Abstract
Inspired by the electromagnetic duality, we propose an approach to realize the fractional angular momentum by using a cold atom which possesses a permanent magnetic dipole momentum. This atom interacts with two electric fields and is trapped by a harmonic potential which enable the motion of the atom to be planar and rotationally symmetric. We show that eigenvalues of the canonical angular momentum of the cold atom can take fractional values when the atom is cooled down to its lowest kinetic energy level. The fractional part of canonical angular momentum is dual to that of the fractional angular momenta realized by using a charged particle. Another approach of getting the fractional angular momentum is also presented. The differences between these two approaches are investigated.
Highlights
In three-dimensional space, the Hamiltonian which governs the dynamics of a neutral particle possessing a permanent magnetic dipole momentum in the background of an electric field is given by
In which λ is charges per unit length on the long filament, 0 is the permittivity of vacuum, r is the distance between the atom and the long filament and er is the unit vector along the radial direction on the plane where the atom moves
Aharonov and Bohm predicted that a charged particle would generate a topology phase when it circled around a long-thin flux-carried solenoid
Summary
By comparing the energy gaps (9) for Landau levels and Eq (8) for a neutral particle which possesses a permanent magnetic dipole momentum in the background of the electric field, one reproduces the duality relation (6). [18–25], the authors solved energy spectra of particles possessing non-vanishing electric or magnetic dipole momenta in the background of electromagnetic fields analytically in various configurations. The fractional part is proportional to the magnetic flux inside the magnetic solenoid Both AB effect and Landau levels are related with the charged particles and magnetic potentials. Their electromagnetic dualities are all concerned with the neutral particles and electric fields.
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