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Two electrons interacting at a mesoscopic beam splitter


The nonlinear response of a beam splitter to the coincident arrival of interacting particles enables numerous applications in quantum engineering and metrology. Yet, it poses considerable challenges to control interactions on the individual particle level. Here, we probe the coincidence correlations at a mesoscopic constriction between individual ballistic electrons in a system with unscreened Coulomb interactions and introduce concepts to quantify the associated parametric nonlinearity. The full counting statistics of joint detection allows us to explore the interaction-mediated energy exchange. We observe an increase from 50% up to 70% in coincidence counts between statistically indistinguishable on-demand sources and a correlation signature consistent with the independent tomography of the electron emission. Analytical modelling and numerical simulations underpin the consistency of the experimental results with Coulomb interactions between two electrons counterpropagating in a quadratic saddle potential. Coulomb repulsion energy and beam splitter dispersion define a figure of merit, which in this experiment is demonstrated to be sufficiently large to enable future applications, such as single-shot in-flight detection and quantum logic gates.

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Fig. 1: Measuring coincidence correlations with single-electron precision.
Fig. 2: Signatures of correlation and interaction in the partitioning of two electrons.
Fig. 3: Dependence on interarrival time and energy detuning: quantitative comparisons with simulation results.
Fig. 4: Signatures of energy exchange and simulation results.

Data availability

The data that support the graphs of this work are available in the Zenodo repository at

Code availability

The code producing the figures is available from the corresponding author upon reasonable request.


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We acknowledge discussions with P. W. Brouwer, J. Fletcher and M. Kataoka, and we thank P. Degiovanni for suggesting the phase shift considerations. E.P., M.K., G.B. and V.K. are supported by the Latvian Council of Science (lzp-2021/1-0232). P.G.S. and P.R. acknowledge financial support from the Deutsche Forschungsgemeinschaft (German Research Foundation) within the framework of Germany’s Excellence Strategy (EXC-2123 QuantumFrontiers-390837967). This work was supported in part by the Joint Research Project SEQUOIA (17FUN04), which received funding from the European Metrology Programme for Innovation and Research cofinanced by the Participating States and from the European Union’s Horizon 2020 Research and Innovation Programme.

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Authors and Affiliations



L.F. and N.U. designed and performed the experiment. G.B., M.K., V.K. and E.P. developed the analytical model. P.R. and P.G.S. introduced the idea of two-electron propagation in a saddle potential. E.P. and N.U. performed numerical simulations. L.F., V.K., E.P. and N.U. analysed the data. F.H. managed funding and project administration. L.F., T.W. and N.U. designed and fabricated the devices. K.P. supervised the material growth. V.K., E.P. and N.U. wrote the manuscript. All authors contributed to the discussion of results.

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Correspondence to Niels Ubbelohde.

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Nature Nanotechnology thanks Christian Flindt and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Micrograph of the sample.

SEM image showing the layout of the CrAu top gates over the etched mesa channel forming the pumps (red), sidewall depletion (green), quantum dot for charge read out and coupling floating gate (blue, red), and entrance/exit to the detector nodes (yellow). The blue arrows indicate the path of the injected electrons.

Extended Data Fig. 2 Distribution of emission energy and time for the two single-electron sources.

Bivariate normal distributions ρi(ϵi, ti) of emission energy ϵi and time ti for scan electron (i = 1, a) and probe electron (i = 2, b). Contour levels are indicated by white lines at intervals equivalent to the coverage factor. The white dashed line illustrates the correlation of the average emission time 〈t1〉 with the scanned average emission energy 〈ϵ1〉.

Extended Data Fig. 3 Dependence of transmission thresholds and partitioning on average emission energy.

Additional schematic diagrams of electron partitioning equivalent to Fig. 2d, but with the statistical distribution centred at different values of 〈ϵ1〉, corresponding to features A, A\({}^{{\prime} }\), C and D in panels (a)-(d) respectively. The threshold functions \({\epsilon }_{1}^{* }({\epsilon }_{2})\) and \({\epsilon }_{2}^{* }({\epsilon }_{1})\) are the same as only the symmetric slice Δt = 0 of the statistical distribution with 〈Δt〉 = 0 is considered here. In panels (a)–(c) the partitioning domains are labelled by the indices nm of Pnm, whereas in (d) only the net contribution to the first order correlation signal s(1) is indicated, which assumes non-zero values only inside the two hatched areas ( + 1 and − 1, respectively).

Extended Data Fig. 4 Dependence of transmission thresholds and partitioning on interarrival time.

Illustration of the Δt dependence of the transmission thresholds \({\epsilon }_{i}^{* }({\epsilon }_{3-i}^{},{{\Delta }}t)\) and joint probability distribution in the partitioning diagram for 〈ϵ1〉 = 〈Δt〉 = 0 with (a) Δt = − 10 ps and (b) Δt = + 10 ps. (c) Alternative presentation of the partitioning diagram in terms of scaled coordinates corresponding to the phase diagram of two-electron partitioning as introduced in ref. 35. The contour line marks the coverage factor k = 1 of the full joint probability distribution of the scaled coordinates for the most symmetric tuning, 〈ϵ1〉 = 〈Δt〉 = 0. In the scaled coordinates the partitioning domain boundaries are independent of Δt while the joint probability distribution is stretched along the horizontal and vertical axis, as at large Δt electrons in most pairs never approach each other close enough for sufficiently strong interaction.

Extended Data Fig. 5 Additional signature for energy exchange.

The probability P00 that both detectors recorded no electron for the dataset shown in Fig. 3 and Fig. 4.

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Supplementary Information

Supplementary Notes I–VI.

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Ubbelohde, N., Freise, L., Pavlovska, E. et al. Two electrons interacting at a mesoscopic beam splitter. Nat. Nanotechnol. 18, 733–740 (2023).

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