Two-electron quantum walks can probe entanglement and decoherence in an electron microscope

Technical Summary: Two-electron quantum walks probe entanglement and decoherence in an electron microscope

Problem and Motivation
While classical physics often approximates many-body quantum systems, the transition from microscopic quantum interactions to emergent macroscopic phenomena—governed by entanglement and decoherence—remains a fundamental area of study. Although entanglement has been demonstrated between free electrons and photons, and between electrons in specific bound states, the quantum properties of multiple free electrons interacting via Coulomb forces in a free-space gas have remained unexplored. Specifically, it is unclear whether free electrons in a dense gas develop significant entanglement or if decoherence from the unobserved environment (the rest of the gas) suppresses these quantum correlations, leading to classical behavior. This study aims to characterize the degree of coherence and entanglement in a free-space electron gas within an ultrafast electron microscope.

Methodology
The researchers employed a "two-electron quantum walk" as a diagnostic tool for quantum state tomography of electron pairs. The experimental setup involved:

1. Electron Generation: Femtosecond laser pulses (1030 nm) were focused on a sharp metallic needle tip to generate a dense gas of approximately 135 electrons per pulse. These electrons interact via strong Coulomb forces, creating energy and time anti-correlations.
2. Selection: An aperture selected electron pairs from the gas, resulting in an average of ~0.2 electrons per pulse in the beam, categorized into zero-, one-, or two-electron events.
3. Quantum Walk Modulation: The selected electron pairs interacted with a long (1.2 ps), coherent laser pulse at a silicon nitride membrane. This interaction coherently modulated the energy of the electrons, creating sidebands via the absorption or emission of photons. The coupling parameter $g$ was controlled by the laser field strength.
4. Detection: Coincidence maps $P(E_1, E_2)$ were measured using an energy-resolved, number-counting screen (Timepix3 detector).
5. Theoretical Analysis: The team developed a maximum-likelihood density matrix reconstruction approach. They modeled the two-electron state as a mixture of a fully entangled pure state and a separable state with the same diagonal elements (classical correlations). The model included parameters for the entanglement fraction ($f$) and phase terms ($\alpha, \beta, \gamma$) representing dispersion from Coulomb interactions.

Key Results

• Spectral Signatures: The measured effective electron spectra showed distinct differences between one-electron and two-electron events. While one-electron events exhibited Rabi revivals typical of coherent quantum walks, the two-electron zero-loss peak initially showed a minimum intensity, increasing only at higher laser fields, indicating different interference dynamics.
• Fringe Visibility: In energy-domain interferometry, one-electron events showed 70% fringe visibility, whereas two-electron events showed only 17% visibility in the effective spectrum. However, coincidence maps revealed high fringe visibility (68%) in the two-electron correlations, indicating that the pairs maintain combined coherent wave-like features despite Coulomb interactions.
• Entanglement Quantification: Theoretical fitting of the coincidence maps against the experimental data yielded an optimal entanglement fraction of $f = 0.18$. The calculated negativity of entanglement was approximately 7%. The authors state that this deviation from a separable state is not statistically significant within the limits of their experimental sensitivity and the subspace searched.
• Decoherence Mechanism: Analysis of the electron gas dynamics revealed that while the single-electron coherence time is 4.7 fs and the nearest-neighbor distance (300 nm) is comparable to the coherence length, the unobserved electrons in the gas act as an environment. This environment causes decoherence, effectively washing out the entanglement between the selected pair, even though the initial multi-electron wavefunction likely possessed entanglement.

Significance and Claims
The paper claims to provide the first experimental investigation of entanglement and decoherence in a free-space electron gas. The primary contribution is the demonstration that a two-electron quantum walk can distinguish between entangled states, classically correlated extended matter waves, and point-like particles.

The authors conclude that under their specific experimental conditions (hundreds of electrons per pulse), the observed electron pairs are best described as non-entangled but coherent matter waves (a separable state with strong classical correlations). They argue that this result illustrates a regime where classical physics emerges from quantum principles: Coulomb interactions create initial entanglement, but the tracing out of the unmeasured reservoir electrons leads to decoherence and the loss of observable entanglement in the selected pair.

The study establishes a framework for characterizing multi-electron quantum states that interact with laser light. The authors suggest that while their current results show limited entanglement due to decoherence, the method allows for the verification of electron-electron entanglement in future experiments. They propose that optimizing parameters—such as using shorter, weaker photoemission pulses on narrow-band tips to maximize coherence length and gas density while minimizing the total electron number—could potentially produce significantly entangled electron pairs. Such states would be relevant for fundamental studies of many-body quantum physics and potential applications in quantum electron microscopy and low-dose imaging.