Focusing on the Dynamics of the Entanglement in Spin Junction

Abstract

We study the dynamics of entanglement in the one-dimensional spin-1/2 XY model in the presence of a transverse magnetic field. A pair of spins are considered as an open quantum system, while the rest of the chain plays the role of the environment. Our study focuses on the pair of spins in the system, the edge spins, and the environment. It is observed that the entanglement between the pair of spins in the system decreases and it can transfer to the rest of the spins. For a value of anisotropy leading to the Ising model, the entanglement is completely back to the system by passing time. On the other hand, the entanglement can only be seen under certain conditions between edge spins of the system and the environment. The pair of spins on the edge will be entangled very quickly and it will disappear after a very short time. A pair of spins far from the system was chosen to examine the behavior of entanglement in the environment. As expected, the transmission of entanglement from the system to the environment takes notable time.

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Acknowledgments

It is our pleasure to thank T. Mohammad Ali Zadeh.

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Correspondence to S. Mahdavifar.

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Appendix

Appendix

Here, we want to express one part of our calculations. We calculate \(\langle a^{\dag }_{m}(t) a_{m}(t)\rangle \) as an example.

$$\begin{array}{@{}rcl@{}} \langle a^{\dag}_{m}(t) a_{m}(t)\rangle &=& \langle\psi_{0}|e^{iHt}a^{\dag}_{m}e^{-iHt}e^{iHt}a_{m}e^{-iHt}|\psi_{0}\rangle \\ &=&\frac{1}{\sqrt{N}}\sum\limits_{k}e^{i(k_{1}-k_{2})m}\\ &&\langle\psi_{0}|e^{iHt}a^{\dag}_{k_{1}}e^{-iHt}e^{iHt}a_{k_{2}}e^{-iHt}|\psi_{0}\rangle. \\ \end{array} $$
(16)

Using the Bogoliubov operators, Eq. (16) can be written as follows

$$\begin{array}{@{}rcl@{}} \langle a^{\dag}_{m}(t) a_{m}(t)\rangle &=&\frac{1}{\sqrt{N}}\sum\limits_{k}e^{i(k_{1}-k_{2})m}\\ &&\langle\psi_{0}|(\text{cos}(k_{1})e^{it\varepsilon(k_{1})}\beta_{k_{1}}^{\dagger} \\ &&+i \sin(k_{1})e^{-it\varepsilon(-k_{1})}\beta_{-k_{1}}) \\ &&(\text{cos}(k_{2})e^{-it\varepsilon(k_{2})}\beta_{k_{2}}^{\dagger} \\ &&-i\sin(k_{2})e^{it\varepsilon(-k_{2})}\beta_{-k_{2}}^{\dagger})|\psi_{0}\rangle. \end{array} $$
(17)

Finally, the above relation is simplified as

$$\begin{array}{@{}rcl@{}} \langle a^{\dag}_{m}(t) a_{m}(t)\rangle &=& \frac{1}{2 N^{2}}\sum\limits_{k,k^{\prime}}(1+e^{i(k+\phi)}+e^{-i(k^{\prime}+\phi))} \\ &&+e^{i(k-k^{\prime})})\\ &&(e^{-it(\varepsilon(k)-\varepsilon(k^{\prime}))}\text{cos}^{2}(k)\text{cos}^{2}(k^{\prime}) \\ &&+e^{it(\varepsilon(k)+\varepsilon(k^{\prime}))}\sin^{2}(k)\text{cos}^{2}(k^{\prime})\\ &&+e^{-it(\varepsilon(k)+\varepsilon(k^{\prime}))}\sin^{2}(k^{\prime})\text{cos}^{2}(k)\\ &&+ e^{it(\varepsilon(k)-\varepsilon(k^{\prime}))}\sin^{2}(k)\sin^{2}(k^{\prime})\\ &&+\frac{1}{4}\sin(2k)\sin(2k^{\prime})(-e^{it(\varepsilon(k)-\varepsilon(k^{\prime}))}\\ &&+e^{it(\varepsilon(k)+\varepsilon(k^{\prime}))}\\ &&+e^{-it(\varepsilon(k)+\varepsilon(k^{\prime}))}-e^{-it(\varepsilon(k)-\varepsilon(k^{\prime}))}))\\ &&+ \frac{1}{4 N^{2}}\sum\limits_{k,k^{\prime}}(\sin^{2}(2k^{\prime}) (1+\text{cos}(k+\phi))\\ &&(2-e^{2it(\varepsilon(k^{\prime})}-e^{-2it(\varepsilon(k^{\prime})}). \end{array} $$
(18)

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Saghafi, Z., Shadman, Z., Lapasar, E.H. et al. Focusing on the Dynamics of the Entanglement in Spin Junction. J Supercond Nov Magn 32, 2865–2870 (2019). https://doi.org/10.1007/s10948-019-5045-0

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Keywords

  • Spin-1/2 XY chain
  • Open quantum system
  • Entanglement
  • Dynamics