Probing quantum-coherent dynamics with free electrons

TL;DR

The paper addresses probing quantum-coherent dynamics of single emitters using free-electron probes in time-resolved cathodoluminescence and EELS. It develops a fully quantum, time-dependent theory of the coherent interaction between a moving free electron and a two-level emitter in an arbitrary initial state, yielding a complex, time-dependent coupling $g(z,t)$ via the Magnus expansion. Key findings show that a free electron can drive transient population oscillations when the emitter is in a coherent superposition and that the EELS spectrum encodes coherence through a time-dependent zero-loss peak oscillating at the emitter frequency $\omega_0$ and modulated by the relative phase $\phi_r$. These results provide a route to characterize quantum-coherent dynamics with sub-femtosecond temporal and nanometer spatial resolution, guiding future experiments and extensions to more complex multilevel or entangled systems.

Abstract

Recent advances in time-resolved cathodoluminescence have enabled ultrafast studies of single emitters in quantum materials with femtosecond temporal resolution. Here, we develop a quantum theory modeling the dynamics of free electrons interacting with quantum emitters in arbitrary initial states. Our analysis reveals that a free electron can induce transient coherent oscillations in the populations when the system is initially prepared in a coherent superposition of its states. Moreover, the electron energy spectrum exhibits a clear signature of the quantum coherence and sensitivity to the transition frequency of the emitter. These coherence effects manifest themselves as oscillations in the zero-loss peak of the spectral energy-loss probability. Our findings pave the way for characterization of quantum-coherent dynamics of individual quantum emitters by electron-probes.

Figures (3)

• Figure 1: (a) Schematic of the interaction between a two-level emitter and an electron wave packet. (b)–(e) Population (b),(c) and coherence (d),(e) dynamics for a $0.5$ keV electron with impact parameter $r_{\perp}=2$ nm. The emitter has $\boldsymbol{d}_x=\boldsymbol{d}_z=30$ Debye and transition wavelengths of 560 nm in (b),(d) and 1000 nm in (c),(e); the initial state is $(\left|g\right>+\left|e\right>)/\sqrt{2}$. (f) Effect of increasing electron energy on the population dynamics for a 560 nm transition. (g) Effect of initial coherence on the population dynamics for a 1000 nm transition.
• Figure 2: (a),(d) Time-evolution of the EELS probability $dP/dE$ for an electron with kinetic energy $\simeq 115$ keV ($v_{0}=0.578c$) for $t\ge0$ and impact parameter $r_{\perp}=2$nm. The two-level emitter parameters in (a) and (d) are the same as in Fig. 1(b) and Fig. 1(c), respectively. (b),(e) Corresponding EELS probability differences $\Gamma_{net}(E)$ and its absolute value $|\Gamma_{net}(E)|$, showing the coherent oscillations in the zero-loss peak.
• Figure 3: (a)-(d) Dependence of the coherent oscillations in the zero-loss peak on the initial quantum coherence $(\left|g\right>+e^{-i \phi_{r}}\left|e\right>)/\sqrt{2}$. The relative phase effects shown for two impact parameters (a),(c) $r_{\perp}=2$nm and (b),(d) $r_{\perp}=1$nm. The transition wavelengths used in (a),(b) and (c),(d) are the same as in Fig. 2(b) and Fig. 2(e), respectively.