  The generation and evolution of entanglement in quantum many-body systems is
an active area of research that spans multiple fields, from quantum information
science to the simulation of quantum many-body systems encountered in condensed
matter, subatomic physics, and quantum chemistry. Motivated by recent
experiments exploring quantum information processing systems with electrons
trapped above the surface of cryogenic noble gas substrates, we theoretically
investigate the generation of \emph{motional} entanglement between two
electrons via their unscreened Coulomb interaction. The model system consists
of two electrons confined in separate electrostatic traps which establish
microwave frequency quantized states of their motion. We compute the motional
energy spectra of the electrons, as well as their entanglement, by
diagonalizing the model Hamiltonian with respect to a single-particle Hartree
product basis. This computational procedure can in turn be employed for device
design and guidance of experimental implementations. In particular, the
theoretical tools developed here can be used for fine tuning and optimization
of control parameters in future experiments with electrons trapped above the
surface of superfluid helium or solid neon.
