Two-electron quantum walks for probing entanglement and decoherence in an electron microscope

Nature Physics (2026)Cite this article Revealing and quantifying entanglement of particles is central for understanding the foundations of quantum mechanics and its implications for modern technology. Recent works have demonstrated entanglement of free electrons and photons; however, the quantum properties of multiple free electrons, and the extent of their entanglement, remain largely unexplored. Here we investigate the degree of coherence and entanglement in a free-space electron beam in an ultrafast electron microscope. We introduce a two-electron quantum walk that transforms the quantum state into different bases for quantum state tomography of entangled or partially entangled electron–electron pairs. The method can distinguish point-like particles from delocalized two-electron matter waves with or without classical correlations or entanglement. As a first application, we study multiparticle quantum effects in short pulses of hundreds of electrons under strong Coulomb correlations. Pairs of postselected electrons are delocalized matter waves with correlation between different parts of the two-electron state. The degree of entanglement is less than 7% due to a limited purity of the initial states and decoherence effects from unmeasured reservoir electrons. 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Zenodo https://doi.org/10.5281/zenodo.18762359 (2026).Download referencesWe thank R. Haindl for raw data on electric fields at needle tips17 and C. Ropers for helpful, critical remarks.

This research was supported by the German Research Foundation (DFG) through SFB-1432. This project was funded by the European Union’s ERC COG, QinPINEM, Project No. 101125662.

This research was also funded by the Gordon and Betty Moore Foundation, through Grant GBMF11473. During preparation of this manuscript, we became aware of independent research on related topics49.Solid State Institute, Technion – Israel Institute of Technology, Haifa, IsraelOffek Tziperman, Ron Ruimy, Ethan Nussinson, Alexey Gorlach, Aviv Karnieli & Ido KaminerFachbereich Physik, Universität Konstanz, Konstanz, GermanyDavid Nabben, Jacob Holder, Yiqi Fang, Daniel Kazenwadel & Peter BaumAndrew and Erna Viterbi Department of Electrical and Computer Engineering, Technion – Israel Institute of Technology, Haifa, IsraelAviv KarnieliDepartment of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa, IsraelIdo KaminerSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarO.T. and I.K. proposed the idea for the experiment. O.T., R.R., A.G., E.N. and A.K. derived the theory. O.T., D.N. and Y.F. performed experiments and analysed the data. J.H. and D.K. made the numerical point-particle simulations. P.B. and O.T. interpreted the results and wrote the manuscript with input from all authors.Correspondence to Ido Kaminer or Peter Baum.The authors declare no competing interests.Nature Physics thanks Christoph Lienau and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.(a–c) Measured data (left column) compared to best-fit theory considering entanglement (middle column) for laser powers of 1 mW, 2 mW and 4 mW, respectively. The residual is small (right column). (d–f) Measured data (left column) compared to best-fit theory considering zero entanglement for laser powers of 1 mW, 2 mW and 4 mW, respectively. The residuals (right column) are slightly larger and have a systematic shape. (g–i) Experimental data when subtracting 75% of separable (no-entanglement) theory. The residuals (left panels) compare slightly better to fully entangled theory (right panels) than to non-entangled theory (above), but overall, there are no significant signs of entanglement, and the state is best described by a coherent matter wave.(a) Time-resolved energy spectrum of single electrons in a non-stretched laser beam49. (b) Time-resolved energy spectrum of two-electron events.Measured electron coincidence maps as a function of time delay for shorter laser pulses (450 fs) that cover only part of the two-electron state. For large negative delays (second panel), mostly only the faster of the two electrons disperses into peaks. For large positive delays (fifth panel), mostly only the slower of the two electrons disperses into peaks.Panel (a) shows the geometry of the emitter section of our electron microscope (black) with calculated electric fields (grey). The scatter plot shows the electron distribution at different times after laser excitation (see color bar). Panel (b) shows the measured and simulated electron energy distributions at an excitation power of ~20 mW. Both spectra show a similar amount of Boersch effect. The dotted lines show the interquartile widths. In (c), we plot a snapshot of the electron gas taken 400 fs after emission from the tip.We compare the laser modulated correlation maps for: (a) a tensor product state, (b) a correlated, non-entangled state, and (c) the experimental results. The two-electron state in our experiment cannot be described as the product of two single electron states.We compare the measured spectra of: (a) single-electron events, (b) two-electron events, and (c) two-electron energy differences for short (violet) and long (yellow) photoemission pulses at the emitter tip. In panel (c) to compute energy differences, we measure the difference in energy for each two-electron event, and derive a histogram. Panels (d, e) show measured correlation maps for long and short photoemission pulses, respectively, at (left to right) 0 mW, 1 mW, and 4 mW of modulation power.(a, b) The amplitude and phase of the wave function. (c) Schmidt coefficients. Multiple nonzero values demonstrate the presence of entanglement.Extracted density matrix (top row), compared to the ideal pure density matrix \({\rho }_{{\rm{pure}}}=|\psi \rangle \langle \psi |\) (bottom row). (a, b) Diagonal elements of the density matrix, which are insufficient for entanglement detection, are identical. The full density matrix is a complex function of four parameters, \({E}_{1},{E}_{2},{E}_{1}^{{\prime} },{E}_{2}^{{\prime} }\), we therefore flatten it to display it in two dimensions. (c, d) Real parts of \(\rho\). (e, f) Imaginary part of \(\rho\). We extract a purity \(\mathrm{Tr}\left({\rho }^{2}\right)\approx 0.03\) and an entanglement negativity of ≈0.07. All color bars stretch from -0.003 to +0.003.Panels (a–d) show the four primary regimes of electron-electron pairs. If two electrons overlap within their coherence lengths, or if they are close enough such that their interaction varies across their coherence length, they may form an entangled pair (first row). If the coherence time is reduced by interaction with the environment, yet remains longer than the wavelength of the laser modulation, the electrons become correlated matter waves (second row), like in our experiment (Fig. 3). When the interaction is too weak, the electrons are uncorrelated matter waves with no relation between their energies (third row). When the coherence length is much shorter than a wavelength of the characterizing laser field, the electrons behave classically, like point particles (fourth row). Panel (e) shows the estimated regimes for electron-electron entanglement in beams with Coulomb effects. Depending on the electron pulse duration at the emitter tip and the number of unmeasured electrons, the resulting two-electron states can be point-like particles (yellow), uncorrelated matter waves (blue), correlated matter waves (red), entangled pairs, or in transitional regimes (black dashed lines). Other, more complex states are also possible but not depicted here. The reported two-electron quantum walk allows us to obtain insight into and clarify the transitions between these regimes. The dots denote the two experiments of Fig. 2 and Extended Data Fig. 6d. The stars mark the theoretical results of Fig. 3 and Extended Data Fig. 5. The parameters at which each regime occurs, the physics at the transitional regions (black dashed lines), and the exact shape of the diagram are currently unknown. Exploring all regimes and transitions between them is an exciting direction for future research.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsTziperman, O., Nabben, D., Ruimy, R. et al. Two-electron quantum walks for probing entanglement and decoherence in an electron microscope. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03254-yDownload citationReceived: 14 May 2025Accepted: 10 March 2026Published: 21 April 2026Version of record: 21 April 2026DOI: https://doi.org/10.1038/s41567-026-03254-yAnyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative