Observing the Quantum Wave Nature of Free Electrons through Spontaneous Emission

We investigate, both experimentally and theoretically, the interpretation of the free-electron wave function using spontaneous emission. We use a transversely wide single-electron wave function to describe the spatial extent of transverse coherence of an electron beam in a standard transmission electron microscope. When the electron beam passes next to a metallic grating, spontaneous Smith-Purcell radiation is emitted. We then examine the effect of the electron wave function transversal size on the emitted radiation. Two interpretations widely used in the literature are considered: (1) radiation by a continuous current density attributed to the quantum probability current, equivalent to the spreading of the electron charge continuously over space; and (2) interpreting the square modulus of the wave function as a probability distribution of finding a point particle at a certain location, wherein the electron charge is always localized in space. We discuss how these two interpretations give contradictory predictions for the radiation pattern in our experiment, comparing the emission from narrow and wide wave functions with respect to the emitted radiation’s wavelength. Matching our experiment with a new quantum-electrodynamics derivation, we conclude that the measurements can be explained by the probability distribution approach wherein the electron interacts with the grating as a classical point charge. Our findings clarify the transition between the classical and quantum regimes and shed light on the mechanisms that take part in general light-matter interactions.

Figure 1

Schematic description of the research question. (a) A classical point charge electron produces highly diverged radiation. (b) The charge-density semi classical approach predicts collimated radiation. (c) The probability semiclassical approach predicts high divergence as in the classical treatment.

Figure 2

Experimental setup. (a) Side view of the viewing chamber of the transmission electron microscope. (b) and (c) top view of the setup for the case of a narrow (b) and wide (c) electron beam. The radiated power was measured in both cases for different values of $\varphi$.

Figure 3

Experimental results. (a) and (b) images of the grating plane as observed for the case of narrow and wide electron beams, respectively, for different values of the azimuthal angle $\varphi$ (marked). (c) Normalized power of SPR as a function of the azimuthal angle $\varphi$ for narrow and wide electron beams, showing no significant difference between the two. The split in the measured stripe in (a) at wide angles is a result of double reflection from the viewing chamber window.

Figure 4

Illustration of spontaneous Smith-Purcell emission from a wide electron. Left, momentum space description for $\varphi =0$. Right, real space. The blue cloud above the grating represents the electron wave function that interacts with the periodic evanescent electromagnetic field (red), thanks to the additional momentum provided by the grating (green).