# The fractional angular momentum realized by a neutral cold atom

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## 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.

In 1984, Aharonov and Casher predicted that there would exist a topology phase when a neutral particle possessing a non-vanishing magnetic dipole momentum moved around a uniformly charged infinitely long filament with its direction paralleling to the filament [1]. It is named Aharonov–Casher (AC) effect.

*m*is the mass of the neutral particle, \({\mathbf {p}} = -i \hbar \mathbf \nabla \) is the canonical momentum, \(\mu \) is the magnitude of the magnetic dipole momentum,

*c*is the speed of light in vacuum, \({\mathbf {n}}\) is the unit vector along the magnetic dipole momentum and \({\mathbf {E}}\) is the electric field. In AC effect setting, the electric field is produced by a uniformly charged infinitely long filament [1]. The explicit form of the electric field in AC effect is

*r*is the distance between the atom and the long filament and \({\mathbf {e}}_r\) is the unit vector along the radial direction on the plane where the atom moves. For AC setting, the last term in Hamiltonian (1) disappears since \(\mathbf \nabla \cdot {\mathbf {E}}^{AC}=0\) for \(r \ne 0\).

*S*. In addition, the energy gaps of Landau levels are also uniform. They can be written as

*B*is the intensity of the magnetic field,

*S*is the area which is perpendicular to the magnetic field through which the flux \(\Phi \) is measured. 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). Thus, the work of [17] can also be regarded as providing a theoretical approach to realize Landau levels by a neutral particle. It may allow us to realize the quantum Hall effect by using neutral atoms and electric fields.

It is worth mentioning that the eigenvalue problem of neutral particles in various backgrounds has attracted much attention since the work of [17]. In Refs. [18, 19, 20, 21, 22, 23, 24, 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.

^{1}Recently, there are renewed interests in the realization of the fractional angular momentum. In [35], the authors find that a pair of bosonic atoms immersed in a fractional quantum Hall state possesses a fractional relative angular momentum provided certain conditions are satisfied. This work was further studied in [36, 37]. In Ref. [28], the author considered a planar ion interacting with a uniform perpendicular magnetic field. Besides this uniform magnetic field, the ion is trapped by a harmonic potential and influenced by an Aharonov–Bohm type magnetic potential, which can be generated by a long-thin magnetic solenoid perpendicular to the plane. The dynamics of the model proposed in [28] is described by the Hamiltonian (Latin indices \(i, \ j\) run from 1 to 2 and the summation convention is used throughout this paper)

The Hamiltonian (10) can be viewed that apart from a harmonic potential, there exists a uniform magnetic field in the AB effect setting if the motion of the particle is confined on the plane perpendicular to the magnetic solenoid. Or, equivalently, besides a harmonic trapping potential, there exists an additional AB type magnetic potentials in the Landau levels setting. The eigenvalues of the canonical angular momentum of this ion are quantized, as expected. When the kinetic energy of the ion is cooled down to its lowest level, however, the author shows that the eigenvalues of the canonical angular momentum could be fractional. 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. A natural question arises: can we realize the fractional angular momentum by using a neutral particle according to electromagnetic duality? In this paper, we will propose a model to realize the fractional angular momentum by using a neutral particle.

*H*is given in (13).

*X*and

*P*, we write the kinetic energy \(\frac{\Pi _i ^2}{2m}\) in Lagrangian (18) as \(\frac{\Pi _i ^2}{2m} = \frac{\mu \rho }{2 mc^2 \epsilon _0}(X^2 + P^2)\), which is analogous to a one-dimensional harmonic oscillator with mass \(M= \frac{mc^2 \epsilon _0}{\mu \rho }\) and frequency

We must emphasis that canonical angular momenta (16) and (27) are the Nöether charges of the rotation symmetry \(x_i \rightarrow x_i ^\prime = x + \delta x_i, \ \delta x_i \sim \epsilon _{ij} x_j\) of the Lagrangian (14) and (22). Therefore, the conservation of (16) and (27) is independent of whether the parameter \(\lambda \) is time-dependent or not.

In order to show that our result is reliable, we show that the result (31) can also be obtained by using an alternative method.

Obviously, the first term on the right-hand side of (35) is analogous to magnetic potentials generated by a long-thin solenoid and the second term is analogous to the magnetic potentials generated by a uniform perpendicular magnetic field.

In summary, based on the electromagnetic duality, we provide a new approach for realizing fractional angular momenta. Different from previous approaches which realized fractional angular momenta by using charged particles, we use a cold neutral atom to archive this aim. Our approach can be regarded as the electromagnetic duality of the approach proposed in [28]. The electromagnetic duality relation (6), which is found in AB and AC effects as well as Landau levels and the model studied in [17], is exactly held for the results of ours and Ref. [28].

## Footnotes

## Notes

### Acknowledgements

We would like to thank referees for making invaluable comments and suggestions, which have improved the manuscript greatly. This work is supported by NSFC with Grant No. 11465006 and partially supported by 20180677-SIP-IPN and the CONACyT under grant No. 288856-CB-2016.

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