}b_2^ * \varGamma _{{\text{R}}22}^ \uparrow + {c_2}c_2^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{44}^{\left( {n - 1} \right)} - {b_2}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{45}^{\left( {n - 1} \right)} \\ &- {b_2}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} + {b_2}b_2^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} + {c_2}c_2^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} - {c_2}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{42}^{\left( {n - 1} \right)}\\ &- {c_2}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{54}^{\left( {n - 1} \right)}] + {\text{H}}{\text{.c}}{\text{. }} \\ \dot \rho _{88}^{\left( n \right)} =\;& \left\langle {{\psi _8}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _8}} \right\rangle = - \frac{1}{2}[(a_3^ * m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_3^ * m_4^ * \varGamma _{{\text{R}}22}^ \downarrow + {a_3}{m_1}\varGamma _{{\text{L}}22}^ \uparrow + {a_3}{m_1}\varGamma _{{\text{L}}22}^ \downarrow + {c_3}{m_4}\varGamma _{{\text{L}}11}^ \uparrow + {c_3}{m_4}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{88}^{\left( n \right)} \\ &+(a_3^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_3^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{78}^{\left( n \right)} + (a_3^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_3^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{98}^{\left( n \right)} \\ &-({a_3}{h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {c_3}{h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{87}^{\left( n \right)} - ({a_3}{n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {c_3}{n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{89}^{\left( n \right)}\\ &+(a_3^ * g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_3^ * g_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{68}^{\left( n \right)} + ({a_3}{g_1}\varGamma _{{\text{L}}22}^ \uparrow + {a_3}{g_1}\varGamma _{{\text{L}}22}^ \downarrow + {c_3}{g_4}\varGamma _{{\text{L}}11}^ \uparrow + {c_3}{g_4}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{86}^{\left( n \right)}] \\ &+ \frac{1}{2}[(a_3^ * {a_3}\varGamma _{{\text{L}}11}^ \uparrow + c_3^ * {c_3}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{00}^{\left( n \right)} - {a_3}c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} + {a_3}a_3^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{22}^{\left( {n - 1} \right)}\\ &- {a_3}c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{53}^{\left( {n - 1} \right)} + ({a_3}a_3^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_3}c_3^ * \varGamma _{{\text{R}}11}^ \uparrow )\rho _{33}^{\left( {n - 1} \right)} - {c_3}a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{23}^{\left( {n - 1} \right)} \\ &+ {c_3}c_3^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} - {c_3}a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{35}^{\left( {n - 1} \right)}] + {\text{H}}{\text{.c}}{\text{. }}\\ \dot \rho _{99}^{\left( n \right)} = \;&\left\langle {{\psi _9}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _9}} \right\rangle = - \frac{1}{2}[(b_4^ * h_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_4^ * h_3^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{79}^{\left( n \right)} + ({b_4}{h_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \downarrow + {c_4}{h_3}\varGamma _{{\text{L}}11}^ \uparrow \\ &+ {c_4}{h_3}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{97}^{\left( n \right)} + (b_4^ * n_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_4^ * n_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \downarrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \uparrow \\ &+{c_4}{n_3}\varGamma _{{\text{L}}11}^ \downarrow )\rho _{99}^{\left( n \right)} + (b_4^ * g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_4^ * g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{69}^{\left( n \right)} + (b_4^ * m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \\ &+ c_4^ * m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{89}^{\left( n \right)} - ({b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow + {c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow )\rho _{96}^{\left( n \right)} - ({b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow\\ &+{c_4}{m_1}{{\text{e}}^{2i\pi \zeta }}\varGamma _{{\text{L}}12}^ \uparrow )\rho _{98}^{\left( n \right)}] + \frac{1}{2}[(b_4^ * {b_4}\varGamma _{{\text{L}}11}^ \downarrow + c_4^ * {c_4}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{00}^{\left( n \right)} - {b_4}c_4^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} \end{split} $

      $ \begin{split} &+ ({b_4}b_4^ * \varGamma _{{\text{R}}22}^ \uparrow + {c_4}c_4^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{44}^{\left( {n - 1} \right)} - {b_4}c_4^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{45}^{\left( {n - 1} \right)} + {b_4}b_4^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} \\ &+{c_4}c_4^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} - {c_4}b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{42}^{\left( {n - 1} \right)} - {c_4}b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{54}^{\left( {n - 1} \right)}] + {\text{H}}{\text{.c}}{\text{. }} \end{split} $

      非对角元项为

      $ \begin{split} \dot \rho _{23}^{\left( n \right)} = \;&\left\langle {{\psi _{ \uparrow \uparrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \uparrow \downarrow }}} \right\rangle = - \frac{1}{2}[(h_3^ * c_2^ * \varGamma _{{\text{R}}11}^ \uparrow + n_3^ * c_4^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \uparrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{23}^{\left( n \right)} - (h_3^ * b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \\ &+n_3^ * b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{43}^{\left( n \right)} - (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{33}^{\left( n \right)} + ({g_4}{c_1}\varGamma _{{\text{R}}11}^ \uparrow\\ &+{m_4}{c_3}\varGamma _{{\text{R}}11}^ \uparrow + {g_1}{a_1}\varGamma _{{\text{R}}22}^ \downarrow + {m_1}{a_3}\varGamma _{{\text{R}}22}^ \downarrow )\rho _{23}^{\left( n \right)} - ({g_4}{a_1}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {m_4}{a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{22}^{\left( n \right)} \\ &-({g_1}{c_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + {m_1}{c_3}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{25}^{\left( n \right)}] + \frac{1}{2}[h_3^ * {g_4}\varGamma _{{\text{L}}11}^ \uparrow \rho _{76}^{\left( n \right)} + h_3^ * {m_4}\varGamma _{{\text{L}}11}^ \uparrow \rho _{78}^{\left( n \right)}\\ &+ n_3^ * {g_4}\varGamma _{{\text{L}}11}^ \uparrow \rho _{96}^{\left( n \right)} + n_3^ * {m_4}\varGamma _{{\text{L}}11}^ \uparrow \rho _{98}^{\left( n \right)} - g_1^ * {g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{66}^{\left( n \right)} - g_1^ * {m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{68}^{\left( n \right)} \\ &- m_1^ * {g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{86}^{\left( n \right)} - m_1^ * {m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{88}^{\left( n \right)} + {g_4}h_3^ * \varGamma _{{\text{L}}11}^ \uparrow \rho _{76}^{\left( n \right)} + {m_4}h_3^ * \varGamma _{{\text{L}}11}^ \uparrow \rho _{78}^{\left( n \right)} \\ &+{g_4}n_3^ * \varGamma _{{\text{L}}11}^ \uparrow \rho _{96}^{\left( n \right)} + {m_4}n_3^ * \varGamma _{{\text{L}}11}^ \uparrow \rho _{98}^{\left( n \right)} - {g_4}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{66}^{\left( n \right)} - {m_4}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{68}^{\left( n \right)} \\ &-{g_4}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{86}^{\left( n \right)} - {m_4}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{88}^{\left( n \right)}]\\ \dot \rho _{67}^{\left( n \right)} =\;& \left\langle {{\psi _6}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _7}} \right\rangle = - \frac{1}{2}[(a_1^ * g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + {c_2}{h_3}\varGamma _{{\text{L}}11}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_2}{h_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_2}{h_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_2}{h_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{67}^{\left( n \right)} \\ &+(a_1^ * m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * m_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{87}^{\left( n \right)} + (a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{77}^{\left( n \right)} + ({c_2}{n_3}\varGamma _{{\text{L}}11}^ \uparrow \\ &+{c_2}{n_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_2}{n_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_2}{n_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{69}^{\left( n \right)} - ({c_2}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_2}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{66}^{\left( n \right)}\\ &- ({c_2}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_2}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{68}^{\left( n \right)} + (a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{97}^{\left( n \right)} \\ &+ ({b_2}{h_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_2}{h_3}\varGamma _{{\text{R}}22}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{L}}11}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_1^ * g_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_1^ * g_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{67}^{\left( n \right)} \\ &+({b_2}{n_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_2}{n_3}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{69}^{\left( n \right)} + ({b_2}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_2}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{66}^{\left( n \right)} + (c_1^ * m_4^ * \varGamma _{{\text{L}}11}^ \uparrow \\ &+ c_1^ * m_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_1^ * m_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_1^ * m_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{87}^{\left( n \right)} - (c_1^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{77}^{\left( n \right)} \\ &-(c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{97}^{\left( n \right)} + a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{97}^{\left( n \right)} + ({b_2}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_2}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{68}^{\left( n \right)}] \\ &+\frac{1}{2}[{c_2}c_1^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} - {c_2}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} + {c_2}c_1^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{54}^{\left( {n - 1} \right)} + {b_2}a_1^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{35}^{\left( {n - 1} \right)}\\ &-({c_2}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {b_2}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{34}^{\left( {n - 1} \right)} + {b_2}a_1^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} - {b_2}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{55}^{\left( {n - 1} \right)}\\ &+ (a_1^ * {c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + c_1^ * {b_2}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{00}^{\left( n \right)} + c_1^ * {c_2}\varGamma _{{\text{R}}11}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} - (c_1^ * {b_2}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \\ &+a_1^ * {c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{34}^{\left( {n - 1} \right)} + c_1^ * {c_2}\varGamma _{{\text{R}}11}^ \downarrow \rho _{54}^{\left( {n - 1} \right)} - c_1^ * {b_2}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} - a_1^ * {c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} \\ &+a_1^ * {b_2}\varGamma _{{\text{R}}22}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} + a_1^ * {b_2}\varGamma _{{\text{R}}22}^ \downarrow \rho _{35}^{\left( {n - 1} \right)} + ({b_2}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {c_2}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{00}^{\left( n \right)}] .\end{split} $

      $ \begin{split} \dot \rho _{24}^{\left( n \right)} = \;&\left\langle {{\psi _{ \uparrow \uparrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \uparrow }}} \right\rangle = - \frac{1}{2}[(h_3^ * c_2^ * \varGamma _{{\text{R}}11}^ \uparrow + n_3^ * c_4^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \uparrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{24}^{\left( n \right)} - (h_3^ * b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \\ &+ n_3^ * b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{44}^{\left( n \right)} - (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{34}^{\left( n \right)} + ({h_3}{c_2}\varGamma _{{\text{R}}11}^ \downarrow \\ &+ {n_3}{c_4}\varGamma _{{\text{R}}11}^ \downarrow + {h_2}{b_2}\varGamma _{{\text{R}}22}^ \uparrow + {n_2}{b_4}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{24}^{\left( n \right)} - ({h_3}{b_2}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {n_3}{b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{25}^{\left( n \right)}\\ &-({h_2}{c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + {n_2}{c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{22}^{\left( n \right)}] + \frac{1}{2}[ - h_3^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{77}^{\left( n \right)} \\ &- h_3^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{79}^{\left( n \right)} - n_3^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{97}^{\left( n \right)} - n_3^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \rho _{99}^{\left( n \right)} \\ &+ g_1^ * {h_2}\varGamma _{{\text{L}}22}^ \uparrow \rho _{67}^{\left( n \right)} + g_1^ * {n_2}\varGamma _{{\text{L}}22}^ \uparrow \rho _{69}^{\left( n \right)} + m_1^ * {h_2}\varGamma _{{\text{L}}22}^ \uparrow \rho _{87}^{\left( n \right)} + m_1^ * {n_2}\varGamma _{{\text{L}}22}^ \uparrow \rho _{89}^{\left( n \right)} \\ &-{h_2}h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{77}^{\left( n \right)} - {n_2}h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{79}^{\left( n \right)} - {h_2}n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{97}^{\left( n \right)} - {n_2}n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow \rho _{99}^{\left( n \right)}\\ &+\left( {{h_2}g_1^ * \varGamma _{{\text{L}}22}^ \uparrow \rho _{67}^{\left( n \right)} + {n_2}g_1^ * \varGamma _{{\text{L}}22}^ \uparrow \rho _{69}^{\left( n \right)} + {h_2}m_1^ * \varGamma _{{\text{L}}22}^ \uparrow \rho _{87}^{\left( n \right)} + {n_2}m_1^ * \varGamma _{{\text{L}}22}^ \uparrow \rho _{89}^{\left( n \right)}} \right)] \end{split} $

      $ \begin{split} \dot \rho _{68}^{\left( n \right)} =\;& \left\langle {{\psi _6}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _8}} \right\rangle = - \frac{1}{2}[(a_1^ * g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_3}{m_4}\varGamma _{L11}^ \uparrow + {c_3}{m_4}\varGamma _{L11}^ \downarrow + {a_3}{m_1}\varGamma _{L22}^ \uparrow + {a_3}{m_1}\varGamma _{L22}^ \downarrow )\rho _{68}^{\left( n \right)} \\ &+(a_1^ * m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * m_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{88}^{\left( n \right)} + (a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * h_2^ * {e^{ - 2i\pi \zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{78}^{\left( n \right)} + ({c_3}{g_4}\varGamma _{{\text{R}}11}^ \uparrow\\ &+ {c_3}{g_4}\varGamma _{{\text{R}}11}^ \downarrow + {a_3}{g_1}\varGamma _{{\text{L}}22}^ \uparrow + {a_3}{g_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{66}^{\left( n \right)} + (a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{98}^{\left( n \right)}\\ &-({c_3}{h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {a_3}{h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{67}^{\left( n \right)} + ({a_3}{m_1}\varGamma _{{\text{R}}11}^ \uparrow + {c_3}{m_4}\varGamma _{{\text{R}}22}^ \downarrow + c_1^ * g_4^ * \varGamma _{{\text{L}}11}^ \uparrow \\ &+ c_1^ * g_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_1^ * g_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_1^ * g_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{68}^{\left( n \right)} + ({a_3}{g_1}\varGamma _{{\text{R}}11}^ \uparrow + {c_3}{g_4}\varGamma _{{\text{R}}22}^ \downarrow )\rho _{66}^{\left( n \right)} \\ &+({a_3}{h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {c_3}{h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{67}^{\left( n \right)} + (c_1^ * m_4^ * \varGamma _{{\text{L}}11}^ \uparrow + c_1^ * m_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_1^ * m_1^ * \varGamma _{{\text{L}}22}^ \uparrow \\ &+ a_1^ * m_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{88}^{\left( n \right)} + ({a_3}{n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {c_3}{n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{69}^{\left( n \right)} - (c_1^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \\ &+a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{78}^{\left( n \right)}] + \frac{1}{2}[({c_3}{n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {a_3}{n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{69}^{\left( n \right)} + {c_3}c_1^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} \\ &-{c_3}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{35}^{\left( {n - 1} \right)} + (a_1^ * {a_3}\varGamma _{{\text{L}}11}^ \uparrow + c_1^ * {c_3}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{00}^{\left( n \right)} - {a_3}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} \\ &+ {a_3}a_1^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} - {a_3}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{53}^{\left( {n - 1} \right)} + ({a_3}a_1^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_3}c_1^ * \varGamma _{{\text{R}}11}^ \uparrow )\rho _{33}^{\left( {n - 1} \right)} \\ &-{c_3}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{23}^{\left( {n - 1} \right)} + (c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{98}^{\left( n \right)} + c_1^ * {c_3}\varGamma _{{\text{R}}11}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} \\ &-a_1^ * {c_3}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{35}^{\left( {n - 1} \right)} + ({a_3}a_1^ * \varGamma _{{\text{L}}11}^ \uparrow + {c_3}c_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{00}^{\left( n \right)} - c_1^ * {a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} \\ &+a_1^ * {a_3}\varGamma _{{\text{R}}22}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} - c_1^ * {a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{53}^{\left( {n - 1} \right)} + \left( {a_1^ * {a_3}\varGamma _{{\text{R}}22}^ \downarrow + c_1^ * {c_3}\varGamma _{{\text{R}}11}^ \uparrow } \right)\rho _{33}^{\left( {n - 1} \right)}\\ &- a_1^ * {c_3}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{23}^{\left( {n - 1} \right)}] \end{split} $

      $ \begin{split} \dot \rho _{25}^{\left( n \right)} = \;&\left\langle {{\psi _{ \uparrow \uparrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \downarrow }}} \right\rangle = - \frac{1}{2}[(h_3^ * c_2^ * \varGamma _{{\text{R}}11}^ \uparrow + n_3^ * c_4^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \uparrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {g_4}{c_1}\varGamma _{{\text{R}}11}^ \downarrow \\ &+ {m_3}{c_3}\varGamma _{{\text{R}}11}^ \downarrow + {h_2}{b_2}\varGamma _{{\text{R}}22}^ \downarrow + {n_2}{b_4}\varGamma _{{\text{R}}22}^ \downarrow )\rho _{25}^{\left( n \right)} - (h_3^ * b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \\ &+ n_3^ * b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{45}^{\left( n \right)} - (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{35}^{\left( n \right)} + ({g_4}{c_1}\varGamma _{{\text{R}}11}^ \downarrow \\ &+({g_4}{a_1}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {m_4}{a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{23}^{\left( n \right)} - ({h_2}{c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + {n_2}{c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{24}^{\left( n \right)}] \\ \dot \rho _{69}^{\left( n \right)} =\;& \left\langle {{\psi _6}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _9}} \right\rangle = - \frac{1}{2}[(a_1^ * g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{69}^{\left( n \right)} \\ &+ ({c_4}{h_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_4}{h_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{67}^{\left( n \right)} + (a_1^ * m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_1^ * m_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{89}^{\left( n \right)}\\ &+ (a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{79}^{\left( n \right)} + (a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{99}^{\left( n \right)} \\ &- ({c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{66}^{\left( n \right)} - ({c_4}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{68}^{\left( n \right)} \\ &+ (c_1^ * g_4^ * \varGamma _{{\text{L}}11}^ \uparrow + c_1^ * g_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_1^ * g_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_1^ * g_1^ * \varGamma _{{\text{L}}22}^ \downarrow + {b_4}{n_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_4}{n_3}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{69}^{\left( n \right)}\\ &+({b_4}{h_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_4}{h_3}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{67}^{\left( n \right)} + ({b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{66}^{\left( n \right)} + (c_1^ * m_4^ * \varGamma _{{\text{L}}11}^ \uparrow \\ &+c_1^ * m_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_1^ * m_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_1^ * m_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{89}^{\left( n \right)} + ({b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_4}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{68}^{\left( n \right)} \\ &-(c_1^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + a_1^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{79}^{\left( n \right)}] + \frac{1}{2}[{c_4}c_1^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} - {c_4}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} \\ &+ {c_4}c_1^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{54}^{\left( {n - 1} \right)} - ({c_4}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {b_4}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{34}^{\left( {n - 1} \right)} + {b_4}a_1^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} \\ &-{b_4}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} + {b_4}a_1^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{35}^{\left( {n - 1} \right)} + (a_1^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + c_1^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{00}^{\left( n \right)} \\ &+(c_1^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{97}^{\left( n \right)} + a_1^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{99}^{\left( n \right)} + ({b_4}c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {c_4}a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{00}^{\left( n \right)}\\ &+c_1^ * {c_4}\varGamma _{{\text{R}}11}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} - (c_1^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + a_1^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{34}^{\left( {n - 1} \right)} + c_1^ * {c_4}\varGamma _{{\text{R}}11}^ \downarrow \rho _{54}^{\left( {n - 1} \right)}\\ &- c_1^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} - a_1^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} + a_1^ * {b_4}\varGamma _{{\text{R}}22}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} + a_1^ * {b_4}\varGamma _{{\text{R}}22}^ \downarrow \rho _{35}^{\left( {n - 1} \right)}] \end{split} $

      $ \begin{split} \dot \rho _{32}^{\left( n \right)} =\;& {\left( {\dot \rho _{23}^{\left( n \right)}} \right)^ * },\;\;\dot \rho _{42}^{\left( n \right)} = {\left( {\dot \rho _{24}^{\left( n \right)}} \right)^ * },\;\;\dot \rho _{52}^{\left( n \right)} = {\left( {\dot \rho _{25}^{\left( n \right)}} \right)^ * }, \\ \dot \rho _{43}^{\left( n \right)} =\;& {\left( {\dot \rho _{34}^{\left( n \right)}} \right)^ * },\;\;\dot \rho _{53}^{\left( n \right)} = {\left( {\dot \rho _{35}^{\left( n \right)}} \right)^ * },\;\;\dot \rho _{54}^{\left( n \right)} = {\left( {\dot \rho _{45}^{\left( n \right)}} \right)^ * }.\\ \dot \rho _{34}^{\left( n \right)} = \;&\left\langle {{\psi _{ \uparrow \downarrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \uparrow }}} \right\rangle = - \frac{1}{2}[(g_4^ * c_1^ * \varGamma _{{\text{R}}11}^ \uparrow + m_4^ * c_3^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \downarrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{34}^{\left( n \right)} \\ &-(g_4^ * a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + m_4^ * a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{24}^{\left( n \right)} - (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{54}^{\left( n \right)} \\ &+({h_3}{c_2}\varGamma _{{\text{R}}11}^ \downarrow + {n_3}{c_4}\varGamma _{{\text{R}}11}^ \downarrow + {h_2}{b_2}\varGamma _{{\text{R}}22}^ \uparrow + {n_2}{b_4}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{34}^{\left( n \right)} - ({h_2}{c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \\ &+{n_2}{c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{32}^{\left( n \right)} - ({h_3}{b_2}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {n_3}{b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{35}^{\left( n \right)}] + \frac{1}{2}[ - (g_4^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \\ &+g_1^ * {h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{67}^{\left( n \right)} - (g_4^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + g_1^ * {n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{69}^{\left( n \right)} - (m_4^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \\ &+m_1^ * {h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{87}^{\left( n \right)} - (m_4^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + m_1^ * {n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{89}^{\left( n \right)} \\ &- ({h_2}g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {h_3}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{67}^{\left( n \right)} - ({n_2}g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {n_3}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{69}^{\left( n \right)}\\ &-({h_2}m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {h_3}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{87}^{\left( n \right)} - ({n_2}m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow + {n_3}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow )\rho _{89}^{\left( n \right)}] \\ \dot \rho _{78}^{\left( n \right)} =\;& \left\langle {{\psi _7}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _8}} \right\rangle = - \frac{1}{2}[(b_2^ * h_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_2^ * h_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {c_3}{m_4}\varGamma _{{\text{L}}11}^ \uparrow + {c_3}{m_4}\varGamma _{{\text{L}}11}^ \downarrow + {a_3}{m_1}\varGamma _{{\text{L}}22}^ \uparrow + {a_3}{m_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{78}^{\left( n \right)} \\ &+(b_2^ * n_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_2^ * n_3^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{98}^{\left( n \right)} + (b_2^ * g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_2^ * g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{68}^{\left( n \right)} + ({c_3}{g_4}\varGamma _{{\text{L}}11}^ \uparrow \\ &+{c_3}{g_4}\varGamma _{{\text{L}}11}^ \downarrow + {a_3}{g_1}\varGamma _{{\text{L}}22}^ \uparrow + {a_3}{g_1}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{76}^{\left( n \right)} + (b_2^ * m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_2^ * m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{88}^{\left( n \right)}\\ &-({c_3}{h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {a_3}{h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{77}^{\left( n \right)} + ({a_3}{m_1}\varGamma _{{\text{R}}11}^ \uparrow + {c_3}{m_4}\varGamma _{{\text{R}}22}^ \downarrow + c_2^ * h_3^ * \varGamma _{{\text{L}}11}^ \uparrow + c_2^ * h_3^ * \varGamma _{{\text{L}}11}^ \downarrow \\ &+ b_2^ * h_2^ * \varGamma _{{\text{L}}22}^ \uparrow + b_2^ * h_2^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{78}^{\left( n \right)} + ({a_3}{h_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {c_3}{h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{77}^{\left( n \right)} + ({a_3}{n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \\ &+ {c_3}{n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{79}^{\left( n \right)} + ({a_3}{g_1}\varGamma _{{\text{R}}11}^ \uparrow + {c_3}{g_4}\varGamma _{{\text{R}}22}^ \downarrow )\rho _{76}^{\left( n \right)} + (c_2^ * n_3^ * \varGamma _{{\text{L}}11}^ \uparrow + c_2^ * n_3^ * \varGamma _{{\text{L}}11}^ \downarrow + b_2^ * n_2^ * \varGamma _{{\text{L}}22}^ \uparrow \\ &+ b_2^ * n_2^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{98}^{\left( n \right)} - (c_2^ * g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + b_2^ * g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{68}^{\left( n \right)} - (c_2^ * m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow \\ &+b_2^ * m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{88}^{\left( n \right)}] + \frac{1}{2}[({c_3}{n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {a_3}{n_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{79}^{\left( n \right)}{c_3}c_2^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{23}^{\left( {n - 1} \right)}\\ &-({c_3}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + {a_3}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{43}^{\left( {n - 1} \right)} + {c_3}c_2^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{45}^{\left( {n - 1} \right)} - {c_3}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} \\ &- {a_3}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} + {a_3}b_2^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{42}^{\left( {n - 1} \right)} + {a_3}b_2^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{53}^{\left( {n - 1} \right)} + (b_2^ * {c_3}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \\ &+c_2^ * {a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{00}^{\left( n \right)} + c_2^ * {c_3}\varGamma _{{\text{R}}11}^ \uparrow \rho _{23}^{\left( {n - 1} \right)} - (b_2^ * {c_3}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + c_2^ * {a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{43}^{\left( {n - 1} \right)} \\ &+c_2^ * {c_3}\varGamma _{{\text{R}}11}^ \downarrow \rho _{45}^{\left( {n - 1} \right)} - b_2^ * {c_3}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} - c_2^ * {a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} + b_2^ * {a_3}\varGamma _{{\text{R}}22}^ \uparrow \rho _{42}^{\left( {n - 1} \right)} \\ &+ b_2^ * {a_3}\varGamma _{{\text{R}}22}^ \downarrow \rho _{53}^{\left( {n - 1} \right)} + ({c_3}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow + {a_3}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow )\rho _{00}^{\left( n \right)}] \end{split} $

      $ \begin{split} \dot \rho _{35}^{\left( n \right)} =\;& \left\langle {{\psi _{ \uparrow \downarrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \downarrow }}} \right\rangle = - \frac{1}{2}[(g_4^ * c_1^ * \varGamma _{{\text{R}}11}^ \uparrow + m_4^ * c_3^ * \varGamma _{{\text{R}}11}^ \uparrow + g_1^ * a_1^ * \varGamma _{{\text{R}}22}^ \downarrow + m_1^ * a_3^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{35}^{\left( n \right)} - (g_4^ * a_1^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \\ &+ m_4^ * a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow )\rho _{25}^{\left( n \right)} - (g_1^ * c_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + m_1^ * c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{55}^{\left( n \right)} + ({g_4}{c_1}\varGamma _{{\text{R}}11}^ \downarrow\\ &+ {m_4}{c_3}\varGamma _{{\text{R}}11}^ \downarrow + {h_2}{b_2}\varGamma _{{\text{R}}22}^ \downarrow + {n_2}{b_4}\varGamma _{{\text{R}}22}^ \downarrow )\rho _{35}^{\left( n \right)} - ({h_2}{c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + {n_2}{c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{34}^{\left( n \right)}\\ &-({g_4}{a_1}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {m_4}{a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{33}^{\left( n \right)}] + \frac{1}{2}[ - g_1^ * {g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{66}^{\left( n \right)}\\ &-g_1^ * {m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{68}^{\left( n \right)} + g_1^ * {h_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{67}^{\left( n \right)} + g_1^ * {n_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{69}^{\left( n \right)} - m_1^ * {g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{86}^{\left( n \right)}\\ &-m_1^ * {m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{88}^{\left( n \right)} + m_1^ * {h_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{87}^{\left( n \right)} + m_1^ * {n_2}\varGamma _{{\text{L}}22}^ \downarrow \rho _{89}^{\left( n \right)} - {g_4}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{66}^{\left( n \right)}\\ &-{m_4}g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{68}^{\left( n \right)} + {h_2}g_1^ * \varGamma _{{\text{L}}22}^ \downarrow \rho _{67}^{\left( n \right)} + {n_2}g_1^ * \varGamma _{{\text{L}}22}^ \downarrow \rho _{69}^{\left( n \right)} - {g_4}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{86}^{\left( n \right)} \\ &- {m_4}m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{88}^{\left( n \right)} + {h_2}m_1^ * \varGamma _{{\text{L}}22}^ \downarrow \rho _{87}^{\left( n \right)} + {n_2}m_1^ * \varGamma _{{\text{L}}22}^ \downarrow \rho _{89}^{\left( n \right)}] \end{split} $

      $ \begin{split} \dot \rho _{79}^{\left( n \right)} = \;&\left\langle {{\psi _7}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _9}} \right\rangle = - \frac{1}{2}[(b_2^ * h_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_2^ * h_3^ * \varGamma _{{\text{R}}22}^ \uparrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{79}^{\left( n \right)} \\ &+ (b_2^ * n_2^ * \varGamma _{{\text{R}}11}^ \downarrow + c_2^ * n_3^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{99}^{\left( n \right)} + (b_2^ * g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_2^ * g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{69}^{\left( n \right)} + ({c_4}{h_3}\varGamma _{{\text{L}}11}^ \uparrow \\ &+ {c_4}{h_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{77}^{\left( n \right)} + (b_2^ * m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + c_2^ * m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{89}^{\left( n \right)} \\ &-({c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{76}^{\left( n \right)} + ({b_4}{n_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_4}{n_3}\varGamma _{{\text{R}}22}^ \uparrow + c_2^ * h_3^ * \varGamma _{{\text{L}}11}^ \uparrow \\ &+c_2^ * h_3^ * \varGamma _{{\text{L}}11}^ \downarrow + b_2^ * h_2^ * \varGamma _{{\text{L}}22}^ \uparrow + b_2^ * h_2^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{79}^{\left( n \right)} + ({b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{76}^{\left( n \right)} \\ &+ ({b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_4}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{78}^{\left( n \right)} + ({b_4}{h_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_4}{h_3}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{77}^{\left( n \right)} + (c_2^ * n_3^ * \varGamma _{{\text{L}}11}^ \uparrow \\ &+c_2^ * n_3^ * \varGamma _{{\text{L}}11}^ \downarrow + b_2^ * n_2^ * \varGamma _{{\text{L}}22}^ \uparrow + b_2^ * n_2^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{99}^{\left( n \right)} - (c_2^ * g_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + b_2^ * g_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{69}^{\left( n \right)}\\ &-(c_2^ * m_1^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + b_2^ * m_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{89}^{\left( n \right)}] + \frac{1}{2}[({c_4}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{78}^{\left( n \right)}\\ &+ (b_2^ * {b_4}\varGamma _{{\text{L}}11}^ \downarrow + c_2^ * {c_4}\varGamma _{{\text{L}}22}^ \uparrow )\rho _{00}^{\left( n \right)} - {b_4}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} + {b_4}b_2^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} \\ &+ {c_4}c_2^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} - {b_4}c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{45}^{\left( {n - 1} \right)} - {c_4}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{42}^{\left( {n - 1} \right)} - {c_4}b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{54}^{\left( {n - 1} \right)}\\ &+ ({b_4}b_2^ * \varGamma _{{\text{R}}22}^ \uparrow + {c_4}c_2^ * \varGamma _{{\text{R}}11}^ \downarrow )\rho _{44}^{\left( {n - 1} \right)} + ({b_4}b_2^ * \varGamma _{{\text{L}}11}^ \downarrow + {c_4}c_2^ * \varGamma _{{\text{L}}22}^ \uparrow )\rho _{00}^{\left( n \right)} - c_2^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{24}^{\left( {n - 1} \right)}\\ &- c_2^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{45}^{\left( {n - 1} \right)} + b_2^ * {b_4}\varGamma _{{\text{R}}22}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} + c_2^ * {c_4}\varGamma _{{\text{R}}11}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} - b_2^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{42}^{\left( {n - 1} \right)} \\ &- b_2^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{54}^{\left( {n - 1} \right)} + (b_2^ * {b_4}\varGamma _{{\text{R}}22}^ \uparrow + c_2^ * {c_4}\varGamma _{{\text{R}}11}^ \downarrow )\rho _{44}^{\left( {n - 1} \right)}] \end{split} $

      $ \begin{split} \dot \rho _{45}^{\left( n \right)} = \;&\left\langle {{\psi _{ \downarrow \uparrow }}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _{ \downarrow \downarrow }}} \right\rangle = - \frac{1}{2}[(h_3^ * c_2^ * \varGamma _{{\text{R}}11}^ \downarrow + n_3^ * c_4^ * \varGamma _{{\text{R}}11}^ \downarrow + h_2^ * b_2^ * \varGamma _{{\text{R}}22}^ \uparrow + n_2^ * b_4^ * \varGamma _{{\text{R}}22}^ \uparrow )\rho _{45}^{\left( n \right)} - (h_3^ * b_2^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \\ &+ n_3^ * b_4^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{55}^{\left( n \right)} - (h_2^ * c_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow + n_2^ * c_4^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{25}^{\left( n \right)} + ({g_4}{c_1}\varGamma _{{\text{R}}11}^ \downarrow\\ &+ {m_4}{c_3}\varGamma _{{\text{R}}11}^ \downarrow + {h_2}{b_2}\varGamma _{{\text{R}}22}^ \downarrow + {n_2}{b_4}\varGamma _{{\text{R}}22}^ \downarrow )\rho _{45}^{\left( n \right)} - ({h_2}{c_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow + {n_2}{c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{44}^{\left( n \right)}\\ &- ({g_4}{a_1}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {m_4}{a_3}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow )\rho _{43}^{\left( n \right)}] + \frac{1}{2}[h_3^ * {g_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{76}^{\left( n \right)} + h_3^ * {m_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{78}^{\left( n \right)} \\ &- h_3^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{77}^{\left( n \right)} - h_3^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{79}^{\left( n \right)} + n_3^ * {g_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{96}^{\left( n \right)} + n_3^ * {m_4}\varGamma _{{\text{L}}11}^ \downarrow \rho _{98}^{\left( n \right)}\\ &- n_3^ * {h_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{97}^{\left( n \right)} - n_3^ * {n_2}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{99}^{\left( n \right)} + {g_4}h_3^ * \varGamma _{{\text{L}}11}^ \downarrow \rho _{76}^{\left( n \right)} + {m_4}h_3^ * \varGamma _{{\text{L}}11}^ \downarrow \rho _{78}^{\left( n \right)}\\ &- {h_2}h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{77}^{\left( n \right)} - {n_2}h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{79}^{\left( n \right)} + {g_4}n_3^ * \varGamma _{{\text{L}}11}^ \downarrow \rho _{96}^{\left( n \right)} + {m_4}n_3^ * \varGamma _{{\text{L}}11}^ \downarrow \rho _{98}^{\left( n \right)}\\ &-{h_2}n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{97}^{\left( n \right)} - {n_2}n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow \rho _{99}^{\left( n \right)}] \\ \dot \rho _{89}^{\left( n \right)} =\;& \left\langle {{\psi _8}} \right|{{\dot \rho }^{\left( n \right)}}\left| {{\psi _9}} \right\rangle = - \frac{1}{2}[(a_3^ * m_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_3^ * m_4^ * \varGamma _{{\text{R}}22}^ \downarrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_4}{n_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{n_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{89}^{\left( n \right)} \\ &+ ({c_4}{h_3}\varGamma _{{\text{L}}11}^ \uparrow + {c_4}{h_3}\varGamma _{{\text{L}}11}^ \downarrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \uparrow + {b_4}{h_2}\varGamma _{{\text{L}}22}^ \downarrow )\rho _{87}^{\left( n \right)} + (a_3^ * g_1^ * \varGamma _{{\text{R}}11}^ \uparrow + c_3^ * g_4^ * \varGamma _{{\text{R}}22}^ \downarrow )\rho _{69}^{\left( n \right)}\\ &+(a_3^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_3^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{79}^{\left( n \right)} + (a_3^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow + c_3^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{99}^{\left( n \right)} \\ &- ({c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{86}^{\left( n \right)} - ({c_4}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + {b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{88}^{\left( n \right)} \\ &+({b_4}{n_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_4}{n_3}\varGamma _{{\text{R}}22}^ \uparrow + c_3^ * m_4^ * \varGamma _{{\text{L}}11}^ \uparrow + c_3^ * m_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_3^ * m_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_3^ * m_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{89}^{\left( n \right)}\\ &+ (c_3^ * g_4^ * \varGamma _{{\text{L}}11}^ \uparrow + c_3^ * g_4^ * \varGamma _{{\text{L}}11}^ \downarrow + a_3^ * g_1^ * \varGamma _{{\text{L}}22}^ \uparrow + a_3^ * g_1^ * \varGamma _{{\text{L}}22}^ \downarrow )\rho _{69}^{\left( n \right)} + ({b_4}{h_2}\varGamma _{{\text{R}}11}^ \downarrow + {c_4}{h_3}\varGamma _{{\text{R}}22}^ \uparrow )\rho _{87}^{\left( n \right)}\\ &+({b_4}{g_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_4}{g_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{86}^{\left( n \right)} + ({b_4}{m_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow + {c_4}{m_1}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{88}^{\left( n \right)} \\ &- (c_3^ * h_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + a_3^ * h_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{79}^{\left( n \right)} - (c_3^ * n_2^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow \rho _{97}^{\left( n \right)} + a_3^ * n_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{99}^{\left( n \right)}] \\ &+ \frac{1}{2}[{c_4}c_3^ * \varGamma _{{\text{R}}11}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} - {c_4}a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} + {c_4}c_3^ * \varGamma _{{\text{R}}11}^ \downarrow \rho _{54}^{\left( {n - 1} \right)} - ({c_4}a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \\ &+ {b_4}c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow )\rho _{34}^{\left( {n - 1} \right)} + {b_4}a_3^ * \varGamma _{{\text{R}}22}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} - {b_4}c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} + {b_4}a_3^ * \varGamma _{{\text{R}}22}^ \downarrow \rho _{35}^{\left( {n - 1} \right)} \\ &+(a_3^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \uparrow + c_3^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \downarrow )\rho _{00}^{\left( n \right)} + c_3^ * {c_4}\varGamma _{{\text{R}}11}^ \uparrow \rho _{32}^{\left( {n - 1} \right)} - (c_3^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \uparrow \end{split} $

      $ \begin{split} &+ a_3^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \downarrow )\rho _{34}^{\left( {n - 1} \right)} + c_3^ * {c_4}\varGamma _{{\text{R}}11}^ \downarrow \rho _{54}^{\left( {n - 1} \right)} - c_3^ * {b_4}{{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}12}^ \downarrow \rho _{55}^{\left( {n - 1} \right)} - a_3^ * {c_4}{{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{R}}21}^ \uparrow \rho _{22}^{\left( {n - 1} \right)} \\ &+ a_3^ * {b_4}\varGamma _{{\text{R}}22}^ \uparrow \rho _{24}^{\left( {n - 1} \right)} + a_3^ * {b_4}\varGamma _{{\text{R}}22}^ \downarrow \rho _{35}^{\left( {n - 1} \right)} + ({b_4}c_3^ * {{\text{e}}^{ - 2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}12}^ \downarrow + {c_4}a_3^ * {{\text{e}}^{2{\text{i\pi }}\zeta }}\varGamma _{{\text{L}}21}^ \uparrow )\rho _{00}^{\left( n \right)}]\\ \dot \rho _{76}^{\left( n \right)} =\;& {\left( {\dot \rho _{67}^{\left( n \right)}} \right)^ * },\dot \rho _{86}^{\left( n \right)} = {\left( {\dot \rho _{68}^{\left( n \right)}} \right)^ * },\dot \rho _{96}^{\left( n \right)} = {\left( {\dot \rho _{69}^{\left( n \right)}} \right)^ * }, \\ \dot \rho _{87}^{\left( n \right)} =\;& {\left( {\dot \rho _{78}^{\left( n \right)}} \right)^ * },\dot \rho _{97}^{\left( n \right)} = {\left( {\dot \rho _{79}^{\left( n \right)}} \right)^ * },\dot \rho _{98}^{\left( n \right)} = {\left( {\dot \rho _{89}^{\left( n \right)}} \right)^ * }. \end{split} $

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