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.

Fig. 1

Electron tomography scheme using a modulated barrier. a An unknown Wigner distribution $W\left(E,t\right)$ of a periodic electron source electron can be filtered using a linear-in-time threshold energy barrier set at height $E_T$. The transmitted and reflected part, labelled $P_T$ and $1-P_T$ result in a proportionate transmitted and reflected currents. A marginal projection of this distribution in the energy, time plane can be measured by fixing the ramp rate of the barrier ${\beta }_E$, which sets $E_T$, then moving the threshold boundary along the axis $S$ in increments $dS$, while measuring the resulting changes in transmitted current. Repeating the experiment at different ramp rates (which sets the angle $\theta$) gives enough information for a numerical reconstruction of the distribution. b False-colour scanning electron micrograph of device identical to that measured (see methods for details). The electron pump (left, highlighted green) injects pump current $I_p$. The barrier (right, highlighted red) selectively blocks electrons giving transmitted current $I_T\le I_P$. The path between these is indicated with a line. The gates along the path (controlled by $V_{\mathrm{G4}}$) depletes the underlying electron gas but do not block the high energy electrons. c Typical time-dependent control voltages for pump $V_{\mathrm{G1}}$ and probe barrier $V_{\mathrm{G3}}$ (each has a DC offset—see methods). d Electron potential $U\left(x\right)$ along the electron path between source and probe barrier at three representative stages for pumping (left) and blocking (right).

Fig. 2

Angle dependent projection of single electron density. a Colour plot: projected electronic density (sinogram) at various angles $\theta$ in the energy–time plane. Colour scale corresponds to $d{I}_T∕dS$, where $dS$ is an incremental step in the (normalised) energy–time plane. Depending on $\theta$, the projection axis $S$ corresponds to an energy projection, time projection or a mixture. The left axis is appropriate for $\theta =\pm 90^{\circ }$ and the right hand for $\theta =0^{\circ }$. Selected projections are shown at angles, where $S$ corresponds to b time projection, c energy projection and a mixed projection d. e Inverse Radon transform of the data in a giving the Wigner phase-space density in units of $h^{-1}$.

Fig. 3

Tomography of excitations produced under different ejection conditions. a Coloured points show different operating points within the one electron/cycle pump current plateau whose boundaries are visible in this plot of $d{I}_p∕d{V}_{\mathrm{G2}}$.  b Circular symbols indicate measured time of arrival, $t_0$ at each of the $V_{\mathrm{G1}}$ operating points in a. Square symbols show the estimated barrier sweep rate at these times. c Single electron tomography for each pump operating points in a, as indicated by the coloured dots (top left is the fastest sweep rate, bottom right is the slowest). Dashed lines are the ejection trajectories calculated with a semiclassical model (See Supplementary Note 4 and Supplementary Figs. 5–7). d single electron chirp rate $dE∕dt$ as measured from fits to the backprojection at each sweep rate on the pumping gate (the solid line is a linear fit). e Solid symbols are the measured peak phase space density $\approx P_1$. Open symbols are from a model that accounts for energy broadening.