Quantum Wave-Particle Duality in Free-Electron–Bound-Electron Interaction

We present a comprehensive relativistic quantum-mechanical theory for interaction of a free electron with a bound electron in a model, where the free electron is represented as a finite-size quantum electron wave packet (QEW) and the bound electron is modeled by a quantum two-level system (TLS). The analysis reveals the wave-particle duality nature of the QEW, delineating the point-particle-like and wavelike interaction regimes and manifesting the physical reality of the wave function dimensions when interacting with matter. This QEW size dependence may be used for interrogation and coherent control of superposition states in a TLS and for enhancement of cathodoluminescence and electron energy-loss spectroscopy in electron microscopy.

Figure 1

Two-level system (TLS) quantum interaction model of quantum electron wave packet interaction with a bound electron.

Figure 2

Temporal evolution of excitation probability $P_2$ and the free-electron energy decrement $\mathrm{\Delta }{E}_F$ for different ${\sigma }_{et}$. (a) TLS initially in the ground state. (b) TLS initially in an equatorial superposition state.

Figure 3

(a) Numerically computed $P_2$ as a function of ${\sigma }_{et}$ and $\zeta$ with the TLS initially in a superposition state. (b) Numerical (solid lines) and analytical (dashed lines) transition probability $P_2$ as a function of $\zeta$ for ${\sigma }_{et}/T_{2,1}=0.1$ and 0.3. (c) Numerical (solid lines) and analytical (dashed lines) $P_2$ as a function of ${\sigma }_{et}$ for different relative incidence phases.

Figure 4

EELS spectrum of long size QEWs (${\sigma }_{et}/T_{2,1}=1$) exhibiting a negative energy recoil ($-E_{2,1}$) shift single sideband for the initial TLS ground state $|1⟩$, symmetric $\pm E_{2,1}$ sidebands for the equatorial TLS states $(|1⟩+e^{i\phi }|2⟩)/\sqrt{2}$, and asymmetric net positive energy gain sidebands for initial states on the 60° latitude angle of the Bloch sphere $(|1⟩+\sqrt{3}{e}^{i\phi }|2⟩)/2$.