Continuous-variable tomography of solitary electrons

J. D. Fletcher,P. W. Brouwer,D. A. Ritchie,I. Farrer,M. Kataoka,E. Locane,V. Kashcheyevs,N. Johnson,J. P. Griffiths,P. See

doi:10.1038/s41467-019-13222-1

Abstract

A method for characterising the wave-function of freely-propagating particles would provide a useful tool for developing quantum-information technologies with single electronic excitations. Previous continuous-variable quantum tomography techniques developed to analyse electronic excitations in the energy-time domain have been limited to energies close to the Fermi level. We show that a wide-band tomography of single-particle distributions is possible using energy-time filtering and that the Wigner representation of the mixed-state density matrix can be reconstructed for solitary electrons emitted by an on-demand single-electron source. These are highly localised distributions, isolated from the Fermi sea. While we cannot resolve the pure state Wigner function of our excitations due to classical fluctuations, we can partially resolve the chirp and squeezing of the Wigner function imposed by emission conditions and quantify the quantumness of the source. This tomography scheme, when implemented with sufficient experimental resolution, will enable quantum-limited measurements, providing information on electron coherence and entanglement at the individual particle level.

Highlights

A method for characterising the wave-function of freely-propagating particles would provide a useful tool for developing quantum-information technologies with single electronic excitations
This is only possible in a restricted phase-space volume close to the Fermi energy. It is not possible where there is no Fermi sea, as in the case of isolated electrons travelling in a depleted lattice space without conduction-band electrons nearby[15,26]. In these cases it is possible to interrogate the beam with a barrier in the beam path[13,15,27], an approach which can enable a different method of tomographic reconstruction of the Wigner function[28]
Periodic electron sources can act as a sensitive probe of on-chip signals[39], with an energy–time resolution set by the electronic phase space distribution

Summary

Introduction

A method for characterising the wave-function of freely-propagating particles would provide a useful tool for developing quantum-information technologies with single electronic excitations. We show that a wide-band tomography of single-particle distributions is possible using energy-time filtering and that the Wigner representation of the mixed-state density matrix can be reconstructed for solitary electrons emitted by an on-demand single-electron source These are highly localised distributions, isolated from the Fermi sea. The Wigner function WðE; tÞ is not directly measurable, but projections along specific trajectories in the phase space of non-commuting variables (position–momentum, energy–time) can be accessed, enabling a tomographic reconstruction[19] somewhat like X-ray tomography Such measurements require a scheme to create and readout projections at different trajectories or mixing angles, for instance via free space evolution of the transverse wavefunction of atomic beams[20,21] or by mixing of photons with a local optical field[4,22]. In these cases it is possible to interrogate the beam with a barrier in the beam path[13,15,27], an approach which can enable a different method of tomographic reconstruction of the Wigner function[28]

Methods

Results

Conclusion