Structured quantum waves

Harris, Jérémie; Grillo, Vincenzo; Mafakheri, Erfan; Gazzadi, Gian Carlo; Frabboni, Stefano; Boyd, Robert W.; Karimi, Ebrahim

doi:10.1038/nphys3404

Cited by 137 publications

(109 citation statements)

References 67 publications

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“…Evidently, this is the nonrelativistic spin in the electron rest frame instead of the relativistic momentum-dependent spin (7). Similarly, the spin-to-orbital AM conversion (10) in nonparaxial electron vortex beams is missing for the NWFW operators,…”

Section: B Newton-wigner-foldy-wouthuysen Operatorsmentioning

confidence: 99%

“…Calculating other characteristics requires structured Dirac-electron solutions, such as electron vortex beams [3][4][5][6][7][8]. A monochromatic beam with welldefined energy E > 0, propagating along the z axis, can be constructed as a Fourier superposition of multiple plane waves with momenta p distributed around the propagation direction,…”

Section: Spin-orbit Interactionmentioning

confidence: 99%

“…These electron vortices are currently attracting considerable attention in several areas of physics, including electron microscopy, quantum theory, and high-energy physics (see [7,8] for reviews). Free-electron vortex states were first described using a simplified model of a scalar nonrelativistic electron [3].…”

Section: Introductionmentioning

confidence: 99%

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Position, spin, and orbital angular momentum of a relativistic electron

2017

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Motivated by recent interest in relativistic electron vortex states, we revisit the spin and orbital angular momentum properties of Dirac electrons. These are uniquely determined by the choice of the position operator for a relativistic electron. We consider two main approaches discussed in the literature: (i) the projection of operators onto the positive-energy subspace, which removes the Zitterbewegung effects and correctly describes spin-orbit interaction effects, and (ii) the use of Newton-Wigner-Foldy-Wouthuysen operators based on the inverse Foldy-Wouthuysen transformation. We argue that the first approach [previously described in application to Dirac vortex beams in K. Y. Bliokh et al., Phys. Rev. Lett. 107, 174802 (2011)] has a more natural physical interpretation, including spin-orbit interactions and a nonsingular zero-mass limit, than the second one [S. M.

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Section: B Newton-wigner-foldy-wouthuysen Operatorsmentioning

confidence: 99%

Section: Spin-orbit Interactionmentioning

confidence: 99%

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Position, spin, and orbital angular momentum of a relativistic electron

2017

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“…Following the suggestion of [8], such electron states were realized experimentally a few years ago [9], for a recent review see [10]. Since the interference occurs in free space, it is free from medium effects and can reveal novel features of the fundamental scattering process.…”

mentioning

confidence: 99%

“…Such states are known as Bessel vortex states, or twisted electrons, [8,10]. The parameter can be adjusted experimentally and plays the role of approximately conserved orbital angular momentum with respect to the average propagation direction.…”

mentioning

confidence: 99%

Double-slit experiment in momentum space

Иванов¹,

Seipt²,

Surzhykov³

et al. 2016

EPL

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-Young's classic double-slit experiment demonstrates the reality of interference when waves and particles travel simultaneously along two different spatial paths. Here, we propose a double-slit experiment in momentum space, realized in the free-space elastic scattering of vortex electrons. We show that this process proceeds along two paths in momentum space, which are well localized and well separated from each other. For such vortex beams, the (plane-wave) amplitudes along the two paths acquire adjustable phase shifts and produce interference fringes in the final angular distribution. We argue that this experiment can be realized with the present day technology. We show that it gives experimental access to the Coulomb phase, a quantity which plays an important role in all charged particle scattering but which usual scattering experiments are insensitive to.

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Generation with phase‐and‐amplitude electron holograms of Laguerre‐Gauss beams with orbital angular momentum up to 200ħ

Venturi

Balboni

Gazzadi

et al. 2016

European Microscopy Congress 2016: Proceedings

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Phase‐and‐amplitude electron holograms provide a flexible way to encode an arbitrary wavefunction by modulating only the hologram phase [1] [2]. This is an innovative step in the direction of novel experiments with structured electron waves [3]. The most interesting application example is the generation of Laguerre‐Gaussian (LG) beams as they can be used to match exactly a single Landau state of electrons inside the objective lens of a transmission electron microscope (TEM). LG beams are a solution of the paraxial Schrödinger equation. They are mainly characterised by two indexes: l, the azimuthal index, representing the orbital angular momentum (OAM), and p, the radial index, where p+1 is the number of intensity radial nodes. They will be referred to with the contract notation LG(l,p). Landau states, on the other hand, are the quantized eigenstates of a charged particle with OAM in a magnetic field. Remarkably, LG beams have, at a given plane, the same form of quantized Landau states [4]. They only differ in the z evolution: while in a magnetic field Landau states are non‐diffractive, LG beams in vacuum expand with defocus but maintain the same intensity shape. We fabricated the holograms with Focused Ion Beam (FIB) on Si 3 N 4 membranes. As a first check of fabrication accuracy, the Energy Filtered‐TEM thickness map is taken. Thickness maps can be considered as a measure of the phase of the electron wavefunction after the hologram. In the first row of figure 1 thickness maps of the holograms LG(0,10), LG(10,0) and LG(10,10) are shown. The second row shows on top the experimentally acquired intensity at the Fraunhofer plane, and at the bottom the intensity and phase (represented by the colour hue) calculated with the software STEM_CELL starting from the thickness maps. The beams with l=10 show indeed the well‐known azimuthal phase ramp of vortex beams while the beams with p=10 show 11 intensity nodes in radial direction with alternating phase as prescribed for LG beams. This allows us to say that we were able to produce LG beams with arbitrary l, p indexes. If the LG beam was generated inside the magnetic field of the objective lens of a TEM, this would permit the visualization of exact Landau states. A more direct test of the LG character of these beams is obtained by visualising their shape invariance after propagation. To this aim, we investigated the propagation behaviour of the beam LG(10,0) with simulations, with w 0 is the beam waist and z R = π W 0 2 /λ is the Rayleigh range for electrons with wavelength λ. Simulations are shown in the first row of figure 2, reporting the beam intensity shape propagated over Δz distance. When propagating, the beam width increases but the circular intensity shape remains the same. This is not true in general for all vortex beams. In order to make a comparison, we generated a vortex beam with 10ħ OAM (named L=10). The second row of figure 2 shows the experimental intensity of the beam L=10, acquired at the Fraunhofer plane (Δz = 0), and with different defocuses. The external intensity ripples are due to the abrupt intensity profile of the hologram (which in this case was not made with the phase‐and‐amplitude technique) and have the same character at all defocuses. The shape of the vortex, which is the circle with maximum intensity, with increasing defocus develops some internal ripples: this beam is therefore not shape invariant after propagation. As a further instance of the improvements due to the phase‐and‐amplitude scheme with respect to the previous ones, in figure 3 a phase‐and‐amplitude LG(200,0) beam (a) and its radial profile (b) is compared with an ordinary hypergeometric‐Gauss beam with L=200ħ (c) and its profile (d). The striking feature is the transverse confinement of the intensity in the LG beam, with respect to L=200 that shows many external ripples. Generating beams with phase‐and‐amplitude holograms shows clear advantages, from the suppression of the unwanted beam intensity ripples to the control of radial and azimuthal degrees of freedom of LG beams, and is of great importance in order to generate shape‐invariant LG beams that can match exactly a single Landau state.

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Structured quantum waves

Cited by 137 publications

References 67 publications

Position, spin, and orbital angular momentum of a relativistic electron

Position, spin, and orbital angular momentum of a relativistic electron

Double-slit experiment in momentum space

Generation with phase‐and‐amplitude electron holograms of Laguerre‐Gauss beams with orbital angular momentum up to 200ħ

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