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Post-Markovian dynamics of quantum correlations: entanglement versus discord

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Abstract

Dynamics of an open two-qubit system is investigated in the post-Markovian regime, where the environments have a short-term memory. Each qubit is coupled to separate environment which is held in its own temperature. The inter-qubit interaction is modeled by XY–Heisenberg model in the presence of spin–orbit interaction and inhomogeneous magnetic field. The dynamical behavior of entanglement and discord has been considered. The results show that quantum discord is more robust than quantum entanglement, during the evolution. Also the asymmetric feature of quantum discord can be monitored by introducing the asymmetries due to inhomogeneity of magnetic field and temperature difference between the reservoirs. By employing proper parameters of the model, it is possible to maintain nonvanishing quantum correlation at high degree of temperature. The results can provide a useful recipe for studying dynamical behavior of two-qubit systems such as trapped spin electrons in coupled quantum dots.

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Acknowledgements

The author wishes to thank The Office of Graduate Studies and Research Vice President of The University of Isfahan for their support.

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Authors and Affiliations

  1. Department of Physics, University of Isfahan, Isfahan, Iran

    Hamidreza Mohammadi

  2. Quantum Optics Group, University of Isfahan, Isfahan, Iran

    Hamidreza Mohammadi

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  1. Hamidreza Mohammadi
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Correspondence to Hamidreza Mohammadi.

Appendix 1: Elements of matrix P

Appendix 1: Elements of matrix P

The elements of matrix \(P(t)=[p_{ij}]_{4\times 4}\) in Eq. (18) can be written as follows:

$$\begin{aligned} p_{1\,1}= & {} \frac{1}{X_{1}Y_{2}} \left[ \frac{X_{1}^{-}Y_{2}^{+}({\hbox {e}}^{-t\gamma _{0}}X_{1} -{\hbox {e}}^{-tX_{1}}\gamma _{0})}{X_{1}-\gamma _{0}}+ \frac{X_{1}^{+}Y_{2}^{-}\left( {\hbox {e}}^{-t\gamma _{0}}Y_{2}- {\hbox {e}}^{-tY_{2}}\gamma _{0}\right) }{Y_{2}-\gamma _{0}}\right. \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{-}Y_{2}^{-}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t\left( X_{1}+Y_{2}-\gamma _{0}\right) }\gamma _{0}\right) }{X_{1}+Y_{2}-\gamma _{0}}+X_{1}^{+}Y_{2}^{+}\right] ,\\ p_{1\,2}= & {} \frac{X_{1}^{+}Y_{2}^{+}}{X_{1}Y_{2}}\left[ \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}Y_{2}(X_{1}+Y_{2}-2\gamma _{0})}{(X_{1}-\gamma _{0})(X_{1}+Y_{2}-\gamma _{0})(\gamma _{0}-Y_{2})}+\frac{{\hbox {e}}^{-tX_{1}}\gamma _{0}}{X_{1}-\gamma _{0}}+\frac{{\hbox {e}}^{-tY_{2}}\gamma _{0}}{Y_{2}-\gamma _{0}}\right. \\&\left. - \frac{{\hbox {e}}^{-t(X_{1}+Y_{2})}\gamma _{0}}{X_{1}+Y_{2}-\gamma _{0}}+1\right] \\ p_{1\,3}= & {} \frac{X_{1}^{+}}{X_{1}Y_{2}} \left[ -\frac{\left( {\hbox {e}}^{-tX_{1}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}} X_{1}\right) Y_{2}^{+}}{\gamma _{0}-X_{1}}+\frac{Y_{2}^{-} \left( {\hbox {e}}^{-tY_{2}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}Y_{2}\right) }{ \gamma _{0}-Y_{2}}\right. \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}Y_{2}^{-} \left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+Y_{2}-\gamma _{0})} \gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+Y_{2}^{+}\right] ,\\ p_{1\,4}= & {} \frac{Y_{2}^{+}}{X_{1}Y_{2}} \left[ -\frac{\left( {\hbox {e}}^{-tY_{2}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}} Y_{2}\right) X_{1}^{+}}{\gamma _{0}-Y_{2}}+\frac{X_{1}^{-} \left( {\hbox {e}}^{-tX_{1}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}X_{1}\right) }{\gamma _{0}-X_{1}}\right. \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{-}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+Y_{2} -\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+X_{1}^{+}\right] , \end{aligned}$$
$$\begin{aligned} p_{2\,1}= & {} \frac{X_{1}^{-}Y_{1}^{-}}{X_{1}Y_{2}}\left[ \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}Y_{2}(X_{1}+Y_{2}-2\gamma _{0})}{(X_{1}-\gamma _{0})(X_{1}+Y_{2}-\gamma _{0})(\gamma _{0}-Y_{2})}+\frac{{\hbox {e}}^{-tX_{1}}\gamma _{0}}{X_{1}-\gamma _{0}}+\frac{{\hbox {e}}^{-tY_{2}}\gamma _{0}}{Y_{2}-\gamma _{0}}\right. \\&\left. - \frac{{\hbox {e}}^{-t(X_{1}+Y_{2})}\gamma _{0}}{X_{1}+Y_{2}-\gamma _{0}}+1\right] ,\\ p_{2\,2}= & {} \frac{1}{X_{1}Y_{2}}\left[ \frac{X_{1}^{+}Y_{2}^{-}({\hbox {e}}^{-t\gamma _{0}}X_{1}-{\hbox {e}}^{-tX_{1}}\gamma _{0})}{X_{1}-\gamma _{0}}+\frac{X_{1}^{-}Y_{2}^{+}\left( {\hbox {e}}^{-t\gamma _{0}}Y_{2}-{\hbox {e}}^{-tY_{2}}\gamma _{0}\right) }{Y_{2}-\gamma _{0}}\right. \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{+}Y_{2}^{+}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t\left( X_{1}+Y_{2}-\gamma _{0}\right) }\gamma _{0}\right) }{X_{1}+Y_{2}-\gamma _{0}}+X_{1}^{-}Y_{2}^{-}\right] ,\\ p_{2\,3}= & {} \frac{Y_{2}^{-}}{X_{1}Y_{2}} \left[ \frac{X_{1}^{+}\left( {\hbox {e}}^{-tX_{1}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}X_{1}\right) }{\gamma _{0}-X_{1}}-\frac{\left( {\hbox {e}}^{-tY_{2}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}Y_{2}\right) X_{1}^{-}}{\gamma _{0}-Y_{2}}\right. \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{+}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+Y_{2}-\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+X_{1}^{-}\right] ,\\ p_{2\,4}= & {} \frac{X_{1}^{-}}{X_{1}Y_{2}} \left[ -\frac{\left( {\hbox {e}}^{-tX_{1}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}} X_{1}\right) Y_{2}^{-}}{\gamma _{0}-X_{1}}+\frac{Y_{2}^{+} \left( {\hbox {e}}^{-tY_{2}}\gamma _{0}{\hbox {e}}^{-t\gamma _{0}}Y_{2}\right) }{\gamma _{0} -Y_{2}}\right. \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}Y_{2}^{+}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+Y_{2}-\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+Y_{2}^{-}\right] , \end{aligned}$$
$$\begin{aligned} p_{3\,1}= & {} \frac{X_{1}^{-}}{X_{1}Y_{2}} \left[ -\frac{\left( {\hbox {e}}^{-tX_{1}}\gamma _{0} -{\hbox {e}}^{-t\gamma _{0}}X_{1}\right) Y_{2}^{+}}{\gamma _{0}-X_{1}}+ \frac{Y_{2}^{-}\left( {\hbox {e}}^{-tY_{2}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}} Y_{2}\right) }{\gamma _{0}-Y_{2}}\right. \nonumber \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}Y_{2}^{-}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+ Y_{2}-\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+Y_{2}^{+}\right] , \nonumber \\ p_{3\,2}= & {} \frac{Y_{2}^{+}}{X_{1}Y_{2}}\left[ \frac{X_{1}^{+}\left( {\hbox {e}}^{-tX_{1}}\gamma _{0} -{\hbox {e}}^{-t\gamma _{0}}X_{1}\right) }{\gamma _{0}-X_{1}}-\frac{\left( {\hbox {e}}^{-tY_{2}} \gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}Y_{2}\right) X_{1}^{-}}{\gamma _{0}-Y_{2}}\right. \nonumber \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{+}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t (X_{1}+Y_{2}-\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+ \gamma _{0}}+X_{1}^{-}\right] ,\nonumber \\ p_{3\,3}= & {} \frac{1}{X_{1}Y_{2}} \left[ \frac{X_{1}^{+}Y_{2}^{+}({\hbox {e}}^{-t\gamma _{0}}X_{1} -{\hbox {e}}^{-tX_{1}}\gamma _{0})}{X_{1}-\gamma _{0}}+ \frac{X_{1}^{-}Y_{2}^{-}\left( {\hbox {e}}^{-t\gamma _{0}}Y_{2}- {\hbox {e}}^{-tY_{2}}\gamma _{0}\right) }{Y_{2}-\gamma _{0}}\right. \nonumber \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{+}Y_{2}^{-}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t\left( X_{1}+Y_{2}-\gamma _{0}\right) }\gamma _{0}\right) }{X_{1}+Y_{2}-\gamma _{0}}+X_{1}^{-}Y_{2}^{+}\right] ,\nonumber \\ p_{3\,4}= & {} \frac{X_{1}^{-}Y_{2}^{+}}{X_{1}Y_{2}} \left[ \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}Y_{2}(X_{1}+Y_{2}-2 \gamma _{0})}{(X_{1}-\gamma _{0})(X_{1}+Y_{2}-\gamma _{0}) (\gamma _{0}-Y_{2})}+\frac{{\hbox {e}}^{-tX_{1}}\gamma _{0}}{X_{1}-\gamma _{0}}+ \frac{{\hbox {e}}^{-tY_{2}}\gamma _{0}}{Y_{2}-\gamma _{0}}\right. \nonumber \\&\left. - \frac{{\hbox {e}}^{-t(X_{1}+Y_{2})}\gamma _{0}}{X_{1}+ Y_{2}-\gamma _{0}}+1\right] \nonumber \\ p_{4\,1}= & {} \frac{Y_{2}^{-}}{X_{1}Y_{2}}\left[ -\frac{\left( {\hbox {e}}^{-tY_{2}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}Y_{2}\right) X_{1}^{+}}{\gamma _{0}-Y_{2}}+\frac{X_{1}^{-}\left( {\hbox {e}}^{-tX_{1}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}X_{1}\right) }{\gamma _{0}-X_{1}}\right. \nonumber \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{-}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+Y_{2}-\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+X_{1}^{+}\right] ,\nonumber \\ p_{4\,2}= & {} \frac{X_{1}^{+}}{X_{1}Y_{2}}\left[ -\frac{\left( {\hbox {e}}^{-tX_{1}}\gamma _{0}-{\hbox {e}}^{-t\gamma _{0}}X_{1}\right) Y_{2}^{-}}{\gamma _{0}-X_{1}}+\frac{Y_{2}^{+}\left( {\hbox {e}}^{-tY_{2}}\gamma _{0}{\hbox {e}}^{-t\gamma _{0}}Y_{2}\right) }{\gamma _{0}-Y_{2}}\right. \nonumber \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}Y_{2}^{+}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t(X_{1}+Y_{2}-\gamma _{0})}\gamma _{0}\right) }{-X_{1}-Y_{2}+\gamma _{0}}+Y_{2}^{-}\right] ,\nonumber \\ p_{4\,3}= & {} \frac{X_{1}^{+}Y_{2}^{-}}{X_{1}Y_{2}} \left[ \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}Y_{2}(X_{1}+Y_{2}-2\gamma _{0})}{(X_{1}-\gamma _{0})(X_{1}+Y_{2}-\gamma _{0})(\gamma _{0}-Y_{2})}+\frac{{\hbox {e}}^{-tX_{1}}\gamma _{0}}{X_{1}-\gamma _{0}}+\frac{{\hbox {e}}^{-tY_{2}}\gamma _{0}}{Y_{2}-\gamma _{0}}\right. \nonumber \\&\left. - \frac{{\hbox {e}}^{-t(X_{1}+Y_{2})}\gamma _{0}}{X_{1}+Y_{2}-\gamma _{0}}+1\right] \nonumber \\ p_{4\,4}= & {} \frac{1}{X_{1}Y_{2}}\left[ \frac{X_{1}^{-}Y_{2}^{-}({\hbox {e}}^{-t\gamma _{0}}X_{1}-{\hbox {e}}^{-tX_{1}}\gamma _{0})}{X_{1}-\gamma _{0}}+\frac{X_{1}^{+}Y_{2}^{+}\left( {\hbox {e}}^{-t \gamma _{0}}Y_{2}-{\hbox {e}}^{-tY_{2}}\gamma _{0}\right) }{Y_{2}-\gamma _{0}}\right. \nonumber \\&\left. + \frac{{\hbox {e}}^{-t\gamma _{0}}X_{1}^{-}Y_{2}^{+}\left( X_{1}+Y_{2}-{\hbox {e}}^{-t\left( X_{1}+Y_{2}-\gamma _{0}\right) }\gamma _{0}\right) }{X_{1}+Y_{2}-\gamma _{0}}+X_{1}^{+}Y_{2}^{-}\right] . \end{aligned}$$
(21)

Here we have defined \(X_{\mu }=X_{\mu }^{+}+X_{\mu }^{-}\) and \(Y_{\mu }=Y_{\mu }^{+}+Y_{\mu }^{-}\).

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Mohammadi, H. Post-Markovian dynamics of quantum correlations: entanglement versus discord. Quantum Inf Process 16, 39 (2017). https://doi.org/10.1007/s11128-016-1451-4

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