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Communications in Theoretical Physics

Fractional Angular Momentum of an Atom on a Noncommutative Plane*

Jian Jing, Qiu-Yue Zhang, Qing xiang Wang, Zheng‐Wen Long, Shi‐Hai Dong

Published Nov 1, 2019• 4 citations
Publisher

Abstract

Abstract The mechanism of obtaining the fractional angular momentum by employing a trapped atom which possesses a permanent magnetic dipole moment in the background of two electric fields is reconsidered by using an alternative method. Then, we generalize this model to a noncommutative plane. We show that there are two different mechanisms, which include cooling down the atom to the negligibly small kinetic energy and modulating the density of electric charges to the critical value to get the fractional angular momentum theoretically .

DOI
10.1088/0253-6102/71/11/1353

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Abstract The mechanism of obtaining the fractional angular momentum by employing a trapped atom which possesses a permanent magnetic dipole moment in the background of two electric fields is reconsidered by using an alternative method. Then, we generalize this model to a noncommutative plane. We show that there are two different mechanisms, which include cooling down the atom to the negligibly small kinetic energy and modulating the density of electric charges to the critical value to get the fractional angular momentum theoretically .
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.
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.
The model of a planar atom which possesses a nonvanishing electric dipole moment interacting with magnetic fields in a specific setting is studied. Energy spectra of this model and its reduced model, which is the limit of cooling down the atom to the negligible kinetic energy, are solved exactly. We show that energy spectra of the reduced model cannot be obtained directly from the full ones by taking the same limit. In order to get the energy spectra of the reduced model from the full model, we must regularize energy spectra of the full model properly when the limit of the negligible kinetic energy is taken. It is one of the characteristics of the Chern–Simons quantum mechanics. Besides this, the canonical angular momentum of the reduced model will take fractional values although the full model can only take integers. It means that it is possible to realize the Chern–Simons quantum mechanics and fractional angular momentum simultaneously by this model.
We show that it is possible to simulate an anyon by a trapped atom which possesses an induced electric dipole moment in the background of electric and magnetic fields in a specific configuration. The electric and magnetic fields we applied contain a magnetic and two electric fields. We find that when the atom is cooled down to the limit of the negligibly small kinetic energy, the atom behaves like an anyon because its angular momentum takes fractional values. The fractional part of the angular momentum is determined by both the magnetic and one of the electric fields. Roles electric and magnetic fields played are analyzed.
We show that it is possible to simulate an anyon by a trapped atom which possesses an induced electric dipole moment in the background of electromagnetic fields with a specific configuration. The electromagnetic fields we applied contain a magnetic and two electric fields. We find that when the atom is cooled down to the limit of the negligibly small kinetic energy, the atom behaves like an anyon because its angular momentum takes fractional values. The fractional part of the angular momentum is determined by both the magnetic and one of the electric fields. Roles two electromagnetic fields played are analyzed.
The possibility of testing spatial noncommutativity via Rydberg atoms is explored. An atomic di-pole of a cold Rydberg atom is arranged in appropriate electric and magnetic fields, so that the motion of the dipole is constrained to be planar and rotationally symmetric. Spatial noncommutativity leads the canonical angular momentum to possess fractional values. In the limit of vanishing kinetic energy, the dominate value of the lowest canonical angular momentum takes variant Planck's over h/2. Furthermore, in the limit of eliminating the magnetic field, the dominate value of the lowest canonical angular momentum changes from variant Planck's over h/2 to variant Planck's over h/4. This result is a clear signal of spatial noncommutativity. An experimental verification of this prediction is suggested.