Two-electron coherence and its measurement in electron quantum optics

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Since pioneering works of Hanbury-Brown and Twiss, intensity-intensity correlations have been widely used in astronomical systems, for example to detect binary stars. They reveal statistics effects and two-particle interference, and offer a decoherence-free probe of the coherence properties of light sources. In the quantum Hall edge channels, the concept of quantum optics can transposed to electrons, and an analogous two-particle interferometry can be developed, in order to characterize single-electron states. We review in this article the recent experimental and theoretical progress on this topic.

Section: Discussionmentioning

confidence: 99%

“…These theoretical tools rely on the formal analogy between the electric field operator

{\hat{E}}^{+} false(x, t false)

(that annihilates photons at time t and position x ), and the electron field operator

\overset{Ψ}{true} false(x, t false)

Section: General Principles Of Two‐particle Interferometrymentioning

confidence: 99%

Two‐particle interferometry in quantum Hall edge channels

Marguerite

Bocquillon

Berroir

et al. 2016

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“…Its physical meaning can be obtained by computing the two‐electron coherence for the state

false| Ψ_{N} 〉

defined in Section . The result is a sum over all the two‐electron wavefunctions

ϕ_{k, l} false(x, y false) = ϕ_{k} false(x false) ϕ_{l} false(y false) - ϕ_{k} false(y false) ϕ_{l} false(x false)

(

k < l

) that can be extracted from the N single‐particle wavefunctions

{false(ϕ_{k} false)}_{k \in {1, \dots, N}}

G_{false| Ψ_{N} 〉}^{false(2 e false)} false(x_{1}, x_{2} false| x_{1}^{'}, x_{2}^{'} false) = \sum_{k < l} ϕ_{k, l} false(x_{1}, x_{2} false) ϕ_{k, l} false(x_{1}^{'}, x_{2}^{'} false) .

Two‐electron coherence at a given position is a function of four times

false(t_{1}, t_{2}; t_{1}^{'}, t_{2}^{'} false)

.…”

Section: Two‐electron Coherencementioning

confidence: 99%

“…The intrinsic two‐electron coherence emitted by a source can be defined from the second‐order electronic coherence by subtracting not only the Fermi sea contribution but also all processes contributing to two‐electron detection and involving the excess single‐electron coherence of the source. These involve classical contributions as well as quantum exchange contributions :

\begin{matrix} right & center & left {scriptG}_{ρ}^{(2 e)} (1, 2 | 1^{'}, 2^{'}) = {scriptG}_{F}^{(2 e)} (1, 2 | 1^{'}, 2^{'}) \end{matrix}

\begin{matrix} right & center & left + {scriptG}_{F}^{(e)} (1 | 1^{'}) Δ {scriptG}_{ρ}^{(e)} (2 | 2^{'}) + {scriptG}_{F}^{(e)} (2 | 2^{'}) Δ {scriptG}_{ρ}^{(e)} (1 | 1^{'}) \end{matrix}

\begin{matrix} right & center & left - {scriptG}_{F}^{(e)} (1 | 2^{'}) Δ {scriptG}_{ρ}^{(e)} (2 | 1^{'}) - {scriptG}_{F}^{(e)} (2 | 1^{'}) Δ {scriptG}_{ρ}^{(e)} (1 | 2^{'}) \end{matrix}

…”

Section: Two‐electron Coherencementioning

confidence: 99%

“…3). These current correlations are directly related to the inter‐channel two‐electron coherence right after the beam splitter since fermionic fields within different channels anticommute:

\begin{matrix} left & left - 12 italicpt 〈 i_{1 out} (t_{1}) i_{2 out} (t_{2}) 〉 = & left e^{2} v_{F}^{2} Δ {scriptG}_{normalout}^{(2 e)} (1, t_{1}; 2, t_{2} | 1, t_{1}^{'}; 2, t_{2}^{'}) . \end{matrix}

Remarkably, this outgoing two‐electron coherence is proportional to the incoming two‐electron coherence :

\begin{matrix} left & left - 12 italicpt Δ {scriptG}_{normalout}^{(2 e)} (1, t_{1}; 2, t_{2} | 1, t_{1}^{'}; 2, t_{2}^{'}) = & left RT Δ {scriptG}_{S}^{(2 e)} (t_{1}, t_{2} | t_{1}^{'}, t_{2}^{'}) . \end{matrix}

Measuring outgoing inter‐channel current correlations in the HBT setup thus directly probes the intrinsic excess coherence of the source, as in photon quantum optics.…”

Section: Two‐electron Coherencementioning

confidence: 99%

See 1 more Smart Citation

Electron quantum optics as quantum signal processing

Roussel

Cabart

Fève

et al. 2017

Self Cite

The recent developments of electron quantum optics in quantum Hall edge channels have given us new ways to probe the behavior of electrons in quantum conductors. It has brought new quantities called electronic coherences under the spotlight. In this paper, we explore the relations between electron quantum optics and signal processing through a global review of the various methods for accessing single-and two-electron coherences in electron quantum optics. We interpret electron quantum optics interference experiments as analog signal processing, converting quantum signals into experimentally observable quantities such as current averages and correlations.This point of view also gives us a procedure to obtain quantum information quantities from electron quantum optics coherences. We illustrate these ideas by discussing two-mode entanglement in electron quantum optics. We also sketch how signal processing ideas may open new perspectives for representing electronic coherences in quantum conductors and understand the properties of the underlying many-body electronic state.

Electronic quantum optics beyond the integer quantum Hall effect

Ferraro

Jonckheere

Rech

et al. 2016

The analog of two seminal quantum optics experiments are considered in a condensed matter setting with single electron sources injecting electronic wave packets on edge states coupled through a quantum point contact. When only one electron is injected, the measurement of noise correlations at the output of the quantum point contact corresponds to the Hanbury-Brown and Twiss setup. When two electrons are injected on opposite edges, the equivalent of the Hong-Ou-Mandel collision is achieved, exhibiting a dip as in the coincidence measurements of quantum optics. The Landauer-Buttiker scattering theory is used to first review these phenomena in the integer quantum Hall effect, next, to focus on two more exotic systems: edge states of two dimensional topological insulators, where new physics emerges from time reversal symmetry and three electron collisions can be achieved; and edges states of a hybrid Hall/superconducting device, which allow to perform electron quantum optics experiments with Bogoliubov quasiparticles.Comment: 13 pages, 10 figures, invited contribution for a focus issue on "Single-electron control in solid-state devices