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Coincidence of the oscillations in the dipole transition and in the persistent current of narrow quantum rings with two electrons

The fractional Aharonov–Bohm oscillation (FABO) of one-dimensional (1D) quantum rings with two electrons has been studied. The analysis is based on the separability of the Hamiltonian into collective and internal motions. The inherent nodal structures of the internal states have been clarified; thereby these states can be classified. Some relations among them have been found. Based on the study, the evolution of the period and amplitudes of the FABO against the magnetic field B can be exactly and analytically described. Furthermore, an additional ‘rule of selection’ imposed on the dipole transition of the ground state has been found. Consequently, the photon energies emitted (absorbed) during the transition have only two choices. The energy difference of these two choices appears as a new kind of oscillation that matches the oscillation of the persistent current exactly. A number of analytical expressions relating the observable and dynamic parameters have been found. These equalities are helpful in the experimental determination of relevant physical quantities. The 1D model is a good approximation for the 2D rings with very narrow width. Quantitative comparison of these two models has been made.