Method for rapid estimation of the energy-time covariance matrix of single electrons

The ability to emit and control single electrons in a dynamical manner enables their use in electron quantum optics and sensing. To characterize the electron states emitted with energy far above the Fermi energy, a dynamic barrier has been used. In this work, we extract the energy-time covariance matrix of single electrons by analyzing the energy variance obtained from the transconductance through the dynamic barrier. This method enables efficient and precise characterization of electron states, especially when a sinusoidal waveform is used. An effective phase-space area and the elliptical distribution are constructed from the covariance matrix and qualitatively compared with the distribution reconstructed from a tomographic method. The area constrained by the uncertainty relation serves as a measure of proximity to the quantum limit. Our results demonstrate an efficient method for characterizing electron states, paving the way for their application in quantum technologies.