🔬 optics

Quantum tomography of free electrons

This paper introduces a universal method for quantum tomography of free electrons using interfering laser-induced quantum paths to fully characterize arbitrary continuous-variable quantum states, enabling the measurement of hidden correlations and the optimization of electron beams for advanced quantum applications.

Original authors: Y. Fang, J. Kuttruff, Z. Zhao, L. Moehrle, P. Hommelhoff, P. Baum

Published 2026-06-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Y. Fang, J. Kuttruff, Z. Zhao, L. Moehrle, P. Hommelhoff, P. Baum

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

The Big Picture: Taking a "Quantum X-Ray" of a Free Electron

Imagine you have a single electron flying through a vacuum. In the world of quantum physics, this electron isn't just a tiny ball; it's a "wave packet" with a specific shape, a specific rhythm, and a hidden internal structure. Scientists call this its quantum state.

For a long time, we could measure the energy of electrons (how fast they are going), but we couldn't see their full "quantum face." We knew they were messy and incoherent (like a crowd of people shouting at different times), but we couldn't map out exactly how they were messy.

This paper introduces a new method to take a complete "quantum X-ray" or a full map of an electron's state. It allows scientists to see not just the electron's energy, but also its hidden timing, its internal correlations, and how it interacts with other electrons.

The Problem: The Electron is a "Blurry Crowd"

Think of the electrons coming out of the microscope's source like a group of runners starting a race.

The Ideal: You want every runner to start at the exact same time, running at the exact same speed, in perfect sync. This is a "pure" quantum state.
The Reality: In real life, the runners start at slightly different times, run at slightly different speeds, and some are jostled by the crowd around them (other electrons). This creates a "mixed state"—a blurry, messy crowd.

Until now, we didn't have a tool to look at this crowd and say, "Okay, here is exactly how much they are out of sync, and here is exactly how the jostling changed their rhythm."

The Solution: The "Two-Tone Laser Ruler"

The researchers built a clever tool to measure this mess. Imagine you are trying to measure the shape of a spinning fan blade, but it's moving too fast to see. You shine two different colored lasers at it.

The Setup: They shoot the electron beam through a special membrane while hitting it with two laser beams that are almost the same color, but just a tiny bit different (like two tuning forks that are slightly out of tune).
The Interaction: When the electron hits these lasers, it can absorb a "kick" of energy from either laser. This creates two different "paths" for the electron to take.
The Interference: Because the electron is a wave, these two paths interfere with each other, creating a pattern of ripples (like waves in a pond meeting).
The Magic: By slowly changing the timing between the electron and the lasers, and by slightly adjusting the difference in color between the two lasers, the researchers can watch how these ripples move and change.

The Analogy: Imagine trying to figure out the shape of a hidden object by shining two flashlights at it from slightly different angles and watching how the shadows dance. By analyzing the dance of the shadows (the interference pattern), they can reconstruct the exact 3D shape of the object (the electron's quantum state).

What They Found: The "Chirp" and the "Jitter"

Using this new "ruler," they mapped the electron's state and found two main things:

The "Chirp" (The Stretch): They found that the electron wave is "chirped." Imagine a violin string that starts with a low note and slides up to a high note very quickly. The electron's energy isn't flat; it's stretched out in time. This happens naturally as the electron flies through the microscope, much like a sound wave stretching out as it travels through the air.
The "Jitter" (The Noise): They measured how much the electrons are "jittering" (randomly changing speed or timing). They found that while the group of electrons is very messy (a wide range of speeds), the individual electron inside that group is actually quite neat and orderly.

The "Space Charge" Surprise: The Crowd Effect

One of the most interesting parts of the paper is what happens when you have many electrons flying together.

The Scenario: Imagine a single electron is like a lone runner. Now imagine 150 runners packed tightly together. Because electrons repel each other (they have the same negative charge), they push against one another. This is called "space charge."
The Expectation: You might think that if 150 electrons push against each other, they would all become a total mess, destroying their delicate quantum nature.
The Discovery: The researchers found something surprising. While the group of electrons gets very messy and their overall timing gets scrambled, the individual electron inside that group remains surprisingly stable.
- Analogy: Imagine a crowded dance floor where everyone is bumping into each other. The overall dance floor is chaotic. But if you zoom in on one specific dancer, they are still doing their specific dance steps perfectly. The crowd messes up the group rhythm, but it doesn't ruin the individual dancer's internal rhythm.

Why This Matters (According to the Paper)

This technique is a "universal diagnostic tool." It doesn't just tell us the electron is messy; it gives us a complete map of why it is messy.

For Microscopy: If we want to take super-sharp pictures of tiny things (like viruses or atoms) using electrons, we need to know exactly how the electron beam is behaving. This tool tells us how to fix the beam to make it sharper.
For Quantum Tech: If we want to use electrons for quantum computing or "quantum walks" (where electrons act like bits of information), we need to know if they are still "pure" or if they have become "mixed." This method lets us check that status instantly.

Summary

The paper presents a new way to "see" the invisible quantum nature of free electrons. By using two slightly different laser colors to create a ripple effect, the scientists can map out the electron's energy, timing, and internal structure. They discovered that while a crowd of electrons gets messy due to repulsion, the individual electrons inside remain surprisingly coherent, offering a new way to optimize electron beams for high-tech imaging and future quantum technologies.

Technical Summary: Quantum Tomography of Free Electrons

Problem Statement

Determining the quantum state of a quantum-mechanical system is a fundamental task in physics. While quantum-state tomography has been pivotal for quantum optics and bound charges in materials, free electrons present a unique challenge. Free electrons are central to high-resolution microscopy, electron-based quantum optics, ultrafast electron microscopy, and particle accelerators. However, they are intrinsically incoherent due to the physics of emission, voltage fluctuations, and statistical Coulomb interactions. Consequently, the phase space volume of free electrons often exceeds that of underlying pure states, resulting in mixed states with unknown correlations between the ensemble and individual coherent parts. Currently, there is a lack of a broadly applicable method to measure and control the quantum state of free electrons beyond special cases involving discrete energy sidebands. The most general quantity required to describe these states is the density matrix, which remains largely inaccessible for continuous variables in free-electron beams.

Methodology

The authors report a universal approach to measure arbitrary free-electron quantum states in continuous variables using spectral shearing interferometry.

Experimental Setup: The experiment utilizes a transmission electron microscope (TEM) with a Schottky field-emitter source operating at 180 keV. Femtosecond electron pulses are generated via photoemission.
Interaction Mechanism: The electron pulses interact with two phase-stabilized, quasi-monochromatic laser waves with slightly detuned wavelengths ( $\lambda_1$ and $\lambda_2$ ) and ultra-narrow linewidths (< 0.2 nm). These lasers are focused onto a silicon nitride membrane where the light-electron interaction occurs.
Measurement Principle: The interaction induces absorption or emission of photons, creating energy sidebands. The simultaneous interaction with two detuned fields creates two quantum paths that interfere. The resulting electron spectrum, $S(\epsilon, \Delta t, \Delta E_{ph})$ , depends on the time delay ( $\Delta t$ ) between the electron pulse and the laser field, and the spectral separation ( $\Delta E_{ph}$ ) between the two lasers.
Data Acquisition: By scanning $\Delta t$ and $\Delta E_{ph}$ , the researchers sample the density matrix. The interference pattern's contrast reveals the magnitude of the off-diagonal density matrix elements $|\rho(\epsilon, \epsilon')|$ , while the phase of the interference fringes reveals the argument $\arg[\rho(\epsilon, \epsilon')]$ .
Reconstruction: A full reconstruction of the density matrix in the time-energy domain is achieved by systematically scanning the frequency difference and time delay, followed by a maximum-likelihood estimation fit to close gaps in the discrete measurement data.

Key Results

1. Proof-of-Principle and Density Matrix Reconstruction
The authors successfully reconstructed the density matrix of free electrons generated from a Schottky emitter.

Interferogram Characteristics: The measured interferograms showed a transition from slow modulation at small detunings to rapid oscillation at larger detunings, directly reflecting the oscillating term $\Delta E_{ph}\Delta t/\hbar$ .
Phase Structure: The reconstructed phase of the density matrix revealed a continuous, non-oscillatory structure with a quadratic spectral phase (linear chirp), consistent with vacuum dispersion and space charge effects.
Mixed State Characterization: The reconstructed amplitude showed an elliptical structure with an aspect ratio $\eta \approx 8 \times 10^{-3}$ $η \approx 8 \times 1 0^{- 3}$ , significantly smaller than that of a pure state ( $\eta=1$ $η = 1$ ).
- Pure State Properties: The individual electrons were found to be quantum-limited wavepackets with a pure energy bandwidth of $0.13 \pm 0.04$ eV and a temporal coherence length of $14 \pm 5$ fs.
- Ensemble Properties: The beam exhibited an incoherent energy spread of $0.88 \pm 0.03$ eV and a temporal jitter of $\sim 277$ fs.
- Conclusion: The electrons exist in a partially mixed state where the temporal coherence length is roughly twenty times shorter than the pulse duration, and the pure spectrum is ten times narrower than the incoherent bandwidth.

2. Simulation of Cat States
To demonstrate the method's versatility, the authors simulated the tomography of a free-electron "cat state" (a superposition of two energy states: $|E_{el} + \Delta E\rangle + |E_{el} - \Delta E\rangle$ ).

The simulation showed that small detunings probe coherence within each component, while larger detunings probe the coherence between the components.
The method successfully reconstructed the density matrix and the Wigner distribution, including the characteristic negative values indicative of non-classicality, despite realistic noise and spectral resolution limits.

3. Multi-Electron Coulomb Interactions
As a primary application, the study investigated how many-body Coulomb interactions (space charge) affect the quantum state of a single, post-selected electron.

Observation: As the number of electrons in the pulse increased (from ~1 to ~150), the slope of the density matrix phase increased, indicating a stronger ensemble chirp.
Simulation vs. Experiment: A semiclassical model using the Hamilton-Jacobi equation and Feynman's path integral reproduced the experimental data.
Key Finding: While the ensemble chirp increased significantly due to Coulomb acceleration/deceleration, the microscopic chirp and coherence properties of individual pure wavepackets remained largely unaffected. The pure energy bandwidth and temporal coherence length of single electrons were robust against moderate space charge effects.

Significance and Claims

The paper claims that this spectral shearing interferometry technique provides a universally applicable diagnostic tool for free-electron quantum optics.

Completeness: It allows for the measurement of the complete density matrix in continuous time-energy variables, revealing hidden correlations between coherent and incoherent widths.
Optimization: The ability to reconstruct arbitrary quantum states enables the controlled optimization of exceptional quantum states for applications such as free-electron quantum optics and quantum electron microscopy.
Robustness: The finding that single-electron coherence is robust against space charge effects suggests that applications relying on temporal coherence (e.g., attosecond microscopy, single-electron quantum walks, and free-electron qubits) are less susceptible to degradation from Coulomb interactions than previously assumed.
Generality: The method is extensible to other degrees of freedom (e.g., transverse dimensions) and can be adapted for continuous electron beams or multi-electron entangled states, providing a necessary foundation for the growing field of free-electron quantum optics.