## Periodic Reporting for period 5 - EQuO (Electron Quantum optics in quantum Hall edge channels)

Reporting period: 2021-04-01 to 2022-03-31

Quantum effects have been studied on photon propagation in the context of quantum optics since the second half of the last century. In particular, using single photon emitters, fundamental tests of quantum mechanics were explored by manipulating single to few photons in Hanbury-Brown and Twiss and Hong Ou Mandel experiments. In nanophysics, there is a growing interest to translate these concepts of quantum optics to electrons propagating in nanostructures. Single electron emitters have been realized such that single elementary electronic excitations can now be manipulated in the analog of quantum optics experiments. Electron quantum optics goes beyond the mere reproduction of optical setups using electron beams, as electrons, being interacting fermions, differ strongly from photons. Contrary to optics, understanding the propagation of an elementary excitation requires replacing the single body description by a many body one.

The purpose of this proposal was to explore the emergence of many body physics in quantum conductors using the setups and concepts of electron quantum optics. More specifically, this project aimed at unveiling the birth, life and death scenarii of elementary Landau quasiparticles. Secondly, the project addressed the generation of entangled few electrons quantum coherent states and study how they are affected by interactions. Finally, the project aimed at applying electron quantum optics concepts to a regime where the ground state itself is a strongly correlated state of matter. In such a situation, elementary excitations are no longer electrons but anyons carrying a fractional charge and obeying fractional statistics. Experimental evidence of the fractional statistics of anyons remained elusive for a long time.

At the end of this proposal, several of its objectives have been fulfilled. The decoherence scenario of single electronic excitations has been completely characterized and understood by combining experiments and theory. We have also been able to implement tomography protocols for electronic states bases on two-particle interferometry in collider geometries. In these setups, one can fully extract an elementary description of quantum electrical currents in terms of elementary electron and hole wavefunctions with their emission probabilities. Regarding the characterization of two-electron entanglement, the project investigated theoretically the possibility to measure two-electron coherence and entanglement using the electronic analog of the Fransson interferometer. Finally the most important outcomes of the project are related to electron quantum optics experiments in the fractional quantum effect. In particular, we have been able to demonstrate the fractional statistics of anyons, which are the elementary excitations of fractional quantum Hall fluids, using two-particle interferences in the geometry of a collider.

The purpose of this proposal was to explore the emergence of many body physics in quantum conductors using the setups and concepts of electron quantum optics. More specifically, this project aimed at unveiling the birth, life and death scenarii of elementary Landau quasiparticles. Secondly, the project addressed the generation of entangled few electrons quantum coherent states and study how they are affected by interactions. Finally, the project aimed at applying electron quantum optics concepts to a regime where the ground state itself is a strongly correlated state of matter. In such a situation, elementary excitations are no longer electrons but anyons carrying a fractional charge and obeying fractional statistics. Experimental evidence of the fractional statistics of anyons remained elusive for a long time.

At the end of this proposal, several of its objectives have been fulfilled. The decoherence scenario of single electronic excitations has been completely characterized and understood by combining experiments and theory. We have also been able to implement tomography protocols for electronic states bases on two-particle interferometry in collider geometries. In these setups, one can fully extract an elementary description of quantum electrical currents in terms of elementary electron and hole wavefunctions with their emission probabilities. Regarding the characterization of two-electron entanglement, the project investigated theoretically the possibility to measure two-electron coherence and entanglement using the electronic analog of the Fransson interferometer. Finally the most important outcomes of the project are related to electron quantum optics experiments in the fractional quantum effect. In particular, we have been able to demonstrate the fractional statistics of anyons, which are the elementary excitations of fractional quantum Hall fluids, using two-particle interferences in the geometry of a collider.

Overview of the project’s results:

-we have performed a quantitative analysis of the decoherence and relaxation of a single electronic excitation propagating along one dimensional edge channels

-we have realized the tomography of single electron excitations

-we have experimentally demonstrated a new method for measuring the fractional charge of the elementary excitations of the fractional quantum Hall effect using high frequency noise measurements.

-we have experimentally demonstrated the fractional statistics of anyons using two-particle interferometry in a collider geometry

We have also worked on theoretical problems connected to the project

-we have developed an algorithm for the extraction of the wavefunctions and of their emission probabilities of the electronic excitations carried by an electrical current.

-we have investigated the generation of single electron excitations from a gate capacitively coupled to a one-dimensional conductor.

-we have investigated the protection of single electronic excitations for Coulomb interaction induced decoherence and relaxation.

-we have investigated how two electron coherence can be measured using noise measurements in the geometry of an electronic Franson interferometer.

-we have performed a quantitative analysis of the decoherence and relaxation of a single electronic excitation propagating along one dimensional edge channels

-we have realized the tomography of single electron excitations

-we have experimentally demonstrated a new method for measuring the fractional charge of the elementary excitations of the fractional quantum Hall effect using high frequency noise measurements.

-we have experimentally demonstrated the fractional statistics of anyons using two-particle interferometry in a collider geometry

We have also worked on theoretical problems connected to the project

-we have developed an algorithm for the extraction of the wavefunctions and of their emission probabilities of the electronic excitations carried by an electrical current.

-we have investigated the generation of single electron excitations from a gate capacitively coupled to a one-dimensional conductor.

-we have investigated the protection of single electronic excitations for Coulomb interaction induced decoherence and relaxation.

-we have investigated how two electron coherence can be measured using noise measurements in the geometry of an electronic Franson interferometer.

Three main achievement of the project pushing the field beyond its current state of the art:

1) the extraction of the elementary excitations carried by any quantum electrical current as well as their wavefunctions and emission probabilities. In a combined experimental and theoretical work, we have been able to compute, from noise measurements, all the elementary excitations carried by any electrical current. In particular, our procedure fully takes into account thermal effects, deviations from perfect single particle emission or effects of thermal excitations. Taking into account all these effect, it can quantify how far a source deviates from the emission of a pure single electron quantum state and more generally provide a simple description of any electrical current in term of its elementary building blocks: single electron and hole excitations with well-defined wavefunctions.

2) the quantitative analysis of the decoherence and relaxation (also called fractionalization) of a single electron propagating in a one-dimensional conductor. ) In one-dimensional conductors, Coulomb interaction leads to the emergence of collective excitations such that single particle excitations (electrons) are eventually destroyed along propagation. We were able to quantitatively analyze this process of destruction of the quasiparticle by generating a single electron from a dynamically driven quantum dot in a one-dimensional edge channel. Using two-particle interferometry, we were able to probe the coherence of this excitation after a few microns of propagation along the one-dimensional channel. We observed that due to interactions, the electron fractionalizes in two pulses carrying respectively the charge and the spin. This process leads to the relaxation and decoherence of the electronic excitation by the emission of collective excitations (electron/hole pairs).

3) the development of a new tool to quantitatively measure the fractional statistics of anyons, elementary excitations of the fractional quantum Hall effect. Anyons have been predicted to exist more than 40 years ago. They consist in elementary excitations of a 2 dimensional system obeying quantum statistics in between that of fermions and that of bosons. In particular, for abelian anyons, the phase associated with the exchange between two particles can take any value between 0 and pi. The elementary excitations of the fractional quantum Hall effect have been predicted in the 80’s to have anyonic statistics. Some of their properties, such as their fractional charge, have been demonstrated in the 90’s but it has proven much more difficult to experimentally demonstrate their fractional statistics. Using quantum point contacts as random emitters of anyonic excitations and another quantum point contact as a beam-splitter, we have been able to demonstrate the fractional statistics of anyons in the geometry of an anyon collider. In this geometry, fermions (electrons) and anyons exhibit strikingly different behaviors. Fermions exclude eachother as expected from the Pauli exclusion principle. On the contrary, anyons are allowed to bunch in packets of charge which shows up as strongly negative cross-correlations between the current fluctuations at the output of the collider.

1) the extraction of the elementary excitations carried by any quantum electrical current as well as their wavefunctions and emission probabilities. In a combined experimental and theoretical work, we have been able to compute, from noise measurements, all the elementary excitations carried by any electrical current. In particular, our procedure fully takes into account thermal effects, deviations from perfect single particle emission or effects of thermal excitations. Taking into account all these effect, it can quantify how far a source deviates from the emission of a pure single electron quantum state and more generally provide a simple description of any electrical current in term of its elementary building blocks: single electron and hole excitations with well-defined wavefunctions.

2) the quantitative analysis of the decoherence and relaxation (also called fractionalization) of a single electron propagating in a one-dimensional conductor. ) In one-dimensional conductors, Coulomb interaction leads to the emergence of collective excitations such that single particle excitations (electrons) are eventually destroyed along propagation. We were able to quantitatively analyze this process of destruction of the quasiparticle by generating a single electron from a dynamically driven quantum dot in a one-dimensional edge channel. Using two-particle interferometry, we were able to probe the coherence of this excitation after a few microns of propagation along the one-dimensional channel. We observed that due to interactions, the electron fractionalizes in two pulses carrying respectively the charge and the spin. This process leads to the relaxation and decoherence of the electronic excitation by the emission of collective excitations (electron/hole pairs).

3) the development of a new tool to quantitatively measure the fractional statistics of anyons, elementary excitations of the fractional quantum Hall effect. Anyons have been predicted to exist more than 40 years ago. They consist in elementary excitations of a 2 dimensional system obeying quantum statistics in between that of fermions and that of bosons. In particular, for abelian anyons, the phase associated with the exchange between two particles can take any value between 0 and pi. The elementary excitations of the fractional quantum Hall effect have been predicted in the 80’s to have anyonic statistics. Some of their properties, such as their fractional charge, have been demonstrated in the 90’s but it has proven much more difficult to experimentally demonstrate their fractional statistics. Using quantum point contacts as random emitters of anyonic excitations and another quantum point contact as a beam-splitter, we have been able to demonstrate the fractional statistics of anyons in the geometry of an anyon collider. In this geometry, fermions (electrons) and anyons exhibit strikingly different behaviors. Fermions exclude eachother as expected from the Pauli exclusion principle. On the contrary, anyons are allowed to bunch in packets of charge which shows up as strongly negative cross-correlations between the current fluctuations at the output of the collider.