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Entanglement rebirth of multi-trapped ions with trap phonon modes: entanglement sudden death with recovery

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Abstract

We consider a multi-qubit system consisting of two trapped ions coupled in a laser field. The ions are identical three-level electronic systems which interact with one another through the phonon modes of their relative or center of mass motions, and the system is tuned so that two-phonon processes dominate the electronic transitions. The resulting evolution of the system is studied theoretically with a focus on the entanglement properties of the system. A method of quantifying the entanglement is discussed, and the time dependence of these quantifications is determined. The cases of the two ions coupled to the same phonon field and to two different isolated phonon fields are compared for Fock cavity modes. Instances of the entanglement sudden death recovery are identified in these various systems.

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Acknowledgments

ARM wishes to thank the Department of Physics and Astronomy at University of California, Riverside, for the use of its library facilities.

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Authors and Affiliations

  1. Laboratoire Collisions Agrégats Réactivité - UMR5589, Université Paul Sabatier, 31062, Toulouse Cedex 09, France

    M. Abdel-Aty & M. Bouchene

  2. University of Science and Technology, Zewail City, Egypt

    M. Abdel-Aty

  3. Mathematics Department, Faculty of Science, Sohag University, Sohâg, Egypt

    M. Abdel-Aty

  4. Department of Physics, Western Michigan University, 1903 W. Michigan Ave., Kalamazoo, MI, 49008-5252, USA

    A. R. McGurn

Authors
  1. M. Abdel-Aty
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  2. M. Bouchene
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  3. A. R. McGurn
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Corresponding author

Correspondence to A. R. McGurn.

Appendix

Appendix

The matrix elements of \(\rho (t)\) in Eq. (12) are obtained directly from Eq. (11) and are given by:

$$\begin{aligned} B&= \sum _{n=0}^\infty |A_1(n,t)+A_8(n+4,t)|^2, \end{aligned}$$
(17)
$$\begin{aligned} C&= \sum _{n=0}^\infty |A_2(n-2,t)+A_6(n+2,t)|^2, \end{aligned}$$
(18)
$$\begin{aligned} D&= \sum _{n=0}^\infty |A_3(n-2,t)+A_7(n+2,t)|^2, \end{aligned}$$
(19)
$$\begin{aligned} E&= \sum _{n=0}^\infty |A_4(n-4,t)+A_5(n,t)|^2, \end{aligned}$$
(20)
$$\begin{aligned} B_1&= \sum _{n=0}^\infty \left[ A_1(n,t)+A_8(n+4,t)\right] \left[ A_2^*(n-2,t)+A_6^*(n+2,t)\right] \!, \end{aligned}$$
(21)
$$\begin{aligned} B_2&= \sum _{n=0}^\infty \left[ A_1(n,t)+A_8(n+4,t)\right] \left[ A_3^*(n-2,t)+A_7^*(n-2,t)\right] \!, \end{aligned}$$
(22)
$$\begin{aligned} B_3&= \sum _{n=0}^\infty \left[ A_1(n,t)+A_8(n+4,t)\right] \left[ A_4^*(n-4,t)+A_5^*(n,t)\right] \!, \end{aligned}$$
(23)
$$\begin{aligned} C_1&= \sum _{n=0}^\infty [A_2(n-2,t)+A_6(n+2,t)][A_3^*(n-2,t)+A_7^*(n+2,t)], \end{aligned}$$
(24)
$$\begin{aligned} C_2&= \sum _{n=0}^\infty \left[ A_2(n-2,t)+A_6(n+2,t)\right] \left[ A_4^*(n-4,t)+A_5^*(n,t)\right] \!, \end{aligned}$$
(25)
$$\begin{aligned} D_1&= \sum _{n=0}^\infty \left[ A_3(n-2,t)+A_7(n+2,t)\right] \left[ A_4^*(n-4,t)+A_5^*(n,t)\right] \!. \end{aligned}$$
(26)

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Abdel-Aty, M., Bouchene, M. & McGurn, A.R. Entanglement rebirth of multi-trapped ions with trap phonon modes: entanglement sudden death with recovery. Quantum Inf Process 13, 1937–1950 (2014). https://doi.org/10.1007/s11128-014-0787-x

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  • DOI: https://doi.org/10.1007/s11128-014-0787-x

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