Simulation of physical phenomena which occur originally in charged particles by neutral ones is an interesting subject. An example is the simulation of Aharononv–Bohm (AB) effect by neutral particles. The AB effect predicts that a charged particle will accumulate a geometrical phase when it moves around a long-thin magnetic-flux carried solenoid.^[1]

The simulation of AB effect by a neutral particle which possesses a permanent magnetic dipole moment was proposed by Aharonov and Casher. In Ref. [2], they predicted that a neutral particle with a permanent magnetic dipole moment would acquire a geometrical phase if it moved around a uniformly electric charged long filament with the direction of the magnetic dipole moment parallel to the filament. It is the Aharonov–Casher (AC) effect.

The simulation of AB effect by a neutral particle with permanent electric dipole moment was proposed in Refs. [3,4]. It was predicted that a neutral particle with a permanent electric dipole moment would receive a geometrical phase if it circled around a uniformly magnetic charged long filament. It is named He–Mckellar–Wilkens (HMW) effect. The observation of HMW effect in experiments is difficult since the magnetic field in HMW effect is produced by magnetic monopoles.^[5,6]

In order to avoid this difficulty, the authors in Ref. [7] proposed an alternative method to observe the HMW effect. Instead of using a neutral particle which possesses a permanent electric dipole moment, they proposed to use a neutral particle with an induced electric dipole moment interacting with an electric field and a magnetic field. Compared with HMW effect, the magnetic field in the proposal^[7] is easily prepared in experiments.

Another example is Landau levels. Landau levels are eigenvalues of a charged planar particle interacting with a uniform perpendicular magnetic field. In Ref. [8], the authors showed that Landau levels could be simulated by an atom which possesses a permanent magnetic dipole moment in the background of an electric field. Since then, there are many research works concerning the analogy between Landau levels and spectra of neutral particles which possess permanent electric or magnetic dipoles interacting with electric and magnetic fields.^[9–21]

We shall show that anyons,^[22,23] which were mostly realized by charged particles before, can also be simulated by a neutral particle with an induced electric dipole moment. As is known, eigenvalues of the canonical angular momentum must be quantized in the three-dimensional space.^[24,25] However, in the two-dimensional space, eigenvalues of the canonical angular momentum can take fractional values.^[26,27] The reason is that the rotation group in three-dimensional space is a non-Abelian one while it is Abelian in the two-dimensional space. Particles which have the fractional angular momentum (FAM) are named anyons.^[22,23] Anyons play important roles in understanding quantum Hall effects^[28] and high T_c superconductivity.^[29] There are several ways to realize anyons. Because of the dynamical properties of the Chern–Simons gauge field, in the absence of the Maxwell term, one can realize anyons by coupling charged particles to the Chern–Simons gauge field in (2+1)-dimensional space–time.^[30–33] Recently, anyons receive renewed interests.^[34–36]

Reference [37] proposed an alternative approach to realize anyons. The author coupled an ion to two magnetic fields. One is a uniform magnetic field and the other is generated by a long-thin magnetic solenoid. Provided the kinetic energy of this ion is cooled down to its lowest level by using the cold atomic technologies, the author found that eigenvalues of the canonical angular momentum of this charged particle can take fractional values. The fractional part is determined by the magnetic flux inside the magnetic solenoid.

In this paper, we propose to simulate anyons by coupling neutral particles, for example, atoms, which possess an induced electric dipole moment to electric and magnetic fields. The electric and magnetic fields we applied contain a magnetic field and two electric fields. The organization of this paper is as follows: in the next section, we introduce our model. Then, we quantize the model canonically and pay attention to its rotation property. Although the canonical angular momentum of this model only can take integer values, we show that the canonical angular momentum of the reduced model, which is obtained by cooling down the kinetic energy of the atom to the negligibly small, takes fractional values. The fractional part of the angular momentum depends on the intensity of the magnetic and only one of the electric fields explicitly. In Section 3, we analyze the roles two electric fields played in the simulation of anyons. We prove that both of the electric fields are necessary to simulate anyons. Summations and conclusions will be given in the last section.

The model we considered is an atom which possesses an induced electric dipole moment (with no permanent electric or magnetic dipole moments) interacting with electric and magnetic fields. The electric and magnetic fields we applied consist of a pair of electric fields E⁽¹⁾, E⁽²⁾ and a uniform magnetic field B.

The atom moves in a cylinder in which a uniform volume charge density ρ is distributed. A long filament with uniform electric charge per length is along the symmetry axis of the cylinder. The magnetic field is uniform and parallel to the symmetry axis of the cylinder which we take to be the z-axis.

The electric fields E⁽¹⁾ and E⁽²⁾ are produced by the long electric charged filament and the uniformly distributed electric charges interior the cylinder, respectively. As a result, these two electric fields are in the radial direction. Explicitly, the electric and magnetic fields we considered are

In Ref. [7], the authors showed that a neutral atom with an induced electric dipole moment would receive a topological phase if it moves around this uniformly electric charged filament in the presence of the magnetic field (2). By confining the atom on a rigid circle, the authors of Ref. [38] investigated the eigenvalue problem of the model considered here.

Due to the electric fields, the atom will be polarized, i.e., it will induce an electric dipole moment

Taking the electric and magnetic fields (1) and (2) into account and trapping the atom by a harmonic potential, we get the Lagrangian which describes the dynamics of the atom. It is

Substituting the expression into the above Lagrangian and confining the motion of the atom in the plane perpendicular to the magnetic field inside the cylinder, we simplify the Lagrangian (4) to the form (the Latin indexes i, j take values 1, 2 and the summation convention is applied throughout the present paper)

We should quantize the model (5) before studying its quantum properties. To this end, we define the canonical momenta with respect to variables x_i

The canonical angular momentum is

Now, we consider the reduced model which is the limit of taking the kinetic energy in Eq. (5) to be negligibly small. This may be realized in experiments by cooling down the atom to a slower velocity so that the effective kinetic energy can be neglected (in an experiment carried out in the early of 1990s, the velocity of atoms can be cooled down to ∼ 1 m⋅s^−1[40]). This kind of reduction is first considered in Ref. [41] during the studies of the Chern–Simons quantum mechanics.

The reduced model is described by the Lagrangian

The righthand side terms of Eq. (12) do not contain velocities, thus, they are in fact the primary constraints in the terminology of Dirac.^[42] We label them as

The canonical angular momentum in this reduced model has the same expression as Eq. (9), i.e., J = ε_ijx_i p_j. Since there are constraints which lead to the dependence among canonical variables x_i, p_i, we rewrite the canonical angular momentum by substituting the constraints (14) into J = ε_ijx_ip_j, and obtain

Taking into account the above commutator, it is clear to see that apart from the term αBk, the canonical angular momentum (17) is equivalent to a one-dimensional harmonic oscillator. With the help of the commutators (20), one can write down the eigenvalues of the canonical angular momentum (12) immediately. They are

Therefore, it shows that eigenvalues of the canonical angular momentum will take fractional values when its kinetic energy is cooled down to the negligibly small. The fractional part is determined by both the intensity of the applied magnetic field and the electric field E⁽¹⁾.

From the eigenvalues of the canonical angular momentum (22), it seems that the electric field E⁽²⁾ does not have any influences on the FAM since the parameter ρ does not appear in Eq. (22) explicitly. In fact, the electric field E⁽²⁾ also plays important roles in producing the FAM. In the next section, we analyze the roles that two electric fields E⁽¹⁾ and E⁽²⁾ played.

As we showed that besides the intensity of the magnetic field, the fractional part of the canonical angular momentum only contains the parameter k. Thus it seems that only the electric field E⁽¹⁾ contributes to the FAM. In the following, we show that the electric field E⁽²⁾ also plays important roles in producing the FAM since the FAM will not appear in the absence of either of the electric fields.

First of all, we consider the case that the electric field E⁽¹⁾ is turned off. In this case, the dynamics is determined by the Lagrangian

The canonical momenta with respective to x_i are given by

However, when the atom is cooled down to the negligibly small kinetic energy, their difference appears. To see it clearly, we set the effective kinetic energy term to zero in Lagrangian (23) in this limit. Therefore, the Lagrangian (23) reduces to

On the contrary, if we turn off the electric field E⁽²⁾ and let E⁽¹⁾ alone, the Lagrangian (5) becomes

The reduced model of the Lagrangian (29) turns out to be

By applying the consistency condition to the primary constraints , we get

It means that when we turn off the electric field E⁽²⁾, the reduced model (29) does not have dynamical degrees of freedom. Thus, the electric field E⁽²⁾ plays important roles in producing the FAM: although it does not contribute to the fractional part of the angular momentum directly, the FAM will not appear in its absence.

In this paper, we propose to simulate an anyon which was usually realized by a charged particle originally by using a trapped cold atom. This atom possesses an induced electric dipole moment interacting with electric and magnetic fields. The electromagnetic fields we applied contain a uniform magnetic field and two electric fields.

We prove that the canonical angular momentum of the model (4) can only take integer values. However, its reduced model which is obtained by cooling down the atom to the limit of the negligibly small kinetic energy produces the FAM. The magnitude of the FAM can be modulated by two parameters, i.e., the intensity of the applied magnetic field and the electric field E⁽¹⁾. Apart from the fractional part, it is also interesting to observe that the differences between eigenvalues of canonical angular momentum are half integers. It is one of the characteristics of Chern–Simons quantum mechanics. In Ref. [44] the author proposed to realize the Chern–Simons quantum mechanics model by a cold Rydberg atom.

All the electric and magnetic fields play important roles in the simulation of FAM. The effect of the electric field E⁽¹⁾ is evident since the magnitude of the FAM is proportional to the parameter k, which is the strength of electric field E⁽¹⁾. Roles the electric field E⁽²⁾ played are subtle. At the first glance, the electric field E⁽²⁾ does not contribute to the fractional part of the angular momentum. However, it does influence the results since the FAM will not appear in its absence.

Besides the contribution to the effective mass, roles the magnetic field played will be more transparent if we introduce the effective gauge potentials and rewrite the Lagrangian (5) as