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Attosecond electron pulse trains and quantum state reconstruction in ultrafast transmission electron microscopy


Ultrafast electron and X-ray imaging and spectroscopy are the basis for an ongoing revolution in the understanding of dynamical atomic-scale processes in matter. The underlying technology relies heavily on laser science for the generation and characterization of ever shorter pulses. Recent findings suggest that ultrafast electron microscopy with attosecond-structured wavefunctions may be feasible. However, such future technologies call for means to both prepare and fully analyse the corresponding free-electron quantum states. Here, we introduce a framework for the preparation, coherent manipulation and characterization of free-electron quantum states, experimentally demonstrating attosecond electron pulse trains. Phase-locked optical fields coherently control the electron wavefunction along the beam direction. We establish a new variant of quantum state tomography—‘SQUIRRELS’—for free-electron ensembles. The ability to tailor and quantitatively map electron quantum states will promote the nanoscale study of electron–matter entanglement and new forms of ultrafast electron microscopy down to the attosecond regime.

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The authors acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) (Schwerpunktprogramm (SPP) 1840 ‘Quantum Dynamics in Tailored Intense Fields’, Sonderforschungsbereich (SFB) 1073 ‘Atomic Scale Control of Energy Conversion’, project A05, and SFB 755 ‘Nanoscale Photonic Imaging’, projects C08 and C09), support by the Lower Saxony Ministry of Science and Culture, and funding of the instrumentation by the DFG and VolkswagenStiftung. The authors thank O. Kfir for discussions. T.H. thanks A. Fischer (Göttingen) and K.-C. Toh (Singapore) for discussions on semidefinite programming.

Author information

K.E.P. built the two-colour interferometer set-up, conducted the two-colour experiments with contributions from A.F., analysed the data, and tested the reconstruction algorithm. Ch.R. conducted measurements of the attosecond pulse trains and analysed the data, both with contributions from K.E.P. S.V.Y. and Cl.R. devised the quantum state reconstruction scheme, which was implemented by S.V.Y. The reconstruction algorithm using regularization and semidefinite programming (SDP) was developed by T.H. with contributions from S.V.Y. The manuscript was written by K.E.P. and Cl.R., with contributions from all authors. Cl.R. and S.S. conceived and directed the study. All authors discussed the results and the interpretation.

Competing interests

The authors declare no competing financial interests.

Correspondence to Claus Ropers.

Electronic supplementary material

  1. Supplementary Information

    Supplementary Figures 1–7.

  2. Experiment code

    Code for the SQUIRRELS algorithm used to reconstruct the free-electron density matrix.

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Further reading

Fig. 1: Experimental scheme.
Fig. 2: SQUIRRELS reconstruction of the free-electron quantum state.
Fig. 3: Application of SQUIRRELS to spatially separated optical near-fields.
Fig. 4: Experimental demonstration of attosecond electron pulse trains.