## Periodic Reporting for period 4 - EQuO (Electron Quantum optics in quantum Hall edge channels)

Reporting period: 2020-04-01 to 2021-03-31

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 pioneer 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 is to specifically explore the emergence of many body physics and its effects on electronic propagation using the setups and concepts of electron quantum optics. The motivations are numerous: firstly single particle emission initializes a simple and well controlled state. This project will unveil the birth, life and death scenarii of Landau quasiparticles and observe the emergence of many-body physics. Secondly, the project will address the generation of entangled few electrons quantum coherent states and study how they are affected by interactions. Finally, the project aims 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 carry a fractional charge and obey fractional statistics. No manipulation of single quasiparticles has been reported yet and the determination of some quasiparticle characteristics, such as the fractional statistics remains elusive.

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 pioneer 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 is to specifically explore the emergence of many body physics and its effects on electronic propagation using the setups and concepts of electron quantum optics. The motivations are numerous: firstly single particle emission initializes a simple and well controlled state. This project will unveil the birth, life and death scenarii of Landau quasiparticles and observe the emergence of many-body physics. Secondly, the project will address the generation of entangled few electrons quantum coherent states and study how they are affected by interactions. Finally, the project aims 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 carry a fractional charge and obey fractional statistics. No manipulation of single quasiparticles has been reported yet and the determination of some quasiparticle characteristics, such as the fractional statistics remains elusive.

An important part of the working time of the reporting period has been devoted to the installation of the new lab and new dilution fridge. The installation of the fridge took more time than expected (see technical problems) due to two technical issues with the fridge. The fridge has been equipped with several rf and dc lines including several amplifiers for high frequency noise and current noise measurements.

Once the fridge was properly working, several experimental tasks of the project have been realized:

-we have performed a quantitative analysis of the decoherence and relaxation of a single electronic excitation propagating along the one dimensional chiral edge channel of the integer quantum Hall regime

-we have realized the tomography of single electron excitations generated by voltage pulses applied to the edge channels of the integer quantum Hall effect

-we are testing new samples designed for the study of electron quantum optics experiments in the fractional quantum Hall regime

We have also worked on theoretical problems connected to the project

-development of an algorithm for the extraction from any electrical current of the wavefunctions and their emission probabilities from two particle interferometry

-generation of single electron excitations from a gate capacitively coupled to a one-dimensional conductor

-protection of single electronic excitations for Coulomb interaction induced decoherence and relaxation

Once the fridge was properly working, several experimental tasks of the project have been realized:

-we have performed a quantitative analysis of the decoherence and relaxation of a single electronic excitation propagating along the one dimensional chiral edge channel of the integer quantum Hall regime

-we have realized the tomography of single electron excitations generated by voltage pulses applied to the edge channels of the integer quantum Hall effect

-we are testing new samples designed for the study of electron quantum optics experiments in the fractional quantum Hall regime

We have also worked on theoretical problems connected to the project

-development of an algorithm for the extraction from any electrical current of the wavefunctions and their emission probabilities from two particle interferometry

-generation of single electron excitations from a gate capacitively coupled to a one-dimensional conductor

-protection of single electronic excitations for Coulomb interaction induced decoherence and relaxation

We have solved two important problems of time dependent quantum electronics:

1) the extraction of the elementary excitations carried by any quantum electrical current as well as their wavefunctions and emission probabilities

2) the quantitative analysis of the decoherence and relaxation (also called fractionalization) of a single electron propagating in a one-dimensional conductor

1) On the theoretical level, it was known that specific types of voltage pulses (Lorentzian) or the use of dynamical quantum dots were able to generate a single excitation above the Fermi sea. However, finite temperature effects, imperfections in the drive signal of simply deformations related to the Coulomb interaction which is always present in electrical conductors would lead to the emission of spurious excitations that were impossible to take into account. These theoretical works were extended to process the many-body state generated at zero temperature by any voltage drive applied to a one-dimensional conductor. However, here also finite temperature effects were neglected and more importantly, the procedure to extract the generated state from experimental measurements was missing. On the experimental level, pioneer work were able to extract the single electron wavefunction carried by a Lorentzian voltage pulse assuming that a single excitation was present which is not exactly the case due to the above mentioned spurious effects. In a combined experimental and theoretical work, we have been able to able to extend the above described state of the art by computing, 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 an simple description of any electrical current in term of its elementary building blocks: single electron and hole excitations with well-defined wavefunctions.

2) It is known that in one-dimensional conductor, Coulomb interaction lead 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 porcess leads to the relaxation and decoherence of the electronic excitation by the emission of collective excitations (electron/hole pairs).

We expect additional results until the end of the project. The techniques developed so far (two-particle interferometry) will be applied to investigate two-electron entanglement and how it propagates in one dimensional conductor. By extending these techniques to the fractional quantum Hall regime, we will study the exchange statistics to the excitations of the fractional quantum Hall regime which are predicted to be neither fermionic nor bosonic (anyonic statistics).

1) the extraction of the elementary excitations carried by any quantum electrical current as well as their wavefunctions and emission probabilities

2) the quantitative analysis of the decoherence and relaxation (also called fractionalization) of a single electron propagating in a one-dimensional conductor

1) On the theoretical level, it was known that specific types of voltage pulses (Lorentzian) or the use of dynamical quantum dots were able to generate a single excitation above the Fermi sea. However, finite temperature effects, imperfections in the drive signal of simply deformations related to the Coulomb interaction which is always present in electrical conductors would lead to the emission of spurious excitations that were impossible to take into account. These theoretical works were extended to process the many-body state generated at zero temperature by any voltage drive applied to a one-dimensional conductor. However, here also finite temperature effects were neglected and more importantly, the procedure to extract the generated state from experimental measurements was missing. On the experimental level, pioneer work were able to extract the single electron wavefunction carried by a Lorentzian voltage pulse assuming that a single excitation was present which is not exactly the case due to the above mentioned spurious effects. In a combined experimental and theoretical work, we have been able to able to extend the above described state of the art by computing, 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 an simple description of any electrical current in term of its elementary building blocks: single electron and hole excitations with well-defined wavefunctions.

2) It is known that in one-dimensional conductor, Coulomb interaction lead 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 porcess leads to the relaxation and decoherence of the electronic excitation by the emission of collective excitations (electron/hole pairs).

We expect additional results until the end of the project. The techniques developed so far (two-particle interferometry) will be applied to investigate two-electron entanglement and how it propagates in one dimensional conductor. By extending these techniques to the fractional quantum Hall regime, we will study the exchange statistics to the excitations of the fractional quantum Hall regime which are predicted to be neither fermionic nor bosonic (anyonic statistics).