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arXiv:1901.10985 · Nature Communications

Continuous-variable tomography of solitary electrons

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

Published Nov 22, 2019• 54 citations
arXivHTMLPDF

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.

DOI
10.1038/s41467-019-13222-1

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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.
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.
Initializing and measuring the wave-function of single freely-propagating particles are challenging but fundamental tasks for applications in quantum information processing and enhanced sensing. Recently, continuous-variable quantum tomography techniques developed for atomic beams and photonic modes have been adapted for on-demand electronic excitations but are limited to the region in energy-time phase space immediately around the Fermi level. Here we present a general, broad-band method for mapping the single-particle phase space Wigner distribution. We reconstruct the Wigner representation of the mixed-state density matrix for \textit{solitary} electrons emitted by an on-demand source. This reveals highly localised distributions isolated from the Fermi sea. Our method can be used to test the degree of `quantumness' of an electron source from measurements of the phase-space density. While both source purity and detector fidelity are suppressed by classical fluctuations, here we resolve fine structure of the wavepackets (a chirp controlled by emission conditions) and find that the resolution should be sufficient to enable quantum-limited measurements. Our technique will provide key information required to master electron coherence and entanglement at the individual particle level, for instance to create excitations from multiple sources that are indistinguishable.
In this work we have explored few tools in Quantum State Tomography for Continuous Variable Systems. The concept of quantum states in phase space representation is introduced in a simple manner by using a few statistical concepts. Unlike most texts of Quantum information in which the Wigner function for a single mode is often more used, in this text the multi-modes state Wigner function is also approached. Our numerical investigations indicate that the reconstructed method using back-projection add some error due the choice of cutoff frequency, therefore it is necessary to use data post-processing, like the semi-definite programs, which provides sufficient conditions correctly estimate the state. Once the information about the state is recovered, important features such as entanglement can also be investigated.
Summary form only given. The goal of any pulse characterization protocol is to describe how the spectral components of the electric field are changing in time. Apart from linear methods such as streak cameras, characterization techniques based on nonlinear processes have been developed. Those processes require broad spectral acceptance bandwidths, i.e. to be capable of resolving ultrashort features of the pulse. For many of those techniques, the pulses are reconstructed by iterative or deconvolution algorithms where the electric field is assumed to be fully coherent and do not account for the possibility of reduced coherence from pulse to pulse, or within a single pulse. In general, the chronocyclic Wigner function should be reconstructed in order to provide full information about an ensemble of partially coherent pulses. To make a parallel with quantum mechanics, most quantum systems cannot be considered as closed and interact with the environment, and subsequently, undergo decoherence. Therefore, a noisy quantum state cannot be described by a single pure state only but as a classical mixture of pure states.
Determining the quantum state of a given quantum-mechanical system is a fundamental task in physics. Quantum-state tomography has been pivotal for establishing quantum optics [1-4] and for revealing the properties of bound charges in materials [5-7]. An emerging other object for studying and utilizing quantum effects are free electrons, elementary particles that are central to high-resolution microscopy [8,9], electron-based quantum optics [10-17], ul-trafast electron microscopy [18-24] and particle accelerators [25-27]. However, free electrons are intrinsically incoherent, and we lack a broadly applicable method to measure and control their quantum state beyond special cases with discrete energy sidebands [28,29]. Here, we report a universal approach to measure arbitrary free-electron quantum states in continuous variables. Two monochromatic but spectrally shifted laser waves produce interfering quan-tum paths that directly reveal the density matrix and thus all essential properties of the pure wavepackets, the ensemble, and their interlinks. As a first application, we show how the quantum state of a single electron is modified by many-body Coulomb interactions of a sur-rounding electron gas. The reported concepts and results provide insight into otherwise hid-den correlations in electron beams and enable the controlled optimization of exceptional quantum states for free-electron quantum optics or quantum electron microscopy.
We present the experimental reconstruction of the Wigner function of an individual electronic spin qubit associated with a nitrogen-vacancy center in diamond at room temperature. This spherical Wigner function contains the same information as the density matrix for arbitrary spin systems. As an example, we exactly characterize the quantum state of a single qubit undergoing a nearly pure dephasing process by the Wigner function. The fidelities and purities during this process are extracted from the experimentally reconstructed Wigner functions, whose dynamics agree with the theoretical prediction.