Electron Orientation Flips Due to Magnetic Fields

A weak magnetic field induces a surprising instability in the behaviour of twisted electrons, similar to the unexpected reorientation of a spinning tennis racket. S. S. Baturin and colleagues at ITMO University show that this ‘Dzhanibekov’ instability occurs within the orbital pseudospin dynamics of electrons travelling in a solenoidal field. The findings identify a key mechanism causing recurrent reversals in electron beam properties, potentially enabling new control over electron beams using techniques found in transmission electron microscopes. The team’s theoretical framework, based on an effective Hamiltonian, predicts conversions between different electron beam profiles and defines the conditions for observing this phenomenon.

Induced Dzhanibekov instability controls electron beam reversal within solenoidal fields

Electrons travelling in a solenoidal field now exhibit Dzhanibekov instability, a phenomenon akin to a tennis racket flipping, with a reversal timescale of approximately 82kρ2H, or 37 centimetres for a 100 keV electron beam. This timescale is a sharp reduction from previous limitations, where such instabilities were not observable in wave-dynamical systems lacking mechanical rotation. ITMO University scientists induced this instability by applying a weak nonlinear magnetic field at the entrance to the solenoidal field, a technique easily implemented using standard octupole correctors found in transmission electron microscopes.

Detailed analysis reveals that the instability’s growth timescale is determined by the Lewis-Ermakov breathing of realistic wave packets, which modulates the nonlinear strength. A weak nonlinear magnetic field at the entrance induces a tennis-racket, or Dzhanibekov, instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. The team discovered that, in the symmetric aligned limit, an exact linear eigenstate, a Laguerre-Gaussian vortex state, becomes a hyperbolic fixed point, resulting in recurrent reversals of the mean pseudospin projection.

These reversals manifest in real space as repeated conversions between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed transverse profiles. The tennis racket, or Dzhanibekov effect, describes an instability of rotational motion where rotation about the intermediate principal axis is unstable, and trajectories near the separatrix undergo recurrent flips. Quantum counterparts have been discussed as Bloch-sphere analogies in driven two-level systems and as genuine flipping dynamics of asymmetric quantum rotors. This instability presents as observable changes in the electron beam’s shape, transitioning between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed profiles, and offers a potential new route to controlling these beams, important for advanced techniques like high-resolution microscopy and materials analysis.

Wave-dynamical instabilities in twisted electron transport

The required regime is estimated to be accessible with standard octupole correctors in a transmission electron microscope. Twisted electrons in magnetic fields provide a setting for this instability without any mechanical rigid-body rotation. Structured electron states in solenoidal fields support controlled transport and reshaping of orbital modes. When axial symmetry of the entrance field is broken, the transverse post-edge dynamics reduces to an exact two-dimensional problem, with fixed-shell sectors carrying a Schwinger su algebra or equivalently, and SO dynamics on the orbital Poincaré sphere.

This description is natural within the geometric-phase theory of structured waves, where effective two-state sectors are represented on Poincaré-type spheres by pseudospin variables, rather than literal real space angular-momentum operators. Consequently, researchers term the shell-resolved degree of freedom an orbital pseudospin, an internal mode variable acting within a degenerate Landau-level shell, not a directly measurable orbital angular momentum vector. Only in the symmetric aligned limit is one component proportional to the canonical operator Lz. On a fixed shell, the poles of the orbital Poincaré sphere correspond to Laguerre-Gaussian vortex states, while the equatorial belt corresponds to Hermite-Gaussian states, directly analogous to the familiar first-order structured light sphere, now realised for an electron wavepacket.

A reversal of the mean orbital pseudospin projection therefore appears not as a change in measured angular momentum, but as a characteristic mode conversion between vortex-like LG and multi-lobed HG profiles, accompanied by a global restructuring of the wavepacket phase. In the linear theory, the magnetic edge fixes the post-entry transport generator and induces only rigid precession of the orbital pseudospin about an axis set by the entrance geometry. Rigid precession alone cannot produce a hyperbolic fixed point, a separatrix, or recurrent reversals.

Because the edge field enters the gauge-fixed post-edge Hamiltonian, a weak nonlinear magnetic boundary can imprint a persistent anisotropic quadratic correction. The current paper presents a Maxwell-consistent thin-edge extension of the solenoidal field that achieves this. In the symmetric aligned limit, a Laguerre-Gaussian state, an exact linear eigenstate, becomes a hyperbolic fixed point of the large-shell dynamics. The resulting motion is the direct wave-dynamical analogue of the Dzhanibekov instability, where trajectories near the separatrix exhibit recurrent reversals of the mean orbital pseudospin projection, manifested in real space as repeated LG to HG mode conversion.

Numerical trajectories display clean recurrent flips, and microscope-scale estimates show that the required regime is accessible with standard octupole correctors. Twisted-electron transport in a nominally uniform magnetic field thus provides a concrete, controllable wave-dynamical realisation of the Dzhanibekov effect on an orbital Poincaré sphere. Throughout the paper, natural units ħ= c = 1 are used. The analysis begins with the thin-edge solenoidal model introduced in Ref. Consider the step-like longitudinal field B z = ∆⊥ Φ2θ( z), supplemented by the divergence-free transverse discontinuity that models the transverse field components at the entrance of the coil, B x = −∂ x Φ2δ( z), B y = −∂ y Φ2δ( z), where the quadratic scalar potential function Φ2 is defined as Φ2 = B0 (1 −β) 2 x2 + β 2 y2 and β ∈ is the symmetry parameter.

The corresponding near-axis vector potential can be gauge-fixed so that only the transverse components depend on the transverse coordinates AT = (−∂ y Φ2θ( z), ∂ x Φ2θ( z), 0). The explicit form of the transverse part reads AT ⊥= B0 (−βθ( z) y, (1 −β)θ( z) x) , Az = 0. This gauge is in Coulomb form, ∇· A = 0 and yields a strictly two-dimensional Schrödinger-like transverse setup in the Foldy-Wouthuysen representation under the paraxial reduction (k⊥≪kz) with spin neglected. The analysis now switches to normalized coordinates x = x/ρH, y = y/ρH, z = z/(kρ2 H) and considers the paraxial reduction of the Dirac equation with spin neglected in the form i∂zψ = H⊥ψ, H⊥= [p⊥−sgn(e)A⊥]2 2 . With the canonical rescaling x = p 2(1 −β) x, y = p 2β y, px = px p 2(1 −β) , py = py √2β, the Hamiltonian assumes the Schwinger form H⊥= Hs + Has, Hs = p2 ⊥ 2 + ρ2 2, Has(α) = −sgn(e) cos(2α) H1 + sin(2α) H3, where sin2(α) = β, ρ2 = x2 + y2 and Li = Hi 2, [ Li, Lj] = iεijk Lk. Here Li are operators of orbital pseudospin projections or the generators of rotation about three orthogonal axes on the orbital Poincaré sphere. For clarity, the analysis specializes to the fully symmetric aligned case α = π 4, β = 1 2, for which the canonical rescaling is trivial and a physical Laguerre-Gaussian (LG) mode is an exact eigenstate of the linear problem.

Hence, bars are dropped in further calculations. In that case the shell Hamiltonian for the electron is e For clarity, the latter representation is adopted and the body Hamiltonian is kept fully symmetric. Thus, the incoming state is a slightly rotated extremal LG state, |ψin⟩= e−iδ L2 |j, j⟩, 0 For experimentally accessible topological charges |l| = 200, 2000 (j = 100, 1000) the window j−1/2 ≪δ ≪1 is comfortably satisfied by δ ≈0.1, 0.2, a value readily achieved by a slight intentional misalignment of the incoming mode or by residual imperfections of the holographic phase plate.

A nonlinear thin-edge octupole perturbation is introduced to move beyond the linear model. This perturbation preserves the two-dimensional structure of the post-edge Hamiltonian while enabling substantially richer dynamics. This can be accomplished by adding a thin normal octupole magnet at the edge, routinely used as a corrector in an electron microscope, through a quartic potential function harmonic in the transverse plane, Φ = Φ2 + Φ(n) 4, with Φ(n) 4 = κ(x4 −6x2y2 + y4), κ = κnρ2 H 2B0R2c and Rc being the bore radius of the octupole and κn = BtipL/Rc with Btip the magnetic field at the pole tip and L is the length (thickness) of the octupole.

A weak nonlinear magnetic entrance edge induces a tennis-racket (Dzhanibekov) instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. Starting from a Maxwell-consistent thin-edge extension of the entrance field, an effective fixed-shell Hamiltonian is derived where linear Schwinger pseudospin precession acquires an anisotropic quadratic correction. In the symmetric aligned limit, an exact linear eigenstate, a Laguerre-Gaussian vortex state, becomes a hyperbolic fixed point of the large-shell dynamics, producing recurrent reversals of the mean pseudospin projection.

These reversals appear in real space as repeated conversions of the transverse profile between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed states. The unavoidable Lewis-Ermakov breathing of realistic wave packets modulates the nonlinear strength and sets the growth timescale, rather than acting as a separate mechanism. Microscope-scale estimates suggest that the required conditions are accessible with standard octupole correctors in a transmission electron microscope.

A weak nonlinear magnetic entrance edge induces a tennis-racket (Dzhanibekov) instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. Starting from a Maxwell-consistent thin-edge extension of the entrance field, an effective fixed-shell Hamiltonian is derived in which linear Schwinger pseudospin precession acquires an anisotropic quadratic correction. In the symmetric aligned limit, an exact linear eigenstate, a Laguerre-Gaussian vortex state, becomes a hyperbolic fixed point of the large-shell dynamics, producing recurrent reversals of the mean pseudospin projection.

These reversals appear in real space as repeated conversions of the transverse profile between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed states. The unavoidable Lewis-Ermakov breathing of realistic wave packets does not create a separate mechanism; it naturally modulates the nonlinear strength and sets the growth timescale. Microscope-scale estimates indicate that the required regime is accessible with standard octupole correctors in a transmission electron microscope.

Twisted electron beams exhibit unexpected instability through a ‘tennis-racket’ mechanism

A surprising instability in electron behaviour has been unlocked, demonstrating a ‘tennis-racket’ effect within twisted electron beams. This theoretical advance could enable finer control of electrons, potentially revolutionising high-resolution microscopy and materials science in the coming decade. However, the current work relies heavily on theoretical modelling and has not yet been demonstrated experimentally.

Researchers have shown that a weak magnetic field can induce an instability in twisted electrons, resembling the unpredictable motion of a tennis racket. This instability arises from the interplay between the electron’s spin and orbital motion within a solenoidal field, causing reversals in its projected spin. The study demonstrates that the natural behaviour of electron wave packets modulates this instability, establishing its timescale. Authors suggest this regime is achievable using existing technology within a transmission electron microscope.