Entanglement dynamics of two interacting qubits under the influence of local dissipation

Abstract

We investigate the dynamics of entanglement given by the concurrence of a two-qubit system in the non-Markovian setting. A quantum master equation is derived, which is solved in the eigenbasis of the system Hamiltonian for X-type initial states. A closed formula for time evolution of concurrence is presented for a pure state. It is shown that under the influence of dissipation non-zero entanglement is created in unentangled two-qubit states which decay in the same way as pure entangled states. We also show that under real circumstances, the decay rate of concurrence is strongly modified by the non-Markovianity of the evolution.

Appendix A

In this appendix we write the explicit forms of the various functions used in the main text.

$$\begin{array}{@{}rcl@{}} &&{\kern-.7pc}\eta(t) = \frac{{\Gamma}_{\mathrm{M}} \lambda}{\lambda^{2}\! +\! J^{2}}\! \left[ \lambda({\kern-.7pt}1\! -\! {\mathrm{e}}^{\lambda t}\! \cos{\kern-.7pt} Jt{\kern-.7pt})\! +\! J{\mathrm{e}}^{-\lambda t}\sin{\kern-.7pt}J{\kern-.7pt}t\! \right]. \end{array} $$

(A1)

$$\begin{array}{@{}rcl@{}} &&{\kern-.8pc}{\Sigma}(t)\! =\! \frac{\gamma_{0} \lambda}{\lambda^{2}\! +\! 9 J^{2}}\! \left[ \lambda{\kern-.7pt}({\kern-.5pt}1\! -\! {\mathrm{e}}^{\lambda t}\! \cos 3Jt{\kern-.5pt})\! +\! 3J{\mathrm{e}}^{-\lambda t}\sin 3Jt \right].\\ \end{array} $$

(A2)

$$\begin{array}{@{}rcl@{}} {\Gamma}_{-}(t)\! &=&\! \frac{1}{2} {{\int}_{0}^{t}} \!\!{\mathrm{d}}z~\eta(z) \end{array} $$

(A3)

$$\begin{array}{@{}rcl@{}} &=&\! \frac{{\Gamma}_{\mathrm{M}} \lambda}{2[\lambda^{2} + J^{2}]}\left[ \lambda t\! -\! \frac{\lambda^{2}-J^{2}}{\lambda^{2}+J^{2}}(1-{\mathrm{e}}^{-\lambda t}\cos Jt) \right.\\ && ~~~~~~~~~~~~~~~~~~~~\left. - \frac{2\lambda J}{\lambda^{2}+J^{2}}{\mathrm{e}}^{-\lambda t} \sin Jt \right]. \end{array} $$

(A4)

$$\begin{array}{@{}rcl@{}} {\Gamma}_{+}(t)\! &=&\! \frac{1}{2} {{\int}_{0}^{t}} \!\!{\mathrm{d}}z~{\Sigma}(z) \end{array} $$

(A5)

$$\begin{array}{@{}rcl@{}} &=&\!\frac{{\Gamma}_{\mathrm{M}} \lambda}{2[\lambda^{2}\! +\! 9J^{2}]}\! \left[\! \lambda t\! -\! \frac{\lambda^{2}\! -\! 9J^{2}}{\lambda^{2}\! +\!9J^{2}}({\kern-.5pt}1\! -\! {\mathrm{e}}^{-\lambda t}\cos 3Jt)\right.\\ && ~~~~~~~~~~~~~~~~~~~~\left. - \frac{6\lambda J}{\lambda^{2}\! +\! 9J^{2}}{\mathrm{e}}^{-\lambda t} \sin 3Jt\! \right]{\kern-.7pt}. \end{array} $$

(A6)

$$\begin{array}{@{}rcl@{}} S_{-}(t)\! &=&\!\frac{{\Gamma}_{\mathrm{M}} \lambda}{2[\lambda^{2} + J^{2}]}\!\left[ J t\! +\! \frac{2J}{\lambda^{2}+J^{2}}(1\! -\! {\mathrm{e}}^{-\lambda t}\cos Jt) \right.\\ && ~~~~~~~~~~~~~~~~~~\left. + \frac{\lambda^{2}-J^{2}}{\lambda^{2}+J^{2}}{\mathrm{e}}^{-\lambda t} \sin Jt \right]. \end{array} $$

(A7)

$$\begin{array}{@{}rcl@{}} S_{+}(t)\! &=&\! \frac{{\Gamma}_{\mathrm{M}} \lambda}{2{\kern-.5pt}[{\kern-.5pt}\lambda^{2}\! +\! 9{\kern-.5pt}J^{2}{\kern-.5pt}]}\! \left[\! 3{\kern-.5pt}J{\kern-.5pt} t\! +\!\frac{6J}{\lambda^{2}\! +\! 9J^{2}}{\kern-.5pt}({\kern-.5pt}1\! -\!{\mathrm{e}}^{-\lambda t}{\kern-.5pt}\cos{\kern-.5pt}3{\kern-.5pt}J{\kern-.5pt}t{\kern-.5pt}) \right.\\ && ~~~~~~~~~~~~~~~~~~~~\left. + \frac{\lambda^{2}-9J^{2}}{\lambda^{2}+9J^{2}}{\mathrm{e}}^{-\lambda t} \sin3 Jt \right] .\end{array} $$

(A8)