Probing quantum-coherent dynamics with free electrons

In the microscopic world of quantum materials, matter does not simply sit still; it exists in a state of constant, delicate potential. Atoms and defects within solids can hold energy in specific ways, and when they do, they can exist in a superposition, a state where they are effectively in two different energy configurations at the same time. This quantum coherence is the engine behind many emerging technologies, from ultra-fast computers to secure communication networks. However, watching these states change is notoriously difficult. They evolve on timescales so short that they vanish in a fraction of a trillionth of a second, and they are easily disturbed by the very tools used to observe them. For decades, scientists have relied on beams of electrons to image materials with incredible precision, using these charged particles to map out the structure of matter. Yet, while these electron beams are excellent at revealing static structures or simple energy losses, they have struggled to capture the fleeting, coherent dance of quantum states as they evolve in real time.

A new theoretical study by researchers at Christian Albrechts University in Kiel, Germany, proposes a way to overcome this hurdle by treating the free electron not just as a probe, but as a dynamic partner in the quantum interaction. The researchers developed a detailed mathematical model to simulate what happens when a single, fast-moving electron passes near a quantum emitter, such as a defect in a crystal, that has been prepared in a coherent superposition. Their work suggests that the passing electron does more than just knock energy out of the system; it can actually induce a temporary, rhythmic oscillation in the emitter's state. This interaction leaves a distinct fingerprint in the energy of the electron itself. By analyzing the spectrum of energy lost or gained by the electron as it flies past, the model predicts that scientists could see clear, rhythmic patterns that reveal the quantum coherence of the emitter, effectively allowing the electron to "listen" to the quantum state's heartbeat.

The core of this discovery lies in how the electron interacts with the quantum system over time. In the researchers' simulations, they modeled a quantum emitter as a simple two-level system, capable of being in a ground state or an excited state, and prepared it in a balanced mix of both. When a free electron, traveling at a significant fraction of the speed of light, passes this emitter at a specific distance, the electric field of the electron couples with the emitter's transition dipole moment. This coupling is not a simple, one-time exchange. Instead, if the electron's wave packet—the spread of its position in space—is long enough to overlap with the natural oscillation period of the emitter, it triggers a transient coherent oscillation. The population of the emitter, meaning the probability of finding it in the excited state versus the ground state, begins to swing back and forth. These swings are not random; they are directly tied to the initial phase of the superposition, a hidden variable that determines the starting point of the quantum rhythm.

Crucially, the study finds that this quantum rhythm is imprinted onto the electron's own energy spectrum. As the electron interacts with the emitter, it can lose or gain energy, creating peaks in the spectrum known as electron energy-loss spectroscopy. The researchers found that the zero-loss peak—the part of the spectrum where the electron has not changed its energy—does not remain a static line. Instead, it exhibits coherent oscillations that mirror the dynamics of the emitter. These oscillations are sensitive to the transition frequency of the emitter and the relative phase of its initial quantum state. For instance, changing the phase of the initial superposition acts like a control knob, shifting the timing and amplitude of the oscillations observed in the electron's energy spectrum. This sensitivity suggests that by carefully measuring the electron's energy after the interaction, one could deduce the quantum coherence properties of the emitter without needing to shape the electron beam into complex forms beforehand.

The researchers also explored the limits of this effect, noting that it is most prominent when the duration of the electron's wave packet is comparable to or longer than the oscillation period of the quantum transition. If the electron passes too quickly, the interaction is too brief to establish these coherent oscillations, and the effect is suppressed. However, for electrons with energies in the range of a few kiloelectronvolts, and for emitters with transition frequencies in the optical or near-infrared range, the effect remains robust. The simulations indicate that these coherent features survive for several femtoseconds, a timescale that is short but measurable with modern ultrafast techniques. The study further connects these theoretical predictions to a potential experimental setup involving electron-driven photon sources, where a laser-like pulse generated by an electron beam could prepare the emitter, and a subsequent electron probe could measure the resulting dynamics.

This work does not claim to have already observed these effects in a laboratory, but rather provides a rigorous theoretical framework that predicts exactly what to look for. The authors emphasize that their model relies on weak interactions and specific conditions, such as the electron passing at a distance of a few nanometers from the emitter. They suggest that materials with strong dipole moments, like certain defects in two-dimensional materials or quantum dots, would be ideal candidates for future experiments. By identifying these specific signatures in the electron energy spectrum, the study opens a path toward characterizing the quantum-coherent dynamics of individual emitters with both high spatial and temporal resolution. If realized, this approach would allow scientists to track the evolution of quantum states in their natural environment, offering a new window into the ultrafast world where quantum coherence dictates the behavior of matter.