Free-Electron Quantum Optics 

Free electrons, for example the electrons in the beam of an electron microscope, can be efficiently manipulated by electromagnetic fields and interact with other quantum systems via their electromagnetic near field. These unique properties can be employed for quantum optics.

A particular example utilizing the first property is the Free-Electron Laser (FEL), in which electrons are accelerated to ultrarelativistic kinetic energies and travel through arrays of magnetic fields with alternating orientations, such that coherent braking radiation is emitted which can be used, for example, to analyse the structure of material samples. While the mechanism of all existing FELs can be described by considering electrons as classical point particles, ideas have been developed how quantum properties of electrons can be used to achieve better beam properties of the generated radiation [1,2].

The direct interaction of the electrons' electromagnetic near field with material samples is employed, for example, in Electron Energy Loss Spectroscopy (EELS) [3], where the kinetic energy of the passing electrons is transferred to the sample and the distribution of energy loss measured after the interaction allows conclusions about the structure and composition of the sample. There are several proposals to exploit the interaction of free electrons with material samples in the quantum domain, in particular by modulating the electron beam current.

It has also been shown theoretically that quantum transitions in the optical spectrum can be coherently controlled when electron beams are modulated in the optical frequency range [5-8]. This appraoch could be used, for example, to measure coherence times of quantum systems on the nano-scale with a spatial resolution far exceeding that of state-of-the-art techniques based on electromagnetic radiation [8]. The necessary electron beam modulations have already been achieved in various setups, where they correspond to modulations of the beam electrons’ wave functions (e.g. [9]–[11]).

However, the direct control of quantum systems is just one particular application of electron beam modulation in the optical domain. Several others have been proposed, such as optical coherence transfer [12], creation of special photon states [13-15] and modifying quantum electrodynamical scattering amplitudes [16].

[1] Bonifacio, R., N. Piovella, and G. R. M. Robb. "Quantum theory of SASE FEL." Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 543.2-3 (2005): 645-652 doi.org/10.1016/j.nima.2005.01.324 

[2] Kling, Peter, et al. "What defines the quantum regime of the free-electron laser?." New Journal of Physics 17.12 (2015): 123019.  doi.org/10.1088/1367-2630/17/12/123019 

[3] Egerton, Ray F. "Electron energy-loss spectroscopy in the TEM." Reports on Progress in Physics 72.1 (2008): 016502. 

[4] D. Rätzel, D. Hartley, O. Schwartz, P. Haslinger, Phys. Rev. Research 3, 023247 (2021), doi.org/10.1103/PhysRevResearch.3.023247;  Preprint arxiv.org/abs/2004.10168

[5] Garcia de Abajo, F. Javier, and Valerio Di Giulio. "Optical excitations with electron beams: Challenges and opportunities." ACS photonics 8.4 (2021): 945-974 doi.org/10.1021/acsphotonics.0c01950 

[6] A. Gover and A. Yariv, “Free-Electron--Bound-Electron Resonant Interaction,” Phys. Rev. Lett., vol. 124, no. 6, p. 064801, Feb. 2020, doi.org/10.1103/PhysRevLett.124.064801 

[7] Z. Zhao, X.-Q. Sun, and S. Fan, “Quantum entanglement and modulation enhancement of free-electron-bound-electron interaction,” Physical Review Letters 126.23 (2021): 233402, doi.org/10.1103/PhysRevLett.126.233402; Preprint: arxiv.org/abs/2010.11396 

[8] R. Ruimy, A. Gorlach, C. Mechel, N. Rivera, and I. Kaminer, “Towards atomic-resolution quantum measurements with coherently-shaped free electrons,”  doi.org/10.1103/PhysRevLett.126.233403; Preprint:  arxiv.org/abs/2011.00348.

[9] A. Feist, K. E. Echternkamp, J. Schauss, V. S. Yalunin, S. Schäfer, and C. Ropers, “Quantum coherent optical phase modulation in an ultrafast transmission electron microscope,” Nature, vol. 521, no. 7551, pp. 200–203, 2015, doi.org/10.1038/nature14463 

[10] C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science, vol. 352, no. 6284, pp. 429–433, 2016. doi.org/10.1126/science.aae0003 

[11] N. Schönenberger, A. Mittelbach, P. Yousefi, J. McNeur, U. Niedermayer, and P. Hommelhoff, “Generation and Characterization of Attosecond Microbunched Electron Pulse Trains via Dielectric Laser Acceleration,” Phys. Rev. Lett., vol. 123, no. 26, p. 264803, 2019. doi.org/10.1103/PhysRevLett.123.264803; Preprint: arxiv.org/abs/1910.14343   

[12] O. Kfir, V. Di Giulio, F. J. G. de Abajo, and C. Ropers, “Optical coherence transfer mediated by free electrons,” doi.org/10.1126/sciadv.abf6380; Preprint: arxiv.org/abs/2010.14948 

[13] A. B. Hayun, O. Reinhardt, J. Nemirovsky, A. Karnieli, N. Rivera, and I. Kaminer, “Shaping Quantum Photonic States Using Free Electrons,” Preprint: arxiv.org/abs/2011.01315.

[14] V. Di Giulio, O. Kfir, C. Ropers, and F. J. G. de Abajo, “Modulation of Cathodoluminescence Emission by Interference with External Light,” ACS Nano, p. acsnano.1c00549, Mar. 2021, doi.org/10.1126/sciadv.abe4270; https://doi.org/10.1021/acsnano.1c00549 

[15] F. J. García de Abajo and V. Di Giulio, “Optical Excitations with Electron Beams: Challenges and Opportunities,” ACS Photonics, vol. 8, no. 4, pp. 945–974, Apr. 2021, doi.org/10.1021/acsphotonics.0c01950 

[16] L. J. Wong et al., “Control of quantum electrodynamical processes by shaping electron wavepackets,” Nat. Commun., vol. 12, no. 1, Art. no. 1, Mar. 2021, doi.org/10.1038/s41467-021-21367-1