Relativistic quantum dynamics of twisted electron beams in arbitrary electric and magnetic fields
aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b Relativistic quantum dynamics of twisted electron beams inarbitrary electric and magnetic fields
Alexander J. Silenko , , , ∗ Pengming Zhang , , † and Liping Zou , ‡ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia Research Institute for Nuclear Problems,Belarusian State University, Minsk 220030, Belarus and University of Chinese Academy of Sciences,Yuquanlu 19A, Beijing 100049, China (Dated: February 20, 2019)
Abstract
Relativistic quantum dynamics of twisted (vortex) Dirac particles in arbitrary electric and mag-netic fields is constructed for the first time. This allows to change the controversial contemporarysituation when the nonrelativistic approximation is used for relativistic twisted electrons. The rela-tivistic Hamiltonian and equations of motion in the Foldy-Wouthuysen representation are derived.A critical experiment for a verification of the results obtained is proposed. The new importanteffect of a radiative orbital polarization of a twisted electron beam in a magnetic field resulting ina nonzero average projection of the intrinsic orbital angular momentum on the field direction ispredicted. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] ~ ) havebeen recently obtained [11]. At present, much attention is also devoted to the interactionsof such beams with atoms, nuclei and a laser field [12]. The dynamics of the intrinsic OAMin external magnetic and electric fields has been studied in Refs. [2, 13–18]. However, wenote in advance that our result is different from the equation of motion of the intrinsic OAMin an electric field previously found in Ref. [2]. The general description of the relativisticdynamics of an intrinsic OAM in arbitrary electric and magnetic fields in the framework ofclassical physics has been made in Ref. [19]. In particular, the correction of the equationof motion of the intrinsic OAM in the electric field previously obtained in Ref. [3] has beenfulfilled in this work. The methods of the manipulation of beams developed in Ref. [19]use not only the magnetic field but also the electric one. The great importance of twistedelectrons requires detailed investigations of fundamental quantum-mechanical properties oftwisted electron beams. Such investigations have been carried out in many publications(see Refs. [3, 4] and references therein). However, only the nonrelativistic approximationhas been formerly used. The correct quantum dynamics of a twisted electron in an electric field has not been elaborated at all. Therefore, some fundamental properties need a furtherinquiry. The quantum-mechanical approach used differs the present work from our recentstudy [19] where the classical approach has been applied.In the present work, the system of units ~ = 1 , c = 1 is used. We include ~ and c explicitly when this inclusion clarifies the problem. The square and curly brackets, [ . . . , . . . ]and { . . . , . . . } , denote commutators and anticommutators, respectively.Since vortex electrons are relativistic quantum objects admitting also a semiclassicaldescription, a construction of a relativistic Schr¨odinger-like dynamics of such particles isnecessary. It is important that one mainly observes a motion of charged centroids and itis instructive to mirror this circumstance in an appropriate quantum description. We solvethis problem in the present work. We should also notice the previous quantum-mechanicalanalysis mirrored in the reviews [3, 4].A twisted electron is a single pointlike particle. However, its wave function is a superpo-sition of states with different momentum directions. While the twisted electron in vacuum2as a nonzero intrinsic OAM and a nonzero component of the momentum in the plane or-thogonal to the direction of its resulting motion, it can be described by the Dirac equationfor a free particle. Quantum mechanics (QM) of the twisted electron in external electromag-netic fields is also governed by the usual Dirac equation. We can disregard the anomalousmagnetic moment of the electron because its g factor is close to 2. The Schr¨odinger form ofthe relativistic QM is provided by the relativistic Foldy-Wouthuysen (FW) transformation.The relativistic extensions of the FW method [20] have been first proposed in Refs. [21, 22].There are many other methods of the relativistic FW transformation (see Refs. [23–30]and references therein). The results obtained by different methods agree because of theuniqueness of the FW representation proven in Ref. [31]. In all previous studies of twistedparticles in external fields, only the nonrelativistic FW transformation has been fulfilled(see the reviews [3, 4]). The relativistic FW transformation has been formerly used only forfree twisted particles [32, 33]. Thus, the nonrelativistic approximation has been used for adescription of relativistic objects (the kinetic energy of twisted electrons, 200 −
300 keV, iscomparable with their rest energy, 511 keV). As a result, the present state of the theory oftwisted electron beams is controversial.The exact relativistic Hamiltonian in the FW representation (the FW Hamiltonian) fora Dirac particle in a magnetic field has been first obtained in Ref. [21] and its validity hasbeen confirmed in other works [25, 34, 35]. It is given by H F W = β √ m + π − e Σ · B , (1)where π = p − e A is the kinetic momentum, B = ∇ × A is the magnetic induction, and β and Σ are the Dirac matrices. This Hamiltonian is valid for a twisted and a untwistedparticle. The spin angular momentum operator is equal to s = ~ Σ /
2. The magneticfield is, in general, nonuniform but time-independent. Unlike all previous investigations, ourquantum description of a twisted Dirac particle is fully relativistic and correctly defines bothelectric and magnetic interactions. Our precedent study [19] has shown the importance ofthe electric field for the manipulation of twisted electron beams.The FW Hamiltonian acts on the bispinor wave function Ψ
F W = φ . Since both thenonrelativistic and the relativistic FW Hamiltonians commute with the operators π and s z , their eigenfunctions coincide. In the uniform magnetic field B = B e z , they have the3orm of non-diffracting Laguerre-Gauss beams [13, 36, 37]. It is convenient to present theexact energy spectrum obtained by different methods [21, 34, 35, 38–41] as follows: E = p m + (2 n + 1 + | l z | + l z + 2 s z ) | e | B, (2)where n = 0 , , , . . . is the radial quantum number and l = r × p = L + L ( e ) is the totalOAM operator being the sum of the intrinsic ( L ) and extrinsic ( L ( e ) ) OAMs. The sameenergy spectrum but for the squared energy operator has been obtained in Refs. [15, 16].We consider the conventional canonical OAMs. The difference between the dynamics of thecanonical and kinetic (mechanical) OAMs has been investigated in Ref. [42]. All energylevels of twisted particles in the uniform magnetic field belong to the Landau levels. Therelativistic approach (unlike the nonrelativistic one) demonstrates that the Landau levelsare not equidistant for any field strength. This property has not been mentioned in Refs.[15, 16, 21, 34, 35, 38–41] while it has been noted in Ref. [39] that the energy spectrumbecomes quasicontinuous when the quantum number n becomes very large. A nonzerointrinsic OAM increases a degeneracy multiplicity of the Landau levels (cf. Refs. [3, 13]).As a rule, we can use the weak-field approximation and can suppose that the de Brogliewavelength, ~ /p , is much smaller than the characteristic size of the nonuniformity region ofthe external field. In this case, the Hamiltonian (1) takes the form H F W = β √ m + π − e (cid:26) ǫ ′ , Π · B (cid:27) = βǫ ′ − β e (cid:20) ǫ ′ ( l + Σ ) · B + B · ( l + Σ ) 1 ǫ ′ (cid:21) ,ǫ ′ = p m + p , (3)where [ ǫ ′ , l ] = 0. Equation (3) agrees with the result obtained in Ref. [37]. The small term mω L r / in the magnetic field . This term originates from e A / (2 m ) and is omitted in the presentstudy because it is proportional to B .It is necessary to take into account that a twisted electron is a charged centroid [2, 3]. Todescribe observable quantum-mechanical effects, we need to present the Hamiltonian in termsof the centroid parameters. The centroid as a whole is characterized by the center-of-chargeradius vector R and by the kinetic momentum π ′ = P − e A ( R ), where P = − i ~ ∂/ ( ∂ R ).The intrinsic motion is defined by the kinetic momentum π ′′ = ppp − e [ A ( r ) − A ( R )]. Here4 pp = − i ~ ∂/ ( ∂ rrr ), rrr = r − R , π ′ + π ′′ = π , P + ppp = p . Since A ( r ) = A ( R ) + 12 B ( R ) × rrr , the operator π takes the form π = π ′ + ppp − e L · B ( R ) + B ( R ) · L ] + π ′ · π ′′ + π ′′ · π ′ . After summing over partial waves with different momentum directions, < π ′ · π ′′ + π ′′ · π ′ > =0. More precisely, the operator π ′ · π ′′ + π ′′ · π ′ has zero expectation values for any eigenstatesof the operator π and, therefore, it can be omitted. It can be added that this summing canbe performed for the squared Hamiltonian H F W .The FW Hamiltonian summed over the partial waves [3] takes the form H F W = βǫ − β e (cid:20) ǫ Λ · B ( R ) + B ( R ) · Λ ǫ (cid:21) ,ǫ = q m + π ′ + ppp , Λ = L + Σ . (4)The momentum and the intrinsic OAM can have different mutual orientations in differentLorentz frames [19, 43]. Such a geometry of twisted waves has been described in Ref. [44].The acceleration of the twisted electron in a uniform magnetic field does not dependon Λ . However, such a dependence takes place in a nonuniform magnetic field due to theStern-Gerlach-like force defined by the operator F SGl = β e (cid:18) ǫ ∇ (cid:2) Λ · B ( R ) (cid:3) + ∇ (cid:2) B ( R ) · Λ (cid:3) ǫ (cid:19) , (5)where ∇ ≡ ∂/ ( ∂ R ). This force is an analog of the Stern-Gerlach one affecting the spin butit is much (approximately, 2 L times) stronger. The operators of the magnetic and electricdipole moments, µ and d , are defined by H ( int ) F W = − (cid:2) µ · B ( R ) + B ( R ) · µ + d · E ( R ) + E ( R ) · d (cid:3) , (6)where H ( int ) F W is the interaction Hamiltonian. The operator of the magnetic dipole momentof a moving centroid obtained from Eq. (4) is given by µ = β e ( L + 2 s )2 ǫ . (7)This equation agrees with Refs. [15, 19, 45, 46]. When the weak-field approximation is notused, it can be obtained that the magnetic dipole moment of a twisted particle in a magnetic5eld is proportional to the kinetic OAM r × π and is affected by the additional current.This current is proportional to − e A and leads to a weak diamagnetic effect [2, 3, 47].To perform a general quantum description of a twisted Dirac particle in external fields, weneed to add terms dependent on the electric field. For a pointlike Dirac particle, these termshave been obtained in Refs. [24, 29, 48]. If one disregards spin effects, one needs to addonly the term e Φ( r ). It has been proven in Ref. [49] that the passage to the classical limitin the FW representation reduces to a replacement of the operators in quantum-mechanicalHamiltonians and equations of motion with the corresponding classical quantities. Themotion of the intrinsic OAM causes the electric dipole moment d which interaction withthe electric field should be taken into account. Certainly, d (0) = 0 in the rest frame of thetwisted electron. The results obtained in Refs. [19, 50] show that in the classical limit d = β × µ , β = V c ≡ ˙ R c . (8)The centroid velocity operator, V , can be obtained from the FW Hamiltonian: V = i [ H F W , R ] = β (cid:26) ǫ , π ′ (cid:27) . Since we use the weak-field approximation, we neglect the correction to this formula pro-portional to the electric field. A comparison with Refs. [24, 29, 48] allows us to obtain theformula for the operator of the electric dipole moment: d = β e (cid:18) ǫ β × L − L × β ǫ (cid:19) = e (cid:18) ǫ π ′ × L − L × π ′ ǫ (cid:19) . (9)Thus, the general FW Hamiltonian for a relativistic twisted particle in electric and mag-netic fields is given by H F W = βǫ + e Φ − β e (cid:20) ǫ L · B ( R ) + B ( R ) · L ǫ (cid:21) + e (cid:26) ǫ L · [ π ′ × E ( R )] − [ E ( R ) × π ′ ] · L ǫ (cid:27) . (10)In this equation, spin effects are disregarded because they can be neglected on the conditionthat L ≫
1. The term e Φ does not include the interaction of the intrinsic OAM with theelectric field.Equation (10) exhaustively describes the quantum dynamics of the intrinsic OAM inthe general case of a twisted Dirac particle in arbitrary electric and magnetic fields. In6articular, the equation of motion of the intrinsic OAM has the form d L dt = i [ H F W , L ] = 12 ( Ω × L − L × Ω ) , Ω = − β e (cid:26) ǫ , B ( R ) (cid:27) + e (cid:20) ǫ π ′ × E ( R ) − E ( R ) × π ′ ǫ (cid:21) . (11)When E ( R ) = 0, we obtain the relativistic quantum-mechanical equation for the Larmorprecession. A comparison with results obtained in the Dirac representation shows strengthsof the FW one. In Refs. [15, 16], a relativistic description of twisted electron beams in a uni-form magnetic field has been given in the former representation. However, the correspondingequation of motion of the OAM has not been obtained in these works.Another important problem is the relativistic quantum dynamics of the kinetic momen-tum. It is defined by the force operator: F = d π ′ dt = ∂ π ′ ∂t + i [ H F W , π ′ ]= e E ( R ) + β e ( ǫ , (cid:16) π ′ × B ( R ) − B ( R ) × π ′ (cid:17)) + F SGl . (12)Beam splitting in nonuniform electric and magnetic fields is conditioned by the Stern-Gerlach-like force operator F SGl = β e (cid:26) ǫ ∇ [ L · B ( R )] + ∇ [ B ( R ) · L ] 1 ǫ (cid:27) − e (cid:26) ǫ ∇ ( L · [ π ′ × E ( R )]) − ∇ ([ E ( R ) × π ′ ] · L ) 1 ǫ (cid:27) . (13)The force is exerted to the center of charge of the centroid. Equations (5) and (13) demon-strate an importance of a general description which includes particle interactions withnonuniform fields. Like Eq. (5), Eq. (13) presents the Stern-Gerlach-like force in termsof the centroid parameters. It is important to mention that p m + ppp is an effective massof the twisted particle.We can conclude that the quantum-mechanical equations obtained in the present workagree with the corresponding classical results given in Ref. [19].In the present work, we consider two important applications of the results obtained anddescribe new effects which experimental study allows one to ascertain fundamental propertiesof twisted electron beams. Rotation of the intrinsic OAM in crossed electric and magnetic fields. – Previously fulfilledexperiments used coherent superpositions of two twisted beams moving in a longitudinal7agnetic field ( z axis). Landau modes with equal amplitudes, the same radial index n ,and opposite projections of the intrinsic longitudinal OAMs undergo the rotation with theLarmor frequency [3, 13, 37]. This rotation is similar to the Faraday effect in optics [37]. Wepropose a similar and a simpler experiment in crossed electric and magnetic fields satisfyingthe relation E = − β × B , where E ⊥ B ⊥ β and β is the normalized beam velocity. Suchfields characterizing the Wien filter do not affect a beam trajectory. We suppose the fields E and B to be uniform.In the considered case, the classical limit of the relativistic equation for the angularvelocity of precession of the intrinsic OAM is given by Ω ( W ) = − e ( m + ppp )2 ǫ B . (14)Since ppp ≪ m , Eq. (14) agrees with the corresponding equation obtained in Ref. [19].While the experiment proposed is similar to the above-mentioned experiment carried outin Refs. [13, 37], it needs a simpler experimental setup. It is necessary to use a single twistedelectron beam possessing a standard orbital polarization collinear to the beam momentum ( z axis). The direction of the magnetic field B and the quasimagnetic one β × E is transversal( x axis). Therefore, the intrinsic OAM rotates in the yz plane with the angular frequency Ω ( W ) and reverses its direction with the angular frequency 2 Ω ( W ) . Since twisted electronbeams are relativistic, a quantitative verification of Eq. (14) can be fulfilled. Thus, theexperiment proposed is a critical experiment for a verification of the main equations for theFW Hamiltonian and the intrinsic-OAM dynamics obtained in the present work. Radiative orbital polarization of twisted electron beams in a magnetic field. – The radiationfrom twisted electrons is one of their fundamental properties. It is caused by their acceler-ation in external magnetic and electric fields and by the time dependence of their magneticmoments. The latter effect conditions the magnetic dipole radiation [51]. Its intensity isstandardly much smaller than that of the electric dipole radiation due to the particle accel-eration. The magnetic dipole radiation proportional to the intrinsic OAMs is important forthe Cherenkov radiation and the transition one [51]. Since the particle motion in a magneticfield is accelerated, we can consider only the electric dipole radiation. For particles closed instorage rings, it is conditioned by radiative transitions between Landau levels and is calledthe synchrotron radiation. As a rule, magnetic focusing is used. We expect that a storageand an acceleration of twisted electrons in cyclotrons (electron rings) will be carried out in8 nearest future. Evidently, these processes cannot vanish the intrinsic OAM due to theconservation of angular momentum. A natural process of an orbital depolarization is causedby a loss of the beam coherence and is not important for the considered problem. Theinitial orbital polarization of twisted electrons is longitudinal ( z axis) while their expectedfinal polarization is vertical and antiparallel to the main magnetic field ( y axis). The beamincoherence does not influence the vertical orbital polarization.We predict the new effect of a radiative orbital polarization of twisted electron beams in amagnetic field – orbital Sokolov-Ternov effect . The well-known effect is the radiative spinpolarization of electron/positron beams in storage rings caused by the synchrotron radiation(Sokolov-Ternov effect [39]). It consists in the radiative spin polarization which is acquired byunpolarized electrons and is opposite to the direction of the main magnetic field. The reasonof the effect is a dependence of spin-flip transitions from the initial particle polarization.The standard analysis [39, 52] can be extended on the twisted particles. It follows from theresults obtained in Refs. [12, 53] that quantum-electrodynamics effects are rather similarfor twisted and untwisted particles. In particular, the amplitude of elastic scattering of twovortex electrons is well approximated by two plane-wave scattering amplitudes with differentmomentum transfers, which interfere and give direct experimental access to the Coulombphase [53].However, an influence of the synchrotron radiation on the orbital polarization of twistedrelativistic particles needs a detailed separate study. In this work, we restrict ourselves toa consideration of some aspects of this problem. First of all, we should note the evidentsimilarity between interactions of the spin and the intrinsic OAM with the magnetic field[see Eq. (4)]. In particular, energies of stationary states depend on projections of the spinand the intrinsic OAM on the field direction. This similarity validates the existence ofthe effect of the radiative orbital polarization. As well as the radiative spin polarization,the corresponding orbital polarization acquired by unpolarized twisted electrons should beopposite to the direction of the main magnetic field. The effect is conditioned by transitionswith a change of a projection of the intrinsic OAM. The probability of such transitions islarge enough if the electron energy is not too small. Similarly to the spin polarization, theorbital one is observable when electrons are accelerated up to the energy of the order of 1GeV. The acceleration can depolarize twisted electrons but cannot vanish L . During theprocess of the radiative polarization, the average energy of the electrons should be kept9nchanged. For additional explanations, see Supplemental Material.Thus, a discovery of the fundamental property of the radiative orbital polarization needsmuch higher energies than usual energies of twisted electron beams (about 300 keV). Weconsider this as a positive factor because the twisted (vortex) states of particles can play animportant role in high-energy physics.In this letter, we have studied the special properties of relativistic twisted particles mov-ing in nonuniform electric and magnetic fields. The general FW Hamiltonian has beenderived and a Stern-Gerlach-like force has been presented in terms of the centroid parame-ters. Furthermore, a benchmark experiment has been proposed to confirm the dynamics ofthe intrinsic OAM of twisted electrons with a very simple experimental setup. At last, wehave predicted a new ”orbital Sokolov-Ternov effect” for a twisted electron beam in a mag-netic field, which can be measured with a high-energy twisted electron beam. The presentstate of quantum mechanics of twisted electrons is controversial because it uses the non-relativistic approximation for a description of relativistic objects. Our relativistic approachsubstantiates (and generalizes) the most part of previous results. We have correctly intro-duced the interaction of the intrinsic OAM with the electric field into quantum mechanicalequations for the first time and have corrected the equation of its motion in the electric fieldpreviously obtained in Ref. [2]. The relativistic approach shows that the Landau levels inthe uniform magnetic field are not equidistant.This work was supported by the Belarusian Republican Foundation for FundamentalResearch (Grant No. Φ18D-002), by the National Natural Science Foundation of China(Grants No. 11575254 and No. 11805242), by the National Key Research and DevelopmentProgram of China (No. 2016YFE0130800), and by the Heisenberg-Landau program of theGerman Federal Ministry of Education and Research (Bundesministerium f¨ur Bildung undForschung). A. J. S. also acknowledges hospitality and support by the Institute of ModernPhysics of the Chinese Academy of Sciences. The authors are grateful to I. P. Ivanov forhelpful exchanges. [1] M. Uchida, A. Tonomura, Generation of electron beams carrying orbital angular momentum,Nature , 737 (2010); J. Verbeeck, H. Tian, P. Schattschneider, Production and application f electron vortex beams, Nature , 301 (2010).[2] K. Bliokh, Y. Bliokh, S. Savel’ev, and F. Nori, Semiclassical Dynamics of Electron WavePacket States with Phase Vortices, Phys. Rev. Lett. , 190404 (2007).[3] K. Y. Bliokh, I. P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. B´ech´e, R. Juchtmans,M. A. Alonso, P. Schattschneider, F. Nori, and J. Verbeeck, Theory and applications of free-electron vortex states, Phys. Rep. , 1 (2017).[4] S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, Electron vortices: Beams withorbital angular momentum, Rev. Mod. Phys. , 035004 (2017).[5] S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, Electromagnetic Vortex Fields,Spin, and Spin-Orbit Interactions in Electron Vortices, Phys. Rev. Lett. , 254801 (2012).[6] J. Rusz, S. Bhowmick, M. Eriksson, N. Karlsson, Scattering of electron vortex beams ona magnetic crystal: Towards atomic-resolution magnetic measurements, Phys. Rev. B ,134428 (2014).[7] A. Edstr¨om, A. Lubk, J. Rusz, Elastic scattering of electron vortex beams in magnetic matter,Phys. Rev. Lett. , 127203 (2016).[8] A. B´ech´e, R. Juchtmans, J. Verbeeck, Efficient creation of electron vortex beams for highresolution STEM imaging, Ultramicroscopy , 12 (2017).[9] V. Grillo, T. R. Harvey, F. Venturi, J. S. Pierce, R. Balboni, F. Bouchard, G. C. Gazzadi,S. Frabboni, A. H. Tavabi, Z. Li, R. E. Dunin-Borkowski, R. W. Boyd, B. J. McMorran, E.Karimi, Observation of nanoscale magnetic fields using twisted electron beams, Nat. Commun. , 689 (2017).[10] B. J. McMorran, A. Agrawal, P. A. Ercius, V. Grillo, A. A. Herzing, T. R. Harvey, M. Linckand J. S. Pierce, Origins and demonstrations of electrons with orbital angular momentum,Phil. Trans. R. Soc. A , 20150434 (2017).[11] V. Grillo, G. C. Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, HighlyEfficient Electron Vortex Beams Generated by Nanofabricated Phase Holograms, Appl. Phys.Lett. , 043109 (2014); V. Grillo, G. C. Gazzadi, E. Mafakheri, S. Frabboni, E. Karimi,and R. W. Boyd, Holographic Generation of Highly Twisted Electron Beams, Phys. Rev.Lett. , 034801 (2015); E. Mafakheri, A. H. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C.Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V.Grillo, Realization of electron vortices with large orbital angular momentum using miniature olograms fabricated by electron beam lithography, Appl. Phys. Lett. , 093113 (2017).[12] A. G. Hayrapetyan, O. Matula, A. Aiello, A. Surzhykov, and S. Fritzsche, Interaction of Rel-ativistic Electron-Vortex Beams with Few-Cycle Laser Pulses, Phys. Rev. Lett. , 134801(2014); O. Matula, A. G. Hayrapetyan, V. G. Serbo, A. Surzhykov, and S. Fritzsche, Ra-diative capture of twisted electrons by bare ions, New J. Phys. , 053024 (2014); D. Seipt,A. Surzhykov, and S. Fritzsche, Structured x-ray beams from twisted electrons by inverseCompton scattering of laser light, Phys. Rev. A , 012118 (2014); D. Chowdhury, B. Basu,P. Bandyopadhyay, Relativistic electron vortex beams in a laser field, Phys. Rev. Lett. ,194801 (2015); V. Serbo, I. P. Ivanov, S. Fritzsche, D. Seipt, and A. Surzhykov, Scatteringof twisted relativistic electrons by atoms, Phys. Rev. A , 012705 (2015); V. A. Zaytsev, V.G. Serbo, and V. M. Shabaev, Radiative recombination of twisted electrons with bare nuclei:Going beyond the Born approximation, Phys. Rev. A , 012702 (2017); A. ˘Cerki´c, M. Busu-lad˘zi´c, and D. B. Milo˘sevi´c, Electron-ion radiative recombination assisted by a bichromaticelliptically polarized laser field, Phys. Rev. A , 063401 (2017).[13] K. Y. Bliokh, P. Schattschneider, J. Verbeeck, F. Nori, Electron vortex beams in a magneticfield: A new twist on Landau levels and Aharonov-Bohm states, Phys. Rev. X , 041011(2012).[14] G. M. Gallatin, B. McMorran, Propagation of vortex electron wave functions in a magneticfield, Phys. Rev. A , 012701 (2012); C. R. Greenshields, R. L. Stamps, S. Franke-Arnold,S. M. Barnett, Is the angular momentum of an electron conserved in a uniform magneticfield? Phys. Rev. Lett. , 033812 (2015).[15] K. van Kruining, A. G. Hayrapetyan, and J. B. G¨otte, Nonuniform currents and spins ofrelativistic electron vortices in a magnetic field, Phys. Rev. Lett. , 030401 (2017).[16] A. Rajabi and J. Berakdar, Relativistic electron vortex beams in a constant magnetic field,Phys. Rev. A , 063812 (2017).[17] Y. D. Han, T. Choi, Classical understanding of electron vortex beams in a uniform magneticfield, Phys. Lett. A , 1335 (2017).[18] M. Babiker, J. Yuan, and V. E. Lembessis, Electron vortex beams subject to static magneticfields, Phys. Rev. A , 013806 (2015); J. Qiu, C. Ren, and Z. Zhang, Precisely measuring theorbital angular momentum of beams via weak measurement, Phys. Rev. A
19] A. J. Silenko, Pengming Zhang and Liping Zou, Manipulating Twisted Electron Beams, Phys.Rev. Lett. , 243903 (2017).[20] L. L. Foldy, S. A. Wouthuysen, On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit, Phys. Rev. , 29 (1950).[21] K. M. Case, Some Generalizations of the Foldy-Wouthuysen Transformation, Phys. Rev. ,1323 (1954).[22] E. I. Blount, Extension of the Foldy-Wouthuysen Transformation, Phys. Rev. , 2454(1962).[23] L. G. Suttorp, S. R. de Groot, Covariant equations of motion for a charged particle with amagnetic moment, Nuovo Cim. A , 245 (1970).[24] A. J. Silenko, Dirac equation in the Foldy-Wouthuysen representation describing the interac-tion of spin-1/2 relativistic particles with an external electromagnetic field, Teor. Mat. Fiz. , 46 (1995) [Theor. Math. Phys. , 1224 (1995)].[25] A. J. Silenko, Foldy-Wouthuysen transformation for relativistic particles in external fields, J.Math. Phys. , 2952 (2003).[26] K. Y. Bliokh, Topological spin transport of a relativistic electron, Europhys. Lett. , 7 (2005);On the Hamiltonian nature of semiclassical equations of motion in the presence of an electro-magnetic field and Berry curvature, Phys. Lett. A , 123 (2006).[27] P. Gosselin and H. Mohrbach, Diagonal representation for a generic matrix valued quantumHamiltonian, Eur. Phys. J. C , 495 (2009).[28] D. Peng and M. Reiher, Local relativistic exact decoupling, J. Chem. Phys. , 244108(2012).[29] A. J. Silenko, Foldy-Wouthyusen transformation and semiclassical limit for relativistic particlesin strong external fields, Phys. Rev. A , 012116 (2008); Comparative analysis of direct and“step-by-step” Foldy-Wouthuysen transformation methods, Teor. Mat. Fiz. , 189 (2013)[Theor. Math. Phys. , 987 (2013)]; Energy expectation values of a particle in nonstationaryfields, Phys. Rev. A, , 012111 (2015); General method of the relativistic Foldy-Wouthuysentransformation and proof of validity of the Foldy-Wouthuysen Hamiltonian, Phys. Rev. A ,022103 (2015).[30] D.-W. Chiou and T.-W. Chen, Exact Foldy-Wouthuysen transformation of the Dirac-PauliHamiltonian in the weak-field limit by the method of direct perturbation theory, Phys. Rev. , 052116 (2016).[31] E. Eriksen, Foldy-Wouthuysen Transformation. Exact Solution with Generalization to theTwo-Particle Problem, Phys. Rev. , 1011 (1958).[32] S. M. Barnett, Relativistic Electron Vortices, Phys. Rev. Lett. , 114802 (2017).[33] K. Y. Bliokh, M. R. Dennis, and F. Nori, Position, spin, and orbital angular momentum of arelativistic electron, Phys. Rew. A , 023622 (2017).[34] W. Tsai, Energy eigenvalues for charged particles in a homogeneous magnetic field – anapplication of the Foldy-Wouthuysen transformation, Phys. Rev. D , 1945 (1973).[35] A. J. Silenko, Connection between wave functions in the Dirac and Foldy-Wouthuysen repre-sentations, Phys. Part. Nucl. Lett. , 501 (2008).[36] L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory , 3rd ed. (Perga-mon Press, Oxford, 1977).[37] C. Greenshields, R. L. Stamps, and S. Franke-Arnold, Vacuum Faraday effect for electrons,New J. Phys. , 103040 (2012).[38] I. M. Ternov, V. G. Bagrov, and V. Ch. Zhukovsky, Synchrotron radiation of electron withvacuum magnetic moment, Vestn. Mosk. Univ., Ser. Fiz. Astron., No. 1, 30 (1966) [Mosc.Univ. Phys. Bull., No. 1, 21 (1966)].[39] A. A. Sokolov and I. M. Ternov, Radiation from relativistic electrons , 2nd ed. (AIP, New York,1986).[40] R. F. O’Connell, Motion of a relativistic electron with an anomalous magnetic moment in aconstant magnetic field, Phys. Lett. A , 391 (1968).[41] V. Canuto and H. Y. Chiu, Quantum Theory of an Electron Gas in Intense Magnetic Field,Phys. Rev. , 1210 (1968).[42] C. R. Greenshields, R. L. Stamps, S. Franke-Arnold, S. M. Barnett, Is the angular momentumof an electron conserved in a uniform magnetic field? Phys. Rev. Lett. , 240404 (2014);M. Wakamatsu, Y. Kitadono, P.-M. Zhang, The issue of gauge choice in the Landau problemand the physics of canonical and mechanical orbital angular momenta, Ann. Phys. (N. Y.) , 287 (2018).[43] K. Y. Bliokh and F. Nori, Relativistic Hall Effect, Phys. Rev. Lett. , 120403 (2012).[44] K. Y. Bliokh and F. Nori, Spatiotemporal vortex beams and angular momentum, Phys. Rev.A , 033824 (2012).
45] A. O. Barut and A. J. Bracken, Magnetic-moment operator of the relativistic electron, Phys.Rev. D , 3333 (1981).[46] K. Y. Bliokh, M. R. Dennis, and F. Nori, Relativistic Electron Vortex Beams: Angular Mo-mentum and Spin-Orbit Interaction, Phys. Rev. Lett. , 174802 (2011).[47] C. Greenshields, S. Franke-Arnold, R. L. Stamps, Parallel axis theorem for free-space electronwavefunctions, New J. Phys. , 093015 (2015).[48] A. J. Silenko, Quantum-mechanical description of the electromagnetic interaction of relativisticparticles with electric and magnetic dipole moments, Russ. Phys. J. , 788-792 (2005).[49] A. J. Silenko, Classical limit of equations of the relativistic quantum mechanics in the Foldy-Wouthuysen representation, Pis’ma Zh. Fiz. Elem. Chast. Atom. Yadra , 144 (2013) [Phys.Part. Nucl. Lett. , 91 (2013)].[50] A. J. Silenko, Spin precession of a particle with an electric dipole moment: contributions fromclassical electrodynamics and from the Thomas effect, Phys. Scripta , 065303 (2015).[51] A. S. Konkov, A. P. Potylitsin, V. A. Serdyutskii, Interference of the transient radiation fieldsproduced by an electric charge and a magnetic moment, Russ. Phys. J. , 1249 (2012); I.P. Ivanov and D. V. Karlovets, Detecting Transition Radiation from a Magnetic Moment,Phys. Rev. Lett. , 264801 (2013); I. P. Ivanov and D. V. Karlovets, Polarization radiationof vortex electrons with large orbital angular momentum, Phys. Rev. A , 043840 (2013);A. S. Konkov, A. P. Potylitsyn, and M. S. Polonskaya, Transition Radiation of Electronswith a Nonzero Orbital Angular Momentum, JETP Lett. , 421 (2014); I. P. Ivanov, V. G.Serbo, and V. A. Zaytsev, Quantum calculation of the Vavilov-Cherenkov radiation by twistedelectrons, Phys. Rev. A , 053825 (2016); I. Kaminer, M. Mutzafi, A. Levy, G. Harari, H. H.Sheinfux, S. Skirlo, J. Nemirovsky, J. D. Joannopoulos, M. Segev, and M. Solja˘ci´c, Quantum˘Cerenkov Radiation: Spectral Cutoffs and the Role of Spin and Orbital Angular Momentum,Phys. Rew. X , 011006 (2016).[52] V. B. Berestetskii, E. M. Lifshitz, L. P. Pitaevskii, Quantum Electrodynamics , 2nd ed. (Perg-amon Press, Oxford, 1982).[53] I. P. Ivanov, Measuring the phase of the scattering amplitude with vortex beams, Phys. Rev.D , 076001 (2012); I. P. Ivanov, D. Seipt, A. Surzhykov, and S. Fritzsche, Elastic scatteringof vortex electrons provides direct access to the Coulomb phase, Phys. Rev. D , 076001(2016).
54] See Supplemental Material. upplemental Material to “Relativistic quantum dynamics oftwisted electron beams in arbitrary electric and magnetic fields” In this Supplemental Material, we present some additional explanations of the orbital
Sokolov-Ternov effect (a radiative orbital polarization of twisted electron beams in a mag-netic field). We briefly consider an origin of the Sokolov-Ternov effect consisting in a ra-diative spin polarization following Ref. [39]. Due to the latter effect, electron spins acquirethe vertical polarization antiparallel to the main magnetic field. The maximum value of thespin polarization in the Sokolov-Ternov effect is 8 √ / ≈ . B = B e z , it is defined by W = − µ · B , (S1)where the operator µ is given by Eq. (7). For electrons ( e = −| e | ), the predominantdirection of the spins and intrinsic OAMs is antiparallel to the z axis.The probability of spin-flip transitions has the form [39] w = 12 τ √ ζ ! , (S2)where the relaxation time is equal to τ = 8 ~ √ mce (cid:18) mc ǫ (cid:19) (cid:18) B B (cid:19) , (S3) ζ = ± z axis, and B = m c / ( | e | ~ ) = 4 . × G.Certainly, there are also radiative transitions with a conservation of the spin projection.The radiative transitions with a change of the vertical projection of the intrinsic OAM, L z = m ~ can be described in a similar way. The selection rule for electric dipole transitions is∆ m = 0 , ±
1. Transitions between levels with | ∆ m | > with a change of the quantum number m can bepresented in the form w m = 1 τ ′ ( a m + χb m ) , (S4)1here τ ′ is the relaxation time, a m , b m are some positive coefficients dependent on the initialquantum number m , and χ = m − m ′ is the difference between the initial and final quantumnumbers. Transitions with a decrease of the vertical projection of the intrinsic OAM ( χ > L + 1 trans-port equations. For the spin-flip transitions, the two transport equations are given by [39] dn − dt = n + w ζ =1 − n − w ζ = − ,dn + dt = n − w ζ = − − n + w ζ =1 , n − + n + = n = const, (S5)where the plus and minus signs denote the states with the positive and negative spin projec-tions. For twisted electrons, the transitions changing the quantum number m are describedby the following transport equations: dn − L dt = n − L +1 w − L +1 , χ =1 − n − L w − L, χ = − ,dn m dt = n m +1 w m +1 , χ =1 + n m − w m − , χ = − − n m ( w m , χ =1 + w m , χ = − ) , − L < m < L,dn L dt = n L − w L − , χ = − − n L w L, χ =1 , n − L + n − L +1 + · · · + n L = n = const. (S6)While the transport equations for the intrinsic OAM are much more cumbersome thanthose for the spin, the dynamics of the intrinsic OAM and the spin is similar. The result isthe final orbital polarization antiparallel to the zz