Quantum tomography of inelastic electron scattering \emph{via} orbital angular momentum states

AI summary

Overview

• Research area: Quantum physics and electron microscopy — specifically quantum state tomography of inelastically scattered electrons using orbital angular momentum (OAM) states.
• Technical level: Intermediate (assumes familiarity with density matrices, quantum coherence, and electron energy-loss spectroscopy, but is accessible to graduate students in physics or materials science).
• Scope: The paper introduces OAM-restricted quantum state tomography as a practical way to reconstruct the density matrix of an electron probe after inelastic plasmon scattering in a transmission electron microscope, demonstrated on a carbon film.

What This Paper Is About

Fully describing a quantum state after inelastic scattering requires reconstructing its density matrix, but in electron microscopy this has been impractical because the number of required measurements grows quadratically with the dimensionality of phase space. The authors sidestep this by restricting tomography to the electron's orbital angular momentum subspace, using an electron OAM sorter to turn a continuous phase space into a finite set of measurable states. The goal is to recover the density matrix of a structured electron probe after it excites volume plasmons in amorphous carbon, and thereby read out coherence loss, symmetry breaking, and the selection rules of the interaction.

Key Contributions

1. OAM-restricted quantum state tomography for electrons: A simplified tomography scheme that traces over the radial degree of freedom and works in a discrete OAM basis truncated at |ℓ| ≤ 5 (d = 2ℓ+1 = 11), yielding an 11 × 11 density matrix.
2. Use of an electrostatic OAM sorter as the measurement device: The sorter performs a conformal log-polar mapping that turns the azimuthal coordinate into a linear coordinate and OAM into a conjugate linear momentum, enabling projective measurements in OAM and in angular superpositions.
3. A constrained reconstruction algorithm: A modified maximum-likelihood estimation (MLE) routine with Lagrange multipliers enforces soft constraints — positivity, trace normalisation, approximate m-symmetry, and proximity to the elastic-state density matrix — to solve an otherwise under-determined inverse problem.
4. Eigenstate decomposition of the measured density matrix as a new EELS observable: Diagonalisation of the reconstructed density matrix links eigenstates to post-scattering channels of Fermi's golden rule, recovering dipole (Δm = ±1) and weaker quadrupole (Δm = ±2) selection rules without imposing them as priors.

Main Findings

• Strong coherence loss after inelastic scattering: Purity dropped from 𝒫 = 0.54 for the elastic probe state to 𝒫 = 0.21 after inelastic interactions in a 10 eV energy window.
• The initial probe is not strictly pure: Even before scattering, averaging over unresolved radial degrees of freedom produces partial decoherence in the OAM basis; purity would approach unity only if radial and azimuthal modes were measured simultaneously.
• OAM purity degrades with energy loss: The zero-loss spectrum shows dominant peaks at m = ±4 with minimal population elsewhere, while at higher energy losses intermediate m values gain intensity, consistent with larger momentum transfer at larger energy loss.
• Angular fringes persist but fine structure is suppressed: Both elastic and inelastic angular profiles show a primary oscillatory pattern with seven visible fringes (one of the eight nominal lobes of the |m| = 4 petal beam is occluded by the sorter's needle electrode), plus a four-fold amplitude modulation from hologram imperfections. Higher-order modulations visible in the elastic case are suppressed in the inelastic case.
• Dominant eigenstate resembles the input probe: The leading eigenstate closely matches the input probe state, confirming that this mode survives the interaction with relatively high coherence.
• Dipole transitions recovered: The third, fourth, and fifth eigenstates show single-OAM-step changes (Δm = ±1) from each dominant petal-beam component, matching dipole selection rules and indicating predominantly dipolar electron–plasmon coupling. Weaker Δm = ±2 quadrupole signatures appear in higher-order eigenstates (reported in the Supplementary Information).
• Near-degeneracy of the second and third eigenvalues: The experiment shows an approximate degeneracy; any splitting may lie within reconstruction uncertainty or reflect the finite spatial extent of the plasmonic excitation.
• Simulation reproduces the picture: A Monte Carlo model treating inelastic scattering as exchange of plane waves with momentum cutoff k_p,max(Δℰ) produces degenerate eigenstates that are rotated versions of one another, resolving the apparent tension between rotational symmetry and OAM exchange.
• Residual asymmetry: A slight asymmetry between positive and negative ℓ components remains despite the weak symmetry constraint, attributed to incomplete optimisation convergence or experimental misalignment.
• Under-determined inverse problem: The 11 × 11 density matrix contains 120 independent real parameters, while only 11 coefficients come from OAM-resolved EELS and approximately 21 from the angular intensity profile — motivating the constrained MLE approach.
• Errors and statistics: Reported errors correspond to 3-times the standard deviation over 10 reconstructed density matrices.

Methodology in Plain English

The team formed a "petal beam" — a coherent superposition of electron vortex beams with equal and opposite OAM, |ℓ| = 4 — using an eight-region phase hologram placed in the condenser aperture. This beam has eight-fold rotational symmetry and was fired at an amorphous carbon film, where it excited volume plasmons.

To measure the OAM content of the electrons after scattering, they used an electrostatic OAM sorter made of two phase elements in Fourier-conjugate planes, the first implemented by a needle-shaped electrode protruding toward the optical axis. The sorter unwraps the azimuthal coordinate into a straight line, so that OAM values appear as separated positions in the detector plane — like turning a circular race track into a straight sprint lane. By adjusting diffraction lens excitation, the team could send either the OAM spectrum or the azimuthal coordinate into an energy filter, combining the sorter with electron energy-loss spectroscopy (OAM-EELS) and with energy-filtered transmission electron microscopy (EFTEM).

Diagonal OAM terms came directly from measured mode probabilities; off-diagonal coherence terms required projecting onto carefully chosen superposition states with systematically varied relative phases, analogous to higher-dimensional Stokes parameters. Because the data alone could not pin down all 120 parameters, the researchers added physically motivated soft constraints and solved for the density matrix with a modified maximum-likelihood algorithm. They then diagonalised the result to read out the mixture of pure states making up the scattered beam, and compared against a simple Monte Carlo model of momentum exchange.

Why This Matters

• Research impact: The work turns EELS from an intensity-measuring technique into a full quantum-state measurement, giving a basis-independent description of inelastic scattering that unifies selection rules, symmetry, and coherence in one framework. It also shows that the OAM sorter's conformal transformation remains valid for partially coherent beams.
• Real-world applications:
  • Characterising decoherence and coherence loss in electron beams used for quantum-free-electron experiments and electron-based quantum technologies.
  • Recovering multipolar selection rules in electron–matter interactions without prior assumptions, useful for spectroscopy of collective excitations.
  • Post-selecting electron OAM to prepare correlated vortex states in plasmonic fields, with surface plasmon polaritons flagged as a promising target due to their longer coherence lengths.
  • Providing a compact, low-measurement-count tomography protocol transferable to structured-light optics and other wave-based probes.
• Industry relevance: Semiconductor metrology and electron-microscope manufacturers (the author list includes Thermo Fisher Scientific) stand to benefit from a technique that extracts richer information from existing microscope hardware, particularly through electrostatic OAM sorter designs compatible with standard columns.

Future Directions

• Improving OAM sorter resolution and reducing cross-talk between adjacent modes, since the needle electrode's field discontinuity and hard spatial cutoff currently limit discrimination; more sophisticated designs are cited as pathways forward.
• Extending the reconstruction beyond the truncated |ℓ| ≤ 5 basis to capture fine angular features that currently lie beyond the accessible resolution.
• Testing the predicted ability to post-select Δm = +1 while suppressing Δm = −1, which would imply generating a plasmon with non-zero topological charge, and pursuing this in surface plasmon polaritons.
• Resolving whether the small splitting between the second and third eigenvalues is genuine symmetry breaking from the finite spatial extent of the plasmonic excitation, or an artifact of misalignment and optimisation convergence.
• Systematically identifying higher multipolar contributions (beyond the Δm = ±2 quadrupole signatures already seen) in future high-resolution experiments.

Target Audience

Physicists and graduate students working in electron microscopy, quantum optics, and quantum metrology; researchers in electron energy-loss spectroscopy and plasmonics interested in coherence and selection rules; and instrument developers working on orbital angular momentum sorting, structured electron beams, or quantum state reconstruction algorithms.

Authors’ abstract

The physical properties of a quantum system, whether pure or mixed, are described fully by its density matrix. Recovery of the density matrix through projective measurements -- referred to as quantum state tomography -- is a cornerstone of quantum optics and metrology. The implementation of this approach in transmission electron microscopy, in particular for the characterisation of an electron beam after inelastic scattering, has remained a longstanding challenge as a result of the complexity of scanning high-dimensional phase spaces, with the number of required measurements growing quadratically with space dimensionality. Here, we introduce a simplified approach by restricting tomography to the electron orbital angular momentum (OAM) subspace. By using an electron optical device known as an OAM sorter, we discretise the phase space into a finite set of measurable states, thus significantly reducing the experimental and computational burden. The resulting measurements suffice to probe essential features of inelastic scattering. We demonstrate the technique by studying the inelastic scattering of a structured electron probe exciting volume plasmons in a carbon film. The combined use of a structured beams and OAM-resolved quantum tomography reveals symmetry-breaking effects and offers insight into the coherence and evolution of the scattered quantum states. Analysis of the diagonalised density matrices further reveals the nature of the induced state transitions, demonstrating the power of the approach for quantum tomography of electron scattering.