Momentum disequilibrium and quantum entanglement of Rydberg multidimensional states

The quantum entanglement of the two components of a hydrogenic system with dimensionality $$D\ge 2$$ is investigated for the ground and excited states from first principles, that is, in terms of the Coulomb potential parameters (the dimensionality and the nuclear charge) and the state’s hyperquantum numbers. To quantify this multidimensional entanglement, we use an heuristic quantifier and a practical genuine entanglement measure which are closely related to the variance and disequilibrium of the system in momentum space, respectively. Then, our interest is focused on the multidimensional entanglement of highly excited (Rydberg) states, obtaining at the leading order a simple dependence on the dimensionality and the principal hyperquantum number n which characterizes the state. Applications to various specific low-lying and high-lying hydrogenic states are shown. In particular, it is rigorously shown that the momentum disequilibrium and the entanglement for the Rydberg multidimensional states follow a scaling law of $$n^2\log \,n$$ and $$n^{2(D-1)}$$ type for two-dimensional and D-dimensional ( $$D>2$$ ) systems, respectively.