⚛️ quantum physics

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

This paper introduces a simplified quantum state tomography method for inelastic electron scattering by restricting measurements to the orbital angular momentum subspace using an OAM sorter, which significantly reduces experimental complexity while enabling the characterization of symmetry-breaking effects and state transitions in volume plasmon excitations.

Original authors: Amir H. Tavabi, Alessio D'Errico, Paolo Rosi, Giovanni Bertoni, Enzo Rotunno, Luca Belsito, Alberto Roncaglia, Stefano Frabboni, Gian Carlo Gazzadi, Peter Tiemeijer, Rafal E. Dunin-Borkowski, Ebrahim

Published 2026-08-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amir H. Tavabi, Alessio D'Errico, Paolo Rosi, Giovanni Bertoni, Enzo Rotunno, Luca Belsito, Alberto Roncaglia, Stefano Frabboni, Gian Carlo Gazzadi, Peter Tiemeijer, Rafal E. Dunin-Borkowski, Ebrahim Karimi, Vincenzo Grillo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to take a perfect photograph of a spinning top. If the top is spinning in a perfectly predictable way, you can just snap a picture and know exactly where it is and how fast it's going. But what if the top is made of fog? Or what if it's bumping into other invisible things that make it wobble and lose its rhythm? In the world of quantum physics, particles like electrons are like these tops. They don't just have a position; they have a "quantum state," which is a fancy way of describing all their properties, including how they spin and wiggle. Scientists use a mathematical tool called a "density matrix" to map out this state. Think of the density matrix as a complete instruction manual for the particle's behavior. If the particle is pure and calm, the manual is short and simple. But if the particle gets messy or bumps into things (a process called "inelastic scattering"), the manual becomes a giant, confusing book full of mixed-up instructions.

For a long time, trying to read this messy manual for electrons was like trying to count every single grain of sand on a beach while a hurricane is blowing. The math gets incredibly complicated, and the number of measurements needed to figure out what's going on grows so fast that it becomes impossible to do in a real lab. This is a big problem because understanding how electrons change when they hit materials could help us build better microscopes, new computers, and deeper insights into how matter works. Scientists have been looking for a shortcut—a way to simplify the problem without losing the important details. They needed a way to peek at the electron's "spin" without getting overwhelmed by all the other chaos happening around it.

This is where the new research comes in. The team, led by scientists from institutions in Germany, Canada, Italy, and the Netherlands, decided to stop trying to measure the electron's entire universe and instead focus on just one specific feature: its "Orbital Angular Momentum" (OAM). You can think of OAM as the electron's internal twist or spiral. Just like a corkscrew or a spiral staircase, an electron can twist in different ways, and this twist is quantized, meaning it comes in specific, whole-number steps. The researchers realized that if they only looked at these twists, they could turn the impossible task of measuring a continuous, messy beach into a much simpler task of counting a finite set of distinct steps.

To do this, they built a special device called an "OAM sorter." Imagine a magical funnel that takes a swirling electron beam and sorts the different types of spirals into separate lanes, like a toll booth that directs cars based on their license plate color. By using this sorter, they could take a complex, messy electron beam that had just bounced off a thin carbon film and sort out exactly how much "twist" each electron had lost or gained. They didn't just look at the energy the electrons lost (which is standard in electron microscopy); they looked at how the pattern of their twist changed.

The experiment involved firing a specially shaped electron beam, which the scientists called a "petal beam" because it looked like a flower with eight petals, at a thin film of carbon. When this beam hit the carbon, it excited "plasmons," which are like collective waves of electrons sloshing around inside the material. This collision is an "inelastic" event, meaning the electron gave up some energy and got a bit messy. The team then used their OAM sorter to take a "snapshot" of the electron's quantum state after the crash.

What they found was fascinating. Before the crash, the electron beam was very orderly, with its twist concentrated on two specific values (like a top spinning at exactly 4 rotations per second, both clockwise and counter-clockwise). After hitting the carbon, the beam became a bit more chaotic. The "purity" of the state dropped from 0.54 to 0.21, meaning the electron lost some of its perfect coordination and became a mix of different twists. However, it didn't become a total mess. The researchers discovered that the electron didn't just lose its twist randomly; it followed specific rules. The data suggested that the electron mostly changed its twist by just one step at a time (a "single-OAM-step change"), which hints that the interaction was governed by simple, dipole-like forces, much like how a magnet might nudge a compass needle.

By using this new method, the team was able to reconstruct the "density matrix" of the scattered electron. This allowed them to see not just the average result, but the full range of possibilities for what the electron became after the collision. They found that while the original "petal" shape was mostly preserved, the collision introduced a slight rotation and some new, smaller twists. This suggests that even when electrons get messy, they still hold onto some of their original structure and coherence. The study shows that by focusing on the orbital angular momentum, scientists can bypass the usual complexity of quantum tomography and get a clear, detailed picture of how electrons interact with matter. It's a new way of listening to the electron's story, one that reveals the subtle symmetries and broken patterns that happen when the quantum world gets a little bump.

Technical Summary: Quantum Tomography of Inelastic Electron Scattering via Orbital Angular Momentum States

The Problem
The full characterization of a quantum system, whether pure or mixed, requires the reconstruction of its density matrix. While quantum state tomography (QST) is a cornerstone of quantum optics, its application to transmission electron microscopy (TEM), particularly for characterizing electron beams after inelastic scattering, has remained a significant challenge. Inelastic scattering introduces mutual incoherence between scattered components, transforming the system from a pure state (describable by a wavefunction) into a mixed state that necessitates a full density matrix representation.

Standard inverse methods for phase retrieval, such as multi-plane phase retrieval or electron ptychography, often rely on assumptions that the probe and sample remain in pure states or utilize envelope functions to model decoherence. These approaches struggle with genuine inelastic events where complex interactions blur such assumptions. Furthermore, reconstructing a density matrix in a high-dimensional phase space is computationally and experimentally prohibitive, as the number of required measurements grows quadratically with the dimensionality of the space. Existing methods like orbital mapping or center-of-mass mapping provide only partial access to the quantum state, typically capturing diagonal elements (probabilities) while failing to resolve the mutual coherences essential for full characterization.

Methodology
To address these challenges, the authors introduce a simplified approach by restricting QST to the electron orbital angular momentum (OAM) subspace. This strategy exploits the inherent quantization of OAM, where the quantum state space is spanned by a discrete set of eigenstates $|m\rangle$ (where $m$ is the topological charge). By tracing over the radial degree of freedom, the problem is reduced from estimating a continuous function over phase space to determining a finite number of matrix elements in a compact $d \times d$ density matrix, where $d = 2\ell + 1$ .

The experimental implementation utilizes an electron optical device known as an OAM sorter. This device employs two consecutive phase elements in Fourier-conjugate planes to perform a conformal mapping of the transverse wavefunction onto a log-polar coordinate system. This transformation maps the azimuthal coordinate $\theta$ to a linear coordinate and the OAM quantum number $m$ to a conjugate linear momentum. The setup allows for two independent measurement configurations:

OAM-resolved Electron Energy-Loss Spectroscopy (OAM-EELS): To measure the discrete OAM spectrum at specific energy losses.
Energy-Filtered Transmission Electron Microscopy (EFTEM): To map the azimuthal spatial coordinate (the discrete Fourier transform of the OAM spectrum) after energy filtering.

The experiment utilized a structured electron probe in the form of a "petal beam," created as a coherent superposition of vortex beams with opposite OAM values ( $|m|=4$ , specifically $|+4\rangle + |-4\rangle$ ). This beam was directed through an amorphous carbon film to excite volume plasmons. The resulting inelastic scattering was analyzed within a 10 eV energy-loss window.

Key Contributions and Results
The study successfully demonstrates the reconstruction of the density matrix for an electron beam undergoing inelastic scattering. Key findings include:

Density Matrix Reconstruction: Using a constrained maximum-likelihood estimation (MLE) algorithm, the authors reconstructed the $11 \times 11$ density matrix (for $|m| \le 5$ ) from experimental data. The algorithm incorporated soft constraints, including positivity, trace normalization, approximate symmetry between positive and negative $m$ values, and proximity to the known elastic scattering state.
Quantification of Decoherence: The purity of the quantum state, defined as $P = \text{Tr}(\hat{\rho}^2)$ , was measured to decrease significantly from $P = 0.54$ for the elastic probe to $P = 0.21$ after inelastic scattering. This confirms a substantial loss of coherence, though a non-negligible degree of coherence persists.
Symmetry Breaking and State Transitions: Analysis of the diagonalized density matrix revealed the nature of induced state transitions. The dominant eigenstate closely matched the input probe, validating the reconstruction. However, subsequent eigenstates revealed transitions characterized by single-OAM-step changes ( $\Delta m = \pm 1$ ). This observation aligns with dipole selection rules, suggesting that the inelastic electron-plasmon interaction is governed predominantly by dipolar coupling.
Angular Momentum Redistribution: The data showed that as energy loss increases, linear momentum transfer associated with plasmon excitation leads to a degradation of OAM purity and a redistribution of angular momentum to intermediate $m$ values.
Validation of the Sorter: The study confirmed that the conformal transformation performed by the OAM sorter remains valid for partially coherent beams, supporting its utility in quantum state tomography.

Significance
The paper claims that this approach offers a powerful tool for probing essential features of inelastic scattering that are inaccessible through elastic imaging alone. By restricting tomography to the OAM subspace, the method significantly reduces the experimental and computational burden while providing full access to both the amplitudes and phases of the quantum state.

The primary significance lies in the ability to reveal symmetry-breaking effects and gain insight into the coherence and evolution of scattered quantum states. The diagonalization of the density matrix provides a novel framework for analyzing Electron Energy-Loss Spectroscopy (EELS) data: rather than observing only integrated signals, one can retrieve the full ensemble of post-interaction states. This allows for a direct connection between observed electron state transitions and the underlying sample transitions, governed by conservation laws and interaction potentials. The work establishes a foundation for future investigations into coherence-preserving and coherence-breaking mechanisms in structured electron beams interacting with complex materials.

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