Dynamics and protection of quantum correlations in a qubit–qutrit system subjected locally to a classical random field and colored noise

Abstract

This paper investigates the evolution and protection of quantum correlations (entanglement and quantum discord (QD) in particular), in a hybrid system formed by a qubit subjected to an external random field and a qutrit subjected to a colored noise generated either by a single or collection of many random bistable fluctuators (RBFs). Two different input states namely the one- and two-parameter class of states of qubit–qutrit are investigated. Entanglement and QD are quantified by means of negativity and geometric QD and their protection investigated by recourse to the weak measurement (WM) and weak measurement reversal (WMR) technique. It is shown that the immunity of entanglement and QD against decoherence can be efficiently increased by properly adjusting the parameters of the input states, no matter the spectrum of the colored noise and the number of RBFs considered. Moreover, it is shown that the WM and WMR technique fails in protecting entanglement in the studied system, but can effectively protect QD from the detrimental impacts of the decoherence. In fact, it is shown that performing a WM followed by a WMR on the qubit of the system allows to shield the QD of the system from decoherence even when the decoherence process is strengthen by considering a large number of RBFs affecting the qutrit.

Appendices

Appendix A: time-evolved density matrices of the system

When the system is initially prepared in the one-parameter class of states of qubit–qutrit, its final state after interacting with the noises takes the following form:

$$\begin{aligned} \rho ^{1p} \left( t\right)= & {} {\mathcal {Y}}\left( t\right) \left( {\left| 00 \right\rangle } {\left\langle 02 \right| } +{\left| 02 \right\rangle } {\left\langle 00 \right| } +{\left| 10 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 10 \right| } \right) \nonumber \\&+\, {\mathcal {E}}\left( t\right) \left( {\left| 00 \right\rangle } {\left\langle 10 \right| } +{\left| 10 \right\rangle } {\left\langle 00 \right| } +{\left| 02 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 02 \right| } \right) \nonumber \\&+\, {\mathcal {A}}\left( t\right) \left( {\left| 00 \right\rangle } {\left\langle 00 \right| } +{\left| 12 \right\rangle } {\left\langle 12 \right| } \right) +{\mathcal {B}}\left( t\right) \left( {\left| 01 \right\rangle } {\left\langle 01 \right| } +{\left| 11 \right\rangle } {\left\langle 11 \right| } \right) \nonumber \\&+\, {\mathcal {C}}\left( t\right) \left( {\left| 02 \right\rangle } {\left\langle 02 \right| } +{\left| 10 \right\rangle } {\left\langle 10 \right| } \right) +{\mathcal {F}}\left( t\right) \left( {\left| 00 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 00 \right| } \right) \nonumber \\&+\, {\mathcal {J}}\left( t\right) \left( {\left| 01 \right\rangle } {\left\langle 11 \right| } +{\left| 11 \right\rangle } {\left\langle 01 \right| } \right) +{\mathcal {N}}\left( t\right) \left( {\left| 02 \right\rangle } {\left\langle 10 \right| } +{\left| 10 \right\rangle } {\left\langle 02 \right| } \right) , \end{aligned}$$

(A1)

with

$$\begin{aligned} {\mathcal {A}}\left( t\right)= & {} \frac{1}{4} \left( 3p-1\right) \vartheta \left( t\right) \xi \left( t\right) +\frac{\left( 1-3p\right) }{16} \vartheta _{2} \left( t\right) +\frac{1}{16} \left( 3-p\right) ,\\ {\mathcal {F}}\left( t\right)= & {} \frac{\left( 1-p\right) }{16} \vartheta _{2} \left( t\right) \xi \left( t\right) +\frac{\left( 3p-1\right) }{4} \vartheta \left( t\right) +\frac{3\left( 1-p\right) }{16} \xi \left( t\right) ,\\ {\mathcal {E}}\left( t\right)= & {} \left( 1-p\right) \frac{\xi \left( t\right) \left( \vartheta _{2} \left( t\right) -1\right) }{16}; {\mathcal {Y}}\left( t\right) =\frac{\left( 3p-1\right) }{16} \left( 1-\vartheta _{2} \left( t\right) \right) ,\\ {\mathcal {N}}\left( t\right)= & {} \frac{\left( 1-3p\right) }{4} \vartheta \left( t\right) +\frac{3\left( 1-p\right) }{16} \xi \left( t\right) +\frac{\left( 1-p\right) }{16} \vartheta _{2} \left( t\right) \xi \left( t\right) ,\\ {\mathcal {C}}\left( t\right)= & {} \frac{\left( 1-3p\right) }{4} \vartheta \left( t\right) \xi \left( t\right) +\frac{\left( 1-3p\right) }{16} \vartheta _{2} \left( t\right) +\frac{\left( 3-p\right) }{16},\\ {\mathcal {B}}\left( t\right)= & {} \frac{\left( 3p-1\right) }{8} \vartheta _{2} \left( t\right) +\frac{1}{8} \left( p+1\right) ,\\ {\mathcal {J}}(t)= & {} \frac{\left( p-1\right) }{8} \left( \vartheta _{2}\left( t\right) -1\right) \xi (t). \end{aligned}$$

In the case of two-parameter class of states the following form is obtained:

$$\begin{aligned} \rho ^{2p} \left( t\right)= & {} {\mathcal {A}}\left( t\right) {\left| 00 \right\rangle } {\left\langle 00 \right| } +{\mathcal {B}}\left( t\right) {\left| 01 \right\rangle } {\left\langle 01 \right| } +{\mathcal {C}}\left( t\right) {\left| 02 \right\rangle } {\left\langle 02 \right| }\nonumber \\&+\, {\mathcal {Y}}\left( t\right) {\left| 10 \right\rangle } {\left\langle 10 \right| } +{\mathcal {E}}\left( t\right) +{\left| 11 \right\rangle } {\left\langle 11 \right| } +{\mathcal {F}}\left( t\right) {\left| 12 \right\rangle } {\left\langle 12 \right| }\nonumber \\&+\,{\mathcal {G}}\left( t\right) \left( {\left| 00 \right\rangle } {\left\langle 02 \right| } +{\left| 02 \right\rangle } {\left\langle 00 \right| } \right) +{\mathcal {I}}\left( t\right) \left( {\left| 00 \right\rangle } {\left\langle 11 \right| } +{\left| 11 \right\rangle } {\left\langle 00 \right| } \right) \nonumber \\&+\, {\mathcal {J}}\left( t\right) \left( {\left| 01 \right\rangle } {\left\langle 10 \right| } +{\left| 10 \right\rangle } {\left\langle 01 \right| } \right) +{\mathcal {K}}\left( t\right) \left( {\left| 01 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 01 \right| } \right) \nonumber \\&+\,{\mathcal {L}}\left( t\right) \left( {\left| 02 \right\rangle } {\left\langle 11 \right| } +{\left| 11 \right\rangle } {\left\langle 02 \right| } \right) +{\mathcal {N}}\left( t\right) \left( {\left| 10 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 10 \right| } \right) , \end{aligned}$$

(A2)

with

$$\begin{aligned} {\mathcal {F}}\left( t\right)= & {} \frac{1}{32} \Big ( 4\alpha \left( 3+4\vartheta \left( t\right) +\vartheta _{2} \left( t\right) \right) - \left( -5+4\vartheta \left( t\right) +\vartheta _{2} \left( t\right) \right) \times \\&\times \,\left( 3\beta +\mu \right) + \left( -1+4\vartheta \left( t\right) -3\vartheta _{2} \left( t\right) \right) \left( \beta -\mu \right) \xi \left( t\right) \Big ),\\ {\mathcal {G}}\left( t\right)= & {} \frac{1}{32} \left( \vartheta \left( t\right) _{2} -1\right) \left( 4\alpha +3\beta \left( \xi \left( t\right) -1\right) -\mu \left( 1+3\xi \left( t\right) \right) \right) ,\\ {\mathcal {A}}\left( t\right)= & {} \frac{1}{32} \Big ( 4\alpha \left( 3-4\vartheta \left( t\right) +\vartheta _{2} \left( t\right) \right) +\left( 5+4\vartheta \left( t\right) -\vartheta _{2} \left( t\right) \right) \times \\&\times \,\left( 3\beta +\mu \right) +\left( 1+4\vartheta \left( t\right) +3\vartheta _{2} \left( t\right) \right) \left( \beta -\mu \right) \xi \left( t\right) \Big ),\\ {\mathcal {B}}\left( t\right)= & {} \frac{1}{16} \Big ( -4\alpha \left( \vartheta _{2} \left( t\right) -1\right) +\left( 3+\vartheta _{2} \left( t\right) \right) \left( 3\beta +\mu \right) -\\&\,\left( 1+3\vartheta _{2} \left( t\right) \right) \left( \beta -\alpha \right) \xi \left( t\right) \Big ),\\ {\mathcal {I}}\left( t\right)= & {} \frac{1}{8} \left( \beta -\mu \right) \left( -\vartheta _{2} \left( t\right) +\vartheta \left( t\right) \left( \xi \left( t\right) -1\right) +\xi \left( t\right) \right) ,\\ {\mathcal {J}}\left( t\right)= & {} \frac{1}{8} \left( \beta -\mu \right) \left( \vartheta \left( t\right) +\vartheta _{2} \left( t\right) +\xi \left( t\right) +\vartheta \left( t\right) \xi \left( t\right) \right) ,\\ {\mathcal {K}}\left( t\right)= & {} -\frac{1}{8} \left( \beta -\mu \right) \left( \vartheta \left( t\right) -\vartheta _{2} \left( t\right) +\xi \left( t\right) \left( \vartheta \left( t\right) -1\right) \right) ,\\ {\mathcal {C}}\left( t\right)= & {} \frac{1}{32} \Big (4\alpha \left( 3+4\vartheta \left( t\right) +\vartheta _{2} \left( t\right) \right) -\left( -5+4\vartheta \left( t\right) +\vartheta _{2} \left( t\right) \right) \left( 3\beta +\mu \right) \\&-\,\left( -1+4\vartheta \left( t\right) -3\vartheta _{2} \left( t\right) \right) \left( \beta -\mu \right) \xi \left( t\right) \Big ),\\ {\mathcal {Y}}\left( t\right)= & {} \frac{1}{32} \Big (4\alpha \left( 3-4\vartheta \left( t\right) +\vartheta _{2} \left( t\right) \right) +\left( 5+4\vartheta \left( t\right) -\vartheta _{2} \left( t\right) \right) \left( 3\beta +\mu \right) \\&-\,\left( 1+4\vartheta \left( t\right) +3\vartheta _{2} \left( t\right) \right) \left( \beta -\mu \right) \xi \left( t\right) \Big ),\\ {\mathcal {N}}\left( t\right)= & {} \frac{1}{32} \left( \vartheta _{2} \left( t\right) -1\right) \left( 4\alpha -3\beta \left( 1+\xi \left( t\right) \right) +\mu \left( 3\xi \left( t\right) -1\right) \right) ,\\ {\mathcal {L}}\left( t\right)= & {} -\frac{1}{8} \left( \beta -\mu \right) \left( \vartheta _{2} \left( t\right) +\vartheta \left( t\right) \left( \xi \left( t\right) -1\right) -\xi \left( t\right) \right) ,\\ {\mathcal {E}}\left( t\right)= & {} \frac{1}{16} \Big (-4\alpha \left( \vartheta _{2} \left( t\right) -1\right) +\left( 3+\vartheta _{2} \left( t\right) \right) \left( 3\beta +\mu \right) \\&+\,\left( 1+3\vartheta _{2} \left( t\right) \right) \left( \beta -\mu \right) \xi \left( t\right) \Big ). \end{aligned}$$

where the time-dependent function $\vartheta _{n}(t) (n=1,2)$ and $\xi (t)$ are the decoherence factors whose explicit expressions can be written as follows:

$$\begin{aligned} \xi (t)= & {} \exp \left( -\dfrac{\Gamma \tau _{c}}{2}\left[ \dfrac{t}{\tau _{c}} +\exp \left( -\dfrac{t}{\tau _{c}}\right) -1 \right] \right) \end{aligned}$$

(A3)

$$\begin{aligned} \vartheta _{n}(t)= & {} \left[ \int _{\gamma _{1}}^{\gamma _{2}}\Psi _{n}(\gamma ,t){{\,\mathrm{P}\,}}(\gamma )\mathrm{d}\gamma \right] ^{N}; \end{aligned}$$

(A4)

with

$$\begin{aligned} \begin{aligned}&\Psi _{n}(\gamma ,t)= \left\{ \begin{aligned}&\displaystyle \mathrm{e}^{\displaystyle -\gamma t}\left[ \begin{aligned}&\dfrac{\gamma \sinh \left( t\sqrt{\gamma ^{2}-n^{2}\lambda ^{2}}\right) }{\sqrt{\gamma ^{2}-n^{2}\lambda ^{2}}}+\cosh \left( t\sqrt{\gamma ^{2}-n^{2}\lambda ^{2}}\right) \end{aligned} \right] ,\gamma >n\lambda \\ \\&\displaystyle \mathrm{e}^{\displaystyle -\gamma t}\left[ \begin{aligned}&\dfrac{\gamma \sin \left( t\sqrt{n^{2}\lambda ^{2}-\gamma ^{2}}\right) }{\sqrt{n^{2}\lambda ^{2}-\gamma ^{2}}}+ \cos \left( t\sqrt{n^{2}\lambda ^{2}-\gamma ^{2}}\right) \end{aligned} \right] ,\gamma <n\lambda \end{aligned}, \right. \end{aligned} \end{aligned}$$

(A5)

Appendix B: final state of system after performing the WM and WMR control scheme

$$\begin{aligned} \rho ^{1p}(t,m,m_{r})= & {} {\mathcal {A}}(t)\left( {\left| 00 \right\rangle } {\left\langle 00 \right| } +{\left| 12 \right\rangle } {\left\langle 12 \right| } \right) +{\mathcal {B}}(t)\left( {\left| 01 \right\rangle } {\left\langle 01 \right| } +{\left| 11 \right\rangle } {\left\langle 11 \right| } \right) \nonumber \\&+\,{\mathcal {C}}(t)\left( {\left| 02 \right\rangle } {\left\langle 02 \right| } +{\left| 10 \right\rangle } {\left\langle 10 \right| } \right) +{\mathcal {I}}(t)\left( {\left| 02 \right\rangle } {\left\langle 10 \right| } +{\left| 10 \right\rangle } {\left\langle 02 \right| } \right) \nonumber \\&+\, {\mathcal {E}}(t)\left( {\left| 00 \right\rangle } {\left\langle 02 \right| } +{\left| 02 \right\rangle } {\left\langle 00 \right| } +{\left| 10 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 10 \right| } \right) \nonumber \\&+\,{\mathcal {F}}(t)\left( {\left| 00 \right\rangle } {\left\langle 10 \right| } +{\left| 10 \right\rangle } {\left\langle 00 \right| } +{\left| 02 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 02 \right| } \right) \nonumber \\&+\, {\mathcal {G}}(t)\left( {\left| 00 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 00 \right| } \right) +{\mathcal {H}}(t)\left( {\left| 01 \right\rangle } {\left\langle 11 \right| } +{\left| 11 \right\rangle } {\left\langle 01 \right| } \right) ,\nonumber \\ \end{aligned}$$

(B1)

where

$$\begin{aligned} {\mathcal {B}}(t)= & {} \frac{3\left( \left( p-\frac{1}{3} \right) \vartheta _{2}(t) +\frac{1}{3} \left( 1+p\right) \right) \left( 2+\left( \xi (t)-1\right) m\right) \left( m_{r} -1\right) }{\left( 4\xi (t)m_{r} -4m_{r} +8\right) m+8m_{r} -16},\\ {\mathcal {I}}(t)= & {} \frac{3\left( \vartheta _{2}(t) -1\right) \left( 2+\left( \xi (t)-1\right) m\right) \left( m_{r} -1\right) \left( p-\frac{1}{3} \right) }{\left( 16+8\left( \xi (t)-1\right) m_{r} \right) m+16m_{r} -32},\\ {\mathcal {E}}(t)= & {} \frac{\sqrt{1-m_{r} } \left( p-1\right) \xi (t)\left( \vartheta _{2}(t) -1\right) \sqrt{1-m} }{\left( 8+\left( 4\xi (t)-4\right) m\right) m_{r} +8m-16} ,\\ {\mathcal {F}}(t)= & {} \frac{\sqrt{1-m_{r} } \sqrt{1-m} \left( \left( p-1\right) \left( \vartheta _{2}(t) +3\right) \xi (t)+4\vartheta (t)\left( 1-3p\right) \right) }{4\xi (t)mm_{r} -4\left( m_{r} -2\right) \left( m-2\right) } ,\\ {\mathcal {G}}(t)= & {} \frac{-\sqrt{1-m_{r} } \left( p-1\right) \xi (t)\left( \vartheta _{2}(t) -1\right) \sqrt{1-m} }{\left( 4+\left( 2\xi (t)-2\right) m\right) m_{r} +4m-8} ,\\ {\mathcal {H}}(t)= & {} \frac{\sqrt{1-m_{r} } \sqrt{1-m} \left( \left( p-1\right) \left( \vartheta _{2}(t) +3\right) \xi (t)-4\vartheta (t)\left( 1-3p\right) \right) }{4\xi (t)mm_{r} -4\left( m_{r} -2\right) \left( m-2\right) } ,\\ {\mathcal {A}}(t)= & {} \frac{\begin{aligned}-12\Bigg (&\left( \left( \vartheta (t)+\frac{1}{4} \vartheta _{2}(t) +\frac{1}{12} \right) p-\frac{\vartheta _{2}(t) }{12} -\frac{1}{4} -\frac{1}{3} \vartheta (t)\right) \left( \xi (t)-1\right) m-\\ {}&-2\left( p-\frac{1}{3} \right) \vartheta (t)\xi (t)+\left( \frac{1}{6} +\frac{\vartheta _{2}(t) }{2} \right) p-\frac{\vartheta _{2}(t) }{6} -\frac{1}{2}\Bigg )\left( m_{r} -1\right) \end{aligned}}{\left( 8\xi (t)m_{r} -m_{r} +16\right) m+16m_{r} -32},\\ {\mathcal {C}}(t)= & {} \frac{\begin{aligned}12\Bigg (&\left( \xi (t)-1\right) \left( \left( \vartheta (t)-\frac{1}{4} \vartheta _{2}(t) -\frac{1}{12} \right) p+\frac{\vartheta _{2}(t) }{12} +\frac{1}{4} -\frac{1}{3} \vartheta (t)\right) m-\\ {}&-2\left( p-\frac{1}{3} \right) \vartheta (t)\xi (t)-\left( \frac{1}{6} +\frac{\vartheta _{2}(t) }{2} \right) p+\frac{\vartheta _{2}(t) }{6} +\frac{1}{2}\Bigg )\left( m_{r} -1\right) \end{aligned}}{\left( 8\xi (t)m_{r} -8m_{r} +16\right) m+16m_{r} -32}, \end{aligned}$$

$$\begin{aligned} \rho ^{2p}(t,m,m_{r})= & {} {\mathcal {A}}(t) {\left| 00 \right\rangle } {\left\langle 00 \right| } +{\mathcal {B}}(t) {\left| 01 \right\rangle } {\left\langle 01 \right| } +{\mathcal {C}}(t) {\left| 02 \right\rangle } {\left\langle 02 \right| } +{\mathcal {E}}(t) {\left| 10 \right\rangle } {\left\langle 10 \right| } \nonumber \\&+\,{\mathcal {H}}(t) \left( {\left| 00 \right\rangle } {\left\langle 02 \right| } +{\left| 02 \right\rangle } {\left\langle 00 \right| } \right) +{\mathcal {I}}(t) \left( {\left| 00 \right\rangle } {\left\langle 11 \right| } +{\left| 11 \right\rangle } {\left\langle 00 \right| } \right) \nonumber \\&+\,{\mathcal {J}}(t) \left( {\left| 01 \right\rangle } {\left\langle 10 \right| } +{\left| 10 \right\rangle } {\left\langle 01 \right| } \right) +{\mathcal {K}}(t) \left( {\left| 01 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 01 \right| } \right) \nonumber \\&+\,{\mathcal {L}}(t) \left( {\left| 02 \right\rangle } {\left\langle 11 \right| } +{\left| 11 \right\rangle } {\left\langle 02 \right| } \right) +{\mathcal {M}}(t) \left( {\left| 10 \right\rangle } {\left\langle 12 \right| } +{\left| 12 \right\rangle } {\left\langle 10 \right| } \right) \nonumber \\&+\,{\mathcal {F}}(t) +{\left| 11 \right\rangle } {\left\langle 11 \right| } +{\mathcal {G}}(t) {\left| 12 \right\rangle } {\left\langle 12 \right| }, \end{aligned}$$

(B2)

where

$$\begin{aligned} {{\,\mathrm{T}\,}}= & {} 2\left( \left( \xi (t) -1\right) m_{r} +2\right) m+4\left( m_{r} -2\right) ,\\ {\mathcal {I}}(t)= & {} \frac{ -\Bigg ( \left( \vartheta (t)+1\right) \xi (t)-\vartheta (t)-\vartheta _{2}(t)\Bigg )\left( \beta -\mu \right) \sqrt{\left( 1-m\right) \left( 1-m_{r} \right) } }{2{{\,\mathrm{T}\,}}},\\ {\mathcal {J}}(t)= & {} \frac{-\Bigg ( \left( \vartheta (t)+1\right) \xi (t)\vartheta (t)+\vartheta _{2}(t) \Bigg )\left( \beta -\mu \right) \sqrt{\left( 1-m\right) \left( 1-m_{r} \right) } }{2{{\,\mathrm{T}\,}}},\\ {\mathcal {K}}(t)= & {} \frac{\Bigg (\left( \vartheta (t)-1\right) \xi (t)+\vartheta (t)-\vartheta _{2}(t) \Bigg )\left( \beta -\mu \right) \sqrt{\left( 1-m\right) \left( 1-m_{r} \right) } }{2{{\,\mathrm{T}\,}}} ,\\ {\mathcal {L}}(t)= & {} \frac{\Bigg (\left( \vartheta (t)-1\right) \xi (t)-\vartheta (t)+\vartheta _{2}(t) \Bigg )\left( \beta -\mu \right) \sqrt{\left( 1-m\right) \left( 1-m_{r} \right) } }{2{{\,\mathrm{T}\,}}} ,\\ {\mathcal {A}}(t)= & {} \frac{\left( 1-m_{r}\right) }{{{\,\mathrm{T}\,}}} \Bigg (\left( \xi (t)-1\right) \left( \left( -\frac{1}{2} \vartheta (t)+\frac{3}{8} \vartheta _{2}(t) -\frac{7}{8}\right) \beta \right. \\&\,\left. +\left( -\frac{1}{2} \vartheta (t)-\frac{1}{8} \vartheta _{2}(t) -\frac{3}{8} \right) \mu +\left( \vartheta (t)-\frac{1}{4} \vartheta _{2}(t) -\frac{3}{4} \right) \alpha \right) m\\&-\,\frac{1}{2} \left( \vartheta (t)+\frac{3}{2} \vartheta _{2}(t) +\frac{1}{4} \right) \left( \beta -\mu \right) \xi (t)\\&+\, \left( \frac{3}{4} \vartheta _{2}(t) -\frac{3}{2} \vartheta (t)-\frac{15}{8} \right) \beta +\left( \frac{1}{8} \vartheta _{2}(t) -\frac{1}{2} \vartheta (t)-\frac{5}{8} \right) \mu \\&+\,2\left( -\frac{1}{4} \vartheta _{2}(t) +\vartheta (t)-\frac{3}{4} \right) \alpha \Bigg ),\\ {\mathcal {B}}(t)= & {} \frac{\left( 1-m_{r}\right) }{2{{\,\mathrm{T}\,}}} \Bigg (\left( \xi (t)-1\right) \left( \left( -\frac{3}{2} \vartheta _{2}(t) -\frac{5}{2} \right) \beta \right. \\&\left. +\,\left( \alpha +\frac{1}{2} \mu \right) \left( \vartheta _{2}(t) -1\right) \right) m+\frac{3}{2} \left( \vartheta _{2}(t) +\frac{1}{3} \right) \left( \beta -\mu \right) \xi (t)\\&+\,\left( -\frac{3}{2} \vartheta _{2}(t) -\frac{9}{2} \right) \beta +\left( -\frac{3}{2} -\frac{1}{2} \vartheta _{2}(t) \right) \mu +2\alpha \left( \vartheta _{2}(t) -1\right) \Bigg ),\\ {\mathcal {C}}(t)= & {} \frac{\left( m_{r} -1\right) }{{{\,\mathrm{T}\,}}} \Bigg ( \left( \xi (t)-1\right) \left( \left( -\frac{1}{2} \vartheta (t)-\frac{3}{8} \vartheta _{2}(t) +\frac{7}{8} \right) \beta \right. \\&\left. +\,\left( -\frac{1}{2} \vartheta (t)+\frac{1}{8} \vartheta _{2}(t) +\frac{3}{8} \right) \mu +\left( \vartheta (t)+\frac{1}{4} \vartheta _{2}(t) +\frac{3}{4} \right) \alpha \right) m \\&-\,\frac{1}{2} \left( \vartheta (t)-\frac{3}{2} \vartheta _{2}(t) -\frac{1}{4} \right) \left( \beta -\mu \right) \xi (t) \\&+\,\left( -\frac{3}{8} \vartheta _{2}(t) -\frac{3}{2} \vartheta (t)+\frac{15}{8} \right) \beta +\left( -\frac{1}{8} \vartheta _{2}(t) -\frac{1}{2} \vartheta (t)+\frac{5}{8} \right) \mu \\&+\,2\left( \frac{1}{4} \vartheta _{2}(t) +\vartheta (t)+\frac{3}{4} \right) \alpha \Bigg ), \end{aligned}$$

$$\begin{aligned} {\mathcal {E}}(t)= & {} \frac{1}{6{{\,\mathrm{T}\,}}} \Bigg ( -8\left( \left( -\frac{1}{2} \vartheta (t)+\frac{3}{8} \vartheta _{2}(t) -\frac{7}{8} \right) \beta \right. \\&\left. +\,\left( -\frac{1}{2} \vartheta (t)-\frac{1}{8} \vartheta _{2}(t) -\frac{3}{8} \right) \mu +\left( \vartheta (t)+\frac{1}{4} \vartheta _{2}(t) -\frac{3}{4} \right) \alpha \right) \left( \xi (t)+1\right) m \\&+\,4\left( \vartheta (t)+\frac{3}{4} \vartheta _{2}(t) +\frac{1}{4} \right) \left( \beta -\mu \right) \xi (t)\\&+\, \left( -12\vartheta (t)+3\vartheta _{2}(t) -15\right) \beta +\left( -4\vartheta (t)+\vartheta _{2}(t) -5\right) \mu \\&+\,16\left( \vartheta (t)-\frac{1}{4} \vartheta _{2}(t) -\frac{3}{4} \right) \alpha \Bigg ),\\ {\mathcal {F}}(t)= & {} -\frac{\xi (t)+1}{2{{\,\mathrm{T}\,}}} \Bigg ( \left( -\frac{3}{2} \vartheta _{2}(t) -\frac{5}{2} \right) \beta \\&+\,\left( \alpha +\frac{\mu }{2} \right) \left( \vartheta _{2}(t) -1\right) m-3\left( \vartheta _{2}(t) +\frac{1}{3} \right) \left( \beta -\mu \right) \xi (t)\\&+\,\left( -3\vartheta _{2}(t) -9\right) \beta -\left( \vartheta _{2}(t) +3\right) \mu +4a\left( \vartheta _{2}(t)-1\right) \Bigg ),\\ {\mathcal {G}}(t)= & {} \frac{1}{8{{\,\mathrm{T}\,}}} \Bigg ( 8\left( \left( -\frac{1}{2} \vartheta (t)-\frac{3}{8} \vartheta _{2}(t) +\frac{7}{8} \right) \beta \right. \\&+\,\left( -\frac{1}{2} \vartheta (t)+\frac{1}{8} \vartheta _{2}(t) +\frac{3}{8} \right) \mu \\&\left. +\left( \vartheta (t)+\frac{1}{4} \vartheta _{2}(t) +\frac{3}{4}\right) \alpha \right) \left( \xi (t)+1\right) m \\&-\,4\left( \vartheta (t)-\frac{3}{4} \vartheta _{2}(t) -\frac{1}{4} \right) \left( \beta -\mu \right) \xi (t)\\&+\, \left( 12\vartheta (t)+3\vartheta _{2}(t) -15\right) \beta +\left( 4\vartheta (t)+\vartheta _{2}(t) -5\right) \mu \\&-\,16\left( \vartheta (t)+\frac{1}{4} \vartheta _{2}(t) +\frac{3}{4} \right) \alpha \Bigg ),\\ {\mathcal {H}}(t)= & {} \frac{\left( \left( \xi (t)-1\right) \left( -\frac{3}{2} \beta +\frac{\mu }{2} +\,\alpha \right) m+\left( \frac{3}{2} \beta -\frac{3}{2} \mu \right) \xi (t)+2\alpha -\frac{\mu }{2} -\frac{3\beta }{2} \right) \left( m_{r} -1\right) \left( \vartheta _{2}(t) -1\right) }{4{{\,\mathrm{T}\,}}},\\ {\mathcal {M}}(t)= & {} \frac{\left( \vartheta _{2}(t) -\,1\right) \left( \left( -\frac{3\beta }{2} +\frac{\mu }{2} +\,\alpha \right) \left( \xi (t)+1\right) m+\left( \frac{3\beta }{2} -\,\frac{3\mu }{2} \right) \xi (t)+\frac{3\beta }{2} -\,2\alpha +\frac{\mu }{2} \right) }{4{{\,\mathrm{T}\,}}}. \end{aligned}$$