Protecting bipartite entanglement by collective decay and quantum interferences

Abstract

We study the entanglement dynamics of a pair of identical three-level V-type atoms that are interacting with a common vacuum reservoir. The effect of vacuum-induced coherence in the spontaneous emission of atoms is included in the atomic dynamics. By considering qubits formed from each atom, the entanglement dynamics of the two-qubit system is studied using concurrence as the entanglement measure. We analytically calculate the time evolution of the concurrence for initial entangled states and show that the entanglement among qubits can be preserved in the longtime limit due to vacuum-induced coherence. We demonstrate that the combined effects of vacuum-induced coherence and collective decay of atoms in the common reservoir can protect the entanglement to a better extent in comparison with the case of atoms radiating independently.

Appendix equations of motion

The density matrix elements written in the basis (4) obey the coupled differential equations obtained from the master equation (1). The equations for the time-evolving density matrix elements are given by

$$\begin{aligned} \dot{\rho }_{11}=&-4\gamma _{1}\rho _{11}-\gamma _{12}(\rho _{18}+\rho _{19}+\rho _{81}+\rho _{91}),\nonumber \\ \dot{\rho }_{22}=&-2\gamma _{1}\rho _{22}+2\gamma _{1}\rho _{11}+2\gamma _{2}\rho _{88}-\gamma _{12}\left( \rho _{26}+ \rho _{62} \right. \nonumber \\&\left. -2(\rho _{18}+\rho _{81}) \right) -\Gamma _{1}(\rho _{23}+\rho _{32})-\Gamma _{vc}(\rho _{27}+\rho _{72}),\nonumber \\ \dot{\rho }_{33}=&-2\gamma _{1}\rho _{33}+2\gamma _{1}\rho _{11}+2\gamma _{2}\rho _{99}-\gamma _{12}\left( \rho _{37} +\rho _{73}\right. \nonumber \\&\left. -2(\rho _{19}+\rho _{91})\right) -\Gamma _{1}(\rho _{23}+\rho _{32})-\Gamma _{vc}(\rho _{36}+\rho _{63}),\nonumber \\ \dot{\rho }_{44}=&2\gamma _{1}(\rho _{22}+\rho _{33})+2\gamma _{2}(\rho _{66}+\rho _{77})+2\gamma _{12}\left( \rho _{26} +\rho _{37}\right. \nonumber \\&\left. +\rho _{62}+\rho _{73} \right) +2\Gamma _{1}(\rho _{23}+\rho _{32})+2\Gamma _{2}(\rho _{67}+\rho _{76})\nonumber \\&+2\Gamma _{vc}(\rho _{27}+\rho _{36}+\rho _{63}+\rho _{72}),\nonumber \\ \dot{\rho }_{55}=&-4\gamma _{2}\rho _{55}-\gamma _{12}(\rho _{58}+\rho _{59}+\rho _{85}+\rho _{95}),\nonumber \\ \dot{\rho }_{66}=&-2\gamma _{2}\rho _{66}+2\gamma _{1}\rho _{99}+2\gamma _{2}\rho _{55}-\gamma _{12} \left( \rho _{26} +\rho _{62}\right. \nonumber \\&-2(\rho _{59}+\rho _{95}))-\Gamma _{2}(\rho _{67}+\rho _{76})-\Gamma _{vc}(\rho _{36}+\rho _{63}),\nonumber \\ \dot{\rho }_{88}=&-2(\gamma _{1}+\gamma _{2})\rho _{88}-\gamma _{12}(\rho _{18}+\rho _{58}+\rho _{81} +\rho _{85}),\nonumber \\ \dot{\rho }_{77}=&~\dot{\rho }_{66}|_{6\leftrightarrow 7,8\leftrightarrow 9,2\leftrightarrow 3},~ \dot{\rho }_{99}=~\dot{\rho }_{88}|_{8\leftrightarrow 9},\nonumber \\ \dot{\rho }_{14}=&-2\gamma _{1}\rho _{14}-\gamma _{12}(\rho _{84}+\rho _{94}),\nonumber \\ \dot{\rho }_{15}=&-2(\gamma _{1}+\gamma _{2})\rho _{15}-\gamma _{12}(\rho _{18}+\rho _{19}+\rho _{85} +\rho _{95}),\nonumber \\ \dot{\rho }_{18}=&-(3\gamma _{1}+\gamma _{2})\rho _{18}-\gamma _{12}(\rho _{11}+\rho _{15}+\rho _{88} +\rho _{98}),\nonumber \\ \dot{\rho }_{19}=&~\dot{\rho }_{18}|_{8\leftrightarrow 9},\nonumber \\ \dot{\rho }_{23}=&-2\gamma _{1}\rho _{23}-\gamma _{12}(\rho _{27}+\rho _{63})+2\Gamma _{2}\rho _{89} -\Gamma _{1}\left( \rho _{22}\right. \nonumber \\&\left. +\rho _{33}-2\rho _{11}\right) -\Gamma _{vc}(\rho _{26}+\rho _{73}-2(\rho _{19}+\rho _{81})),\nonumber \\ \dot{\rho }_{26}=&-(\gamma _{1}+\gamma _{2})\rho _{26}-\gamma _{12}(\rho _{22}+\rho _{66}-2(\rho _{15} +\rho _{89}))\nonumber \\&+2\gamma _{1}\rho _{19}+2\gamma _{2}\rho _{85}-\Gamma _{1}\rho _{36}-\Gamma _{2}\rho _{27}\nonumber \\&-\Gamma _{vc}(\rho _{23}+\rho _{76}),\nonumber \\ \dot{\rho }_{27}=&-(\gamma _{1}+\gamma _{2})\rho _{27}-\gamma _{12}(\rho _{23}+\rho _{67})+\Gamma _{1}(2\rho _{18} -\rho _{37})\nonumber \\&+\Gamma _{2}(2\rho _{85}-\rho _{26})-\Gamma _{vc}(\rho _{22}+\rho _{77}-2(\rho _{15}+\rho _{88})),\nonumber \\ \dot{\rho }_{36}=&~\dot{\rho }_{27}|_{2\leftrightarrow 3,7\leftrightarrow 6,8\leftrightarrow 9},~ \dot{\rho }_{37}=~\dot{\rho }_{26}|_{2\leftrightarrow 3,7\leftrightarrow 6,8\leftrightarrow 9},\nonumber \\ \dot{\rho }_{45}=&-2\gamma _{2}\rho _{45}-\gamma _{12}(\rho _{48}+\rho _{49}),\nonumber \\ \dot{\rho }_{48}=&-(\gamma _{1}+\gamma _{2})\rho _{48}-\gamma _{12}(\rho _{41}+\rho _{45}),\nonumber \\ \dot{\rho }_{49}=&~\dot{\rho }_{48}|_{8\leftrightarrow 9},\nonumber \\ \dot{\rho }_{58}=&-(\gamma _{1}+3\gamma _{2})\rho _{58}-\gamma _{12}(\rho _{51}+\rho _{55}+\rho _{88} +\rho _{98}),\nonumber \\ \dot{\rho }_{59}=&~\dot{\rho }_{58}|_{8\leftrightarrow 9},\nonumber \\ \dot{\rho }_{67}=&-2\gamma _{2}\rho _{67}-\gamma _{12}(\rho _{27}+\rho _{63})+2\Gamma _{1}\rho _{98} +2\Gamma _{2}\rho _{55}\nonumber \\&-\Gamma _{2}(\rho _{66}+\rho _{77})-\Gamma _{vc}(\rho _{37}+\rho _{62}) \nonumber \\&+2\Gamma _{vc}(\rho _{58}+\rho _{95}),\nonumber \\ \dot{\rho }_{89}=&-2(\gamma _{1}+\gamma _{2})\rho _{89}-\gamma _{12}(\rho _{19}+\rho _{59}+\rho _{81}+\rho _{85}). \end{aligned}$$

(A.1)

The other density matrix elements are decoupled from the above equations and are identically zero at all times with the initial conditions considered in this paper.