Spin-split excitation gap and spin entanglement of a pair of interacting electrons in a quantum dot

We calculate the energy eigenvalues, the spin-split excitation gap (energy separation between the spin-triplet excited state and the spin-singlet ground state) and the concurrence for two interacting electrons captured in a quantum dot (QD) formed by a gigahertz electron pump which is modeled by harmonic confining potentials. We find from our calculations a peak in the QD size dependence of the energy level for the spin-singlet ground state, indicating the effect due to Coulomb blockade. In addition, we observe a local minimum in the QD size dependence of the spin-split excitation gap for a relatively narrow quasi-one-dimensional (1D) channel formed from an etched wire, but a strong positive peak for the spin-split excitation gap in its QD size dependence with a relatively wide 1D channel. From the existence of a robust spin-split excitation gap against both thermal fluctuation due to finite (low) temperatures and the nonadiabatic effect due to fast barrier variations, we predict a spin-entangled electron pair inside the QD with a weak coupling to external leads. An interference-type experiment which employs a gate-controlled electron pump and a beam splitter is proposed to verify this prediction. For the electron pump, a sinusoidal radio-frequency signal is applied to the entrance gate of a two-gated system over a narrow channel etched in a GaAs/AlGaAs heterostructure, where the measured current within the channel shows plateaus at Nef with N = 1, 2, ... being the number of captured electrons in a QD and f the frequency of the sinusoidal signal