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Physical Review Letters

Cold Rydberg Atoms as Realizable Analogs of Chern-Simons Theory

C. Baxter

Published Jan 23, 1995• 42 citations
Publisher

Abstract

The angular momentum spectrum of an atomic dipole in constant electric and magnetic fields is calculated. The arrangement of the fields are such that the motion of the dipole is planar and rotationally symmetric, with the result that the R\"ontgen term of the Lagrangian takes on the appearance of a Chern-Simons interaction. On eliminating the kinetic energy term of the Lagrangian, the angular momentum spectrum changes from one consisting of integers to one consisting of positive half integers. The possibility of physically achieving such an elimination is discussed in the context of cold Rydberg atoms.

DOI
10.1103/physrevlett.74.514

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The angular momentum spectrum of an atomic dipole in constant electric and magnetic fields is calculated. The arrangement of the fields are such that the motion of the dipole is planar and rotationally symmetric, with the result that the R"ontgen term of the Lagrangian takes on the appearance of a Chern-Simons interaction. On eliminating the kinetic energy term of the Lagrangian, the angular momentum spectrum changes from one consisting of integers to one consisting of positive half integers. The possibility of physically achieving such an elimination is discussed in the context of cold Rydberg atoms.
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.
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.
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.
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.