Quantum Entanglement and Modulation Enhancement of Free-Electron–Bound-Electron Interaction

The modulation and engineering of the free-electron wave function bring new ingredients to the electron-matter interaction. We consider the dynamics of a free-electron passing by a two-level system fully quantum mechanically and study the enhancement of interaction from the modulation of the free-electron wave function. In the presence of resonant modulation of the free-electron wave function, we show that the electron energy loss and gain spectrum is greatly enhanced for a coherent initial state of the two-level system. Thus, a modulated electron can function as a probe of the atomic coherence. We further find that distantly separated two-level atoms can be entangled through interacting with the same free electron. Effects of modulation-induced enhancement can also be observed using a dilute beam of modulated electrons.

Figure 1

Schematic of the interaction between an electron (blue wave packet) and a two-level system (orange circle). The inset shows a schematic of a train of electrons interacting with the two-level atom.

Figure 2

Left: Schematic of the electron ladder for an electron modulated at frequency $\omega$ and the energy levels of the two-level system, where ${\omega }_a=l\omega$ and $l=3$. Right: Maximal $|s|$ and the corresponding drift length for different harmonic orders.

Figure 3

Spectrum change of the free electron interacting with a two-level system in state $|{\mathrm{\Psi }}_a⟩=(|0⟩+|1⟩)/\sqrt{2}$, where the free electron is either a 60 keV Gaussian wave packet with ${\sigma }_p\ll {\omega }_a/v_0$ (a)–(c) or a resonant modulated wave packet with optimal modulation strength $g_m=0.68$, modulation frequency ${\omega }_a$ (620 nm), and drift length 10 mm (d)–(f). (a),(d) The electron energy spectrum before the interaction. (b),(e), and (c),(f) are the spectrum changes after the interaction in the first and second order of $|g|$, respectively, where a typical $g=1\times {10}^{-3}$ is assumed.

Figure 4

Schematic of an electron interacting with two atoms.