Single-photon blockade in a hybrid optomechanical system involving two qubits in the presence of phononic number and coherent states

Abstract

Single-photon generation sources have gained a prime importance in quantum information science and technologies. In this regard, serious efforts have been recently performed in the optomechanical systems in order to generate the single-photon states in a macroscopic scale. In this paper, we consider an optomechanical system contains two optical cavities both coupled with a mechanical membrane which is placed between them. Each optical mode interacts with a qubit (two-level atom). Considering the steady-state solution, we analytically obtain the state vector of the entire system wherein the initial mechanical mode has been prepared as phononic number and coherent states. In the continuation, in order to study the single-photon blockade phenomenon, we evaluate the equal-time second-order correlation function ($g^{2}(0)$) as well as the probability distribution of single photon ($P_1$) to confirm the results obtained from $g^{2}(0)$. In addition, we investigate the effect of photon tunneling, decay rate of photons and coupling between photons and each qubit on the single-photon blockade within one of the cavities. Our numerical results show that the complete single-photon blockade occurs, i.e., $g^{(2)}(0)$ tends to very small amounts of the order of $10^{-8}$ to $10^{-4}$ via increasing the photon tunneling and the coupling between photons and qubits. Furthermore, increasing the dissipation effects attenuates the single-photon blockade occurrence.

Appendix A

In this appendix, we aim to obtain the twelve algebraic equations in the steady-state regime from Eq. (13),

$$\begin{aligned}{} & {} (E_{1,0,m(\beta ),0,0}-i \gamma _{c}) C_{1,0,m(\beta ),0,0} + g C_{0,0,m(\beta ),1,0} e_3 \nonumber \\{} & {} \quad +\, J C_{0,1,m(\beta ),0,0} e_4 +\varepsilon _p (C_{0,0,m(\beta ),0,0} e_3 \nonumber \\{} & {} \quad +\, C_{1,1,m(\beta ),0,0} e_5 + \sqrt{2} C_{2,0,m(\beta ),0,0} e_6) =0, \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} (E_{0,1,m(\beta ),0,0}-i \gamma _{c}) C_{0,1,m(\beta ),0,0} +g C_{0,0,m(\beta ),0,1} e_7 \nonumber \\{} & {} \quad +\,J C_{1,0,m(\beta ),0,0} e_8 + \varepsilon _p (C_{0,0,m(\beta ),0,0} e_7 \nonumber \\{} & {} \quad +\, C_{1,1,m(\beta ),0,0} e_9) + \sqrt{2} C_{0,2,m(\beta ),0,0} e_{10} =0, \end{aligned}$$

(A.2)

$$\begin{aligned}{} & {} (E_{0,0,m(\beta ),1,0}-i \gamma _{a})C_{0,0,m(\beta ),1,0} + g C_{1,0,m(\beta ),0,0} e_{1}\nonumber \\{} & {} \quad +\,\varepsilon _p (C_{1,0,m(\beta ),1,0} e_1 + C_{0,1,m(\beta ),1,0} e_2)=0, \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} (E_{0,0,m(\beta ),0,1}-i \gamma _{a}) C_{0,0,m(\beta ),0,1} + g C_{0,1,m(\beta ),0,0} e_{2}\nonumber \\{} & {} \quad +\,\varepsilon _p (C_{0,1,m(\beta ),0,1} e_2 + C_{1,0,m(\beta ),0,1} e_1)=0, \end{aligned}$$

(A.4)

$$\begin{aligned}{} & {} (E_{1,1,m(\beta ),0,0}- 2 i \gamma _{c}) C_{1,1,m(\beta ),0,0} + g (C_{1,0,m(\beta ),0,1} e_{11} \nonumber \\{} & {} \quad +\, C_{0,1,m(\beta ),1,0} e_{12}) + \sqrt{2} J (C_{2,0,m(\beta ),0,0} e_{13} \nonumber \\{} & {} \quad +\, C_{0,2,m(\beta ),0,0} e_{14}) +\varepsilon _p (C_{1,0,m(\beta ),0,0} e_{11} \nonumber \\{} & {} \quad +\, C_{0,1,m(\beta ),0,0} e_{12}) =0, \end{aligned}$$

(A.5)

$$\begin{aligned}{} & {} (E_{2,0,m(\beta ),0,0}- 2 i \gamma _{c}) C_{2,0,m(\beta ),0,0} + \sqrt{2} g C_{1,0,m(\beta ),1,0} e_{15} \nonumber \\{} & {} \quad +\, \sqrt{2} J C_{1,1,m(\beta ),0,0} e_{16} +\sqrt{2} \varepsilon _p C_{1,0,m(\beta ),0,0} \, e_{15} =0, \end{aligned}$$

(A.6)

$$\begin{aligned}{} & {} (E_{0,2,m(\beta ),0,0}- 2 i \gamma _{c}) C_{0,2,m(\beta ),0,0} + \sqrt{2} g C_{0,1,m(\beta ),0,1} \, e_{17} \nonumber \\{} & {} \quad +\, \sqrt{2} J C_{1,1,m(\beta ),0,0}\, e_{18} +\sqrt{2} \varepsilon _p C_{0,1,m(\beta ),0,0} \, e_{17} =0, \end{aligned}$$

(A.7)

$$\begin{aligned}{} & {} (E_{1,0,m(\beta ),1,0}- i (\gamma _{c}+\gamma _{a})) C_{1,0,m(\beta ),1,0} \nonumber \\{} & {} \quad +\, \sqrt{2} g C_{2,0,m(\beta ),0,0} e_{6} + J C_{0,1,m(\beta ),1,0} e_{4} \nonumber \\{} & {} \quad +\, \varepsilon _p C_{0,0,m(\beta ),1,0} \, e_{3} =0, \end{aligned}$$

(A.8)

$$\begin{aligned}{} & {} (E_{0,1,m(\beta ),0,1}- i (\gamma _{c}+\gamma _{a})) C_{0,1,m(\beta ),0,1} \nonumber \\{} & {} \quad +\, \sqrt{2} g C_{0,2,m(\beta ),0,0} e_{10} + J C_{1,0,m(\beta ),0,1} e_{8} \nonumber \\{} & {} \quad +\, \varepsilon _p C_{0,0,m(\beta ),0,1} \, e_{7} =0, \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} (E_{0,1,m(\beta ),1,0}- i (\gamma _{c}+\gamma _{a})) C_{0,1,m(\beta ),1,0} \nonumber \\{} & {} \quad +\, g C_{0,0,m(\beta ),1,1} e_{7} + J C_{1,0,m(\beta ),1,0} e_{8} \nonumber \\{} & {} \quad +\, \varepsilon _p C_{0,0,m(\beta ),1,0} \, e_{7} =0, \end{aligned}$$

(A.10)

$$\begin{aligned}{} & {} (E_{1,0,m(\beta ),0,1}- i (\gamma _{c}+\gamma _{a})) C_{1,0,m(\beta ),0,1} \nonumber \\{} & {} \quad + g C_{0,0,m(\beta ),1,1} e_{3} + J C_{0,1,m(\beta ),0,1} e_{4} \nonumber \\{} & {} \quad +\, \varepsilon _p C_{0,0,m(\beta ),0,1} \, e_{3} =0, \end{aligned}$$

(A.11)

$$\begin{aligned}{} & {} (E_{0,0,m(\beta ),1,1}- 2 i \gamma _{a}) C_{0,0,m(\beta ),1,1} \nonumber \\{} & {} \quad +\, g (C_{0,1,m(\beta ),1,0} \, e_{2} + C_{1,0,m(\beta ),0,1} \, e_1)=0, \end{aligned}$$

(A.12)

where we have used the following notations,

$$\begin{aligned} e_{1}= & {} e_{3}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(0,0)|{\tilde{m}}({\tilde{\beta }})(1,0)\rangle , \end{aligned}$$

(A.13)

$$\begin{aligned} e_{2}= & {} e_{7}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(0,0)|{\tilde{m}}({\tilde{\beta }})(0,1)\rangle , \end{aligned}$$

(A.14)

$$\begin{aligned} e_{4}= & {} e_{8}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(1,0)|{\tilde{m}}({\tilde{\beta }})(0,1)\rangle , \end{aligned}$$

(A.15)

$$\begin{aligned} e_{5}= & {} e_{11}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(1,0)|{\tilde{m}}({\tilde{\beta }})(1,1)\rangle , \end{aligned}$$

(A.16)

$$\begin{aligned} e_{6}= & {} e_{15}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(1,0)|{\tilde{m}}({\tilde{\beta }})(2,0)\rangle , \end{aligned}$$

(A.17)

$$\begin{aligned} e_{9}= & {} e_{12}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(0,1)|{\tilde{m}}({\tilde{\beta }})(1,1)\rangle , \end{aligned}$$

(A.18)

$$\begin{aligned} e_{10}= & {} e_{17}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(0,1)|{\tilde{m}}({\tilde{\beta }})(0,2)\rangle , \end{aligned}$$

(A.19)

$$\begin{aligned} e_{13}= & {} e_{16}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(1,1)|{\tilde{m}}({\tilde{\beta }})(2,0)\rangle , \end{aligned}$$

(A.20)

$$\begin{aligned} e_{14}= & {} e_{18}^{*}=\langle {\tilde{m}}({\tilde{\beta }})(1,1)|{\tilde{m}}({\tilde{\beta }})(0,2)\rangle . \end{aligned}$$

(A.21)

By applying Eq. 12 and using the above definitions, we obtain,

$$\begin{aligned} e_{1}= & {} e_{6} = e_{7} = e_{9} = e_{11} = e_{17} \nonumber \\= & {} \exp \left( -\frac{G^2}{2 \omega _{m}^2}\right) L_{m}\left( -\frac{G^2}{\omega _{m}^2}\right) , \end{aligned}$$

(A.22)

$$\begin{aligned} e_{4}= & {} e_{14} = e_{16}= \exp \left( -\frac{2 G^2}{ \omega _{m}^2}\right) L_{m}\left( -4\frac{G^2}{\omega _{m}^2}\right) , \end{aligned}$$

(A.23)

for the phononic number state ($|m\rangle$) where $L_{m}(-\frac{G^2}{\omega _{m}^2})$ is the Laguerre polynomial. Similarly, for the phononic coherent states $(|\beta \rangle )$ one may arrive at,

$$\begin{aligned} e_{1}= & {} e_{5} = e_{6} = e_{6} = e_{7} = e_{9} = e_{11} = e_{17} \nonumber \\= & {} \exp \left( -\frac{G^2}{2 \omega _{m}^2}\right) \exp \left[ \frac{G}{ \omega _{m}}(\beta -\beta ^*)\right] , \end{aligned}$$

(A.24)

$$\begin{aligned} e_{4}= & {} e_{14} = e_{16}\nonumber \\= & {} \exp \left( -\frac{2 G^2}{ \omega _{m}^2}\right) \exp \left[ \frac{-2 G}{ \omega _{m}}(\beta -\beta ^*)\right] . \end{aligned}$$

(A.25)

In the weak-driving case, the probability amplitudes can also be classified into various groups of different orders of the small value parameter $\varepsilon _p$. In detail, the amplitude $C_{0,0,m(\beta ),0,0}$ is of the zero-order of $\varepsilon _p$, $C_{1,0,m(\beta ),0,0}$, $C_{0,1,m(\beta ),0,0}$, $C_{0,0,m(\beta ),1,0}$, $C_{0,0,m(\beta ),0,1}$ are of the first-order of $\varepsilon _p$ and $C_{1,1,m(\beta ),0,0}$, $C_{2,0,m(\beta ),0,0}$, $C_{0,2,m(\beta ),0,0}$, $C_{1,0,m(\beta ),1,0}$, $C_{0,1,m(\beta ),0,1}$, $C_{0,1,m(\beta ),1,0}$, $C_{1,0,m(\beta ),0,1}$ and $C_{0,0,m(\beta ),1,1}$ are of the second-order of $\varepsilon _p$. Accordingly, we have ignored the terms of order $\varepsilon _p ^3$ in the equations. So, we eventually arrive at the solution of Eqs. (A.1) to (A.4) as,

$$\begin{aligned} C_{1,0,m(\beta ),0,0}= & {} \left( \frac{J \,e_{4}\, e_{7}- e_{3}\, d}{d^2-J^2 \, e_{4}\, e_{8}}\right) \varepsilon _p C_{0,0,m(\beta ),0,0}, \end{aligned}$$

(A.26)

$$\begin{aligned} C_{0,1,m(\beta ),0,0}= & {} \left( \frac{J \,e_{3}\, e_{8}- e_{7}d}{d^2-J^2 e_{4} e_{8}}\right) \varepsilon _p C_{0,0,m(\beta ),0,0}, \end{aligned}$$

(A.27)

$$\begin{aligned} C_{0,0,m(\beta ),1,0}= & {} C_{0,0,m(\beta ),0,1} \nonumber \\= & {} \frac{g \exp (-\frac{G^2}{\omega _m^2})}{(E_{0,0,m(\beta ),1,0}-i \gamma _{a})(d + J \exp (-\frac{G^2}{\omega _m^2}))}\varepsilon _p C_{0,0,m(\beta ),0,0},\nonumber \\ \end{aligned}$$

(A.28)

where

$$\begin{aligned} d= (E_{1,0,m(\beta ),0,0}-i \gamma _{c})-\frac{g^2\, e_{2} \, e_{7}}{(E_{0,0,m(\beta ),1,0}-i \gamma _{a})}. \end{aligned}$$

(A.29)

Also, from solving Eqs. (A.5) to (A.12) we achieve the other coefficients as below,

$$\begin{aligned} C_{1,1,m(\beta ),0,0}= & {} \frac{ 1}{N_t} \bigg [ [(A_1 X_1 + A_2 X_2)\nonumber \\{} & {} +\, \frac{g}{\varDelta _t}( p_1 e_{4}+ p_2 e_{8}+ q_1 e_{3}+ q_2 e_{7})] \varepsilon _p C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} +\,\frac{\sqrt{2} \varepsilon _p e_{17}}{N^2-N_{m_1} N_{m_2}}[ A_1 N_{m_1}+A_2 N ] C_{0,1,m(\beta ),0,0} \nonumber \\{} & {} -\, \varepsilon _p ( C_{1,0,m(\beta ),0,0} e_{11} + C_{0,1,m(\beta ),0,0} e_{12}) \nonumber \\{} & {} +\,\frac{\sqrt{2} \varepsilon _p e_{15}}{N^2-N_{m_1} N_{m_2}} \left[ A_1 N +A_2 N_{m_2} \right] C_{1,0,m(\beta ),0,0}\bigg ], \end{aligned}$$

(A.30)

$$\begin{aligned} C_{2,0,m(\beta ),0,0}= & {} \left( \frac{1}{N^2-N_{m_1} N_{m_2}}\right) \nonumber \\{} & {} \bigg [ -\sqrt{2}J C_{1,1,m (\beta ),0,0} (e_{16} N + e_{18} N_{m_1}) \nonumber \\{} & {} -[(s_1 + s_2)N + (s^{\prime }_1+s^{\prime }_2)N_{m_1}] \varepsilon _p C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} -\sqrt{2} N_{m_1} \varepsilon _p e_{17} C_{0,1,m(\beta ),0,0} - \sqrt{2} N \varepsilon _p e_{15} C_{1,0,m(\beta ),0,0} \bigg ], \end{aligned}$$

(A.31)

$$\begin{aligned} C_{0,2,m(\beta ),0,0}= & {} \left( \frac{1}{N^2-N_{m_1} N_{m_2}}\right) \nonumber \\{} & {} \bigg [ -\sqrt{2}J C_{1,1,m (\beta ),0,0} (e_{18} N + e_{16} N_{m_2}) \nonumber \\{} & {} -[ (s^{\prime }_1+s^{\prime }_2) N + (s_1 + s_2)N_{m_2}] \varepsilon _p C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} -\sqrt{2} N_{m_2} \varepsilon _p e_{15} C_{1,0,m(\beta ),0,0} - \sqrt{2} N \varepsilon _p e_{17} C_{0,1,m(\beta ),0,0} \bigg ], \end{aligned}$$

(A.32)

$$\begin{aligned} C_{1,0,m(\beta ),1,0}= & {} \frac{1}{M^2- \varDelta _{m_1}\varDelta _{m_2}}\nonumber \\{} & {} \bigg [ (b_1 M -\varDelta _{m_1}a_2)\varepsilon _p e_{7} C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} +\, (b_2 \varDelta _{m_1} -a_1 M)\varepsilon _p e_{3} C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} -\, \sqrt{2}g \varDelta _{m_1} e_{10} C_{0,2,m(\beta ),0,0}\nonumber \\{} & {} - \sqrt{2}g M e_{6} C_{2,0,m(\beta ),0,0} \bigg ], \end{aligned}$$

(A.33)

$$\begin{aligned} C_{0,1,m(\beta ),0,1}= & {} \frac{1}{M^2- \varDelta _{m_1}\varDelta _{m_2}}\nonumber \\{} & {} \bigg [ (b_2 M -\varDelta _{m_2}a_1)\varepsilon _p e_{3} C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} +\, (b_1 \varDelta _{m_2} -a_2 M)\varepsilon _p e_{7} C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} -\, \sqrt{2}g \varDelta _{m_2} e_{6} C_{2,0,m(\beta ),0,0}\nonumber \\{} & {} -\, \sqrt{2}g M e_{10} C_{0,2,m(\beta ),0,0} \bigg ], \end{aligned}$$

(A.34)

$$\begin{aligned} C_{0,1,m(\beta ),1,0}= & {} \frac{1}{\varDelta _{t}} \bigg [ (E_{0,1,m(\beta ),1,0}-i(\gamma _c +\gamma _a)-\varDelta _g) J e_{8}C(1,0,m(\beta ),1,0) \nonumber \\{} & {} +\,(E_{0,1,m(\beta ),1,0}-i(\gamma _c +\gamma _a)-\varDelta _g)\varepsilon _p e_{7} C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} +\,\varDelta _{g_1}J e_{4}C_{0,1,m(\beta ),0,1}+\varDelta _{g_1}\varepsilon _p e_{3} C_{0,0,m(\beta ),1,0} \bigg ], \end{aligned}$$

(A.35)

$$\begin{aligned} C_{1,0,m(\beta ),0,1}= & {} \frac{1}{\varDelta _{t}} \bigg [ (E_{0,1,m(\beta ),1,0}-i(\gamma _c +\gamma _a)-\varDelta _g) J e_{4} C_{0,1,m(\beta ),0,1} \nonumber \\{} & {} +\,(E_{0,1,m(\beta ),1,0}-i(\gamma _c +\gamma _a)-\varDelta _g)\varepsilon _p e_{7} C_{0,0,m(\beta ),1,0} \nonumber \\{} & {} +\,\varDelta _{g_2}J e_{8} C_{1,0,m(\beta ),1,0}+\varDelta _{g_2}\varepsilon _p e_{7} C_{0,0,m(\beta ),1,0}\bigg ], \end{aligned}$$

(A.36)

$$\begin{aligned} C_{0,0,m(\beta ),1,1}= & {} -\frac{g}{E_{0,0,m(\beta ),1,1}-2i \gamma _{a}} \nonumber \\{} & {} \left[ C_{0,1,m(\beta ),1,0} e_2 + C_{1,0,m(\beta ),0,1} e_1\right] , \end{aligned}$$

(A.37)

where we have used the following notations to summarize the amplitudes,

$$\begin{aligned} N_{t}= & {} (E_{1,1,m(\beta ),0,0}-2i\gamma _c)-\frac{\sqrt{2} J}{N^2-N_{m_1}\, N_{m_2}}\\{} & {} \bigg [ A_1 (e_{16} \, N + e_{18}\,N_{m_1}) + A_2 (e_{18}\, N + e_{16}\, N_{m_2}) \bigg ],\\ A_{1}= & {} \sqrt{2}J \,e_{13}\, \\{} & {} +\, \frac{\sqrt{2}g^2 J q_1 \,e_4 \, e_6 \, \varDelta _{m_2} + \sqrt{2}g^2 \,J \,q_2 \,e_8\, e_6 \, M}{(M^2- \varDelta _{m_1}\, \varDelta _{m_2})\varDelta _t}, \\ A_{2}= & {} \sqrt{2}\,J \, e_{14} \\{} & {} +\, \frac{\sqrt{2}g^2 \,J \,q_1\, e_4 \,e_{10} \, M + \sqrt{2}g^2 \, J \, q_2\, e_8 \, e_{10}\, \varDelta _{m_1}}{(M^2- \varDelta _{m_1}\, \varDelta _{m_2})\varDelta _t},\\ X_1= & {} \frac{(s_1 + s_2)N + (s^\prime _1 + s^\prime _2)N_{m_1}}{N^2-N_{m_1}\, N_{m_2}},\\ X_2= & {} \frac{(s^\prime _1 + s^\prime _2)N + (s_1 + s_2)N_{m_2}}{N^2-N_{m_1}\, N_{m_2}}, \\ p_1= & {} \frac{J\, q_1}{M^2- \varDelta _{m_1}\, \varDelta _{m_2}}\bigg [ (b_2 \, M -a_1\, \varDelta _{m_2})\, e_3 \\{} & {} +\,(b_1 \, \varDelta _{m_2} - a_2 \, M) \,e_7 \bigg ],\\ p_2= & {} \frac{J \, q_2}{M^2- \varDelta _{m_1}\, \varDelta _{m_2}}\bigg [ (b_1\, M -a_2\, \varDelta _{m_1})\, e_7 \\{} & {} +\,(b_2 \, \varDelta _{m_1} - a_1 \, M)\, e_3 \bigg ],\\ q_1= & {} (E_{0,1,m(\beta ),1,0}-i(\gamma _c +\gamma _a)-\varDelta _g)e_{11}+ \varDelta _{g_1}\, e_{12},\\ q_2= & {} \varDelta _{g_2}\, e_{11}+(E_{0,1,m(\beta ),1,0}-i(\gamma _c +\gamma _a)-\varDelta _g)\, e_{12}, \\ N= & {} (E_{2,0,m(\beta ),0,0} -2i \gamma _c)-\frac{2g^2 \, M \, e_6 \, e_{15}}{M^2- \varDelta _{m_1}\, \varDelta _{m_2}}, \\ N_{m_1}= & {} \frac{2g^2 \varDelta _{m_1} \, e_{10} \, e_{15}}{M^2- \varDelta _{m_1}\, \varDelta _{m_2}}, \ \ N_{m_2}= \frac{2g^2 \varDelta _{m_2}\, e_6\, e_{17}}{M^2- \varDelta _{m_1} \, \varDelta _{m_2}},\\ s_1= & {} \frac{\sqrt{2} \, g \, e_7 \, e_{15}}{M^2- \varDelta _{m_1} \, \varDelta _{m_2}}(b_1 \, M -a_2 \, \varDelta _{m_1}), \\ s_2= & {} \frac{\sqrt{2} \, g \, e_3 \, e_{15}}{M^2- \varDelta _{m_1}\, \varDelta _{m_1}}(b_2\, M -a_1 \, \varDelta _{m_2}),\\ s^{\prime }_1= & {} \frac{\sqrt{2} \, g \, e_7\, e_{17}}{M^2- \varDelta _{m_1}\, \varDelta _{m_2}}(b_2 \, M -a_1\, \varDelta _{m_2}), \\ s^{\prime }_2= & {} \frac{\sqrt{2} \,g \, e_7 \, e_{17}}{M^2- \varDelta _{m_1}\, \varDelta _{m_2}}(b_1 \, \varDelta _{m_2} -a_2 \, M), \\ a_1= & {} 1-\frac{J \, \varDelta _{g_1}\, e_4}{\varDelta _t}, \ \ a_2 = 1-\frac{J \varDelta _{g_2}e_8}{\varDelta _t}, \\ b_1= & {} \frac{J e_4 (E_{1,0,m(\beta ),1,0}-i(\gamma _c+\gamma _a)-\varDelta _{g})}{\varDelta _t},\\ b_2= & {} \frac{J \, e_8 (E_{1,0,m(\beta ),1,0}-i(\gamma _c+\gamma _a)-\varDelta _{g})}{\varDelta _t}, \\ \varDelta _{m_1}= & {} \frac{J^2 e_4 ^2 \varDelta _{g_1}}{\varDelta _t}, \, \, \varDelta _{m_2}=\frac{J^2 e_8 ^2 \varDelta _{g_2}}{\varDelta _t},\\ M= & {} (E_{1,0,m(\beta ),1,0}-i(\gamma _c+\gamma _a)) \\{} & {} -\,\frac{J^2\, e_4 \, e_8 (E_{1,0,m(\beta ),1,0}-i(\gamma _c+\gamma _a)-\varDelta _{g})}{\varDelta _t},\\ \varDelta _{t}= & {} (E_{1,0,m(\beta ),1,0}-i(\gamma _c+\gamma _a)-\varDelta _{g})^2-\varDelta _{g_1}\varDelta _{g_2},\\ \varDelta _g= & {} \frac{g^2\, e_2 \, e_7}{E_{0,0,m(\beta ),1,1}-2i\gamma _a}, \\ \varDelta _{g_1}= & {} \frac{g^2 e_1 e_7}{E_{0,0,m(\beta ),1,1}-2i\gamma _a}, \\ \varDelta _{g_2}= & {} \frac{g^2 \, e_2 \, e_3}{E_{0,0,m(\beta ),1,1}-2i\gamma _a}, \end{aligned}$$