Quantum entanglement in a soluble two-electron model atom

We investigate the entanglement properties of bound states in an exactly soluble two-electron model, the Moshinsky atom. We present exact entanglement calculations for the ground, first and second excited states of the system. We find that these states become more entangled when the relative inter-particle interaction becomes stronger. As a general trend, we also observe that the entanglement of the eigenstates tends to increase with the states’ energy. There are, however, “entanglement level-crossings” where the entanglement of a state becomes larger than the entanglement of other states with higher energy. In the limit of weak interaction, we also compute (exactly) the entanglement of higher excited states. Excited states with anti-parallel spins are found to involve a considerable amount of entanglement even for an arbitrarily weak (but non zero) interaction. This minimum amount of entanglement increases monotonically with the state’s energy. Finally, the connection between entanglement and the Hartree-Fock approximation in the Moshinsky model is addressed. The quality of the ground-state Hartree-Fock approximation is shown to deteriorate, and the corresponding correlation energy to grow, as the entanglement of the (exact) ground state increases. The present work goes beyond previous related studies because we fully take into account the identical character of the two constituting particles in the entanglement calculations, and provide analytical, exact results both for the ground and the first few excited states.