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2021 IPS Conference
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- IPS Conference 2021

**Abstract:** We present the first observation of the quantum nature of the inverse-Cherenkov effect, by phase-matching light & electron waves. Each electron simultaneously absorbs and emits hundreds of photon quanta by interacting with the light coherently along hundreds of microns. We also propose an extension to the current theory in the field to capture the complete spatiotemporal nature of the strong interaction.

**Text: **The Cherenkov effect (also called Vavilov-Cherenkov effect) has attracted vast interest since its discovery in 1934 [1] and the Nobel Prize of 1958, yet to this day, all experiments on the subject have been perfectly accounted for by classical electrodynamics. Similarly, all demonstrations of analogous effects in a wide range of fields – such as water waves, acoustics, and even phononics [2] – are also explained entirely classically. Likewise, all experiments using the **inverse** Cherenkov effect for electron acceleration [3] and for other electron–laser interactions are described classically [4]. Theoretical work predicts new Cherenkov phenomena coming from quantum electrodynamics that provide new concepts and designs of highly controllable light sources, more efficient accelerators and detectors [4,5].

Here we demonsrate the quantum nature of the inverse Cherenkov effect and observe how the quantum wavefunction of each electron evolves into a coherent plateau, analogous to a frequency comb in ultrashort laser pulses, containing hundreds of quantized energy peaks. By precisely matching the group velocity of the relativistic electron wavefunction and the phase velocity of light, we achieve the Cherenkov phase matching condition so that each point in the electron’s wavefunction experiences a constant electric field.

The experimental setup that we used to demonstrate the coherent resonant interaction is an ultrafast transmission electron microscope (UTEM) [6], where a laser pulse is splits to a pump and a probe. The probe generates electrons from the UTEM’s cathode, while the pump is focused on a BK7 prism wall to interact with the electrons at the Cherenkov angle. We tune the electron velocity to match the laser-pump angle by changing the accelerating voltage to 207.2 kV. The key to the effects is achieving a grazing-angle interaction with the sample surface, where the electrons remain atremain at a distance of a few hundred nanometers from the surface for hundreds of microns. The achievement of such a grazing angle condition is, to the best of our knowledge, realized here for the first time in any transmission electron microscope. This condition maximizes the strength of the electron–laser interaction.

The interaction can be explained by a single dimensionless constant g [6,7], which in our case can be simplified to:

g(x,y,T)=q_e/ℏω ∫_(-∞)^∞ E ̃_z (x,y,z^',T+z^'/v_e )* e^(-iωz^'/v_e ) dz^', (1)

where (x,y,T) describe the electron trajectory (T=t-z\v_e), q_e is the electron charge, ω is the laser frequency, v_e is the electron velocity and E ̃_z is the z component of the electric field phasor. The phase-matching occurs when the z component of the electric field’s wave vector (k_z) equals to ω/v_e and the new temporal dependence captures the dynamics of such extended phase-matched interaction. In this situation, each point in the electron’s wavefunction sees a constant field and g grows linearly with the interaction length → resulting in coherent constructive interference.

The phase-matched interaction in the UTEM opens the door for utilizing quantum electrodynamics for new applications in electron microscopy and in free-electron pump-probe spectroscopy [8]. The free-electron comb we created has been previously observed only in pulsed photoexcitation of *bound electron systems*, as in above-threshold ionization. Free electrons have markedly different physical phenomena and applications as compared to bound electrons and can induce new relativistic effects, such as the Cherenkov effect that we observed in this work. The energy comb spectra of free electrons are expected to generate new kinds of radiation composed of high harmonic orders, with intriguing prospects in attosecond science.

* Figures that clarify our findings can be send upon demand.

** Our work was published recently on Nature Physics (November 2020). "Resonant phase-matching between a light wave and a free-electron wavefunction ": https://www.nature.com/articles/s41567-020-01042-w

**References**

[1] Cherenkov, P. A. Visible emission of clean liquids by action of γ radiation. **Dokl. Akad. Nauk SSSR** 2, 451-454 (1934).

[2] Anderson, T. I. et al. Electron-phonon instability in graphene revealed by global and local noise probes. **Science** 364, 154-157 (2019).

[3] Kimura, W. D. et al. Laser acceleration of relativistic electrons using the inverse Cherenkov effect. **PRL** 74, 546 (1995); Kozák, M. et al. Acceleration of sub-relativistic electrons with an evanescent optical wave at a planar interface. **Opt. Express** 25, 19195-19204 (2017).

[4] Gover, A. et al. Superradiant and stimulated-superradiant emission of bunched electron beams. **Rev. Mod. Phys.** 91, 035003 (2019).

[5] Rivera, N., et al., Light emission based on nanophotonic vacuum forces. accepted **Nature Phys.** (2019).

[6] Zewail, A. H. Four-dimensional electron microscopy. **Science** 328, 187-193 (2010); Feist, A. et al. Ultrafast transmission electron microscopy using a laser-driven field emitter: Femtosecond resolution with a high coherence electron beam. **Ultramicroscopy** 176, 63-73 (2017).

[7] de Abajo, F. J. G. et al. Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields. **Nano Lett.** 10, 1859 (2010); Park, S. T. et al. Photon-induced near-field electron microscopy (PINEM): theoretical and experimental. **NJP** 12, 123028 (2010).

[8] Polman, A., Kociak, M. & de Abajo, F. J. G. Electron-beam spectroscopy for nanophotonics. **Nature Materials** 18, 1158-1171 (2019).

The experimental setup (Fig. 2a) that we used to demonstrate the coherent resonant interaction is an ultrafast transmission electron microscope (UTEM) [6], where a laser pulse is splits to a pump and a probe. The probe generates electrons from the UTEM’s cathode, while the pump is focused on a BK7 prism wall to interact with the electrons at the Cherenkov angle (see Fig. 2b). We tune the electron velocity to match laser-pump angle by changing the accelerating voltage to 207.2 keV. The key to the effects is achieving a grazing-angle interaction with the sample surface, where the electrons remain at a distance of a few hundred nanometers from the surface for hundreds of microns. The achievement of such a grazing angle condition is, to the best of our knowledge, realized here for the first time in any transmission electron microscope. This condition maximizes the strength of the electron–laser interaction.

The interaction can be explained by a single dimensionless constant g [6,7], which in our case can be simplified to:

(1)

where

describe the electron trajectory (

,

is the electron charge,

is the laser frequency,

is the electron velocity and

is the

component of the electric field phasor. The phase-matching occurs when the z component of the electric field’s wave vector (

) equals to

and the new temporal dependence captures the dynamics of such extended phase-matched interaction. In this situation, each point in the electron’s wavefunction sees a constant field and

grows linearly with the interaction length

resulting in *coherent constructive interference*.

The experimental setup (Fig. 2a) that we used to demonstrate the coherent resonant interaction is an ultrafast transmission electron microscope (UTEM) [6], where a laser pulse is splits to a pump and a probe. The probe generates electrons from the UTEM’s cathode, while the pump is focused on a BK7 prism wall to interact with the electrons at the Cherenkov angle (see Fig. 2b). We tune the electron velocity to match laser-pump angle by changing the accelerating voltage to 207.2 keV. The key to the effects is achieving a grazing-angle interaction with the sample surface, where the electrons remain at a distance of a few hundred nanometers from the surface for hundreds of microns. The achievement of such a grazing angle condition is, to the best of our knowledge, realized here for the first time in any transmission electron microscope. This condition maximizes the strength of the electron–laser interaction.

The interaction can be explained by a single dimensionless constant g [6,7], which in our case can be simplified to:

(1)

where

describe the electron trajectory (

,

is the electron charge,

is the laser frequency,

is the electron velocity and

is the

component of the electric field phasor. The phase-matching occurs when the z component of the electric field’s wave vector (

) equals to

and the new temporal dependence captures the dynamics of such extended phase-matched interaction. In this situation, each point in the electron’s wavefunction sees a constant field and

grows linearly with the interaction length

resulting in *coherent constructive interference*.

IPS Conference 2021

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