Spatial entanglement in two-electron atomic systems

Recently, there have been considerable interests to investigate quantum entanglement in two-electron model atoms [1, 2]. Here we investigate quantum entanglement for the ground and excited states of two-electron atomic systems using correlated wave functions, concentrating on the particle-particle entanglement coming from the continuous spatial degrees of freedom. We use the two-electron wave functions constructed by employing $B$-spline basis to calculate the linear entropy of the reduced density matrix $L=1-Tr_A (\rho _A^2 )$ as a measure of the spatial entanglement. Here $\rho _A =Tr_B (\left| \varphi \right\rangle _{AB} { }_{AB}\left\langle \varphi \right|)$ is the one-electron reduced density matrix obtained after tracing the two-electron density matrix over the degrees of freedom of the other electron. Here, we investigate spatial entanglement for two-electron systems with $Z$=1 to $Z$=10. When $Z$ is decreased from $Z$=1.0 to about $Z \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} $ 0.911, the H$^{-}$ ion becomes unbound. This would lead in a situation of one electron bound by the nucleus with the other electron being free. Such a wave function would be expected to have a spatial entanglement of $L $= 1/2. Numerical results will be presented at the meeting. \\[4pt] [1] J. P. Coe and I. D'Amico, \textit{J. Phys.: Conf. Ser.} \textbf{254}, 012010 (2010) \\[0pt] [2] D. Manzano \textit{et. al.}, \textit{J. Phys. A: Math. Theor.} \textbf{43}, 275301 (2010)