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Quantum tomography of an electron

Nature volume 514, pages 603–607 (2014)Cite this article

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Abstract

The complete knowledge of a quantum state allows the prediction of the probability of all possible measurement outcomes, a crucial step in quantum mechanics. It can be provided by tomographic methods1 which have been applied to atomic2,3, molecular4, spin5,6 and photonic states. For optical7,8,9 or microwave10,11,12,13 photons, standard tomography is obtained by mixing the unknown state with a large-amplitude coherent photon field. However, for fermions such as electrons in condensed matter, this approach is not applicable because fermionic fields are limited to small amplitudes (at most one particle per state), and so far no determination of an electron wavefunction has been made. Recent proposals involving quantum conductors suggest that the wavefunction can be obtained by measuring the time-dependent current of electronic wave interferometers14 or the current noise of electronic Hanbury-Brown/Twiss interferometers15,16,17. Here we show that such measurements are possible despite the extreme noise sensitivity required, and present the reconstructed wavefunction quasi-probability, or Wigner distribution function17, of single electrons injected into a ballistic conductor. Many identical electrons are prepared in well-controlled quantum states called levitons18 by repeatedly applying Lorentzian voltage pulses to a contact on the conductor19,20,21. After passing through an electron beam splitter, the levitons are mixed with a weak-amplitude fermionic field formed by a coherent superposition of electron–hole pairs generated by a small alternating current with a frequency that is a multiple of the voltage pulse frequency16. Antibunching of the electrons and holes with the levitons at the beam splitter changes the leviton partition statistics, and the noise variations provide the energy density matrix elements of the levitons. This demonstration of quantum tomography makes the developing field of electron quantum optics with ballistic conductors a new test-bed for quantum information with fermions20,22,23,24. These results may find direct application in probing the entanglement of electron flying quantum bits25, electron decoherence17 and electron interactions. They could also be applied to cold fermionic (or spin-1/2) atoms26.

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Figure 1: Schematics of quantum wave tomography.
Figure 2: Measurement of the diagonal part of the energy density matrix.
Figure 3: Off-diagonal part of energy density matrix.
Figure 4: Wigner function and leviton wavefunction in the time domain.

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Acknowledgements

We acknowledge the ERC Advanced Grant 228273 MeQuaNo and thank P. Jacques for technical help, P. Pari, P. Forget and M. de Combarieu for cryogenic support, and P. Degiovanni and C. Grenier for discussions improving the manuscript.

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Author notes

  1. T. Jullien and P. Roulleau: These authors contributed equally to this work.

Authors and Affiliations

  1. Service de Physique de l’Etat Condensé, IRAMIS/DSM (CNRS URA 2464), CEA Saclay, F-91191 Gif-sur-Yvette, France,

    T. Jullien, P. Roulleau, B. Roche & D. C. Glattli

  2. CNRS, Laboratoire de Photonique et de Nanostructures, Route de Nozay, 91460 Marcoussis, France,

    A. Cavanna & Y. Jin

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  1. T. Jullien
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  6. D. C. Glattli
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Contributions

D.C.G. designed the project. T.J. and P.R. made the measurements and did the data analysis. B.R. contributed to the data analysis. P.R., T.J., B.R. and D.C.G. wrote the article. The sample was provided by Y.J. on wafer from A.C.

Corresponding author

Correspondence to D. C. Glattli.

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The authors declare no competing financial interests.

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This file contains Supplementary Methods, Supplementary Text and Data, Supplementary Figures 1-3 and additional references. (PDF 540 kb)

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Jullien, T., Roulleau, P., Roche, B. et al. Quantum tomography of an electron. Nature 514, 603–607 (2014). https://doi.org/10.1038/nature13821

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