  We propose an approach to entangle spins of electrons floating on liquid
helium by coherently manipulating their spin-orbit interactions. The
configuration consists of single electrons, confined individually on liquid
helium by the microelectrodes, moving along the surface as the harmonic
oscillators. It has been known that the spin of an electron could be coupled to
its orbit (i.e., the vibrational motion) by properly applying a magnetic field.
Based on this single electron spin-orbit coupling, here we show that a
Jaynes-Cummings (JC) type interaction between the spin of an electron and the
orbit of another electron at a distance could be realized via the strong
Coulomb interaction between the electrons. Consequently, the proposed JC
interaction could be utilized to realize a strong orbit-mediated spin-spin
coupling and implement the desirable quantum information processing between the
distant electrons trapped individually on liquid helium.
