Continuous-variable tomography of solitary electrons

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.

Time-energy filtering of single electrons in ballistic waveguides

“…We also show that if the gate voltage V(t) has a linear dependence on time, no modification of the Wigner function is necessary and equation (1) is valid if we take p(E, t) to be the Wigner function of the incoming electron ρ in (E, t). This confirms that the protocol implemented in [40] can be used not only for classical but also for quantum tomography, i.e. quantum state reconstruction.…”

“…In this section we discuss the relevance of our results to the recently proposed and implemented tomography protocol for solitary electrons [40]. [40] consists of measuring (51) for a sufficiently wide range of V 0 and θ to enable numerical computation of the the inverse Radon transform using the standard filtered back-projection algorithm [47].…”

“…A recent work by Fletcher et al [40] addresses the characterization of these 'high-energy' electrons by using an energy barrier that is tuned to match the energy of the quasiparticles. The authors propose and implement an electron tomography protocol that reconstructs the joint energy-time quasi-probability distribution p(E, t) of the incoming electrons by measuring the transmitted charge through an energy barrier whose height is varied linearly in time.…”

“…The two parameters of the linear modulation (the offset in energy V 0 and the slope α in the time-energy plane) provide a two-dimensional map of the charge Q(V 0 , α), that is used to infer p(E, t) using inversion techniques from tomographic image processing [40]. The key relation connecting the properties of the incoming wave-packets with the measured signal can be written as…”

“…For a point-contact geometry sketched in figure 2, this approximation is justified because the two counterpropagating edge states are closest in centre of the constriction where the effect of a sufficiently wide top gate (shaded region) is spatially uniform. Although, strictly speaking, the condition (5) is stronger than the assumption that a (time-independent) shift of the gate voltage V is equivalent to a shift in energy, the relation T(E, V )=T(E−V ) (which is central to the tomography experiment [40]) does not hold for a generic scatterer if the condition (5) is not satisfied. Finally, we set the gate potential to zero sufficiently far away from the constriction,…”