Generating and detecting a prescribed single-electron state is an important step towards solid-state fermion optics. We propose how to generate an electron in a Gaussian state, using a quantum-dot pump with gigahertz operation and realistic parameters. With the help of a strong magnetic field, the electron occupies a coherent state in the pump, insensitive to the details of nonadiabatic evolution. The state changes during the emission from the pump, governed by competition between the Landauer-Buttiker traversal time and the passage time. When the former is much shorter than the latter, the emitted state is a Gaussian wave packet. The Gaussian packet can be identified by using a dynamical potential barrier, with a resolution reaching the Heisenberg minimal uncertainty ℏ/2.

Understanding ultrafast coherent electron dynamics is necessary for application of a single-electron source to metrological standards 1 , quantum information processing 2 , including electron quantum optics 3 , and quantum sensing 4,5 .While the dynamics of an electron emitted from the source has been extensively studied 6-11 , there is as yet no study of the dynamics inside the source. This is because the speed of the internal dynamics is typically higher than 100 GHz, beyond state-of-the-art experimental bandwidth 2 . Here, we theoretically and experimentally demonstrate that the internal dynamics in a silicon singleelectron source comprising a dynamic quantum dot can be detected, utilising a resonant level with which the dynamics is read out as gate-dependent current oscillations. Our experimental observation and simulation with realistic parameters show that an electron wave packet spatially oscillates quantum-coherently at ∼ 200 GHz inside the source. Our results will lead to a protocol for detecting such fast dynamics in a cavity and offer a means of engineering electron wave packets 12 . This could allow high-accuracy current sources [13][14][15][16] , high-resolution and high-speed electromagnetic-field sensing 4 , and high-fidelity initialisation of flying qubits 17,18 .Owing to recent demonstrations of high-accuracy GHz operation 13-16 , a single-electron pump with a tunable-barrier quantum dot (QD) becomes promising for application to ondemand single-electron sources 1 . Because of the fast dynamic movement of the QD, there can occur nontrivial electron dynamics, such as non-adiabatic excitation 19 and subsequent coherent time evolution. While the non-adiabatic excitation could degrade the pumping accuracy, a spatial movement of an electron wave packet due to the coherent time evolution can be used for engineering a wave packet emitted from the QD 12 , which could make possible electron quantum optics experiments and high-speed quantum sensing with high resolution.In addition, understanding of the fast electron dynamics could offer insight into quantum computing with QDs. However, the existing standard measurement technique 20-22 does not have enough bandwidth to detect fast dynamics beyond 100 GHz. In order to overcome the limitation and detect the fast dynamics in the QD, we use a temporal change in a tunnel rate between a resonant level in a tunnel barrier and a QD, which is induced by the dynamic change of the QD potential.First of all, we explain how coherent oscillations of an electron in a single-electron pump 2 occur. A single-electron pump with a tunable-barrier QD consists of the entrance (Fig. 1a, left) and exit (right) potential barriers, formed by applying gate voltages V ent and V exit , respectively 23 . An AC voltage V ac (t) with frequency f in is added to dynamically tune the entrance barrier. The QD energy level is also tuned owing to the cross coupling. When the energy level E qd n (n = 1, 2, · · ·) of the QD with n electrons is lower than the Fermi level E f , electrons can be load...

In recent charge-pump experiments, single electrons are injected into quantum Hall edge channels at energies significantly above the Fermi level. We consider here the relaxation of these hot edge-channel electrons through longitudinal-optical phonon emission. Our results show that the probability for an electron in the outermost edge channel to emit one or more phonons en route to a detector some microns distant along the edge channel suffers a double-exponential suppression with increasing magnetic field. This explains recent experimental observations. We also describe how the shape of the arrival-time distribution of electrons at the detector reflects the velocities of the electronic states post phonon emission. We show how this can give rise to pronounced oscillations in the arrival-time-distribution width as a function of magnetic field or electron energy.

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