Induced Fractional Zero-Point Angular Momentum for Charged Particles of the Bohm-Aharonov System by means of a"Spectator"Magnetic Field

An induced fractional zero-point angular momentum of charged particles by the Bohm-Aharonov (B-A) vector potential is realized via a modified combined trap. It explores a"spectator"mechanism in this type of quantum effects: In the limit of the kinetic energy approaching one of its eigenvalues the B-A vector potential alone cannot induce a fractional zero-point angular momentum at quantum mechanical level in the B-A magnetic field-free region; But when there is a"spectator"magnetic field the B-A vector potential induces a fractional zero-point angular momentum. The"spectator"does not contribute to such a fractional angular momentum, but plays essential role in guaranteeing non-trivial dynamics at quantum mechanical level in the required limit. This"spectator"mechanism is significant in investigating the B-A effects and related topics in both aspects of theory and experiment.

As is well known, quantum states of charged particles can be influenced by electromagnetic effects even if those particles are in a region of vanishing field strength [1,2].
As predicted by Bohm and Aharonov (B-A) [2], experiments [3] showed that in a multiply connected region where field strength is zero everywhere the interference spectrum suffered a shift according to the amount of the loop integral of magnetic vector potential around an unshrinkable loop. Wu and Yang [4] pointed out that the B-A effects is due to the non-trivial topology of the space wehere the magnetic field strength is vanishing.
The B-A effect is purely quantum mechanical one which explores far-reaching consequences of vector potential in quantum theory. This effect has been received much attention for years [5,6,7]. Recently investigations in this topic concentrated on revealing new types of quantum phases: The Aharonov-Casher effect [8], the He-McKellar-Wilkens phase [9] and the Anandan phase [10].
In another aspect a fractional angular momentum originated from the Poynting vector produced by crossing the Coulomb field of a charged particle with an external magnetic field has been predicted by Peshkin, Talmi and Tassie for years [6,11]. There are lots of works concerning fractional angular momentum in B-A dynamics and their fractional statistics (see the reviews [12][13][14][15][16][17] and references therein). Spatial noncommutativity also leads to fractional angular momentum [18,19].
Recently Kastrup [20] considered the question of how to quantize a classical system of the canonically conjugate pair angle and orbital angular momentum. This has been a controversial issue since the founding days of quantum mechanics [21]. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. A crucial point is that the irreducible unitary representations of the euclidean group E(2) or of its covering groups allow for orbital angular momentum l =h(n + δ) where n = 0, ±1, ±2, · · · , and 0 ≤ δ < 1. The case δ = 0 corresponds to fractional zero-point angular momentum. Kastrup investigated the physical possibility of fractional orbital angular momentum in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields, and pointed out that if implementable this would lead to a wealth of new theoretical, experimental and even technological possibilities.
In this paper the induced fractional zero-point angular momentum of charged parti-cles by the B-A vector potential is realized via a modified combined trap. It explores a "spectator" mechanism in this type of quantum effects: In the limit of the kinetic energy approaching one of its eigenvalues the B-A vector potential alone cannot induce a fractional zero-point angular momentum of charged particles at quantum mechanical level in a region of vanishing B-A field strength; But when there is a "spectator" magnetic field the B-A vector potential induces a fractional zero-point angular momentum in the same region. The "spectator" does not contribute to such a fractional angular momentum, but plays essential role in guaranteeing non-trivial dynamics at quantum mechanical level in the required limit. This type of quantum effects is so remarkable that in quantum mechanics the vector potential itself has physical significant meaning and becomes effectively measurable not only in shifts of interference spectra originated from quantum phases but also in physical observables.

Dynamics in a Modified
Combined Trap -We consider ions constrained in a modified combined trap including the B-A type magnetic field. The Paul, Penning, and combined traps share the same electrode structure [22]. A combined trap operates in all of the fields of the Paul and Penning traps being applied simultaneously.
The trapping mechanism in a Paul trap involves an oscillating axially symmetric electric potentialŨ (ρ, φ, z, t) = U(ρ, φ, z)cosΩt with U(ρ, φ, z) = V (z 2 − ρ 2 /2)/2d 2 where ρ, φ and z are cylindrical coordinates, V and d are, respectively, characteristic voltage and length, andΩ is a large radio-frequency. The dominant effect of the oscillating potential is to add an oscillating phase factor to the wave function. Rapidly varying terms of time in Schrodinger equation can be replaced by their average values. Thus for Ω ≫ Ω ≡ √ 2q|V |/µd 2 1/2 we obtain a time-independent effective electric potential [23] V ef f = q 2 ∇U · ∇U/4µΩ 2 = µω 2 P (ρ 2 + 4z 2 )/2 where µ and q(> 0) are, respectively, the mass and charge of the trapped ion, and ω P = Ω 2 /4Ω. A modified combined trap combines the above electrostatic potential and two magnetic fields [24]: a homogeneous magnetic field B c aligned along the z axis in a normal combined trap and a B-A type magnetic field B 0 produced by, for example, an infinitely long solenoid with radius ρ = (x 2 1 + x 2 2 ) 1/2 = a. Inside the solenoid (ρ < a) B 0,in = (0, 0, B 0 ) is homogeneous along the z axis, and outside the solenoid (ρ > a) B 0,out = 0. The vector potential A c of B c is chosen as (Henceforth the summation convention is used) 2). At ρ = a the potential A in passes continuously over into A out . The Hamiltonian of the modified com- This Hamiltonian can be decomposed into a one-dimensional harmonic Hamiltonian H z (z) along the z-axis with the axial frequency ω z = 2ω P and a two-dimensional Hamiltonian H ⊥ (x 1 , x 2 ), . Inside the solenoid the ion's motion is the same as the one with a total magnetic field B c + B 0,in .
In the following we consider the motion outside the solenoid. The two-dimensional Hamiltonian outside the solenoid is [22,23] where ω c = qB c /µc and ω 0 = qB 0 /µc are the cyclotron frequencies corresponding to, respectively, the magnetic fields B c and B 0,in . The Hamiltonian H ⊥ possess a rotational symmetry in (x 1 , x 2 ) -plane. The z-component of the orbital angular momentum J z = ǫ ij x i p j commutes with H ⊥ . They have common eigenstates.
Dynamics in the Limit of the Kinetic Energy Approaching its Lowest Eigenvalue -In this limit the kinetic energy is Here K i is the mechanical momenta corresponding to the vector potentials A c,i and A out,i .
It is worth noting that the B-A vector potential A out,i does not contributes to the commu- The kinetic energy E k is rewritten as the Hamiltonian of a harmonic In a laser trapping field, using a number of laser beams and exploiting Zeeman tuning, the speed of atoms can be slowed to the extent of 1 ms −1 , see [26]. Ions are the common object in cooling and trapping. In order to experimentally realizing the limit of E k → E k0 through laser cooling in a trap ions are used.
In the limit of the kinetic energy approaching its lowest eigenvalue the Hamiltonian H ⊥ in Eq. (1) has non-trivial dynamics [27,28,19]. The Lagrangian corresponding to H ⊥ is In the limit of Constraints -For the reduced system (H 0 , L 0 ) the canonical momenta are Eq. (5) does not determine velocitiesẋ i as functions of p i and x j , but gives relations among p i and x j , that is, such relations are the primary constraints [29, 28, 19] The physical meaning of Eq. (6) is that it expresses the dependence of degrees of freedom among p i and x j . The constraints (6) should be carefully treated [30]. The subject can be treated simply by the symplectic method in [31,32]. In this paper we work in the Dirac formalism. The Poisson brackets of the constraints (6) are From Eq. (7), {ϕ i , ϕ j } = 0, it follows that the conditions of the constraints ϕ i holding at all times do not lead to secondary constraints.
C ij defined in Eq. (7) are elements of the constraint matrix C. Elements of its in- The Dirac brackets of ϕ i with any variables x i and p j are zero so that the constraints (6) are strong conditions. It can be used to eliminate dependent variables. If we select x 1 and x 2 as the independent variables, from the constraints (6) the variables p 1 and p 2 can be represented by, respectively, the independent variables x 2 and x 1 as The Dirac brackets of x 1 and x 2 is Quantum Behavior of the Reduced System -Now we consider quantum behavior of the reduced system (H 0 , L 0 ). By defining the following effective mass and frequency, µ * ≡ µω 2 c /ω 2 P , ω * ≡ ω 2 P /ω c , the Hamiltonian H 0 is represented as H 0 = p 2 /2µ * + µ * ω * 2 x 2 /2 + hω c /2. We introduce an annihilation operator A = µ * ω * /2h x + i 1/2hµ * ω * p and its conjugate one A † . The operators A and A † satisfies [A, A † ] = 1. The eigenvalues of the number operator N = A † A is n = 0, 1, 2, · · · . Using A and A † , the reduced Hamiltonian H 0 is rewritten as H 0 =hω * A † A + 1/2 +hω c /2. Now we consider the angular momentum of the ion. Using Eq. (8) to replace p 1 and p 2 by, respectively, the independent variables x 2 and x 1 , the orbital angular momentum J z = ǫ ij x i p j is rewritten as where Φ 0 = πa 2 B 0 is the total flux of the magnetic field B 0 inside the solenoid. Similarly, using A and A † to rewrite J z , we obtain J z = qΦ 0 /2πc +h A † A + 1/2 . The zero-point angular momentum of J z is J 0 =h/2 + qΦ 0 /2πc. In the above the term [33] is the zero-point angular momentum induced by the AB vector potential. J AB takes fractional values. It is related to the region where the magnetic field B 0,out = 0 but the corresponding vector potential A out = 0.
2. Dynamics in the Case of B c = 0 -It is worth noting that here B c , like a "spectator", does not contribute to J AB . In order to clarify the role played by B c , we consider the case of B c = 0. In this case the modified combined trap is as stable as a Paul trap. The corresponding kinetic energy reduces toẼ k = µẋ iẋi /2 = K 2 1 +K 2 2 /2µ wherẽ In the aboveK i is the mechanical momenta corresponding to the B-A vector potential A out,i .
Unlike the ordinary vector potential, the special feature of the B-A vector potential is that it does not contributes to the commutator [K i ,K j ]. BecauseK i are commuting, behavior of E k is similar to a Hamiltonian of a free particle. Its spectrum is a continuous one. WhenẼ k approaching some constantẼ k ( = 0) the Hamiltonian H ⊥ reduces toH 0 =Ẽ k + µω 2 P x i x i /2. The Lagrangian corresponding toH 0 is FromL 0 we obtain the canonical momentã Now we clarify that the caseẼ k = 0 should be excluded. The limit of the kinetic energy E k = µẋ iẋi /2 → 0 corresponds two possibilities:ẋ i = 0 or µ → 0. In the caseẋ i = 0 the Lagrangian L in Eq. (3) reduces toL ′ 0 = −µω 2 P x i x i /2. The corresponding canonical momentap i = ∂L ′ 0 /∂ẋ i = 0. Therefore there is no dynamics. According to the definition of the frequency Ω the other possibility µ → 0 is forbidden.
Eq. (14) gives the reduced primary constraints Here the special feature is that the corresponding Poisson brackets are zero, According to Dirac's formalism of quantizing constrained systems, there is no way to establish dynamics at quantum mechanical level. This means that the B-A vector potential alone cannot lead to non-trivial dynamics at quantum mechanical level in the required limit, thus does not contribute to the energy spectrum and angular momentum at all.
It is clear that though the vector potential A c,i of the "spectator" magnetic field B c does not contribute to J AB , it plays essential role in guaranteeing non-trivial dynamics at quantum mechanical level in the limit of the kinetic energy approaching one of its eigenvalues. This example reveals that, unlike ordinary vector potential, the physical role played by the B-A vector potential is subtle. This needs to be carefully analyzed at quantum mechanical level. In this case the modified combined trap reduces to a combined trap. The Hamiltonian (1) reduces toĤ ⊥ (x 1 , x 2 ) = (p i + µω c ǫ ij x j /2) 2 /2µ + µω 2 P x 2 i /2. Its kinetic energy isÊ k = (K 2 1 +K 2 2 )/2µ wherê In Eq. (17)K i is the mechanical momenta corresponding to the vector potentials A c,i . The commutation relations betweenK i 's are the same as the ones between K i 's in Eq. (2).
The eigenvalues ofÊ k isÊ kn =hω c (n + 1/2), which are just the Landau levels of charged particles in an external magnetic field.
It leads to the following constraintŝ The Poisson brackets ofφ i are the same as ones of the constraints ϕ i in Eq. (7): From Eq. (21), {φ i ,φ i } = 0, it follows that the conditions of the constraintsφ i holding at all times do not lead to secondary constraints.
By the similar procedure of treating the constraints (6), we find that the reduced system (Ĥ 0 ,L 0 ) has non-trivial dynamics at quantum mechanical level in the limit of ǫ ij x i p j can be represented by the canonical variables x and p asĴ z = (p 2 /2µ + µω 2 c x 2 /2) /ω c . The zero-point angular momentum can be read out from this harmonic-like "Hamiltonian", J 0 =h/2. We note that in this case there is no fractional zero-point angular momentum.
The above results elucidate that A c are essentially different from A 0 : the A c alone can lead to non-trivial dynamics at quantum mechanical level in the limit of the kinetic energy approaching its lowest eigenvalue.
Here H ′ ⊥ is the sameĤ ⊥ . In the limit of the kinetic energy approaching its lowest eigenvalue the corresponding reduced constraints are the sameφ i in Eq. (20). Under the gauge transformation G the (20) to represent p 1 and p 2 by, respectively, the independent variables x 2 and x 1 , the first term in J ′ z reads x 1 p 2 − x 2 p 1 = µω c (x 2 1 + x 2 2 )/2. Thus we obtain J ′ z is the same J z in Eq. (10). This result shows that the fractional zero-point angular momentum induced by the B-A vector potential is a real physical observable which cannot be gauged away by a gauge transformation.
In summary, this paper explores a "spectator" mechanism in B-A effects. It is clarified that the B-A vector potential alone cannot lead to non-trivial dynamics at quantum mechanical level in the limit of the kinetic energy approaching one of its eigenvalues. In such a limit the B-A vector potential alone cannot induce a fractional zero-point angular momentum. When there is a "spectator" magnetic field the B-A vector potential induces a fractional zero-point angular momentum. The induced effect essentially depends upon the participation of a "spectator" magnetic field. The "spectator" vector potential does not contribute to the fractional angular momentum, but plays essential role in guaranteeing non-trivial dynamics at quantum mechanical level in the required limit. The "spectator" mechanism is significant in both aspects of theory and experiment. In the theoretical aspect, it is revealed that, unlike ordinary vector potentials, the physical role played by the B-A vector potential is subtle. This needs to be carefully analyzed at quantum mechanical level. In the experimental aspect, existence of a "spectator" magnetic field is necessary for inducing the fractional angular momentum by the B-A vector potential. As an example, the modified combined trap provides a realistic way to realize this "spectator" mechanism.
The author would like to thank M. Peshkin  consists essentially in a shift of the phase of the original wave function. One can adjust the radio-frequencyΩ to compensate the phase shift, therefore for a modified combined trap the derivation in Ref. [23] also remains valid. The modification ofΩ leads to the corresponding modification of the effective frequecy ω P = Ω 2 /4Ω of V ef f .
In Eq. (1) the ω P means the modified effective frequecy.
[25] M. Peshkin pointed out that in the simplest case of the flux line and no other fields, there can be no wave function whose kinetic energy expectation is zero, hence no zero eigenvalue of the kinetic energy, (a private communication).
[28] Jian-zu Zhang, Phys. Rev. Lett. 77 44 (1996) 44. From the Hamiltonian H 0 = p iẋi −L 0 , using p i = ∂L 0 /∂ẋ i and the Lagragian equatioṅ It indicates that H 0 can be expressed as a function of x i and p i . Thus we obtain Because of the constraints ϕ i (x, p) = 0 of Eq. (6), H 0 plus any linear combination of ϕ i is also a Hamiltonian of the system, i. e., the H 0 can be replaced by