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Analytical solution and applications of three qubits in three coupled modes without rotating wave approximation

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Abstract

We study the dynamics of the three-qubit system interacting with multi-mode without rotating wave approximation (RWA). A physical realization of the system without direct qubits interactions with dephasing bath is proposed. It is shown that non-Markovian characters of the purity of the three qubits and the coupling strength of modes are stronger enough the RWA is no longer valid. The influences of the dephasing of qubits and interactions of modes on the dynamics of genuine multipartite entanglement and bipartite correlations of qubits are investigated. The multipartite and bipartite quantum correlations could be generated faster if we increase the coupling strength of modes and the RWA is not valid when the coupling strength is strong enough. The unitary transformations approach adopted here can be extended to other systems such as circuit or cavity quantum electrodynamic systems in the strong coupling regime.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11365009 and 11065007) and the Scientific Research Foundation of Jiangxi (Grant Nos. 20151BAB202020 and 20161BBE50069).

Author information

Authors and Affiliations

  1. Department of Applied Physics, East China Jiaotong University, Nanchang, 330013, People’s Republic of China

    Jian-Song Zhang & Liu-Juan Zhang

  2. Department of Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, People’s Republic of China

    Ai-Xi Chen

  3. Institute for Quantum Computing, University of Waterloo, Ontario, N2L3G1, Canada

    Ai-Xi Chen

  4. Applied Science University, Sitra, Bahrain

    Mahmoud Abdel-Aty

  5. Zewail City of Science and Technology, Giza, Egypt

    Mahmoud Abdel-Aty

  6. Sohag University, Sohag, Egypt

    Mahmoud Abdel-Aty

Authors
  1. Jian-Song Zhang
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  2. Liu-Juan Zhang
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Corresponding author

Correspondence to Ai-Xi Chen.

Appendix

Appendix

The initial state of the whole system is given in Eq. (13). In the new picture defined by \(U_j\), the initial state of the whole system is

$$\begin{aligned} |\psi '_j(0)\rangle= & {} U_j |\psi (0)\rangle = \sum _{\mu = 1}^8 d_{\mu }|\mu \rangle |\alpha _j^{(\mu )}, r_{j,a}\rangle _a |\beta _j^{(\mu )}, r_{j,b}\rangle _b |\gamma _j^{(\mu )}, r_{j,c}\rangle _c. \end{aligned}$$
(25)

The initial state of three qubits in the interaction picture defined by Eq. (11) is \(\rho ''_{j,ABC}(0) = Tr_{abc}[|\psi _j''(0)\rangle \langle \psi _j''(0)]\). The density matrix of the qubits is given by Eq. (15) with

$$\begin{aligned} \eta _{\mu \mu }= & {} 0,\quad \eta _{\nu \mu } = \eta _{\mu \nu }, \quad \eta _{18} = \eta _{27} = \eta _{36} = \eta _{45} = 3\eta \nonumber \\ \eta _{14}= & {} \eta _{16} = \eta _{17} = \eta _{23} = \eta _{25} = \eta _{28} = \eta _{35} = \eta _{38} = \eta _{46} \nonumber \\= & {} \eta _{47} = \eta _{58} = \eta _{67} = 2\eta , \nonumber \\ \eta _{12}= & {} \eta _{13} = \eta _{15} = \eta _{24} = \eta _{26} = \eta _{34} = \eta _{37} = \eta _{48} \nonumber \\= & {} \eta _{56} = \eta _{57} = \eta _{68} = \eta _{78} = \eta , \nonumber \\ f_{j,\mu \nu }(t)= & {} e^{i\varphi _{j,\mu \nu }} \langle A^{(\nu )}_{j,a}, \varepsilon _{j,a}|A^{(\mu )}_{j,a}, \varepsilon _{j,a}\rangle \langle A^{(\nu )}_{j,b}, \varepsilon _{j,b}|A^{(\mu )}_{j,b}, \varepsilon _{j,b}\rangle \langle A^{(\nu )}_{j,c}, \varepsilon _{j,c}|A^{(\mu )}_{j,c}, \varepsilon _{j,c}\rangle \qquad \quad \end{aligned}$$
(26)
$$\begin{aligned} A^{(\mu )}_{j,a}= & {} -\,\lambda ^{(\mu )}_{j,a} + e^{-i\omega _{j,a} t}\alpha _j^{(\mu )}, \quad A^{(\mu )}_{j,b} = -\lambda ^{(\mu )}_{j,b} + e^{-i\omega _{j,b} t}\beta _j^{(\mu )}, A^{(\mu )}_{j,c} = -\lambda ^{(\mu )}_{j,c}\nonumber \\&+\, e^{-i\omega _{j,c} t}\gamma _j^{(\mu )}, \nonumber \\ \alpha _j^{(\mu )}= & {} (\lambda ^{(\mu )}_{j,a} + \alpha '), \beta _j^{(\mu )} = (\lambda ^{(\mu )}_{j,b} + \beta '), \gamma _j^{(\mu )} = (\lambda ^{(\mu )}_{j,c} + \gamma '), \end{aligned}$$
(27)
$$\begin{aligned} \lambda ^{(1)}_{j,a}= & {} (a_1 + b_1 + c_1)\lambda _{j,a}, \quad \lambda ^{(2)}_{j,a} = (a_1 + b_1 - c_1)\lambda _{j,a},\nonumber \\&\lambda ^{(3)}_{j,a} = (a_1 - b_1 + c_1)\lambda _{j,a}, \end{aligned}$$
(28)
$$\begin{aligned} \lambda ^{(4)}_{j,a}= & {} (a_1 - b_1 - c_1)\lambda _{j,a}, \quad \lambda ^{(5)}_{j,a} = (-a_1 + b_1 + c_1)\lambda _{j,a},\nonumber \\&\lambda ^{(6)}_{j,a} = (-a_1 + b_1 - c_1)\lambda _{j,a},\end{aligned}$$
(29)
$$\begin{aligned} \lambda ^{(7)}_{j,a}= & {} (-a_1 - b_1 + c_1)\lambda _{j,a}, \quad \lambda ^{(8)}_{j,a} = (-a_1 - b_1 - c_1)\lambda _{j,a},\end{aligned}$$
(30)
$$\begin{aligned} \lambda ^{(1)}_{j,b}= & {} (a_2 + b_2 + c_2)\lambda _{j,b}, \quad \lambda ^{(2)}_{j,b} = (a_2 + b_2 - c_2)\lambda _{j,b},\nonumber \\&\lambda ^{(3)}_{j,b} = (a_2 - b_2 + c_2)\lambda _{j,b},\end{aligned}$$
(31)
$$\begin{aligned} \lambda ^{(4)}_{j,b}= & {} (a_2 - b_2 - c_2)\lambda _{j,b}, \quad \lambda ^{(5)}_{j,b} = (-a_2 + b_2 + c_2)\lambda _{j,b},\nonumber \\&\lambda ^{(6)}_{j,b} = (-a_2 + b_2 - c_2)\lambda _{j,b},\end{aligned}$$
(32)
$$\begin{aligned} \lambda ^{(7)}_{j,b}= & {} (-a_2 - b_2 + c_2)\lambda _{j,b}, \quad \lambda ^{(8)}_{j,b} = (-a_2 - b_2 - c_2)\lambda _{j,b},\end{aligned}$$
(33)
$$\begin{aligned} \lambda ^{(1)}_{j,c}= & {} (a_3 + b_3 + c_3)\lambda _{j,c}, \quad \lambda ^{(2)}_{j,c} = (a_3 + b_3 - c_3)\lambda _{j,c},\nonumber \\&\lambda ^{(3)}_{j,c} = (a_3 - b_3 + c_3)\lambda _{j,c},\end{aligned}$$
(34)
$$\begin{aligned} \lambda ^{(4)}_{j,c}= & {} (a_3 - b_3 - c_3)\lambda _{j,c}, \quad \lambda ^{(5)}_{j,c} = (-a_3 + b_3 + c_3)\lambda _{j,c},\nonumber \\&\lambda ^{(6)}_{j,c} = (-a_3 + b_3 - c_3)\lambda _{j,c},\end{aligned}$$
(35)
$$\begin{aligned} \lambda ^{(7)}_{j,c}= & {} (-a_3 - b_3 + c_3)\lambda _{j,c}, \quad \lambda ^{(8)}_{j,c} = (-a_3 - b_3 - c_3)\lambda _{j,c},\end{aligned}$$
(36)
$$\begin{aligned} \varepsilon _{j,a}= & {} e^{-2i\omega _{j,a} t} r_{j,a}, \quad \varepsilon _{j,b} = e^{-2i\omega _{j,b} t} r_{j,b}, \quad \varepsilon _{j,c} = e^{-2i\omega _{j,c} t} r_{j,c}, \end{aligned}$$
(37)
$$\begin{aligned} \alpha '= & {} \cos \theta _1 \alpha + \sin \theta _1 \cos \theta _2 \beta + \sin \theta _1\sin \theta _2 \gamma , \end{aligned}$$
(38)
$$\begin{aligned} \beta '= & {} -\,\sin \theta _1 \alpha + \cos \theta _1\cos \theta _2 \beta + \cos \theta _1\sin \theta _2\gamma ,\end{aligned}$$
(39)
$$\begin{aligned} \gamma '= & {} -\,\sin \theta _2\beta + \cos \theta _2 \gamma , \quad \theta _1 = \arccos \frac{1}{\sqrt{3}}, \quad \theta _2 = \pi /4. \end{aligned}$$
(40)

Here, \(|A^{(\mu )}_{j,a}, \varepsilon _{j,a}\rangle = S(\varepsilon _{j,a})D(A^{(\mu )}_{j,a}) |0\rangle \), \(|A^{(\mu )}_{j,b}, \varepsilon _{j,b}\rangle = S(\varepsilon _{j,b})D(A^{(\mu )}_{j,b}) |0\rangle \), and \(|A^{(\mu )}_{j,c}, \varepsilon _{j,c}\rangle = S(\varepsilon _{j,c})D(A^{(\mu )}_{j,c}) |0\rangle \) are the squeezed coherent states of modes a, b, and c, respectively. \(D(\alpha ) = e^{\alpha a^{\dag } - \alpha ^* a}\) and \(S(\varepsilon ) = e^{\frac{1}{2}(\varepsilon a^{\dag 2} - \varepsilon ^* a)}\) are the displacement and squeezed operators of modes.

The phase factors of \(f_{j,\mu \nu }(t)\) are as follows:

$$\begin{aligned} \varphi _{j,\mu \mu }= & {} 0, \quad \varphi _{j,12} = -t(\omega _0 + 2\xi _{j,AC} + \xi _{j,BC}) - Im(\varTheta _{j,12}), \nonumber \\ \varphi _{j,13}= & {} -\,t(\omega _0 + 2\xi _{j,AB} + 2\xi _{j,BC}) - Im(\varTheta _{j,13}),\nonumber \\&\varphi _{j,14} = -\,2t(\omega _0 + \xi _{j,AB} + \xi _{j,AC}) - Im(\varTheta _{j,14}),\nonumber \\ \varphi _{j,15}= & {} -\,t(\omega _0 + 2\xi _{j,AB} + 2\xi _{j,AC}) - Im(\varTheta _{j,15}),\nonumber \\&\varphi _{j,16} = -\,2t(\omega _0 + \xi _{j,AB} + \xi _{j,BC}) - Im(\varTheta _{j,16}),\nonumber \\ \varphi _{j,17}= & {} -\,2t(\omega _0 + \xi _{j,AC} + \xi _{j,BC}) - Im(\varTheta _{j,17}), \quad \varphi _{j,18} = -\,3\omega _0 t - Im(\varTheta _{j,18}),\nonumber \\ \varphi _{j,23}= & {} -\,2t(\xi _{j,AB} - \xi _{j,AC}) - Im(\varTheta _{j,23}), \quad \nonumber \\ \varphi _{j,24}= & {} -\,t(\omega _0 + 2\xi _{j,AB} - 2\xi _{j,BC}) - Im(\varTheta _{j,24}),\nonumber \\ \varphi _{j,25}= & {} -\,2t(\xi _{j,AB} - \xi _{j,BC}) - Im(\varTheta _{j,25}), \quad \nonumber \\ \varphi _{j,26}= & {} -\,t(\omega _0 + 2\xi _{j,AB} - 2\xi _{j,AC}) - Im(\varTheta _{j,26}),\nonumber \\ \varphi _{j,27}= & {} -\,\omega _0 t - Im(\varTheta _{j,27}), \quad \varphi _{j,28} = -2t(\omega _0 - \xi _{j,AC} - \xi _{j,BC}) - Im(\varTheta _{j,28}),\nonumber \\ \varphi _{j,34}= & {} -\,t(\omega _0 + 2\xi _{j,AC} - 2\xi _{j,BC}) - Im(\varTheta _{j,34}), \quad \nonumber \\ \varphi _{j,35}= & {} -\,2t(\xi _{j,AC} - \xi _{j,BC}) - Im(\varTheta _{j,35}), \nonumber \\ \varphi _{j,36}= & {} -\,\omega _0 t - Im(\varTheta _{j,36}), \quad \varphi _{j,37} = -t(\omega _0 - 2\xi _{j,AB} + 2\xi _{j,AC}) - Im(\varTheta _{j,37}), \nonumber \\ \varphi _{j,38}= & {} -\,2t(\omega _0 - \xi _{j,AB} - \xi _{j,BC}) - Im(\varTheta _{j,38}), \quad \varphi _{j,45} = \omega _0 t - Im(\varTheta _{j,45}), \nonumber \\ \varphi _{j,46}= & {} 2t(\xi _{j,AC} - \xi _{j,BC}) - Im(\varTheta _{j,46}), \quad \nonumber \\ \varphi _{j,47}= & {} 2t(\xi _{j,AB} - \xi _{j,BC}) - Im(\varTheta _{j,47}),\nonumber \\ \varphi _{j,48}= & {} -\,t(\omega _0 - 2\xi _{j,AB} - 2\xi _{j,AC}) - Im(\varTheta _{j,48}), \quad \nonumber \\ \varphi _{j,56}= & {} -\,t(\omega _0 - 2\xi _{j,AC} + 2\xi _{j,BC}) - Im(\varTheta _{j,56}), \nonumber \\ \varphi _{j,57}= & {} -\,t(\omega _0 - 2\xi _{j,AB} + 2\xi _{j,BC}) - Im(\varTheta _{j,57}), \quad \nonumber \\ \varphi _{j,58}= & {} -\,2t(\omega _0 - \xi _{j,AB} - \xi _{j,AC}) - Im(\varTheta _{j,58}), \nonumber \\ \varphi _{j,67}= & {} 2t(\xi _{j,AB} - \xi _{j,AC}) - Im(\varTheta _{j,67}), \quad \varphi _{j,68} \nonumber \\= & {} -\,t(\omega _0 - 2\xi _{j,AB} - 2\xi _{j,BC}) - Im(\varTheta _{j,68}), \nonumber \\ \varphi _{j,78}= & {} -\,t(\omega _0 - 2\xi _{j,AC} - 2\xi _{j,BC}) - Im(\varTheta _{j,78}),\nonumber \\ \varTheta _{j,\mu \nu }= & {} e^{i\omega _{j,a}t}\left( \lambda ^{(\mu )}_{j,a} \alpha _j^{(\mu )} - \lambda ^{(\nu )}_{j,a} \alpha _j^{(\nu )}\right) +e^{i\omega _{j,b}t}\left( \lambda ^{(\mu )}_{j,b} \beta _j^{(\mu )} - \lambda ^{(\nu )}_{j,b} \beta _j^{(\nu )}\right) \nonumber \\&+\,e^{i\omega _{j,c}t}\left( \lambda ^{(\mu )}_{j,c} \gamma _j^{(\mu )} - \lambda ^{(\nu )}_{j,c} \gamma _j^{(\nu )}\right) , \end{aligned}$$
(41)

where Im(c) is the imaginary part of complex number c.

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Zhang, JS., Zhang, LJ., Chen, AX. et al. Analytical solution and applications of three qubits in three coupled modes without rotating wave approximation. Quantum Inf Process 17, 125 (2018). https://doi.org/10.1007/s11128-018-1891-0

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