Resonant phase-matching between a light wave and a free-electron wavefunction


Quantum light–matter interactions of bound electron systems have been studied extensively. By contrast, quantum interactions of free electrons with light have only become accessible in recent years, following the discovery of photon-induced near-field electron microscopy (PINEM). So far, the fundamental free electron–light interaction in all PINEM experiments has remained weak due to its localized near-field nature, which imposes an energy–momentum mismatch between electrons and light. Here, we demonstrate a strong interaction between free-electron waves and light waves, resulting from precise energy–momentum phase-matching with the extended propagating light field. By exchanging hundreds of photons with the field, each electron simultaneously accelerates and decelerates in a coherent manner. Consequently, each electron’s quantum wavefunction evolves into a quantized energy comb, spanning a bandwidth of over 1,700 eV, requiring us to extend the PINEM theory. Our observation of coherent electron phase-matching with a propagating wave is a type of inverse-Cherenkov interaction that occurs with a quantum electron wavefunction, demonstrating how the extended nature of the electron wavefunction can alter stimulated electron–light interactions.

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Fig. 1: Quantum versus classical phase-matching of an electron and light.
Fig. 2: Experimental set-up.
Fig. 3: Comparison of the experimental data with both conventional PINEM theory and our extended theory.
Fig. 4: Conditions for phase-matching and theoretical analysis.
Fig. 5: Experimental results showing the coherent electron energy comb forming a plateau or energy peaks.

Data availability

All data that support the plots and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.


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We thank the IDES Company and especially S.T. Park for illuminating discussions and advice. We are also grateful to I. Goykhman for fruitful discussions. The experiments were performed on the UTEM of the I. K. AdQuanta group installed in the Electron Microscopy Center (MIKA) in the Department of Material Science and Engineering at the Technion. The research was supported by ERC starting grant NanoEP 851780 and the Israel Science Foundation grant 831/19. K.W. is partially supported by a fellowship from the Lady Davis Foundation. I.K. acknowledges the support of the Azrieli Faculty Fellowship.

Author information




R.D. achieved the grazing-angle condition in the transmission electron microscope and led the experimental work including sample preparation. S.N. and R.D. worked on the design before the experiment began. R.D., K.W., M.S., O.B., Y.A. and S.N. carried out the experiments. K.W., S.N., Y.A., R.D., O.R. and Y.K. developed the theory and analysed the results. M.H.L. and X.S. contributed to the discussion of the results. I.K. conceived the research. All authors provided substantial input to all aspects of the project and to the writing of the manuscript.

Corresponding author

Correspondence to Ido Kaminer.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Albert Polman and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Discussion and Supplementary Figs. 1–6.

Supplementary Video 1

Finite-difference time-domain simulation of the setup. The simulation assumes a right-angle glass prism with refractive index 1.513 at 700 nm, 15 μm leg size, and base angle of 45°. The laser undergoes total internal reflection inside the prism and generates an evanescent field that interacts with the electron that passes nearby.

Source data

Source Data Fig. 1

Data used to plot Fig. 1b–d. Experimental data points and theoretical data from the classical and quantum mechanical models.

Source Data Fig. 3

Data used to plot Fig. 3. Experimental data points and data from conventional as well as extended PINEM theory.

Source Data Fig. 4

Data used to plot Fig. 4b–e. Experimental and theoretical data points.

Source Data Fig 5

Data used to plot Fig. 5a,b. Experimental data points and theoretical data from the classical and quantum mechanical models.

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Dahan, R., Nehemia, S., Shentcis, M. et al. Resonant phase-matching between a light wave and a free-electron wavefunction. Nat. Phys. 16, 1123–1131 (2020).

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