The Geometric phase and fractional orbital angular momentum states in electron vortex beams
TThe Geometric phase and fractional orbital angular momentum states in electronvortex beams
Pratul Bandyopadhyay, ∗ Banasri Basu, † and Debashree Chowdhury ‡ Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India Department of Physics, Harish-Chandra Research institute,Chhatnag Road, Jhusi, Allahabad, U. P. 211019,India
We study here fractional orbital angular momentum (OAM) states in electron vortex beams (EVB)from the perspective of geometric phase. We have considered the skyrmionic model of an electron,where it is depicted as a scalar electron orbiting around the vortex line, which gives rise to thespin degrees of freedom. The geometric phase acquired by the scalar electron orbiting aroundthe vortex line induces the spin-orbit interaction. This leads to the fractional OAM states whenwe have non-quantized monopole charge associated with the corresponding geometric phase. Thisinvolves tilted vortex in EVBs. The monopole charge undergoes the renormalization group (RG)flow, which incorporates a length scale dependence making the fractional OAM states unstable uponpropagation. It is pointed out that when EVBs move in an external magnetic field, the Gouy phaseassociated with the Laguerre-Gaussian modes modifies the geometric phase factor and a properchoice of the radial index helps to have a stable fractional OAM state.
I. INTRODUCTION
It is now well known that optical vortex beams [1] car-rying orbital angular momentum(OAM) exhibit an az-imuthal phase structure exp ( i(cid:96)φ ) , where (cid:96) is an integernumber implying OAM of (cid:96) (cid:126) per photon [2]. These canbe produced in the laboratory using spiral phase plates[3] or computer generated holograms [4, 5]. Such opticalbeams are the Laguerre-Gaussian(LG) modes, which im-print a 2 πl step in the phase of the electromagnetic field.However, it is also possible to generate a situation suchthat the phase step is not an integer multiple of 2 π. Thiscorresponds to fractional OAM. Light emerging from afractional phase step in general is unstable on propaga-tion. Gotte et. al. [6] have pointed out that fractionalOAM can be produced through a generic superpositionof light modes with different values of (cid:96) and LG beamswith a minimal number of different Gouy phases increasepropagational stability. These states can be decomposedinto basis of integer OAM states.In recent times electron vortex beams(EVBs) withOAM have been produced experimentally [7–9]. An EVBis generally visualized as a scalar electron orbiting aroundthe vortex line. Bliokh et. al. [10] studied the relativisticEVBs representing the angular momentum eigenstate ofa free Dirac equation and constructed exact Bessel beamsolution. In a recent paper [11], it has been argued thatin the skyrmionic model of an electron, where it is de-picted as a scalar electron rotating around a directionvector(vortex line), which is topologically equivalent toa magnetic flux line corresponding to the spin degrees of ∗ Electronic address: b [email protected], retired Prof. † Electronic address: [email protected] ‡ Electronic address: [email protected], Present address:Department of Physics,Ben-Gurion University,Beer Sheva 84105,Is-rael freedom, EVBs appear as a natural consequence. Evi-dently, the vortex here is a spin vortex. Apart from this,when the orbiting electron carries OAM, EVBs exhibitan azimuthal phase structure exp ( i(cid:96)φ ) . It has been observed that optical and electron vortexbeams carrying OAM share much similarities in their be-haviour. Indeed, just like EVBs, the dynamics of the op-tical vortex beams(OVBs) can also be studied from theperspective of geometric phase acquired by the beam fieldorbiting around the vortex line [12]. In this note, we shallstudy the situation when EVBs carry fractional OAM.When the scalar electron in an EVB orbiting aroundthe vortex line, acquires the Berry phase, which involvesquantized monopole charge, we have screw or edge dislo-cation [11, 12]. The screw dislocation essentially corre-sponds to paraxial beams, where the vortex line is paral-lel to the wave front propagation direction, whereas foredge dislocation, the vortex line is orthogonal to the wavefront propagation direction. Besides, there are situationswhen we have tilted vortex, where the vortex line makesan angle θ with respect to the wave front propagation di-rection such that 0 < θ < π/ , which corresponds to themixed screw-edge dislocation. In this case the associatedBerry phase involves non-quantized monopole chargeand incorporates spin-orbit interaction(SOI), which effec-tively induces fractional OAM. It is here argued that forEVBs the structural stability arises, when these beamsmove in an external magnetic field, where we can haveLG modes such that the monopole charge of the Berryphase is modified by the Gouy phase[13]. This analysissuggests that fractional OAM in vortex beams is a natu-ral phenomenon and can be visualized properly from theperspective of the Berry phase.In sec. 2, we shall recapitulate certain features of theBerry phase in EVBs and its association with the spin-orbit interaction leading to fractional OAM states. In sec.3, we shall discuss the fractional OAM states from theview point of the monopole harmonics. In sec. 4 we shallconsider the stability of such beams when these move in a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec an external magnetic field and the relevant consequences. II. ELECTRON VORTEX BEAMS WITHFRACTIONAL ORBITAL ANGULARMOMENTUM AND THE BERRY PHASE
In a recent paper[11], the geometrodynamics of EVBshave been analysed from the point of view of the geomet-ric phase. To this end, we have considered the skyrmionicmodel of an electron, where it is depicted as a scalarelectron orbiting around a direction vector(vortex line),which gives rise to the spin degrees of freedom [14]. Evi-dently, the scalar electron moving around the vortex lineacquires the Berry phase after forming a closed loop. Ithas been pointed out that for paraxial beams, the Berryphase is vanishing. For non-paraxial beams, correspond-ing to the edge dislocation, where the vortex line is or-thogonal to the wave front propagation direction, theBerry phase involves quantized monopole. However, fortilted vortex, where the vortex line makes an arbitraryangle with the wave-front propagation direction, the cor-responding Berry phase involves non-quantized monopolecharge. It is here noted that in this case electron car-ries fractional OAM. The situation involves spin-orbitinteraction(SOI) and fractional OAM appears as a con-sequence of this.The Berry phase acquired by the scalar electron encir-cling the vortex line is 2 πµ [15], where µ is the monopolecharge. In terms of the solid angle subtended by theclosed circuit at the origin of the unit sphere, where themonopole is located, the Berry phase is given by µ Ω( C ) , where Ω( C ) is the solid angle given byΩ( C ) = (cid:90) C (1 − cosθ ) dφ = 2 π (1 − cosθ ) . (1)Here θ is the polar angle of the vortex line with the quan-tization axis(z axis) and µ corresponds to the monopolecharge, which effectively represents the spin. Indeed,the total angular momentum of a charged particle inthe field of a magnetic monopole of charge µ is givenby (cid:126)J = (cid:126)L − µ ˆ (cid:126)r, where (cid:126)L is the orbital angular momen-tum(OAM). In case, where OAM is vanishing, µ corre-sponds to the total angular momentum of the particle,i.e. the spin angular momentum(SAM) with s z = ± µ .One may note that for µ = 1 / , and the correspondingBerry phase derived from (1) is, φ B = π (1 − cosθ ) . (2)If one considers a reference frame where the scalar elec-tron in the EVB is taken to be fixed and the vortex statemoves in the field of a magnetic monopole around a closedpath, evidently, φ B in (2) will correspond to the Berryphase acquired by the vortex state(spin state) in an EVB.The angle θ represents the deviation of the vortex linefrom the z axis. Equating this phase φ B with 2 πµ, whichis the Berry phase acquired by the scalar electron moving around the vortex line in the closed path, we find thatthe effective monopole charge associated with the corre-sponding vortex line having polar angle θ with the z axisis given by µ = 12 (1 − cosθ ) . (3)Eqn. (3) suggests that the paraxial beams with θ = 0and orthogonal non-paraxial beams with θ = π/ < θ < π , the correspondingmonopole charge µ is non-quantized and involves tiltedvortices. Non-paraxial beams in general incorporatesSOI, which modifies the OAM (cid:104) (cid:126)L (cid:105) as well as spin (cid:104) (cid:126)S (cid:105) .Introducing the mapping (cid:126)L = l ˆ z and (cid:126)S = s ˆ z, we notethat EVBs having tilted vortex, which are characterizedby the Berry phase associated with the non-quantizedmonopole charge, the OAM as well as SAM is modifiedas (cid:104) (cid:126)L (cid:105) = ( l − µ )ˆ z (cid:104) (cid:126)S (cid:105) = ( s + µ )ˆ z. (4)This suggests that the OAM in such a situation cantake any arbitrary fractional value, as the factor µ cantake any value between 0 to 1 . It is observed that thespecific case for the quantized fractional value µ = , corresponds to the non-paraxial beam, where the vortexis orthogonal to the wave-front propagation direction. Inanalogy with the central charge of conformal field the-ory, the monopole charge undergoes the renormalizationgroup (RG) flow [16, 17]. When µ depends on a certainparameter λ, for certain fixed points λ ∗ , µ takes quan-tized values. However, for other values of λ, µ is non-quantized. The RG flow suggests that µ is a decreasingfunction such that L ∂µ∂L ≤ , where L is the length scale.Thus we observe that for non-quantized value of µ, itchanges with a characteristic length scale. This impliesthat EVBs carrying fractional OAM will be unstable onpropagation. However it will be shown here that, whenEVBs move in an external magnetic field, we can at-tain the stability of these states. Indeed in this case theBerry phase factor µ is changed through the incorpora-tion of the Gouy phase associated with the correspondingLG modes. The stability of the fractional OAM state isattained by choosing an appropriate value of the radialindex, which essentially corresponds to the choice of afixed value of the length scale L. III. MONOPOLE HARMONICS ANDFRACTIONAL ORBITAL ANGULARMOMENTUM
It has been shown in some earlier papers [12, 18, 19]that the presence of the spin vector elevates the localiza-tion region of a relativistic massive as well as a masslessparticle from S to S . Indeed noting that S is equiva-lent to SU (2) , we can write S = SU (2) U (1) so that S can beconstructed from S by Hopf fibration. The Abelian field U (1) corresponds to monopole field, which gives rise tothe spin of the system. The magnetic flux line associatedwith this represents the vortex line. The vortex in anEVB corresponds to the spin vortex. Thus the introduc-tion of the vortex line, which gives rise to the spin degreesof freedom extends the localization region from S to S and incorporates an angle χ corresponding to the orien-tation around the vortex line. The situation is exhibitedclearly in the Skyrmion model of an electron, where itis considered that a scalar electron rotates around a di-rection vector(vortex line), which gives rise to the spindegrees of freedom [14, 15]. This leads to the introduc-tion of a field function of the form φ ( x µ , ξ µ ) , where (cid:126)x isthe spatial coordinate and (cid:126)ξ is the 3-vector representingthe direction vector. In this case the wave function takesinto account the polar coordinates(r, θ, φ ) for the spatialcoordinate (cid:126)x and an angle χ to specify the rotational ori-entation around the direction vector (cid:126)ξ. The eigen-valueof the operator i ∂∂χ represents the internal helicity [15].For an extended body represented by the de-Sitter groupSO(4,1), θ , φ and χ just represent the Euler angles. In3 space dimensions these three Euler angles have theircorrespondence to an axisymmetric system, where theanisotropy is introduced along a particular direction andthe components of the linear momentum satisfy a com-mutation relation of the form,[ p i , p j ] = iµ(cid:15) ijk x k r . (5)Evidently µ represents the charge of a magneticmonopole. The monopole harmonics incorporating theterm µ has been extensively studied by Fierz [20], Hurst[21] as well as by Wu and Yang [22]. Following them wecan write y m,µl = (1 + x ) ( m − µ )2 (1 − x ) − ( m + µ )2 d l − m d l − mx (cid:2) (1 + x ) l − m (1 − x ) l + µ (cid:3) e imφ e − iµχ , (6)where x = cosθ . Here l is the orbital angular momentumof a charged particle in the field of a magnetic monopolehaving charge µ and m is the eigen value of the z - com-ponent of L .For specific values of l = , m = ± , µ = ± , theabove expression yields for Y m,µl Y , = sin θ e i ( φ − χ ) / Y − , = cos θ e − i ( φ + χ ) / Y , − = cos θ e i ( φ + χ ) / Y − , − = sin θ e − i ( φ − χ ) / (7) This represents monopole harmonics for half-integralOAM l = with µ = ± . The doublet φ = (cid:18) φ φ (cid:19) , (8)with φ = Y , and φ = Y − , corresponds to a twocomponent spinor. The charge conjugate state is givenby ¯ φ = (cid:18) ¯ φ ¯ φ (cid:19) , (9)with ¯ φ = Y − , − and ¯ φ = Y , − . This shows that afermion can be viewed as a scalar particle moving withhalf-integral OAM in the field of a magnetic monopoleand represents a skyrmion. This suggests that fractionalOAM states have their relevance in the context of mag-netic monopoles.The expression (6) is valid when µ is quantized hav-ing values 0 , ± , ± ......... so that we have integerand fractional OAM with half integer value. Howeverwhen µ is non-quantized, OAM can take any arbitraryfractional value. A quantum state with fractional OAMwith non-quantized monopole charge is given by | M (cid:105) with M = (cid:96) + µ where (cid:96) is an integer and µ is the fractionalpart lying between 0 and 1 . It may be mentioned that wehave considered the states | M (cid:105) such that it is applicableto the quantized value of µ = also. The monopole har-monics contain the term involving µ in the form e − iµχ , where χ denotes the orientation of the vortex line i.e theposition of the discontinuity. The angle χ takes the value0 < χ < π. Generalizing this to the non-quantized val-ues of µ, we note that the term containing the value of µ in the state | M (cid:105) is of the form e − iµχ . Indeed a gener-alised expression for the fractional OAM state | M (cid:105) with M = (cid:96) + µ , (cid:96) ( µ ) being an integer (fraction), can be de-rived where these states are decomposed into the basis ofinteger OAM states. This is given by [6, 23] | M ( χ ) (cid:105) = (cid:88) (cid:96) (cid:48) c (cid:96) (cid:48) | M ( χ ) (cid:105)| (cid:96) (cid:105) , (10)where the coefficients c (cid:96) (cid:48) | M ( χ ) (cid:105) are given by c (cid:96) (cid:48) | M ( χ ) (cid:105) = exp ( − iµχ ) iexp [( i ( M − (cid:96) (cid:48) ) θ ]2 π ( M − (cid:96) (cid:48) ) exp [ i ( M − (cid:96) (cid:48) ) χ ](1 − exp ( iµ π )) . (11)with (cid:96) (cid:48) ∈ Z . It is noted that the expression (11) containsthe term e − iµχ as we have in the monopole harmonicsgiven by the expression (6). In fact such a term ariseshere from the consideration of the single-valuedness cri-terion derived from the position of the branch cut χ fromthe multivalued function e iµθ n where θ n = θ + πn (cid:96) +1 vary-ing from − (cid:96), − (cid:96) + 1 , .... + (cid:96) θ being the starting point ofthe interval [ θ , θ + 2 π ]. Thus we have the dependanceof the fractional OAM states on the dislocation χ .A characteristic feature of the state | M ( χ ) (cid:105) is thatthe probabilities P (cid:96) (cid:48) ( M ) of observing the fractional OAMstate for specific values of (cid:96) (cid:48) given by the modulus squareof the probability amplitudes are independent of the an-gle χ [23]. In fact we have P l (cid:48) ( M ) = | c l (cid:48) M ( χ ) | = sin ( µπ )( M − l (cid:48) ) π (12) . For M = l ∈ Z and l − l (cid:48) (cid:54) = 0, the fractional part µ = 0so that we have vanishing OAM probability. For l (cid:48) , wecan write P l (cid:48) ( M ) = lim µ → P (cid:96) (cid:48) ( l + µ ) = 12 π lim µ → − cos (2 πn )( l + µ − l (cid:48) ) = δ ii (13)This means that for integer M the OAM distributionis singular and is equipped with a single non-vanishingprobability at M = l (cid:48) . For fractional values of M , theprobabilities are peaked around the nearest integer to M .In case of light beams, in cylindrical coordinate thefield amplitude of the Laguerre-Gaussian mode is givenby u (cid:96)p ( ρ, φ, z ) = c (cid:96)p (cid:112) w ( z ) (cid:32) ρ √ w ( z ) (cid:33) | (cid:96) | exp (cid:18) − ρ w ( z ) (cid:19) L | (cid:96) | p (cid:18) ρ w ( z ) (cid:19) exp (cid:18) i ρ w ( z ) zz R (cid:19) exp ( i(cid:96)φ ) exp (cid:20) − i (2 p + | (cid:96) | + 1) tan − ( zz R ) (cid:21) , (14)where w ( z ) is the Gaussian spot size. w ( z ) = (cid:115) z R + z kz R = w (cid:115) z z R . (15)Here k denotes the wave number and z R the Rayleighrange, w the beam waist and L | (cid:96) | p are the Laguerre poly-nomials. The normalization constants are given by c (cid:96)p = (cid:115) p ! π ( | (cid:96) | + p )! . (16)The Gouy phase exp (cid:104) − i (2 p + | (cid:96) | + 1) tan − ( zz R ) (cid:105) de-scribes the phase change as the beam moves through thebeam waist situated at z = 0 . As it has been observedthat the finite superposition of the LG modes is centredaround the nearest integer to M , we can write ψ M ( χ ) ( ρ, φ, z ) = (cid:96) max (cid:88) (cid:96) (cid:48) = (cid:96) min c (cid:96) (cid:48) | M ( χ ) | u (cid:96) (cid:48) p ( ρ, φ, z ) (17) Indeed the distribution of the coefficients | c (cid:96) | shows thatmodes with an OAM index (cid:96) (cid:48) very different from M con-tribute only little to the superposition and for all prac-tical purposes, we can take the finite superposition ofthe LG modes which is centred around the nearest inte-ger to M. It is noted here that for every mode the sum(2 p + | (cid:96) | + 1) is equal to | (cid:96) max | + 1 or (cid:96) max . It is to be mentioned that when the fractional OAMstate is decomposed into the basis of integer OAM states,it is not possible to have the same Gouy phase for allmodes. This follows from the fact that, the superpositioninvolves odd and even values of (cid:96), but p has to be aninteger. However we can restrict the number of Gouyphases to two, such that one is for even (cid:96) and the otheris for odd (cid:96). Indeed when OVBs having fractional OAMstates are generated from the superposition of LG modes,the instability of these states leading to the change uponpropagation arises from the interference between the LGmodes with different Gouy phases. By choosing properradial indices p, for each value of (cid:96), the instability ofthe fractional OAM states can be suppressed [6]. It isnoted that when the fractional value of µ is associatedwith the monopole charge, the instability arises due tothe variation of µ with length scale L according to theRG flow equation. We can relate this length scale withthe radial index p in the Gouy phase so that the changein L will induce fluctuation in the Gouy phase factor.Due to the constraint that the radial index p must be aninteger, it is not possible to have the same Gouy phasefor all modes in the superposition. Thus we note thatin OVBs the change in the monopole charge µ with thelength scale effectively leads to the interference betweenthe LG modes with different Gouy phases giving rise tothe instability. A proper choice of p for each value of L helps to make the OVBs with fractional OAM statesstable upon propagation. IV. FRACTIONAL OAM STATES ANDELECTRON VORTEX BEAMS IN ANEXTERNAL MAGNETIC FIELD
Here we consider the fractional OAM states in EVBswhen these propagate in an external magnetic field. Asmentioned above, fractional OAM states arise when wehave tilted vortex associated with the non-quantizedvalue of the monopole charge µ. The states become un-stable upon propagation, since the RG flow suggests that µ changes with the length scale L. However, stability ofthe beam can be attained when fractional OAM statesare generated from EVBs propagating in a magnetic field.We consider that the magnetic field B ( r ) is generated bya monopole of strength α situated at the origin of a unitsphere. The flux passing through the surface Σ is givenby Φ | Σ = (cid:90) Σ (cid:126)Bd(cid:126)S = 2 πα (18)This effectively corresponds to the Berry phase attainedby a particle moving around the flux in a closed contour C corresponding to the holonomyΦ B = (cid:90) C (cid:126)Ad(cid:126)r (19)If the magnetic field B ( r ) is an axially symmetric lon-gitudinal one, the vortex vector potential can be chosenas (cid:126)A ( (cid:126)r ) = B ( r ) r (cid:126)e φ , (20)where (cid:126)B = (cid:126) ∇ × (cid:126)A. For the magnetic flux φ , we can writethe vortex vector potential as (cid:126)A ( (cid:126)r ) = φ πr ˆ (cid:126)e φ = αr ˆ (cid:126)e φ , (21)with α = φ π being denoted as the magnetic field param-eter.It may be mentioned here that Bliokh et al [24], con-sidered EVBs in the presence of a single magnetic flux line corresponding to an infinitely thin shielded solenoiddirected along the z axis and containing the flux φ. Thesituation involves the vortex vector potential given by (cid:126)A ( (cid:126)r ) = φ πr (cid:126)e a = αr (cid:126)e a , where α represents the vortexcharge of the vector potential (cid:126)A. This gives rise to Besselbeams in contrast to the LG beams. However the Besselbeams here are characterized by the fact that the Besselfunction order is shifted by α, so that the wave functioninvolves J l − α ( kr ) instead of J l ( kr ) . Indeed the expectation value of OAM yields (cid:104) L z (cid:105) = ( l − α ) . (22)This resembles the SOI but here it is caused by the Zee-man type interaction between OAM and magnetic field.However, this does not give rise to Zeeman energy sincethe wave function with (cid:104) L z (cid:105) (cid:54) = 0 is localized outside thearea of the magnetic field. In fact the situation here in-volves Dirac phase instead of the Berry phase when themagnetic field is associated with monopole charge. TheDirac phase is given by φ | D = (cid:90) C (cid:126)A.d(cid:126)r = 2 πα, (23)where C is a closed loop characterized by C = ( r =const. φ ∈ [0 , π ] . It is noted that in the sharp point limit of theskyrmionic model of an electron, we can consider thenon-relativistic situation. The Schroedinger equation incylindrical coordinates for an electron in the magneticfield is given by − (cid:126) m (cid:34) r ∂∂r (cid:18) r ∂∂r (cid:19) + 1 r (cid:18) ∂∂φ + ig r w m (cid:19) + ∂ ∂z (cid:35) ψ = Eψ, (24)where w m = 2 (cid:113) (cid:126) | eB | is the magnetic length parameter, g = sgnB = ± r w m here essentially representsthe monopole charge which follows from (20) and (21).The solution of eqn (24) has the form of non-diffractingLaguerre-Gaussian (LG) beams given by [24] ψ L(cid:96),p (cid:39) (cid:18) rw m (cid:19) | (cid:96) | L | (cid:96) | p (cid:18) r w m (cid:19) exp (cid:18) − r w m (cid:19) exp [ i ( (cid:96)φ + k z z )] . (25)It is noted that Landau LG modes yields exact solution.The wave numbers satisfy the dispersion relation E = k m − Ω (cid:96) + | Ω | (2 p + | (cid:96) | + 1) . (26)where Ω = eB m is the Larmor frequency. Here E || = k m is the energy of the free longitudinal motion. Thetransverse motion energy is given by E ⊥ = E Z + E g , where E Z = Ω (cid:96) represents the Zeeman energy and E g = | Ω | (2 p + | (cid:96) | + 1) , p being the radial quantum number,is associated with the Gouy phase of the diffractive LGmodes. For the Landau state, the expected value of OAMcorresponds to (cid:104) L z (cid:105) = (cid:96) + g (cid:104) r ω m (cid:105) , (27)where g = ± (cid:104) r ω m (cid:105) is given by (cid:104) r ω m (cid:105) = (cid:104) ψ | r ω m | ψ (cid:105)(cid:104) ψ | ψ (cid:105) = (2 p + | (cid:96) | + 1) . (28)This determines the squared spot size of the LG beams.From this we have the relation (cid:104) α (cid:105) = (cid:104) r w m (cid:105) = 2 p + | (cid:96) | + 1 , (29)where p = 0 , , .... is the radial quantum number and | (cid:96) | is the azimuthal quantum number. So fractional OAMstates given by (cid:96) + µ where µ corresponds to the fractionalvalue µ ∈ [0 ,
1] representing the monopole charge in freecase is now modified as µ eff = µ + g (2 p + | (cid:96) | + 1) (30)As it is mentioned, the instability in the fractional OAMstates arises due to the change of µ upon propagationas the RG flow implies that the non-quantized monopolecharge changes with the length scale. However, an appro-priate choice of the radial quantum number p essentiallyfixes a length scale, so that in this case these states be-come stable. Indeed when we use the modified monopolecharge the state | M ( χ ) (cid:105) will be transformed to | ˜ M ( χ ) (cid:105) , where ˜ M ( χ ) = (cid:96) + µ eff . When ˜ M ( χ ) is decomposed intothe basis of integer OAM states, from eqn.(12), we findthat the probability P (cid:96) ( ˜ M ) is now changed to P (cid:96) ( ˜ M ) = | c (cid:96) (cid:48) [ ˜ M ( α )] | = sin [ µ eff π ]( ˜ M − (cid:96) (cid:48) ) π = sin [ µ + (2 p + | (cid:96) | + 1)) π ]( l + µ + (2 p + | (cid:96) | + 1) − (cid:96) (cid:48) ) π . (31)Now noting that sin [( N + µ ) π ] = sin [ µπ ], when N isan integer. We finally can write, P (cid:96) ( ˜ M ) = sin [ µπ ]( (cid:96) + µ + (2 p + | (cid:96) | + 1) − (cid:96) (cid:48) ) π . (32)As is evident from the above eqn., this probability P (cid:96) ( ˜ M )vanishes for integer OAM states ( µ = 0). From eqn. (32)we note that for various values of p, P l ( ˜ M ) changes andas p increases this approaches towards zero. Thus withincreasing values of p we have the situation such that thecontribution of the fractional part to the state | ˜ M ( χ ) (cid:105) gradually vanishes and it approaches towards an integerOAM state, leading to a stable configuration. Thus with Μ P (cid:72) M (cid:142) (cid:76) Μ P (cid:72) M (cid:142) (cid:76) FIG. 1: (Color online): The Probability P ( ˜ M ) for variousvalues of µ and p for (a) (cid:96) = 2 , (cid:96) (cid:48) = 1 and (b) (cid:96) = − , (cid:96) (cid:48) = 1Down:(b) . the appropriate choice of p, we can achieve stability. Thuswe find that when, EVBs move in an external magneticfield, the change in µ as depicted in eqn. (30), effectivelymakes the corresponding fractional OAM states stableupon propagation, when a proper radial index p is chosen.It is to be noted that, unlike OVBs, here we need notrequire a superposition of LG modes to have stable frac-tional OAM state. This suggests that, for EVBs movingin an axisymmetric longitudinal magnetic field, we canhave stable fractional OAM states. In Fig. 1, we haveplotted the probability P ( ˜ M ) for various values of µ and p in (a) with (cid:96) = 2 , (cid:96) (cid:48) = 1 and in (b) with (cid:96) = − , (cid:96) (cid:48) = 1.As the Gouy phase factor 2 p + | (cid:96) | + 1 , essentiallycorresponds to the expectation value of the magneticmonopole charge, the stability of these states can beachieved by properly tuning the magnetic field. Thismakes the situation of EVBs different from OVBs havingstable fractional OAM states. V. DISCUSSION
The EVBs carrying fractional OAM have the specificfeature that the geometric phase acquired by the scalarelectron orbiting around the vortex line involves non-quantized monopole charge. Fractional OAM states arisedue to the spin-orbit coupling associated with the gen-eration of the geometric phase. The RG flow of themonopole charge induces length scale dependence in it,which makes the EVBs unstable on propagation. How-ever, when EVBs propagate in an external magnetic field,the change in the Berry phase factor due to incorporationof the Gouy phase makes the fractional OAM states sta-ble, with a proper choice of the radial index. This makesthe situation different from OVBs, where fractional OAMstates become stable when these are generated by thesynthesis of the LG modes with a minimal number ofdifferent Gouy phases.We have already pointed out in an earlier paper [11]that the temporal variation of the monopole charge in anEVB in free space leads to spin Hall effect. This arisesfrom the anomalous velocity caused by the Berry curva-ture. In the presence of an external magnetic field, themodified Berry curvature leads to spin filtering such thateither positive or negative spin states emerge in spin Hallcurrents with clustering of spin 1 / z → ∞ . However,it does not show the formation of the chain of vortices.Finally, we may add here that fractional OAM statehave already been studied in entangled photon pairs[27, 28]. Indeed as fractional OAM states involvenon-quantized monopole charge, we have to take intoaccount the effect of the Dirac string. However theobservability of the Dirac string can be avoided, when wetake into account an entangled state [17]. In view of thiswe note that just like OVBs it is expected that, thesestates can also be exhibited for entangled electron pairs.A stable fractional OAM state thus become potentiallyimportant in the study of quantum information andfoundation. [1] J.F. Nye and M.V. Berry, Proc R Soc. (London) A ,165 (1974)[2] L. Allen, M. W. Beijersbergen , R. J. C. Spreeuw and J.P. Woerdman, Phys. Rev. A , 8185, (1992).[3] M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensenand J. P. Woerdman, Optics Communications, ,321,(1994).[4] V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, JETPLetters, , 429, (1990).[5] N. R. Heckenberg, R. M. Duff, C.P. Smith and A.G.White, Opt. Lett. , 221 (1992)[6] J.B.Gotte et al., J.Mod.Opt. , 1723, (2006);Optics Ex-press , 993, (2008).[7] M. Uchida and A. Tonomura, Nature , 737, (2010).[8] J. Verbeeck, H.Tian and P. Schattschneider,Nature ,301, (2010).[9] B. J. McMorran et. al, Science, , 192, (2011).[10] K. Y. Bliokh et. al., Phys.Rev.Lett, , 174802, (2011).[11] P. Bandyopadhyay, B. Basu, D. Chowdhury,Proc.R.Soc.(London) A , 20130525, (2014).[12] P. Bandyopadhyay, B. Basu, D. Chowdhury, Phys.Rev. Lett. , 144801,(2016); Phys. Rev. Lett. ,194801,(2015).[13] D. Chowdhury, B. Basu, P. Bandyopadhyay, Phys. Rev.A , 033812 (2015). [14] P. Bandyopadhyay , K. Hajra, J. Math. Phys. , 711,(1987).[15] D. Banerjee, , P. Bandyopadhyay, J. Math. Phys. ,990, (1992).[16] P. Bandyopadhyay, Int. J. Mod Phys. A , 1415, (2000).[17] P. Bandyopadhyay, Proc. R. Soc.(London) A. , 427,(2011).[18] F Bayen and J. Niederlie, Czech. J. Phys. B ,1317,(1981).[19] P. Bandyopadhyay, Int. J. Theo.Phys. , 131, (1987).[20] M. Fierz,Helv.Phys.Acta, ,27,(1944).[21] C.A Hurst, Ann. Phys ,51, (1968).[22] T. T. Wu, C. N. Yang, Nuclear phys B, , 365, (1976).[23] J.B.Gotte et al., J. Mod. Opt, , 1723, (2007).[24] K. Y. Bliokh, P. Schattschneider, J. Verbeeck, and F.Nori, Phys. Rev. X , 041011, (2012).[25] M V Berry, J Opt. A , 259 (2004)[26] J. Leach, E. Yao and M. J Padgett, New Journal ofPhysics, , 71, (2004)[27] S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nien-huis, and J. P. Woerdman, Phys. Rev. Lett. , 217901,(2004).[28] G. F. Calvo, A. Picn, and A. Bramon, Phys. Rev. A75