Vortex electron generated by microwave photon with orbital angular momentum in a magnetic field

With semi-quantum theory, we quantitatively deduce microwave photons radiated from the moving electrons in a magnetic field, as well as the interaction of transmitted microwave photons with the vortex electrons in the magnetic field. It shows that the Orbital Angular Momentum (OAM) transition between microwave photons and vortex electrons in the magnetic field occurs when the relativistic effect is considered. This work indicates an effective way to transfer OAM between microwave photons and vortex electrons theoretically. By the OAM microwave photon resonance absorption, different vortex electrons with radial and magnetic quantum numbers can be generated. Furthermore, vortex electrons can be detected to analyze the OAM carried by microwave photons.


I. INTRODUCTION
An electro-magnetic (EM) wave carries Angular Momentum (AM), which is considered to be a fundamental physical quantity based on both the theories of classical mechanics and quantum mechanics. 1,24][5][6] In addition to OAM in the optical and radio frequencies, 5,7 some methods for generating OAM in x-ray beams and electron beams were also proposed, respectively. 8,9Moreover, the generations of acoustic OAM and γ ray beams with OAM were also proposed, respectively. 10,11he research studies on OAM of the infrared and extreme ultraviolet light were also explored. 12,13A few matter-waves with OAM were also studied.][16] Besides, the twisted neutron beams were also well investigated. 17,18ecause OAM is a fundamental physical quantity, it leads to several applications in communications, 19 detection, 20 astronomy, 21 optical tweezers, 22 manipulation, 23 and quantum information processing. 24ortex electrons were first discussed by Bliokh et al. 25 The AM and magnetic properties of the electron vortex can lead to potential applications in the optical vortex generation and detection. 26,27r example, the OAM of microwave photons can be transited to vortex electrons.The additional magnetic properties caused by OAM can be utilized to detect the OAM of microwave photons by the similar Stern-Gerlach experiment of vortex electron in a gradient magnetic field or OAM Hall effect in a uniform electric field. 28,29nfortunately, there is a low transition probability that the OAM transition between microwave photons and vortex electrons occurs in free space. 30The electron in a magnetic field along the z axis will make a cyclotron motion and form a Landau state. 31hen the speed of the electron is much smaller than the speed of light, the electron can radiate the circularly polarized EM wave. 32,33n addition, when the electron is in the relativistic state, radiation conditions are different.Katoh et al. showed that a relativistic free electron in the circular motion can radiate the EM wave with OAM.More specifically, the radiation field contains harmonic components and the photon of the lth harmonic carries l ̵ h total AM. 34Besides, OAM radio waves and optical vortices have been generated experimentally by the gyrotron and relativistic vortex electron beams, respectively. 27,35n this paper, the OAM transition probability between microwave photons and vortex electrons is evaluated theoretically.
It is demonstrated that the EM wave with OAM can be emitted and absorbed by the relativistic electron in the magnetic field. 2,36,37In other words, the OAM of microwave photons can be transferred to vortex electrons, which reveals that the OAM of vortex electrons can be detected to analyze the OAM of microwave photons.

II. SYSTEM MODEL
As shown in Fig. 1, an electron rotates around the z axis in the xOy plane with an angular velocity ω in the magnetic field B = (0, 0, −B) (picking B to be positive), and the velocity of the electron is v.The acceleration of the electron can be expressed as a 0 = |v|.The radius and the angular frequency of the electron cyclotron motion are R 0 = mev/|eB| and ω = |eB|/me, respectively, where e, me, v, and c denote the charge of the electron, the mass of the electron, the speed of the electron, and the speed of light, respectively (picking e to be negative). 1In the framework of quantum mechanics, 2 the velocity expectation value ⟨v⟩ of the electron satisfies Lorentz's law.By adopting the metric signature (+, −, −, −) and natural unit ̵ h = c = 1, the time-dependent standard Dirac equation can be expressed as where ψ is the 4-component wave function, p is the momentum operator, α and β are the 4 × 4 Dirac matrices, A and Φ are the vector potential and the scalar potential, respectively.According to Maxwell's equations, the vector potential is related to the magnetic field, i.e., B = ∇ × A. Therefore, in the cylindrical coordinate system r = (r, φ, z), the vector potential can be written as A = 1/2(B × r) = 1/2(Br ⃗ φ), and the scalar potential Φ = 0. 38 In the presence of the magnetic field, the two kinds of OAM, the gauge-variant canonical OAM and the gauge-invariant kinetic OAM, are absolutely different. 39All time-independent nonrelativistic approximation solutions of the positive energy electron 2-component wave function in Eq. ( 1) and the corresponding energy expressions are calculated as follows: En,m = 1/2(2n where n is the radial quantum number, m is the magnetic quantum number, L m n ( * ) denotes the generalized Laguerre polynomial, s = ±1/2 denotes the spin state sign, Σ denotes the spin state (up spin Σ ↑ = [1, 0] T and down spin Σ ↓ = [0, 1] T ), and p z = ̵ hkz denotes the linear momentum along the z axis. 2,40ince the direction of the electrons in the magnetic field and the direction of the magnetic field B are always in the right-handed spiral relationship, the kinetic AM direction is consistent with the magnetic field direction (selecting the negative direction).To facilitate theoretical calculations, the gauge is chosen to make m positive, i.e., m > 0. Assuming that electrons move mainly in the xOy plane, ignoring the momentum of the z axis, it can be seen that the timeindependent state of the electron is |n, m⟩, which is related to two quantum numbers.The orthogonality of all states is ⟨n ′ , m ′ |n, m⟩ ∝ δ m ′ m , where δ denotes the Kronecker function.Furthermore, the canonical OAM is expressed as where ℒ z = −i ̵ h∂/∂φ denotes the OAM operator along the z axis.Therefore, all states of the Landau level carry the canonical OAM eigenvalue of m ̵ h.The eigenstate equation is written as ℒ z |n, m⟩ = m ̵ h|n, m⟩.

III. MICROWAVE PHOTONS WITH OAM RADIATED BY RELATIVISTIC VORTEX ELECTRONS
According to Ref. 41, the energy of the relativistic electron is calculated by All energy levels of twisted particles in the uniform magnetic field belong to the Landau levels. 42It is demonstrated in the relativistic approach that the Landau levels are not equidistant for any field strength.According to Refs.40 and 41, the 4-component Dirac positive energy electron wave function (up spin ↑ and down spin ↓) can be written as follows: where r = μr and E+ = me + E (R) .Besides, the symbol E (R) represents the relativistic energy of the state Ψ, i.e., Ψ ↑ and Ψ ↓ .

ARTICLE scitation.org/journal/adv
According to Eq. ( 21) and Refs.43 and 44, the transition matrix elements can be divided into four cases, whose expressions are as follows: where Aμ = (0, −A kl ) denotes the four dimension vector ∫ d 4 x potential, γ μ = (γ 0 , γ x , γ y , γ z ) denotes the gamma matrices, and x denotes the four dimension (time-space) integration.The subscripts d and f denote the initial and final states, respectively.The electric field vector of the OAM wave propagating along the z axis in the cylindrical polar coordinate is , where E 0 is the electric strength, and and ∥ stand for the transverse and propagation vector component, respectively.
∥ .The Hamiltonian of the field is defined by , where a kl and a † kl are the annihilation and generation operators of the EM field with the frequency ω 0 and the OAM l ̵ h, respectively.A kl (r, t) is the vector potential of the single OAM photon, which is expressed as where H.c. denotes the Hermitian conjugate, and a kl ∝ e iω 0 t and a † kl ∝ e −iω 0 t represent the OAM photon radiation and absorption process, respectively.The following transformation of the gamma matrices can be used to calculate Eq. ( 6): Ignoring the electron momentum changes in the z direction, substituting Eqs. ( 7) and (8) into Eq.( 6), the transition matrix can be calculated as follows: where F l±1 (r) is expressed as where Furthermore, E(ΔE), Φ(Δm), and P(Δkz) can be, respectively, calculated by To be specific, E(ΔE), P(Δkz) and Φ(Δm) reveal the conservation of energy, linear momentum and angular momentum in the radiation process, respectively.In other words, not only the energy and the linear momentum, but also the angular momentum can be transferred from the vortex electron in the magnetic field to the radiation microwave photon.According to Eq. ( 10), the radiation probabilities of the microwave photons carrying different OAM modes at different frequencies are different.In order to analyze the radiation probability, X l±1 ( * ) in Eq. ( 10) is necessary to be calculated first.F l±1 (r) is related to the radiation probability of the microwave photon carrying l ̵ h OAM and ± ̵ h SAM with frequency ω 0 , which can be approximated by Besides, X l±1 ( * ) in Eq. ( 10) denotes the integration of the Laguerre polynomial multiplied by the Bessel function, whose expression can be calculated as where Γ( * ) denotes the gamma function and min( * ) denotes the minimum of two numbers.The last term of Eq. ( 13) can be calculated as follows: where W M denotes the Whittaker M function.
It can be seen in Eq. ( 10) that Therefore, the spontaneous radiation of the relativistic Landau electron is also resonant radiation, where the radiation frequency is N times the electron cyclotron frequency and the total AM carried by the radiation microwave photon is N ̵ h.
According to Eqs. ( 9)-( 14), it can be seen that the transition probabilities of all states are not zero, which decrease with the increase of Δm when n = n ′ , m is a constant, and Δm > 0. Because the transition matrix elements are not zero for any m and l, relativistic cyclotron electrons can simultaneously emit a variety of OAM microwave photons.When the electron transits from Ψ d,↑ to Ψ f ,↑ , the reduced energy is approximately Δm ̵ hω.Therefore, there must be Δm ̵ h total AM transition to the radiated photon, which carries ± ̵ h SAM and (Δm ∓ 1) ̵ h = l ̵ h OAM. 34,35

IV. RELATIVISTIC VORTEX ELECTRONS DRIVEN BY MICROWAVE PHOTONS WITH OAM
The relativistic vortex electron driven by the microwave photon with OAM is discussed in this subsection.The process of relativistic vortex electron driven by the microwave photon with OAM is the inverse of the aforementioned radiation process.When the photon radiation process is discussed, the generation operator is utilized to analyze the radiation probability.In other words, the vector potential Aμ of the vortex photons in Eq. ( 6) is chosen to use the generation operators.If the absorption process is discussed, the annihilation operator can be adopted, that is, the Hermitian conjugate of the vector potential Aμ in Eq. ( 6) is utilized.
Similarly, the transition matrix is adopted to explain the vortex electron driven by the microwave photon with OAM.To be specific, the transition matrix from the electron initial state |ψ d ⟩ to the finial state where Δm = m − m ′ .Furthermore, E (a) (ΔE), Φ (a) (Δm), and P (a) (Δkz) can be, respectively, calculated by Equations ( 11) and ( 16) are the only difference between radiation and absorption processes.In the radiation process, i.e., Eq. ( 11), the energy, linear momentum, and angular momentum are transferred from the vortex electron to the radiation microwave photon.Hence, the energy of the final state for the vortex electron is equal to the energy of the initial state for the vortex electron minus that of the radiation microwave photon with OAM in the radiation process, i.e., E The same conclusions can be drawn for the linear momentum and angular momentum in the radiation process.On the contrary, in the absorption process, the energy of the final state for the vortex electron is equal to the energy of the initial state for the vortex electron plus that of the microwave photon with OAM, i.e., E hω 0 .The same conclusions can be drawn for the linear momentum and angular momentum in the absorption process.It is obvious that both the radiation and absorption processes satisfy the conservation of energy, linear momentum, and angular momentum.
Although the calculation results are similar, the explanations of these two processes are slightly different because the spontaneous radiation process of relativistic electrons is usually accompanied by different harmonic components, 45 but the absorption process is generally for photons of a single frequency.For example, when the vortex electron is subjected to the microwave photon with OAM l and frequency (l ± 1)ω, the absorption process occurs.If n and m are the radial and magnetic quantum numbers of the initial state for the vortex electron, respectively, the total change of the summation of the radial and magnetic quantum numbers is l ± 1, i.e., n ′ + m ′ = n + m + l ± 1.When the radial quantum number remains unchanged, m ′ = m + l ± 1, the angular momentum of the microwave photon is transferred to the vortex electron.

V. INTERACTION BETWEEN MICROWAVE PHOTONS AND NON-RELATIVISTIC VORTEX ELECTRONS
When the electron is subjected to the plane microwave photon with the angular frequency ω 0 , the wave vector |k| = ω 0 /c, and the polarization unit vector ϵ = (e 1 + ie 2 )/ √ 2, where e 1 and e 2 are unit vector groups orthogonal to each other in the plane perpendicular to the wave vector.The potential function of the microwave photon is denoted as (A ′ , Φ ′ ) within Coulomb specification ∇ ⋅ A ′ = 0, Φ ′ = 0.According to Ref. 36 and assuming that the photon is monochromatic, the time-dependent perturbation Hamiltonian is Taking the angle between the wave vector k and the field strength B as θ, the perturbation plane wave can be expanded by the dipole approximation.Hence, the approximate expression of the photon vector potential is obtained as A ′ (r, t) = ϵa 0 e −iω 0 t e ik⋅r = ϵa 0 e −iω 0 t e ikz cos θ e ikr sin θ cos(φ−φ ′ ) ≈ ϵa 0 e −iω 0 t e ikz cos θ , (18)   where a 0 denotes the coefficient of the normalization and r = |r | denotes the projection vector of r in the xOy plane, whose angle difference is φ ′ with the projection of the wave vector k in the xOy plane.
Assuming that the electron transits from the state is the Hamiltonian without perturbation, and z is the unit vector along the z axis), the time-dependent perturbation matrix element can be calculated as where When kz is small and the second term D in Eq. ( 19) is ignored, ℋ ′ f d (t) can be simplified.According to the linear combination of the wave function |Ψ(t)⟩ = ∑ ci(t)|ψi⟩ exp(−iEit/ ̵ h) and the time-dependent perturbation theory, 2 the probability of the transition can be obtained as It can be seen from Eq. ( 20) that when the angular frequency ω 0 of the photon is close to the integer multiple of the cyclotron angular frequency ω of the electron: , the transition probability is large.However, in other cases, the probability is almost zero. 37When the perturbation field is regarded as the electric dipole approximation, the transition process has obvious resonance properties.
To calculate the term ⟨ψ f |(ϵ ⋅ r )e ikz cos θ |ψ d ⟩ of Eq. ( 19), we consider the process of spontaneous emission of the electron.More specifically, the electron transits from the initial state |ψ d ⟩ to the final state |ψ f ⟩.It is concluded that the result is the same regardless of the direction of ϵ.Therefore, according to Eq. ( 2), the spontaneous emission transition matrix element  f d can be calculated as 36,37 where x = μ 2 r 2 .The integral of I(x) is zero when m = m ′ and |n − n ′ | > 1.Hence, the transition can occur only when m − 1 = m ′ ,  f d ≠ 0. Besides, the angular frequency of the radiation photons is ωe = (E d − E f )/ ̵ h = ω, and the angular momentum is ̵ h.Moreover, the time-independent perturbation matrix element can be calculated as follows: where ζ is the phase of the microwave photon wave projection vector in the xOy plane, and δ k cos θ+k z ,k ′ z indicates the conservation of the linear momentum.According to Eqs. ( 21) and ( 22), a nonrelativistic electron regardless of its spin sign in a magnetic field cannot directly exchange OAM with plane microwave photons because multi-level transitions of the Landau level cannot occur.For the absorption process, the direction of the microwave vector is not important because the final calculation results are the same because the term δ m−1,m ′ .Besides, the probability of spin flip transition is zero because

VI. RESULTS AND DISCUSSIONS
For OAM microwave photons radiated by relativistic electrons, it consists of multiple frequencies and OAM.The cylindrical waveguide or resonant cavity is a good way to obtain the OAM microwave photon with a single frequency and a single OAM, such as the rotating microwave cylindrical cavity modes TE ℓp (p > 0), whose electric field expression is Er = iE 0 ℓ/(κr)J ℓ (κr)e iℓφ e ik ∥ z e iω ℓ,p t , Eφ = E 0 J ′ ℓ (κr)e iℓφ e ik ∥ z e iω ℓ,p t , (23)   where κ=  ℓ,p /Rc,  ℓ,p is the pth root of the derivative of the ℓorder Bessel function of first kind, J ′ ℓ ( * ) and Rc is the radius of the cylindrical waveguide. 35,46Besides, the single mode operating frequency of the waveguide is ω ℓ,p =  ℓ,p c/Rc, and the operating OAM mode is ℓ − 1.
Therefore, we can select the appropriate waveguide radius Rc to get the radiation microwave frequency Δmω = ω ℓ,p and OAM mode number l = ℓ − 1 in accordance with the working conditions of the waveguide.The radius of the waveguide Rc is equivalent to the wavelength of the radiation OAM microwave when p = 1, and it increases as p increases.Because the waveguide radius is also much larger than the electron cyclotron radius, the influence on the electron radiation process can be neglected.
The numerical results are presented to show the radiation power of the microwave photons with different OAM.When the frequency ω 0 = Δmω of the microwave photons has a big difference from the resonant frequency ω ℓ,p , the radiation power will attenuate with e −α and α ≈ 2. 1,2 As shown in Fig. 2, when ℓ = 2, 4, and 6, SAM of the microwave photons is ̵ h and ω 0 = ω ℓ,p , the normalized power at Δm = 2, 4, 6 is the largest.For the case without cavity, the results are similar to those in Ref. 45.
Afterward, the normalized wave function probability amplitudes are simulated to explain the absorption process.As shown in Fig. 3, the wave function can be drawn in the plane polar coordinate system.Although the kinetic OAM changes with time, the whole process satisfies the conservation of canonical OAM.For example, gyrotrons and undulators can be used to generate microwave carrying OAM. 34,35Compared with undulators, the method of generating OAM microwave using the relativistic electrons in the gyrotron has low requirements on the magnetic field and higher efficiency. 27,46oreover, microwave OAM photons can also be utilized to drive vortex electrons.According to Eqs. ( 6) and ( 9), the remaining transition matrix elements can be calculated as follows: It can be seen that the electron spin flip transition contributes an AM ± ̵ h to the radiated photon.However, the transition probabilities of the spin flip case and the non-spin flip case are different.Moreover, as the relativistic effect decreases, the radiation probabilities of high-order OAM microwave also become finally, the transition is forbidden.
According to Refs.31, 38, and 41, the canonical OAM L can z and kinetic OAM L kin z are, respectively, expressed as where B sign = ±1 denotes the sign of the magnetic field.Although the OAM eigenvalue is m ̵ h, the vector potential changes the observed OAM because of the Zeeman interaction between the magnetic field and the OAM.When relativistic non-vortex electrons absorb OAM microwave photons, their azimuthal quantum number becomes non-zero.In the case of N > Δm, i.e., the frequency of the absorbed photons is greater than the total angular momentum difference times the cyclotron angular frequency of the electron, N − Δm radial quantum number is added to the vortex electrons according to Eqs. ( 9) and (24), which causes the angular momentum variation of vortex electrons.

VII. CONCLUSION
In conclusion, we analyze the OAM transition process between microwave photons and vortex electrons in the magnetic field.In the case of relativity, cyclotron electrons can be utilized to radiate and absorb OAM microwave photons.In the case of radiation, the Nth harmonic of the radiation wave carries N ̵ h total AM.In the case of absorption, microwave photons of different frequencies and OAM can be used to generate vortex electrons with different Landau level quantum numbers.In the future, we may use gyrotrons or superconducting quantum circuits to generate microwave photons carrying OAM and detect them experimentally.Furthermore, our work may be important for future superconducting quantum computing, sensing, communications, and other quantum applications.

FIG. 1 .
FIG. 1.The electron rotates around the z axis in the magnetic field.