Electric quadrupole moment and the tensor magnetic polarizability of twisted electrons and a potential for their measurements

TL;DR

This paper derives the relativistic Hamiltonian for twisted electrons, revealing they have a measurable electric quadrupole moment and tensor magnetic polarizability, with proposed methods for experimental detection and potential applications.

Contribution

The paper introduces the first theoretical prediction that twisted electrons possess a measurable electric quadrupole moment and tensor magnetic polarizability, including methods for their experimental measurement.

Findings

Twisted electrons have a significant electric quadrupole moment.

Tensor magnetic polarizability of twisted electrons can be measured in storage rings.

Proposed methods include freezing orbital angular momentum and resonance techniques.

Abstract

For a twisted (vortex) Dirac particle in nonuniform electric and magnetic fields, the relativistic Foldy-Wouthuysen Hamiltonian is derived including high order terms describing new effects. The result obtained shows for the first time that a twisted spin-1/2 particle possesses a tensor magnetic polarizability and a measurable (spectroscopic) electric quadrupole moment. We have calculated the former parameter and have evaluated the latter one for a twisted electron. The tensor magnetic polarizability of the twisted electron can be measured in a magnetic storage ring because a beam with an initial orbital tensor polarization acquires a horizontal orbital vector polarization. The electric quadrupole moment is rather large and strongly influences the dynamics of the intrinsic orbital angular momentum. Three different methods of its measurements, freezing the intrinsic orbital angular momentum and two resonance methods, are proposed. The existence of the quadrupole moment of twisted electrons can lead to practical applications.

Equations

$$\begin{array}[]{c}{\cal H}_{FW}=\beta\sqrt{m^{2}+\bm{\pi}^{2}-e\bm{\Sigma}\cdot\bm{B}},\end{array}$$

$$\displaystyle\begin{array}[]{c}\bm{\pi}^{2}={\bm{\pi}^{\prime}}^{2}+{\boldsymbol{\mathfrak{p}}}^{2}-\frac{e}{2}\left[\bm{L}\cdot\bm{B}(\bm{R})+\bm{B}(\bm{R})\cdot\bm{L}\right]+\frac{e^{2}}{4}\bigl{[}\bm{B}(\bm{R})\times\boldsymbol{\mathfrak{r}}\bigr{]}^{2},\end{array}$$

$${\mathcal{H}}_{FW}$$

$$ϵ$$

$$W=-\beta {\beta }_T(\mathbf{L}\cdot \mathbf{B}{)}^2,{\beta }_T=\frac{e^2{\hslash }^2}{8m^3}=5.25\times 10^4{\mathrm{fm}}^3.$$

$$Q_0=-e<{\mathfrak{r}}^2>=-e\frac{\int \varrho (\mathfrak{r}){\mathfrak{r}}^{3}d\mathfrak{r}}{\int \varrho (\mathfrak{r})\mathfrak{r}d\mathfrak{r}},$$

$$W=-\frac{1}{2}\bigl{[}\bm{d}\cdot\bm{E}(\bm{R})+\bm{E}(\bm{R})\cdot\bm{d}\bigr{]}-\frac{1}{12}\left\{Q_{ij},\frac{\partial E_{i}(\bm{R})}{\partial X_{j}}\right\},$$

$$Q_{ij}=\frac{3Q_s}{2j(2j-1)}\left[\{j_i,j_j\}-\frac{2}{3}{\delta }_{ij}j(j+1)\right],$$

$$Q_s=\frac{3K^2-j(j+1)}{(j+1)(2j+3)}{Q}_0,$$

$$<r^2>=\frac{w_m^2}{2}\left(2n+|l_z|+1\right),w_m=2\sqrt{\frac{\hslash }{|eB|}}.$$

$$Q_{ij}^{(curr)}=-\frac{1}{2ϵ}\left[3L_i{\mu }_j+3L_j{\mu }_i-2{\delta }_{ij}(\mathbf{L}\cdot \mathbf{\mu })\right],\mathbf{\mu }=\frac{e\mathbf{s}}{ϵ}.$$

$$\Delta \Omega \sim L{\left|\frac{\partial E_i}{\partial X_j}\right|}_{max}\times 10^{-10}{\mathrm{s}}^{-1},$$

$$W$$

$$\mathcal{E}$$

$$W=-\frac{Q_s}{4L^2}{L}_r^2\frac{\partial {\mathcal{E}}_r}{\partial R}.$$

$$\frac{d\mathbf{L}}{dt}$$

$$\mathbf{\Omega }$$

$${\mathbf{B}}_0=\left(\frac{2}{{\tilde{\beta }}^2}-1\right)\tilde{\mathbf{\beta }}\times \mathbf{E},\mathbf{\omega }=-\frac{e{\mathbf{B}}_0}{mc\gamma ({\gamma }^2+1)}.$$

$$R_0=\frac{V}{\omega }=({\gamma }^2+1)\sqrt{{\gamma }^2-1}\frac{m{c}^2}{|e|B_0}.$$

$$\frac{\partial B_z}{\partial R}=-\frac{n{B}_0}{R_0},0<n<1,$$

$$\begin{array}[]{c}\frac{dL_{z}}{dt}=\frac{Q_{s}\widetilde{\beta}nB_{0}}{4L^{2}R_{0}}\{L_{r},L_{\phi}\}.\end{array}$$

$$\sqrt{\mathcal{A}+\mathcal{B}}=\frac{1}{2}\left\{\sqrt{\mathcal{A}(1+x)}\right\}=\frac{1}{2}\left\{\sqrt{\mathcal{A}},\sqrt{1+x}\right\}=\frac{1}{2}\left\{\sqrt{\mathcal{A}},\left(1+\frac{x}{2}-\frac{x^2}{8}\right)\right\},$$

$$[L_i,L_j]=i{e}_{ijk}{L}_k,[S_i,S_j]=i{e}_{ijk}{S}_k,$$

$$P_i$$

$$\begin{array}[]{c}{\cal H}=\bm{\Omega}\cdot\bm{L}+\frac{1}{2}\alpha_{ij}(L_{i}L_{j}+L_{j}L_{i}),\end{array}$$

$$\begin{array}[]{c}\frac{dL_{k}}{dt}=(\bm{\Omega}\times\bm{L})_{k}+\frac{1}{2}\alpha_{ij}\bigl{(}e_{kil}\{L_{l},L_{j}\}+e_{kjl}\{L_{l},L_{j}\}\bigr{)}.\end{array}$$

$$\begin{array}[]{c}P_{\rho}(0)=\sin{\theta}\cos{\psi},~{}~{}~{}P_{\phi}(0)=\sin{\theta}\sin{\psi},~{}~{}~{}P_{z}(0)=\cos{\theta},\end{array}$$

$$\begin{array}[]{c}P_{\rho\rho}(0)=\frac{1}{2}\left(3\sin^{2}{\theta}\cos^{2}{\psi}-1\right),~{}~{}~{}P_{\phi\phi}(0)=\frac{1}{2}\left(3\sin^{2}{\theta}\sin^{2}{\psi}-1\right),\\ P_{zz}(0)=\frac{1}{2}\left(3\cos^{2}{\theta}\cos^{2}{\psi}-1\right),~{}~{}~{}P_{\rho\phi}(0)=\frac{3}{4}\sin^{2}{\theta}\sin{(2\psi)},\\ P_{\rho z}(0)=\frac{3}{4}\sin{(2\theta)}\cos{\psi},~{}~{}~{}P_{\phi z}(0)=\frac{3}{4}\sin{(2\theta)}\sin{\psi}.\end{array}$$

$$\mathbf{P}(0)=0$$

$$\mathcal{H}$$

$$\frac{d{L}_{\rho }}{dt}$$

$$\frac{d{L}_{\varphi }}{dt}$$

Full text

The discovery of twisted (vortex) electron beams UTV whose existence was predicted in Ref. Bliokh2007 has shown that particles can carry an intrinsic orbital angular momentum (OAM). Since twisted electrons possess large magnetic moments, this discovery opens new possibilities in the electron microscopy and investigations of magnetic phenomena (see Refs. BliokhSOI ; Lloyd ; LloydPhysRevLett2012 ; Rusz ; Edstrom ; imaging ; Observation ; OriginDemonstration and references therein). Twisted electron beams with large intrinsic OAMs (up to 1000$\displaystyle\hbar)$ have been recently obtained VGRILLO . Basic properties of free twisted beams have been considered in Refs. Bialynicki-Birula ; Barnett . The dynamics of the intrinsic OAM in external magnetic and electric fields has been studied in Refs. Bliokh2007 ; Bliokhmagnetic ; magnetic ; Greenshields ; classicalmagnetic ; experimentmagnetic . The general relativistic description of the classical and quantum dynamics of the intrinsic OAM in arbitrary electric and magnetic fields has been discussed in Refs. Manipulating and ResonanceTwistedElectrons , respectively. In Ref. ResonanceTwistedElectrons , the relativistic quantum dynamics of twisted (vortex) electrons has been constructed in the Schrödinger form on the basis of the relativistic Foldy-Wouthuysen (FW) transformation JMP ; Classicallimit ; relativisticFW . In the present work, an application of the approach used in Ref. ResonanceTwistedElectrons allows us to obtain new fundamental properties of twisted electron beams. We demonstrate for the first time that a twisted spin-1/2 particle can possess a large measurable (spectroscopic) electric quadrupole moment (EQM) and a tensor magnetic polarizability (TMP). We calculate these new fundamental parameters (caused by the intrinsic OAM) for a twisted electron and develop methods for their measurements.

While a twisted electron is a single pointlike particle described by the standard Dirac equation, its wave function has a nontrivial spatial structure (see the reviews BliokhSOI ; Lloyd ). A particle with an intrinsic OAM is characterized by nontrivial solutions of the Dirac and Schrödinger equations. Such solutions are coherent superpositions of partial plane waves with different momenta BliokhSOI ; Lloyd ; IvanovScattering .

The relativistic FW transformation (see Refs. JMP ; Classicallimit ; relativisticFW and references therein) being the relativistic generalization of the original method FW can be applied to obtain the Schrödinger form of the relativistic quantum mechanics. The exact relativistic Hamiltonian in the FW representation (the FW Hamiltonian) for a twisted or a untwisted Dirac particle $\displaystyle(g=2)$ in a static (in general, nonuniform) magnetic field has been first obtained in Ref. Case and is given by $\displaystyle(\hbar=1,~{}c=1)$ JMP ; Case ; Energy1 ; Energy3

[TABLE]

where $\displaystyle\bm{\pi}=\bm{p}-e\bm{A}$ is the kinetic momentum, $\displaystyle\bm{B}$ is the magnetic induction, and $\displaystyle\beta$ and $\displaystyle\bm{\Sigma}$ are the Dirac matrices. This Hamiltonian acts on the bispinor \displaystyle\Psi_{FW}=\left(\begin{array}[]{c}\phi\\ 0\end{array}\right).

A twisted electron is a charged centroid Bliokh2007 ; BliokhSOI . Needed derivations are similar to those in Ref. ResonanceTwistedElectrons . However, second-order terms in $\displaystyle B$ should be calculated. We can suppose that the de Broglie wavelength, $\displaystyle\hbar/p$, is much smaller than the characteristic size of the nonuniformity region of the external field. Summing over partial waves with different momentum directions brings the operator $\displaystyle\bm{\pi}^{2}$ to the form ResonanceTwistedElectrons

[TABLE]

where $\displaystyle\bm{R}$ and $\displaystyle\bm{\pi}^{\prime}$ are the center-of-charge radius vector and the kinetic momentum of the centroid as a whole, $\displaystyle\boldsymbol{\mathfrak{r}}=\bm{r}-\bm{R}$ and $\displaystyle\boldsymbol{\mathfrak{p}}=-i\hbar\partial/(\partial\boldsymbol{\mathfrak{r}})$ are internal canonical variables, and $\displaystyle\bm{L}\equiv\boldsymbol{\mathfrak{r}}\times\boldsymbol{\mathfrak{p}}$ is the intrinsic OAM ResonanceTwistedElectrons . The straightforward extraction of the square root SM1 brings Eq. (1) into the form

[TABLE]

In Eq. (2), all terms proportional to $\displaystyle 1/\epsilon^{3}$ were not previously taken into account. The curly brackets $\displaystyle\{\dots,\dots\}$ denote anticommutators. The additional terms appearing in the second order expansion of the square root in Eq. (1) in a power series in $\displaystyle 1/\epsilon$ are quadratic or bilinear in $\displaystyle\bm{B}$. Usually, $\displaystyle\epsilon^{2}\sim{m^{2}}=4.41\times 10^{9}$ $\displaystyle e\cdot$T, $\displaystyle L\sim 100,B\sim 1$ T, and these terms are approximately 7 orders less than the main OAM-dependent term. Therefore, our previous results and conclusions ResonanceTwistedElectrons are correct, while the additional OAM-dependent terms define new physical effects. The second to last term in the FW Hamiltonian (2) characterizes the spin – intrinsic OAM coupling in the magnetic field and describes the additional spin precession caused by the intrinsic OAM. Reversing $\displaystyle\bm{L}$ changes the sign of the OAM-dependent correction to the spin precession frequency. The existence of the spin – intrinsic OAM coupling has been previously established in Refs. BliokhSOI ; Lloyd ; Bliokhmagnetic ; magnetic . The last term describing the tensor interaction of the intrinsic OAM with the magnetic field is similar to the corresponding spin tensor interaction. The operator of the latter interaction has the form $\displaystyle W=-\beta_{T}(\bm{S}\cdot\bm{B})^{2}$, where $\displaystyle\beta_{T}$ is the TMP defined in the particle rest frame and $\displaystyle\bm{S}$ is the spin matrix $\displaystyle[\bm{S}^{2}=s(s+1)]$. Thus, the TMP caused by the intrinsic OAM is given by

[TABLE]

The TMP of the twisted electron is much larger than TMPs of particles and nuclei conditioned by the spin interactions. In particular, for pointlike W± bosons $\displaystyle\beta^{(W)}_{T}\sim 10^{-11}$ fm3 PhysRevDspinunit ; PhysRevDunitexact . For the deuteron, the theoretical estimation is $\displaystyle\beta_{T}=0.195$ fm3 BETA_T . In addition, the tensor interaction of the twisted electron is proportional to $\displaystyle L^{2}\sim 10^{4}-10^{6}$.

The operator $\displaystyle\bm{L}$ commutes with $\displaystyle\boldsymbol{\mathfrak{p}}^{2}$. The noncommutation of this operator with \displaystyle\bigl{[}\bm{B}(\bm{R})\times\boldsymbol{\mathfrak{r}}\bigr{]}^{2} does not lead to any important effects because the expectation values of the nonzero commutators $\displaystyle\left[L_{i},(\bm{B}(\bm{R})\times\boldsymbol{\mathfrak{r}})^{2}\right]\,(i=x,y$ when $\displaystyle\bm{B}(\bm{R})=B\bm{e}_{z})$ are equal to zero.

Thus, the evolution of the intrinsic OAM does not reduce to its precession. The same assertion has been made in Refs. BliokhSOI ; Lloyd ; magnetic ; classicalmagnetic . However, Eq. (2) shows the existence of a new interaction caused by the TMP of the twisted Dirac particle. The corresponding relativistic classical equation has the same form as Eq. (1) (except for the spin term). Moreover, a consideration of a twisted centroid leads to the classical equation similar to Eq. (2). The tensor electric and magnetic polarizabilities caused by the spin interactions are common properties of nuclei with the spin $\displaystyle s\geq 1$. Besides this, the exact FW Hamiltonians for pointlike spin-1/2 and spin-1 particles (with $\displaystyle g=2)$ in a uniform magnetic field (see Eq. (1) and Refs. PhysRevDspinunit ; PhysRevDunitexact ) are almost identical (the only difference is the form of the spin matrices). The existence of the TMP for the pointlike spin-1 particles additionally substantiates its existence for the twisted Dirac ones. We also mention that the three-component spin operator $\displaystyle\bm{s}=\hbar\bm{\Sigma}/2$ and the OAM operator $\displaystyle\bm{L}$ are defined in the particle rest frame and in the lab frame, respectively.

While the TMP of the twisted electron is large as compared with that of the deuteron, its measurement is a difficult experimental task. The TMP leads to very small shifts of energy levels and cannot be determined using the magnetic-resonance method. While $\displaystyle(B_{\|}\cos{\omega t})^{2}=B_{\|}^{2}(1+\cos{2\omega t})/2$, an oscillating horizontal $\displaystyle(\|)$ magnetic field $\displaystyle\bm{B}_{\|}$ which frequency is half that of transitions between the Landau levels cannot stimulate any resonance. This occurs because the operator $\displaystyle(\bm{B}_{\|}\cdot\bm{L})^{2}$ (unlike the operator $\displaystyle\bm{B}_{\|}\cdot\bm{L})$ does not mix neighboring energy levels. The same situation takes place for the tensor polarizabilities caused by the spin. In particular, the operators $\displaystyle(\bm{E}_{\|}\cdot\bm{S})^{2}$, $\displaystyle(\bm{B}_{\|}\cdot\bm{S})^{2}$ mix only the levels $\displaystyle s_{z}=\pm 1$ for spin-1 particles PRC2007 ; PRC2008 ; PRC2009 ; JPhysConfSer .

The best possibilities to measure the TMP of the twisted electron are provided by the effects found in Ref. Bar3 and investigated in detail in Refs. PRC2008 ; BarJPhysG . If the TMP is caused by the spin interactions, it produces a spin rotation with two frequencies instead of one, beating with a frequency proportional to $\displaystyle\beta_{T}$, and transitions between vector and tensor polarizations Bar3 ; BarJPhysG . Following Ref. PRC2008 , we propose to use a tensor-polarized twisted electron beam in a magnetic storage ring. In this case, the TMP is the only reason of the appearance of a horizontal orbital vector polarization of the beam. This polarization grows almost linearly in time SM1 ; PRC2008 . The horizontal component of the intrinsic OAM rotates with the Larmor frequency. The experiment can be performed in an electron storage ring or in a Penning trap. It needs a high beam coherency. To reach such a coherency, some methods developed for the electric-dipole-moment experiment PhysRevLett2016 can be applied.

We also consider OAM-dependent interactions proportional to field derivatives. Contrary to the pointlike electron, the twisted electron (centroid) has a highly anisotropic spatial structure. Such an object possesses the EQM, while this property has not been previously mentioned. The Laguerre-Gaussian wave function describing a wave beam BliokhSOI ; Lloyd does not allow a rigorous determination of the electron density shape in the centroid rest frame $\displaystyle(p_{z}\rightarrow 0)$. It is natural to assume that this is a strongly oblate (pancake shaped) spheroid. Twisted electron states in a uniform magnetic field BliokhSOI ; Bliokhmagnetic always have such a shape. In this case, the intrinsic EQM of the centroid is given by $\displaystyle(e=-|e|)$

[TABLE]

where $\displaystyle\varrho(\mathfrak{r})=|\Psi_{FW}|^{2}$ is the electron density and $\displaystyle\mathfrak{r}$ is the radial coordinate of the cylindrical coordinate system introduced relative to the center of charge of the centroid. In the nonrelativistic approximation, the square of the FW wave function reduces to the square of the corresponding Schrödinger wave function.

The interaction of an extended charged particle with a static electric field is defined by the operator

[TABLE]

where $\displaystyle\bm{d}$ and $\displaystyle Q_{ij}$ are the operators of the electric dipole moment (see Ref. ResonanceTwistedElectrons ) and the EQM, respectively. In the centroid rest frame, the EQM interacts only with an electric field. The EQM operator of the twisted electron averaged on states with the specific total angular momentum $\displaystyle\bm{j}=\bm{L}+\bm{s}$ is defined in the centroid rest frame and is given by

[TABLE]

where $\displaystyle Q_{s}$ is the spectroscopic EQM. Its connection with the intrinsic EQM has the form EQM ; EQMreview

[TABLE]

where $\displaystyle K$ is the projection of the total angular momentum onto the symmetry axis of the particle. In the case at hand, $\displaystyle j\gg 1$ and $\displaystyle Q_{s}\approx Q_{0}$.

For the Landau problem, eigenfunctions of the nonrelativistic and relativistic Hamiltonians coincide ResonanceTwistedElectrons . For twisted and untwisted electrons, the mean square of the radial cylindrical coordinate of the pointlike electron is given by Bliokhmagnetic

[TABLE]

Here $\displaystyle w_{m}$ is the beam waist BliokhSOI ; Bliokhmagnetic and $\displaystyle\bm{l}$ is the sum of the intrinsic and extrinsic OAMs. When $\displaystyle B=1$ T, $\displaystyle w_{m}=5.1\times 10^{-8}$ m.

The diameter of the vortex beam depending on the OAM has been determined in Refs. imaging ; OAMdiameter . It is about 10 nm when the topological charge is $\displaystyle\mathfrak{m}=50$ $\displaystyle(L_{z}=\mathfrak{m}\hbar)$ and is proportional to $\displaystyle\mathfrak{m}$ OAMdiameter .

The spin also contributes to the EQM of the twisted electron. An orbital motion of the magnetic moment of a spinning particle leads to the appearance of an electric current quadrupole moment (ECQM) ECQM :

[TABLE]

Its appearance results in a small correction to the spin precession frequency. The sign of this correction is defined by the sign of $\displaystyle L_{z}$. The ECQMs are comparatively small $\displaystyle(Q^{(curr)}\sim eL/m^{2})$. The correction to the spin precession frequency is of the following order:

[TABLE]

where the maximum gradient of the electric field is given in units of V/m2. The ECQMs cause a spin – intrinsic OAM coupling.

Let us analyze the potential for measuring the EQM of twisted electrons in storage rings when $\displaystyle L\gg 1$ and spin effects are neglected. It is convenient to determine the dynamics of $\displaystyle L_{z}$. Relativistic effects in interactions of EQMs of spinning particles with electric and magnetic fields have been described in Refs. PKS ; MovingQ ; Spin1JETP . Since polarization effects conditioned by the spin and intrinsic OAM are similar SM1 , we can use the results PKS ; MovingQ ; Spin1JETP with substituting $\displaystyle\bm{L}$ for $\displaystyle\bm{S}$ and taking into account that $\displaystyle L>>1$ and $\displaystyle\bm{V}\cdot\bm{E}=0$ $\displaystyle(\bm{V}$ is the centroid velocity operator). The quadrupole interaction in the lab frame reads $\displaystyle(\bm{\widetilde{\beta}}=\bm{V}/c=\widetilde{\beta}\bm{e}_{\phi})$

[TABLE]

where $\displaystyle\bm{\mathcal{E}}=\bm{\mathcal{E}}(\bm{R})$ is the quasielectric field. All fields are defined in the lab frame. The noncommutativity of operators is neglected. The intrinsic OAM presented in Eq. (10) should not be confused with the extrinsic OAM. The operator $\displaystyle W$ should be added to the Hamiltonian (2). The use of a nonuniform magnetic field for focusing may be preferable.

The interaction operator $\displaystyle W$ contains the terms proportional to $\displaystyle\partial\mathcal{E}_{z}/\partial Z$ and $\displaystyle\partial\mathcal{E}_{r}/\partial R$ $\displaystyle(R$ is the radial coordinate of the cylindrical coordinate system). The former term commuting with $\displaystyle L_{z}$ can be disregarded. The effect of the latter term is defined by

[TABLE]

The last term in Eq. (6) does not contribute to the interaction operator (11) because $\displaystyle\partial\mathcal{E}_{i}/\partial X_{i}\equiv{\rm div}\,\bm{\mathcal{E}}=0$.

To determine the effect of $\displaystyle W$ on the dynamics of the intrinsic OAM, we can use the fact that the components of the spin and the OAM satisfy equivalent commutation relations. Therefore, the OAM polarization tensor $\displaystyle\{L_{i},L_{j}\}$ rotates in external fields with the same angular velocity as the OAM (the spin polarization tensor possesses the equivalent property JPhysG2015 ). The dynamics of the intrinsic OAM is defined by the large term ResonanceTwistedElectrons

[TABLE]

and by corrections to this term caused by the EQM. Other corrections are defined by Eqs. (2) and (3). In Eq. (12), $\displaystyle\bm{\Omega}$ is the operator of the angular velocity of Larmor precession in external fields. The Larmor precession caused by the vertical magnetic field and by the radial electric one (if the latter field is also used) does not change $\displaystyle L_{z}$.

In Eqs. (2), (11), and (12), a noncommutativity of the operators of coordinate and intrinsic OAM can be ignored because the operators $\displaystyle\boldsymbol{\mathfrak{p}}$ and $\displaystyle\bm{L}$ are defined only by the internal coordinates.

The change of $\displaystyle L_{z}$ caused by the quadrupole interaction (11) is observable only when the Larmor precession is eliminated. This can be done by freezing the intrinsic OAM Manipulating in a specific combination of vertical magnetic and radial electric fields equalizing the angular velocities of the beam rotation and the intrinsic-OAM one $\displaystyle(\bm{\omega}=\bm{\Omega})$. An angle between the intrinsic OAM and the momentum remains unchanged. A similar method of freezing the spin frozenEDM may be applied in electric-dipole-moment experiments. Freezing the intrinsic OAM takes place when Manipulating

[TABLE]

If magnetic focusing is used, the magnetic field $\displaystyle\bm{B}$ is nonuniform and $\displaystyle\bm{B}_{0}$ is an average magnetic field. The forces caused by the electric and magnetic fields are oppositely directed. Electrons move counterclockwise. The ring radius is defined by

[TABLE]

The nonuniformity of $\displaystyle\bm{B}$ leads to a nonuniform electric field in the centroid rest frame and to a turn of the intrinsic OAM in the vertical plane. In this case,

[TABLE]

where $\displaystyle n$ is the field index. In Eq. (11), $\displaystyle\mathcal{E}_{r}=\widetilde{\beta}B_{z}$ and $\displaystyle\partial\mathcal{E}_{r}/\partial R=-\widetilde{\beta}nB_{0}/R_{0}$.

The commutator of the total FW Hamiltonian (including $\displaystyle W)$ with the OAM operator results in the following addition to the equation of motion:

[TABLE]

Therefore, a beam with an initial horizontal orbital polarization acquires a vertical orbital polarization (cf. the similar spin effect PRC2007 ; PRC2008 ; PRC2009 ; JPhysConfSer ; Bar3 ; BarJPhysG ; Bar1 ). The beam can be tensor-polarized (when two beams with opposite orbital polarizations are joined) or vector-polarized. Let $\displaystyle\psi$ be the angle defining the orbital polarization relative to the $\displaystyle\bm{e}_{r}$ and $\displaystyle\bm{e}_{\phi}$ axes. The azimuth $\displaystyle\psi=0$ characterizes the intrinsic OAM directed radially outward. The change of $\displaystyle L_{z}$ is maximum when the direction of the initial horizontal orbital polarization satisfies the condition $\displaystyle|\sin{2\psi}|=1$ (cf. Refs. SM1 ; PRC2007 ; PRC2009 ).

The effect of the EQM on the OAM dynamics is very strong and can be easily observed. When the beam energy is equal to 300 keV and $\displaystyle R_{0}=0.5$ m, $\displaystyle\widetilde{\beta}=0.777$, $\displaystyle E=2.46$ MV/m, $\displaystyle B_{0}=0.0148$ T, $\displaystyle f=\omega/(2\pi)=7.41\times 10^{7}$ Hz. When $\displaystyle L\sim 100$ and the OAM diameter is determined based on the data presented in Ref. OAMdiameter , the quantity $\displaystyle Q_{s}/(|e|R_{0})\sim 10^{-16}$ m is not negligible as compared with the reduced Compton wavelength of the electron $\displaystyle{\lambdabar}_{C}=\hbar/(mc)=3.86\times 10^{-13}$ m. The frequency of the cyclic evolution of the orbital polarization (cf. Refs. PRC2007 ; PRC2008 ; PRC2009 ; Bar3 ; BarJPhysG ; Bar1 ) is 5 orders of magnitude less than the cyclotron frequency $\displaystyle f$ and can be properly measured. The corresponding terms in the FW Hamiltonian also differ by 5 orders of magnitude. Therefore, the main intrinsic-OAM dynamics is correctly described by the equations obtained in Refs. Manipulating ; ResonanceTwistedElectrons , while the new EQM-dependent effect is rather important.

There is a systematical error caused by the small vertical electric field and the corresponding radial magnetic field $\displaystyle<B_{r}>=-<E_{z}>/\widetilde{\beta}$ leading to a vanishing average Lorentz force (cf. Ref. frozenEDM ). This systematical error originates from field misalignments. However, it seems to be small and can be eliminated in measurements at two values of the field index.

The EQM of the twisted electron can also be measured by the magnetic-resonance method. In this case, a constant electric field is not needed. The resonance effect is provided by a nonuniform field oscillating with the angular frequency $\displaystyle\Omega$. The resonance field vanishing in the center of the beam trajectory, $\displaystyle B_{z}=\mathfrak{B}(R-R_{0})\cos{(\Omega t+\varphi)}$ or $\displaystyle E_{r}=\mathfrak{E}(R-R_{0})\cos{(\Omega t+\varphi)}$, is preferable. It is well known that such a field is equivalent to two fields rotating with the angular velocities $\displaystyle\bm{\Omega}$ [see Eq. (12)] and $\displaystyle-\bm{\Omega}$. The first of these creates the resonance effect. In the frame rotating with the angular velocity $\displaystyle\bm{\Omega}$, the intrinsic-OAM dynamics is similar to that when the intrinsic OAM is frozen SM1 .

The magnetic-resonance method provides the less sensitivity than the method of freezing the intrinsic OAM because the resonance field usually covers a small part of the ring circumference. Nevertheless, the effect of the EQM on the intrinsic-OAM dynamics can be properly detected. Otherwise, the magnetic-resonance method allows one to apply a stronger magnetic field, a smaller ring size, and, therefore, a lower number of twisted electrons. Another advantage of this method is a simpler experimental setup.

The third method of measuring the EQM of the twisted electron is based on a standard stimulation of resonance transitions by an oscillating longitudinal magnetic field. This method, unlike the previous one, is sensitive to the quadrupole splitting of Landau levels defined by Eq. (11). The splitting is caused by the focusing magnetic field creating a nonuniform electric field in the electron rest frame. Therefore, the quadrupole splitting is proportional to the field index $\displaystyle n$. The stimulating magnetic field can be conditioned by a usual rf cavity and is longitudinal because such a field does not affect the beam motion. The considered method is similar not only to the magnetic-resonance method but also to the nuclear-quadrupole-resonance one. The resonance frequencies defined by a quadrupole structure of the energy levels depend on $\displaystyle n$.

The three methods considered need an increase in the currently available beam intensity. However, similar experiments can be carried out with a single twisted electron in a Penning trap.

Large EQMs of twisted electrons rather strongly interact with nonuniform electric fields. We expect that the twisted electrons can be successfully used not only in investigations of magnetic properties (see Refs. BliokhSOI ; Lloyd ; LloydPhysRevLett2012 ; Rusz ; Edstrom ; Observation ; OriginDemonstration and references therein) but also for nanoscale measurements of nonuniform electric fields in matter.

In this Letter, we have calculated minor terms in the relativistic FW Hamiltonian describing a twisted Dirac particle in nonuniform electric and magnetic fields. The results presented by Eqs. (2) – (6) have shown for the first time that the twisted electron $\displaystyle(s=1/2)$ possesses a TMP and a spectroscopic EQM. We have calculated the former parameter and have evaluated the latter one. It is still generally accepted that only particles and nuclei with spin $\displaystyle s\geq 1$ are characterized by these parameters. The TMP of the twisted electron is several orders of magnitude bigger than that of the deuteron. It can be measured in a magnetic storage ring because a beam with the initial orbital tensor polarization acquires a horizontal orbital vector polarization. The EQM is rather large and strongly influences the dynamics of the intrinsic OAM. We propose three different methods of its measurements, freezing the intrinsic OAM and two resonance methods. We expect that the existence of the EQM of twisted electrons can find practical applications because the EQM interaction with nonuniform electric fields in matter depends on the intrinsic-OAM direction. All the considered effects also take place for twisted positrons. Additional explanations are presented in the Supplemental Material SM1 .

This work was supported by the Belarusian Republican Foundation for Fundamental Research (Grant No. $\displaystyle\Phi$18D-002), by the National Natural Science Foundation of China (Grants No. 11575254 and No. 11805242), by the National Key Research and Development Program of China (No. 2016YFE0130800), and by the Heisenberg-Landau program of the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung). A. J. S. also acknowledges hospitality and support by the Institute of Modern Physics of the Chinese Academy of Sciences. The authors are grateful to I. P. Ivanov and O. V. Teryaev for helpful exchanges.